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https://infoscience.epfl.ch/record/212744	Infoscience Journal article # Insight on the photocatalytic bacterial inactivation by co-sputtered TiO2-Cu in aerobic and anaerobic conditions Co-sputtered TiO2-Cu polyester (TiO2-Cu-PES) under actinic light induced bacterial reduction of E. coli in the presence of O2 (air) and under anaerobic conditions. The bacterial inactivation/oxidation proceeds in the absence of O2 (air) probably due to the highly oxidative TiO2vb(h+) species and the toxic Cu present. By the choice of suitable scavengers, the presence of highly oxidative radicals was confirmed in aerobic media. The E. coli inactivation in aerobic media proceeds on TiO2-Cu-PES within 30 min and with a slower kinetics of 90 min in anaerobic media. Malondialdehyde generation a product of bacterial inactivation, was observed on the TiO2-Cu-PES in air and in lesser amounts under anaerobic conditions. Repetitive bacterial inactivation cycles show a Cu-release of 2 ppb/cm2 by the TiO2-Cu-PES surface as determined by inductively coupled plasma mass spectrometry (IPC-MS). The Cu released is far below the values reported for the Cu released by TiO2-Cu-PES samples by sputtering Ti and Cu in sequential order from two targets. By X-ray photoelectron spectroscopy (XPS), redox catalysis by the Cu and TiO2-species was observed under anaerobic conditions providing further evidence for processes leading to bacterial inactivation in anaerobic media. A mechanism for the TiO2-Cu-PES bacterial inactivation is suggested consistent with the results reported in this study.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9033128023147583, "perplexity": 12000.732269332284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320049.84/warc/CC-MAIN-20170623100455-20170623120455-00445.warc.gz"}
http://www.emathzone.com/tutorials/geometry/find-equation-of-tangent-line-to-parabola.html	# Find the Equation of the Tangent Line to Parabola Example: Find the equation of the tangent to the parabola ${y^2} = 13x$ parallel to the line $7x - 9y + 11 = 0$. Solution: The given equation of a parabola can be written in the form: Compare this equation of a parabola with the general equation of a parabola ${y^2} = 4ax$. After comparing with the standard equation of a parabola we have $a = \frac{{13}}{4}$ Now to find the slope of a given line we use $m = - \frac{7}{{ - 9}} = \frac{7}{9}$ The required equation of the tangent to the parabola is given as This is the equation of the tangent to the parabola.	{"extraction_info": {"found_math": true, "script_math_tex": 5, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7848798036575317, "perplexity": 97.92995326460428}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822747.22/warc/CC-MAIN-20171018051631-20171018071631-00145.warc.gz"}
https://orinanobworld.blogspot.com/2017/07/	## Friday, July 14, 2017 ### Memory Minimization As I grow older, I'm starting to forget things (such as all the math I ever learned) ... but that's not the reason for the title of this post. A somewhat interesting question popped up on Mathematics StackExchange. It combines a basic sequencing problem (ordering the processing of computational tasks) with a single resource constraint (memory) and a min-max criterion (minimize the largest amount of memory consumed anywhere in the sequence). What differentiates the problem from other resource-constrained scheduling problems I've seen is that the output of each task eats a specified amount of memory and must stay there until every successor task that needs it as input completes, after which the memory can be recycled. In particular, the output of the last task using the result of task $n$ will coexist momentarily with the output of $n$, before $n$'s output can be sent to the bit-recycler. Also, there is no time dimension here (we neither know nor care how much CPU time tasks need), just sequencing. The data for the problem is a precedence graph (directed acyclic graph, one node for each task) plus an integer weight $w_n$ for each task $n$, representing the memory consumption of the value computed by task $n$. There are several ways to attack a problem like this, including (mixed) integer programming (MIP), constraint programming (CP), and all sorts of metaheuristics. MIP and CP guarantee optimal solutions given enough resources (time, memory, computing power). Metaheuristics do not guarantee optimality, but also do not require commercial software to implement and may scale better than MIP and possibly CP. I thought I'd publish my MIP model for the problem, scaling be damned. #### Index sets and functions Let $N$ be the number of tasks (nodes). • I will use $V=\left\{ 1,\dots,N\right\}$ as the set of tasks and $S=\left\{ 1,\dots,N\right\}$ as the set of schedule slots. (Yes, I know they are the same set, but the dual notation may help distinguish things in context.) • $A\subset V\times V$ will represent the set of arcs, where $(u,v)\in A$ means that task $v$ takes the output of task $u$ as an input. • $\pi:V\rightarrow2^{V}$ and $\sigma:V\rightarrow2^{V}$ will map each task to its immediate predecessors (the inputs to that task) and immediate successors (tasks that need its value as inputs) respectively. #### Parameters The only parameter is $w:V\rightarrow\mathbb{N}$, which maps tasks to their memory footprints. #### Variables There are a few ways to express assignments in MIP models, the two most common of which are "does this critter go in this slot" and "does this critter immediately follow this other critter". I'll use the former. • $x_{ij}\in\left\{ 0,1\right\} \ \forall i\in V,\forall j\in S$ will designate assignments, taking the value 1 if task $i$ is scheduled into slot $j$ in the sequence and 0 otherwise. • $r_{ij}\in\left[0,1\right]\ \forall i\in V,\forall j\in S$ will take the value 1 if the output of task $i$ is resident in memory immediately prior to performing the computational task in slot $j$ and 0 otherwise. • $d_{ij}\in\left[0,1\right]\ \forall i\in V,\forall j\in S$ indicates whether all tasks requiring the output of task $i$ have been completed prior to the execution of the task in slot $j$ (1) or not (0). • $z>0$ is the maximum memory usage anywhere in the schedule. We can actually winnow a few of these variables. If task $i$ has any predecessors, then clearly $x_{i1}=0$, since there is no opportunity before the first slot to execute a predecessor task. Similarly, $d_{i1}=0\,\forall i\in V$, since nothing has been completed prior to the first slot. You may think at this point that you have spotted a typo in the second and third bullet points (brackets rather than braces). You haven't. Read on. #### Objective The easiest part of the model is the objective function: minimize $z$. #### Constraints The assignment constraints are standard. Every task is assigned to exactly one slot, and every slot is filled with exactly one task. $$\sum_{j\in S}x_{ij}=1\quad\forall i\in V$$ $$\sum_{i\in V}x_{ij}=1\quad\forall j\in S$$ Precedence constraints are also fairly straightforward. No task can be scheduled before each of its predecessors has completed.$$x_{ij}\le\sum_{k <j}x_{hk}\quad\forall i\in V,\forall j\in S,\forall h\in\pi(i)$$ Next, we define memory consumption. Immediately after executing the task in any slot $j$, the memory footprint will be the combined consumption of the output of that task and the output of any earlier task still in memory when slot $j$ executes.$$z\ge\sum_{i\in V}w(i)\left(r_{ij}+x_{ij}\right)\quad\forall j\in S$$ Defining whether the output of some task $i$ is no longer needed prior to executing the task in some slot $j>1$ is fairly straightforward.$$d_{ij}\le\sum_{k<j}x_{hk}\quad\forall i\in V,\forall 1<j\in S,\forall h\in\sigma(i)$$As noted above, there is no "before slot 1", so we start this set of constraints with $j=2$. Finally, the result of task $i$ remains in memory at the point where the task in slot $j$ executes if task $i$ executed before slot $j$ and the memory has not yet been freed.$$r_{ij}\ge\sum_{k<j}x_{ik}-d_{ij}\quad\forall i\in V,\forall 1<j\in S$$Again, this is nonsensical for $j=1$, so we start with $j=2$. • It has $O(N^2)$ binary variables, so, as I noted above, it will not scale all that gracefully. • As mentioned early, the relaxation of $d_{ij}$ and $r_{ij}$ from binary to continuous in the interval $[0,1]$ was deliberate, not accidental. Why is this justified? $d_{ij}$ is bounded above by an integer quantity, and the solver "wants" it to be $1$: the sooner something can be removed from memory, the better for the solution. Similarly, $r_{ij}$ is bounded below by an integer expression, and the solver "wants" it to be $0$: the less cruft hanging around in memory, the better. So the solution will be binary even if the variables are not. • The solution might be "lazy" about clearing memory. From the model perspective, there is no rush to remove unneeded results as long as they are not contributing to the maximum memory footprint. The maximum memory load $z$ may occur at multiple points in the schedule. At least one would contain nothing that could be removed (else the solution would not be optimal). At points $j$ where memory use is below maximum, and possibly at points where memory use equals the maximum, there may be tasks $i$ for which $d_{ij}=0$ in the solution but could be set to $1$, without however reducing $z$. This can be avoided by adding constraints that force $d_{ij}=1$ wherever possible, but those constraints would enlarge the model (quite possibly slowing the solver) without improving the final solution.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.561714768409729, "perplexity": 687.8009475184957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247500089.84/warc/CC-MAIN-20190221051342-20190221073342-00257.warc.gz"}
https://en.wikipedia.org/wiki/Normal_(mathematics)	# Normal (Redirected from Normal (mathematics)) Normal may refer to: ## Mathematics There are many unrelated notions of "normality" in mathematics. Non-mathematical subjects very frequently refer to Normal (geometry) and Normal Distribution; see also the disambiguation pages for Normal form and Normalization. ### Topology • Normal space (or ${\displaystyle T_{4}}$) spaces, topological spaces characterized by separation of closed sets ## Transport • Metro Normal, a rapid transit railway station in the Mexico City Metro system	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6566988825798035, "perplexity": 7960.0264363311535}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122629.72/warc/CC-MAIN-20170423031202-00510-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/simple-vector-addition.375500/	1. Feb 4, 2010 ### anti404 1. Vector A has a magnitude of 3 m and makes an angle of 10o with the positive x axis. Vector B also has a magnitude of 10 m and is directed along the negative x axis. Enter your answers in distance then angle(in degrees). Find A + B Find A - B 2. R = sqrroot(Cx^2+Cy^2) Cx = Ax+Bx Cy = Ay+By Asin(theta)=x Bsin(theta)=y 3. 3sin10=.5209 = Ay; 3cos10=2.954=Ax basically, I don't know how to get By and Bx values from a vector that is following either axis. if I could get those, then I would become unstuck from this problem very quickly. any conceptual help would be appreciated. Justin 2. Feb 5, 2010 ### hage567 Just draw out the B vector on its own. The Bx and By components have to add up to the total B vector. If the vector is ONLY in the x direction, what do you think the y component would be?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8483359217643738, "perplexity": 686.8913555644001}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187820466.2/warc/CC-MAIN-20171016214209-20171016234209-00248.warc.gz"}
https://iclr.cc/virtual/2021/poster/3359	Certify or Predict: Boosting Certified Robustness with Compositional Architectures Mark Niklas Mueller · Mislav Balunovic · Martin Vechev Virtual Keywords: [ robustness ] [ certified robustness ] [ Provable Robustness ] [ Network Architecture ] [ Adversarial Accuracy ] [ Abstract ] [ [ Paper ] Thu 6 May 1 a.m. PDT — 3 a.m. PDT Abstract: A core challenge with existing certified defense mechanisms is that while they improve certified robustness, they also tend to drastically decrease natural accuracy, making it difficult to use these methods in practice. In this work, we propose a new architecture which addresses this challenge and enables one to boost the certified robustness of any state-of-the-art deep network, while controlling the overall accuracy loss, without requiring retraining. The key idea is to combine this model with a (smaller) certified network where at inference time, an adaptive selection mechanism decides on the network to process the input sample. The approach is compositional: one can combine any pair of state-of-the-art (e.g., EfficientNet or ResNet) and certified networks, without restriction. The resulting architecture enables much higher natural accuracy than previously possible with certified defenses alone, while substantially boosting the certified robustness of deep networks. We demonstrate the effectiveness of this adaptive approach on a variety of datasets and architectures. For instance, on CIFAR-10 with an $\ell_\infty$ perturbation of 2/255, we are the first to obtain a high natural accuracy (90.1%) with non-trivial certified robustness (27.5%). Notably, prior state-of-the-art methods incur a substantial drop in accuracy for a similar certified robustness.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.555622935295105, "perplexity": 2320.31524842156}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570651.49/warc/CC-MAIN-20220807150925-20220807180925-00649.warc.gz"}
http://hackage.haskell.org/package/ChasingBottoms-1.3.0.3/docs/src/Test-ChasingBottoms-Approx.html	{-# LANGUAGE ScopedTypeVariables, FlexibleInstances, UndecidableInstances #-} -- \| -- Module : Test.ChasingBottoms.Approx -- Copyright : (c) Nils Anders Danielsson 2004-2011 -- License : See the file LICENCE. -- -- Stability : experimental -- Portability : non-portable (GHC-specific) -- module Test.ChasingBottoms.Approx ( Approx(..) ) where import Test.ChasingBottoms.Nat import Data.Function import Data.Generics import qualified Data.List as List {-\| 'Approx' is a class for approximation functions as described in The generic approximation lemma, Graham Hutton and Jeremy Gibbons, Information Processing Letters, 79(4):197-201, Elsevier Science, August 2001, <http://www.cs.nott.ac.uk/~gmh/bib.html>. Instances are provided for all members of the 'Data' type class. Due to the limitations of the "Data.Generics" approach to generic programming, which is not really aimed at this kind of application, the implementation is only guaranteed to perform correctly, with respect to the paper (and modulo any bugs), on non-mutually-recursive sum-of-products datatypes. In particular, nested and mutually recursive types are not handled correctly with respect to the paper. The specification below is correct, though (if we assume that the 'Data' instances are well-behaved). In practice the 'approxAll' function can probably be more useful than 'approx'. It traverses down /all/ subterms, and it should be possible to prove a variant of the approximation lemma which 'approxAll' satisfies. -} class Approx a where -- \| @'approxAll' n x@ traverses @n@ levels down in @x@ and replaces all -- values at that level with bottoms. approxAll :: Nat -> a -> a -- \| 'approx' works like 'approxAll', but the traversal and -- replacement is only performed at subterms of the same monomorphic -- type as the original term. For polynomial datatypes this is -- exactly what the version of @approx@ described in the paper above -- does. approx :: Nat -> a -> a instance Data a => Approx a where approxAll = approxAllGen approx = approxGen -- From The generic approximation lemma (Hutton, Gibbons): -- Generic definition for arbitrary datatype \mu F: -- approx (n+1) = in . F (approx n) . out -- Approximation lemma (valid if F is locally continuous), -- for x, y :: \mu F: -- x = y <=> forall n in Nat corresponding to natural numbers. -- approx n x = approx n y approxGen :: Data a => Nat -> a -> a approxGen n \| n == 0 = error "approx 0 = _\|_" \| otherwise = \(x :: a) -> gmapT (mkT (approxGen (pred n) :: a -> a)) x -- We use mkT to only recurse on the original type. This solution is -- actually rather nice! But sadly it doesn't work for nested or -- mutually recursive types... -- Note that the function is defined in the \n -> \x -> style, not -- \n x -> which would mean something subtly different. ------------------------------------------------------------------------ -- Recurses on everything... approxAllGen :: Data a => Nat -> a -> a approxAllGen n \| n == 0 = error "approx 0 = _\|_" \| otherwise = \x -> gmapT (approxAllGen (pred n)) x ------------------------------------------------------------------------ -- Behaves exactly like approxGen. (?) approxGen' :: Data a => Nat -> a -> a approxGen' n \| n == 0 = error "approx 0 = _\|_" \| otherwise = \x -> let d = dataTypeOf x n' = pred n fun childTerm = if dataTypeOf childTerm === d then approxGen' n' childTerm else childTerm in gmapT fun x where (===) = (==) on dataTypeRep	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6608293056488037, "perplexity": 11919.422357602838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662527.91/warc/CC-MAIN-20160924173742-00018-ip-10-143-35-109.ec2.internal.warc.gz"}
https://www.ncbi.nlm.nih.gov/pubmed/17397255?dopt=Abstract	Format Choose Destination PLoS Comput Biol. 2007 Mar 30;3(3):e60. Epub 2007 Feb 14. # Potential energy landscape and robustness of a gene regulatory network: toggle switch. ### Author information 1 Department of Physics and Astronomy, State University of New York Stony Brook, Stony Brook, New York, United States of America. ### Abstract Finding a multidimensional potential landscape is the key for addressing important global issues, such as the robustness of cellular networks. We have uncovered the underlying potential energy landscape of a simple gene regulatory network: a toggle switch. This was realized by explicitly constructing the steady state probability of the gene switch in the protein concentration space in the presence of the intrinsic statistical fluctuations due to the small number of proteins in the cell. We explored the global phase space for the system. We found that the protein synthesis rate and the unbinding rate of proteins to the gene were small relative to the protein degradation rate; the gene switch is monostable with only one stable basin of attraction. When both the protein synthesis rate and the unbinding rate of proteins to the gene are large compared with the protein degradation rate, two global basins of attraction emerge for a toggle switch. These basins correspond to the biologically stable functional states. The potential energy barrier between the two basins determines the time scale of conversion from one to the other. We found as the protein synthesis rate and protein unbinding rate to the gene relative to the protein degradation rate became larger, the potential energy barrier became larger. This also corresponded to systems with less noise or the fluctuations on the protein numbers. It leads to the robustness of the biological basins of the gene switches. The technique used here is general and can be applied to explore the potential energy landscape of the gene networks. PMID: 17397255 PMCID: PMC1848002 DOI: 10.1371/journal.pcbi.0030060 [Indexed for MEDLINE] Free PMC Article	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8366013765335083, "perplexity": 880.9582436550969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317688.48/warc/CC-MAIN-20190822235908-20190823021908-00300.warc.gz"}
https://www.eevblog.com/forum/testgear/tektronix-2465b-oscilloscope-teardown/msg886346/	### Author Topic: Tektronix 2465B oscilloscope teardown (Read 469087 times) 0 Members and 3 Guests are viewing this topic. #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #650 on: February 20, 2016, 04:53:04 am » More on the Dallas IC up-grade. I powered up the 2465B again today and all appears well, so the FM16W08 appears to be holding its memory. But, just in case, I ordered a spare DS1225AD today to copy the scopes cal data to and then put it in storage, just in case. I have a spare FM16W08 and adapter board that I'll do the same with. AT this point, it's cheap & easy backup. Then if we get hit with an EMP or a super solar storm, maybe one will survive, assuming the rest of the scope makes it. DSO - I think there may be some confusion about the data retention of the FM16W08. See data below. The 10 years at 85 Deg. C is hot. At 65 Deg. C (that's 149 F) the data retention is 151 years - according to the Cypress data sheet. So, if I keep it below 75 F, it might even out live us. There was a concern of shorting the pins on the Dallas DS1225Y when removing it. But after looking at the expanded diagram of that IC posted by Holden, I concluded that if they were momentarily shorted, it would be through high impedance paths. So the battery should survive that. And I used the smallest tip on the vacuum desolder tool, so I don't think they ever shorted. After removing the DS1225Y, I was about to plug it into a conductive piece of black foam, but then had second thoughts about it discharging the battery through possible high impedance paths via. the foam. But, then while looking up DSO's concern about the IC's life, I found where the data sheet seems to think it's not a problem - if I understand it correctly. So, you're probably right - not a problem. Also, for the sake of science, I want to excavate through the top of the old IC to the battery, to measure the remaining voltage. If done right, the IC will not be functionally harmed. DSO - When using my vacuum desolder tool, most of the pins desoldered ok. But half dozen or so pins were a challenge. It just would not pull all the solder out. Even turned the board upside down to get help from gravity. Cleaned the tip orifice, but it was clear and the vacuum was working at the tip. Since this is my first real application with it, I wanted to ask what temp you had yours set to? This one was set to 300 C. Another possibility was, the tip might be too small to transfer enough heat through the multi-layer board. Below are photos of the board before and after replacing the Dallas memory IC. #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #651 on: February 20, 2016, 05:01:46 am » Well, the pictures were lost again. This is the most unfriendly picture loader. No way to see in advance what is ready to be posted. Will try to get 3 of them back here. #### MSO • Contributor • Posts: 42 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #652 on: February 22, 2016, 07:18:24 am » OLD-E, On the DS1225Y, the battery is located near the bottom of the chip, not the top (at least that's what I gleaned from Holden's write up). Holden goes on to recommend evacuating pin 7 on the embedded DS1218 memory controller if you need direct access to the battery's positive node. When sucking solder with a Hakko or any other desoldering gun, it's common to have some pin/hole combinations not fully evacuate. The normal course of action is to resolder those pins and then try again. The fresh solder aids the heat transfer from the gun through the pin/hole and usually results in complete evacuation of the solder. It's recommended that when using a desoldering gun to get the pin into the orifice of the gun and apply side pressure on the pin to aid heat transfer. I also try to get the gun to the solder filet itself, but not far enough to actually touch the solder pad. When I see the solder melt, then I apply the vacuum and that usually clears the solder. Some of the pins have been bent over to hold the chip in place during the manufacturing process, so those pins require a two step process; just touch the solder filet with the gun and suck up as much solder as will easily come. Then straighten the pin, apply new solder if needed and then follow the normal sequence. Finally, you will encounter situations where the soldered pad has little thermal relief; the pad is part of a large trace or ground plane that dissipates the heat from the desoldering gun, making it difficult to get an easy flow of solder. In these situations, you may need to raise the temperature of your desoldering gun to get enough heat to get the solder to flow easily. Having said all that, I set Hakko 808 to 350 C when desoldering. That high of a temperature requires a lot of care not to lift a pad. While it's good to touch the pin and the top of the solder filet, you really don't want to touch the pad itself and you don't want to apply the heat for too long. If the joint doesn't fully desolder, move on the next joint and come back to the failed one later. Apply new solder to the failed joints and then try those again. There may be a joint or two (I didn't encounter any when re-capping the 2467B) that simply won't submit to the desoldering gun. With those, you'll either have to heat with an iron while simultaneously lifting the component or, in the worst case, cut the pin off on the component side of the board and then use solder wick to clear the hole. Almost forgot; I used a desoldering tip with a 1.0mm hole for all of the components on the 2467B. The two large 290uf caps C1021 & C1022 on the A3 board had the largest diameter leads and pushed my 1.0 mm tip to its limits, but with a little solder wick to remove the excess solder on the top of board, I was able to use the gun to clear those whales properly. As an aside, I checked all the caps that I pulled out of my 2467B and found only the RIFA caps (those rectangular caps with translucent plastic cases) were out of spec. All the aluminum electrolytics, even those that had started leaking, were still measuring OK, so it looks as though I caught them all in time. It's good to hear that you were able to save your calibration data; I know I sighed a huge sigh of relief when I was able to read mine with my programmer. I was paranoid enough, though, to read the bin file and double check that all the calibrations I had recorded via EXER 02 before starting work on the scope matched those in the bin file. I now have backups of the DS1225Y on another DS1225Y, on my hard drive, on a backup hard drive, on a USB memory stick stored with my manual and on a CD rom. How's that for OCD? #### BravoV • Super Contributor • Posts: 7356 • Country: • +++ ATH1 ##### Re: Tektronix 2465B oscilloscope teardown « Reply #653 on: February 23, 2016, 03:44:08 am » I now have backups of the DS1225Y on another DS1225Y, on my hard drive, on a backup hard drive, on a USB memory stick stored with my manual and on a CD rom. How's that for OCD? Not bad for the OCD , but you forgot putting it into cloud storage, and another usb stick stuck with duct tape inside the scope. « Last Edit: February 23, 2016, 04:33:13 am by BravoV » #### MSO • Contributor • Posts: 42 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #654 on: February 23, 2016, 04:18:44 am » I'll get there Bravo, I'm just getting warmed up. BTW, thanks for all your contributions to this thread. I don't think I would have been successful without the myriad pictures, tips, observations and comments that you and many others have made here. #### BravoV • Super Contributor • Posts: 7356 • Country: • +++ ATH1 ##### Re: Tektronix 2465B oscilloscope teardown « Reply #655 on: February 23, 2016, 05:08:45 am » MSO, my pleasure and thanks to other contributors too that made up this thread. PS : Actually I created this thread just to share this beast teardown photos at 1st, never thought it grows into this giant. #### FireDragon • Regular Contributor • Posts: 62 ##### 2465BCT Repair and Modifications « Reply #656 on: February 23, 2016, 08:26:42 am » I got the latest set of parts in and made the latest round of repairs and modifications. I replaced the feed through fan capacitor (which was shorted) and the fan with a higher capacity fan with about the same noise level. I also added an 0.1uF ceramic bypass capacitor to help suppress noise on the fan input voltage to prevent it from feeding back into the scope. That should reduce the overall temperature inside the scope because the default will be higher airflow. Additionally, I was unhappy with the noise level on the various power supply voltages. All of them were well within specification, but the noise was still much higher than I liked. So, on board A2A1 I replaced the 100uF filtering capacitors - C1260, C1280, C1300, C1330, C1350 - with 150uF capacitors. Additionally, I added a 150uF capacitor across VR1293. The reason for this, is that all of the output voltages have some sort of filtering across them - except for the 10V reference voltage. Its noise specifications are very loose, and any noise on it will be copied by all of the other output voltages because they will attempt to match the reference voltage, noise and all. I also added a 50 ohm resistor between ground and the Option 1E input signal to provide proper termination. Finally, I replaced the 10.000Mhz crystal with a 10.0000Mhz crystal. To support that I changed C2550 and C2551, respectively to 33pF and 39pF (this was actually no change for C2550, but I replaced it anyway). This gives me 10.00014Mhz whereas before it had been somewhere around 9.99280Mhz. I am measuring the frequency with the scope itself and I have reason to believe that it is reading high - but by how much I don't know yet since the scope isn't calibrated yet and I don't have a calibrated frequency reference. Changing the crystal won't improve the scope's performance or accuracy, but it should bring the resulting calibration constants closer to "zero" and so less likely to hit an extreme point. I couldn't easily get the frequency close to 10Mhz. When I had C2551 at 33uF I had 10.00047Mhz, and 39uF gives me 10.00014. The next step up would be 43uF which would probably be too much - especially if the scope is reading the frequency high. The result of the power supply changes is a drastic reduction in noise on the all of the output voltages. I probably should have replaced C1220 and C1240 but I didn't have any higher voltage, high capacity capacitors on hand. Still, most of the output voltages now have total noise at around 1-2mv with no observable line noise. One of the +5v supplies (J119-2) has about 20mv noise which may be line related, but up to 30mv line related noise is allowed and up to 150mv total noise. So, a definite win! I'm getting pretty good at taking this scope apart and putting it back together with all of the practice I'm getting! #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #657 on: February 25, 2016, 06:21:23 am » LOL - - Following the stress, and then the joy, of getting the cal data out of the old DS1225Y, I understand the desire to have back up memories for the backups, etc. So it might be said; the more intense the emotional experience, the more memory backups. My new DS1225AD arrived today, so I can program it along with a spare FM16W08. Then I'll have 3 back ups (counting the hard drive) in addition to the new FM IC installed in the scope. In addition to all that, I still have the old DS1225Y removed from the scope. I excavated through the epoxy of its underside this evening to get to pin 7 on the DS1218 as MSO suggested. It probably was easier than going through the top to reach the top of the battery case. The end result was that I reached the pin without damaging the function of the IC (I think). I was surprised to measure the battery at 3.304 volts! So it appears that it still has considerable life left. Dr H. Holden claims in his article that it remains functional down to ~1.8 volts. But in the process, I ran into an unexpected snag. When I reached the first sign of metal, while following Holden's sketch on page 6 of his article, I tried to read the voltage, but kept getting zero. So I kept excavating and testing with no progress. Finally, returning to square 1 and working back through everything, I found that his illustration was wrong! Important to know for anyone else trying the same. His illustration shows a bottom view of the DS1225Y. But the 2 rows of pins are transposed - move the upper row straight down and the lower row up and all is well. Looking at the illustration, I had the meter lead clipped to the "Gnd" on the lower right pin, and it should have been on the upper right. He has another illustration on page 21 which is a top view of the DS1225Y and those pins are not transposed. Over the last many days, I've powered up the scope to confirm that all is well. But recently the previously mentioned bug occasionally reappears. It always happens on power on when it goes through its health check. Some times it stops with all 1's and dashes where the readouts should be. It has also read "TEST 03 PASSED." One time it stopped reading "ALL 00 FAIL 03." Another time it read "TEST 03 FAIL 02," which the lookup says is "Readout Ram failure." It did this this before I replaced the DS1225Y. But in all cases, pressing "A/B TRIG" returns the scope to normal operation where everything is perfect. I even retuned to test mode and started it looping through all the startup tests and left it run for an hour with no hiccups. I have some remote ideas to look at, and I want to recap the power supply. Thanks to MSO I have his capacitor list for guidance. FireDragon - Thanks for sharing. A lot of good info on this site. Starting at the beginning with BravoB's 1st entry it took a long time to read it all. But like a best seller, it was hard to put down. Thanks DSO for all the tricks in using a vacuum tool. Yes, I was using the small, 1mm tip too. I'll get more experience on the PS board. #### Bryan • Frequent Contributor • Posts: 606 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #658 on: February 25, 2016, 06:58:22 am » I well up in tears when I read users who successfully replaced their DS12257. I tried replacing mine a year or so ago, did everything right in prep, unsoldering properl. Went to read it on the programmer and all the data was 00 or FF, can't remember, something went wrong during the removal process and it erased the memory. Tried all the tricks I could dig up in trying to recover, but I guess the battery in mine was at the point that the removal or memory read brought the voltage down ever so slightly past the threshold and everything was lost. Didn't know at the time that the memory was available on screen, could have taken some pictures and added the data back manually to the new chip with the programmer. Oh well. -=Bryan=- #### casinada • Frequent Contributor • Posts: 600 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #659 on: February 25, 2016, 07:10:31 am » If you lost your data ( I did) you can use somebody else's backup as a starting point, that way your scope might be out of calibration but it will be easier than to start from scratch. In some cases it will come back without showing errors. Calibrating can be lots of fun. I had to do it with my 2465BDM #### Bryan • Frequent Contributor • Posts: 606 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #660 on: February 25, 2016, 07:11:54 am » Yes, that's what I did, had to try a number of backups that are available, I think I came close, but it still will need a full calibration. -=Bryan=- #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #661 on: February 25, 2016, 04:04:30 pm » Bryan - Yes, I remember reading a ways back about losing your memory. Bad news! But maybe - I'm new to all this - but some thoughts are, you might try reinstalling it back into your scope to see if it may still work. If you happen to have installed a socket, it would be an easier install. Another (better) thought is; according to Holden, the battery could go below the threshold level of the embedded DS1218 battery controller. But the memory is not actually gone. In that case, an external battery could be piggy backed on to the DS1225Y to bring the voltage back up. Then it could be read on the programmer. As written below, I just excavated down to pin 7 of the battery controller for the positive battery connection. Then the return line goes to pin 14 on the DS1225Y. That can be done in a hour or so. Another possibility could be that a lack of static control might have caused a non-fixable problem. From what I could see, shorting the pins should not be a big deal. Maybe if shorted for a long time (years) , it might speed up the discharge of the battery. #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #662 on: February 25, 2016, 08:10:08 pm » More thoughts. To avoid soldering to the the dead DS1225 before inserting it into the programmer, one could connect the external battery return wire to pin 14 by just looping it around the top of the pin, or inserting the return wire into the pin receptacle of the zero insertion force socket of the programmer. Likewise, once pin 7 (+battery point) is exposed, one could just hold a sharp pointed probe on the pin during the programmer read time. Again, no soldering to the DS2018 pin. And it is my guess, that even if your battery connections are noisy, the existing battery will take on a certain amount of charge which will produce a constant voltage above the threshold to the DS1225Y during the read time. This is all for a temporary read, so make shifting might be a safer and quicker way to go. Picture below is of my DS1225 after excavation. And I removed more potting than necessary for a temporary connection. The exposed metal pin is visible next to the body of the DS2018. Excavation was accomplished with a pointed X-acto Knife and a needle pointed probe tip. The biggest challenge was in to minimize bending the DS1225Y pins during excavation. #### Bryan • Frequent Contributor • Posts: 606 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #663 on: February 26, 2016, 04:44:51 am » Thanks, yes I had tried that when I originally attempted to replace the Dallas. No luck at the time, may try again, perhaps I may get lucky. But doubt it. The voltage of the battery was less than a volt. -=Bryan=- #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #664 on: February 26, 2016, 04:00:11 pm » Oh - bad news! My condolences. A battery voltage down to 1 volt is probably not enough to keep it alive. I'm curious (for the sake of science) did you happen to measure the battery voltage again after the external battery had been disconnected? Based on my battery experience, I'm betting that the nearly expired battery will take on a charge that will last much longer than needed to read the memory, maybe even for days. If that is true than one would not need a makeshift probe on the underside of the DS when reading. Thanks for sharing. #### Bryan • Frequent Contributor • Posts: 606 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #665 on: February 27, 2016, 09:39:31 am » Oh - bad news! My condolences. A battery voltage down to 1 volt is probably not enough to keep it alive. I'm curious (for the sake of science) did you happen to measure the battery voltage again after the external battery had been disconnected? Based on my battery experience, I'm betting that the nearly expired battery will take on a charge that will last much longer than needed to read the memory, maybe even for days. If that is true than one would not need a makeshift probe on the underside of the DS when reading. Thanks for sharing. The voltage of the battery is after it had been in storage for over a year. Always thought there was a hope of recovery, but nope. Tried again on the programmer and all I get is 0x00 and 0xFf in what seems to be 3-4 banks of memory. I am not sure what went wrong, could have been anything. Still kicking myself for not taking pictures of the CRT that shows the memory allocations. Would have been a lot of work to rebuild, but at least I would have had something. I have a TG501 and a SG503 so think I have the basics for a decent recalibration by myself. Just need to put together some accurate voltage references. Although going by memory the voltage levels have to be a square wave. Can probably use a function generator and calibrate the voltage level. Just needs to be within 3% anyways. I find I don't use the scope much, handy if I want to look at some higher frequencies than what my Rigol DS1052E can do. Afraid the days of the analog scope are becoming closer to a end, just so much more you can do with a digital scope. Although nothing sweeter than the glow of a CRT, one of the reasons I still keep it<g> -=Bryan=- #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #666 on: March 01, 2016, 06:33:28 am » Seems that everything has its share of kinks (trivia reporting). Got together with my programmer friend yesterday to get the stored cal data transferred from my old XP computer, into the 2 spare IC's. I had soldered the spare FM16W08 onto the adapter board and the spare DS1225AD had arrived, so repeating the data transfers should have been an easy task. We started out by reading what was on the FM IC, and that appeared to work. So we tried to perform some simple tests, like entering all ff's for example, and it would not complete the write, Tried transferring real data into it with similar "failed" status. So thinking that maybe the FM IC had a problem, we plugged in the new Dallas IC and had similar problems. After spending a few frustrating hours on trying to get something to work, we decided to at least get a screen print of the data file from the programmer so that I could have a hard copy for my note book. But then could not get the printer to work without freezing up. Decision was finally made to quit for the evening with a plan to start at square one on Monday. So, today I put in a call to my Computer Angel service, that I have available on call, to go through the older XP computer to get it cleaned up and debugged if needed. Once I feel comfortable with that, then the plan is to reload software into the programmer, etc. Eventually we'll get it - I hope. But at least we have a new working FM IC in the scope - whew! Today the computer allowed me to get a hard copy of the cal data printed out so something is strange there. With cal data printed out, I was then able to easily compare that with the hand copied data from the scope CRT. It takes awhile, but it can be done. This turned up about a dozen errors. From that, I compared the errors directly with the digits on the CRT. Bottom line was, in all cases the printed out data from the programmer agreed with the data read off the scope CRT. This is what should be, but just wanted to confirm that. In most cases, the errors were where the spine of a B or the left edge of a 3 was perfectly centered under a vertical reticle line. So using an eye loop and fuzzing out the focus a little, I was able to see the spine of the B showing on both sides of the reticle line. So I think my hand copied data off the scope screen, and the printed data from the programmer is now all correct. So while some things are coming together, it's not without its challenges and set backs. #### MSO • Contributor • Posts: 42 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #667 on: March 03, 2016, 05:56:33 am » OLD-E, I just spent the better part of hour writing a response to your last post and then a made a typo of some sort and lost the entire post. The replay window simply disappeared. That’s probably much better for you as I now, by rewriting the response, I can eliminate some of redundancies and extraneous comments. I would expect some differences to exist between the NVRAM currently installed in the ‘scope and the DS1227 that you replaced. Running time, the On/Off count and the set up in effect the last time you used the scope come to mind. I’m not sure if there would be any changes made to the calibration area of the NVRAM; for instance balancing the channel 1/channel 2 inputs might end up there. When you have doubt about your programmer reading the device properly, you’ll find a tool such as Beyond Compare to be invaluable. Simply read the NVRAM device in the programmer, save it to a binary file then read it again and save to a second binary file. Using Beyond Compare, compare the two files just read. If there are differences, then you know the programmer isn’t reading properly. Be aware though, that many programmers keep the data they last read in an internal buffer and won’t modify that saved data if they can’t read the device a second time. So it’s best to force the programmer to dump the data in its internal buffer before reading the device a second time. If the programmer’s software allows you to directly edit its buffer, insert a series of known values near the start of the buffer; I’ll usually use a few series of AA, 55, AA, 55 values to ensure that the programmer actually read the device a second time. Depending upon your programmer/software, you may have to close the software and restart it again or try loading a different, unrelated binary file between first and second reads. You can use this approach when comparing reads between two different NVRAM devices as well. If one of the devices has more memory than the other, then just ensure that the two devices match up to the size of the smallest device. #### ChunkyPastaSauce • Supporter • Posts: 536 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #668 on: March 03, 2016, 06:09:01 am » Question... I bought one for basically nothing because from a university. Basically nothing because it doesn't work....other than that,very good condition The calibration is probably still there as the date code isn't that far out. It likely died due to a very failed fan. I measured the voltage rails..... they are all over the place. Some of them are low and if I remember correctly....some of them higher than they should be. So my question...... is it worth trying to fix the PSU when some of the rails are higher by like 150%? Or is it likely the higher voltage damaged other parts of the scope and not worth the effort? « Last Edit: March 03, 2016, 06:10:32 am by ChunkyPastaSauce » #### MSO • Contributor • Posts: 42 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #669 on: March 03, 2016, 07:36:16 am » ChunkyPastaSauce, I'm not knowledgeable enough to answer your primary question (is it worth it), but having re-capped both power supply boards and the A5 control board for roughly $50, I'd say it's worth a shot. My suspicion is that the overheating may have damaged a couple of the main board chips, but for$50 and little time and effort (for me it's an enjoyable hobby) you can determine if its worth pursuing further. #### Bryan • Frequent Contributor • Posts: 606 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #670 on: March 04, 2016, 02:01:46 am » The 2465B is pretty rugged and well designed, I would invest in the time and cost to get the power supply fixed and go from there. More than likely the caps are toast and there may be some other issues on the power supply board. If you scour the service manual you will see that much of the other circuitry on the power supply board and other boards have pretty good protection if something goes wrong on the power supply side of things. -=Bryan=- #### ChunkyPastaSauce • Supporter • Posts: 536 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #671 on: March 04, 2016, 03:10:55 am » Ok, thanks to both It looks to be a nice scope, certainly better than the working one I have now (which isn't bad either tek2235). Typically pass on doing anything on PSU high sides.... basically they frighten me. Also the fact there is a CRT. Right now, I'm nervous about getting the board out..... Ive checked the main caps on the PSU for residual charge at the black plastic shield check points, they are empty. Anything else on the PSU board I should be aware of that may have the potential to make me have regrets messing with it? Or generally pretty safe if the charge check points, at the black plastic shield, are around 0V? I've check all the caps for obvious problems that can be checked, without having to pull the PSU boards out. Thanks again. « Last Edit: March 04, 2016, 03:14:23 am by ChunkyPastaSauce » #### Old-E • Regular Contributor • Posts: 57 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #672 on: March 04, 2016, 03:31:56 am » Chunky, I happen to love the fine qualities of my Tektronix 2465B scope (assuming that's what you have). For me, it's worth the dollars & effort to have a reasonably great and high performance scope, not to mention the education and hobby element. From what I've seen on E-bay, these scopes in good clean condition and working perfectly are going for $1,000 -$1,400. I saw a copy of an old brochure (circa 1990) advertising a new scope like mine for $9,200. But if your thoughts are to get it working in order to turn a quick profit, well, that might work if you value your time at 10 cents an hour (exaggerating, well maybe). If your interests are profit (nothing wrong with that), then you might be better off parting it out on E-Bay or selling it as is. My concern would be that the failed fan, followed by failed power supplies, leading to over voltage conditions, could have wiped out many IC's, etc. This could be a costly / time consuming over hall relative to the numbers we're talking about. You say the memory IC has a recent date stamp. But if it's older than 10 years, which is the guaranteed life, then you should be looking at replacing it. The cost of a replacement IC, or later version with no life limits, plus a programmer will total about$150. The other parts, as MSO indicated, are fairly cheap, providing all the mechanical stuff is ok. Then there is quite a bit of labor. But, if you are interested in, and reasonably ok at working with electronics, and would find a project like this rewarding, then this could work well for you. Forums, like this one, are a treasure trove of information and help. Others here may have different thoughts. #### ChunkyPastaSauce • Supporter • Posts: 536 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #673 on: March 04, 2016, 06:06:00 am » Hi Old-E Yep it's the 2465B If I can get it working, not selling it, unless it's not for me for some reason. I bought it for fun at \$35 knowing it would be some level of a project and knowing I might not be able to get it working at my skill level, somewhere beginner-intermediate hobby range (I'm a ME not EE). If I can, great, if I can't then no big deal and Ill probably put it up for anyone willing to pay shipping if they want to have ago at a fairly desirable analog scope or need parts. I went ahead and pulled the inverter and regulator boards out.. Nothing too horrendous looking (the board is very clean by the way, no dust except a cm or 2 at the fan entrance). The rectangular cap packages, a number of those are somewhat bloated...almost all have hairline cracks. The other caps looked ok visually, except possibly some of the silver guys near to bottom but hard to tell (Edit - found some baddies after removing ). I checked the diodes and all of them seem reasonable.. I thought I found some bad ones but after lifting they were fine... got excited for a few. So Ill pull the caps; although I plan on replacing all anyway, ordering ESR meter for fun. On the date code, I was reading about that. I thought it was 20 year life but now looking at the spec sheet..... uhoh lol. We will see... Wouldn't the cal data be kinda useless after repairing the power supply board, it would need a recal anyway? Thanks again « Last Edit: March 04, 2016, 06:35:33 am by ChunkyPastaSauce » #### ChunkyPastaSauce • Supporter • Posts: 536 • Country: ##### Re: Tektronix 2465B oscilloscope teardown « Reply #674 on: March 04, 2016, 09:51:35 pm » Baddies. All of these probably, hairline cracks on case likely ballooned caps « Last Edit: March 04, 2016, 09:53:55 pm by ChunkyPastaSauce » Smf	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2023182362318039, "perplexity": 3446.7923306648277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703531429.49/warc/CC-MAIN-20210122210653-20210123000653-00132.warc.gz"}
https://brilliant.org/practice/partial-fractions-repeated-factors/	× ## Partial Fractions Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum. # Repeated Factors If the following is an identity in $$x$$: $\frac{4x^2+27x+33}{(x-1)(x+3)^2}=\frac{a}{x-1}+\frac{bx+c}{(x+3)^2},$ what is the value of $$a+b+c$$? Suppose $$a$$, $$b$$, and $$c$$ are constants such that the following holds for all real numbers $$x$$ such that all of the denominators are nonzero: $\frac{9}{x(x+1)^2}=\frac{a}{x}+\frac{b}{x+1}+\frac{c}{(x+1)^2}.$ What is the value of $$abc?$$ If the following is an identity in $$x$$: $\frac{17x^3+7x^2-17x-7}{(x-1)^2(x+1)^2}=\frac{A}{x-1}+\frac{B}{x+1},$ what is the value of $$A \times B?$$ If the following is an identity in $$x$$: $\frac{11x^2+90x+99}{(x-3)^2(x+3)^2}=\frac{A}{(x-3)^2}-\frac{B}{(x+3)^2},$ what is the value of $$A \times B?$$ If the following is an identity in $$x$$: $\frac{-13x+32}{(x+1)(x-2)^2}=\frac{a}{x+1}+ \frac{ b} { (x-2) } + \frac{ c} { (x-2)^2} ,$ what is the value of $$a+b+c$$? ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9135594367980957, "perplexity": 150.32241100957683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541170.58/warc/CC-MAIN-20161202170901-00468-ip-10-31-129-80.ec2.internal.warc.gz"}
https://docs.scipy.org/doc/scipy-1.5.3/reference/generated/scipy.integrate.solve_bvp.html	# scipy.integrate.solve_bvp¶ scipy.integrate.solve_bvp(fun, bc, x, y, p=None, S=None, fun_jac=None, bc_jac=None, tol=0.001, max_nodes=1000, verbose=0, bc_tol=None)[source] Solve a boundary value problem for a system of ODEs. This function numerically solves a first order system of ODEs subject to two-point boundary conditions: dy / dx = f(x, y, p) + S * y / (x - a), a <= x <= b bc(y(a), y(b), p) = 0 Here x is a 1-D independent variable, y(x) is an N-D vector-valued function and p is a k-D vector of unknown parameters which is to be found along with y(x). For the problem to be determined, there must be n + k boundary conditions, i.e., bc must be an (n + k)-D function. The last singular term on the right-hand side of the system is optional. It is defined by an n-by-n matrix S, such that the solution must satisfy S y(a) = 0. This condition will be forced during iterations, so it must not contradict boundary conditions. See [2] for the explanation how this term is handled when solving BVPs numerically. Problems in a complex domain can be solved as well. In this case, y and p are considered to be complex, and f and bc are assumed to be complex-valued functions, but x stays real. Note that f and bc must be complex differentiable (satisfy Cauchy-Riemann equations [4]), otherwise you should rewrite your problem for real and imaginary parts separately. To solve a problem in a complex domain, pass an initial guess for y with a complex data type (see below). Parameters funcallable Right-hand side of the system. The calling signature is fun(x, y), or fun(x, y, p) if parameters are present. All arguments are ndarray: x with shape (m,), y with shape (n, m), meaning that y[:, i] corresponds to x[i], and p with shape (k,). The return value must be an array with shape (n, m) and with the same layout as y. bccallable Function evaluating residuals of the boundary conditions. The calling signature is bc(ya, yb), or bc(ya, yb, p) if parameters are present. All arguments are ndarray: ya and yb with shape (n,), and p with shape (k,). The return value must be an array with shape (n + k,). xarray_like, shape (m,) Initial mesh. Must be a strictly increasing sequence of real numbers with x[0]=a and x[-1]=b. yarray_like, shape (n, m) Initial guess for the function values at the mesh nodes, ith column corresponds to x[i]. For problems in a complex domain pass y with a complex data type (even if the initial guess is purely real). parray_like with shape (k,) or None, optional Initial guess for the unknown parameters. If None (default), it is assumed that the problem doesn’t depend on any parameters. Sarray_like with shape (n, n) or None Matrix defining the singular term. If None (default), the problem is solved without the singular term. fun_jaccallable or None, optional Function computing derivatives of f with respect to y and p. The calling signature is fun_jac(x, y), or fun_jac(x, y, p) if parameters are present. The return must contain 1 or 2 elements in the following order: • df_dy : array_like with shape (n, n, m), where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d (y_q)_j. • df_dp : array_like with shape (n, k, m), where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d p_j. Here q numbers nodes at which x and y are defined, whereas i and j number vector components. If the problem is solved without unknown parameters, df_dp should not be returned. If fun_jac is None (default), the derivatives will be estimated by the forward finite differences. bc_jaccallable or None, optional Function computing derivatives of bc with respect to ya, yb, and p. The calling signature is bc_jac(ya, yb), or bc_jac(ya, yb, p) if parameters are present. The return must contain 2 or 3 elements in the following order: • dbc_dya : array_like with shape (n, n), where an element (i, j) equals to d bc_i(ya, yb, p) / d ya_j. • dbc_dyb : array_like with shape (n, n), where an element (i, j) equals to d bc_i(ya, yb, p) / d yb_j. • dbc_dp : array_like with shape (n, k), where an element (i, j) equals to d bc_i(ya, yb, p) / d p_j. If the problem is solved without unknown parameters, dbc_dp should not be returned. If bc_jac is None (default), the derivatives will be estimated by the forward finite differences. tolfloat, optional Desired tolerance of the solution. If we define r = y' - f(x, y), where y is the found solution, then the solver tries to achieve on each mesh interval norm(r / (1 + abs(f)) < tol, where norm is estimated in a root mean squared sense (using a numerical quadrature formula). Default is 1e-3. max_nodesint, optional Maximum allowed number of the mesh nodes. If exceeded, the algorithm terminates. Default is 1000. verbose{0, 1, 2}, optional Level of algorithm’s verbosity: • 0 (default) : work silently. • 1 : display a termination report. • 2 : display progress during iterations. bc_tolfloat, optional Desired absolute tolerance for the boundary condition residuals: bc value should satisfy abs(bc) < bc_tol component-wise. Equals to tol by default. Up to 10 iterations are allowed to achieve this tolerance. Returns Bunch object with the following fields defined: solPPoly Found solution for y as scipy.interpolate.PPoly instance, a C1 continuous cubic spline. pndarray or None, shape (k,) Found parameters. None, if the parameters were not present in the problem. xndarray, shape (m,) Nodes of the final mesh. yndarray, shape (n, m) Solution values at the mesh nodes. ypndarray, shape (n, m) Solution derivatives at the mesh nodes. rms_residualsndarray, shape (m - 1,) RMS values of the relative residuals over each mesh interval (see the description of tol parameter). niterint Number of completed iterations. statusint Reason for algorithm termination: • 0: The algorithm converged to the desired accuracy. • 1: The maximum number of mesh nodes is exceeded. • 2: A singular Jacobian encountered when solving the collocation system. messagestring Verbal description of the termination reason. successbool True if the algorithm converged to the desired accuracy (status=0). Notes This function implements a 4th order collocation algorithm with the control of residuals similar to [1]. A collocation system is solved by a damped Newton method with an affine-invariant criterion function as described in [3]. Note that in [1] integral residuals are defined without normalization by interval lengths. So, their definition is different by a multiplier of h*0.5 (h is an interval length) from the definition used here. New in version 0.18.0. References 1(1,2) J. Kierzenka, L. F. Shampine, “A BVP Solver Based on Residual Control and the Maltab PSE”, ACM Trans. Math. Softw., Vol. 27, Number 3, pp. 299-316, 2001. 2 L.F. Shampine, P. H. Muir and H. Xu, “A User-Friendly Fortran BVP Solver”. 3 U. Ascher, R. Mattheij and R. Russell “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations”. 4 Cauchy-Riemann equations on Wikipedia. Examples In the first example, we solve Bratu’s problem: y'' + k exp(y) = 0 y(0) = y(1) = 0 for k = 1. We rewrite the equation as a first-order system and implement its right-hand side evaluation: y1' = y2 y2' = -exp(y1) >>> def fun(x, y): ... return np.vstack((y[1], -np.exp(y[0]))) Implement evaluation of the boundary condition residuals: >>> def bc(ya, yb): ... return np.array([ya[0], yb[0]]) Define the initial mesh with 5 nodes: >>> x = np.linspace(0, 1, 5) This problem is known to have two solutions. To obtain both of them, we use two different initial guesses for y. We denote them by subscripts a and b. >>> y_a = np.zeros((2, x.size)) >>> y_b = np.zeros((2, x.size)) >>> y_b[0] = 3 Now we are ready to run the solver. >>> from scipy.integrate import solve_bvp >>> res_a = solve_bvp(fun, bc, x, y_a) >>> res_b = solve_bvp(fun, bc, x, y_b) Let’s plot the two found solutions. We take an advantage of having the solution in a spline form to produce a smooth plot. >>> x_plot = np.linspace(0, 1, 100) >>> y_plot_a = res_a.sol(x_plot)[0] >>> y_plot_b = res_b.sol(x_plot)[0] >>> import matplotlib.pyplot as plt >>> plt.plot(x_plot, y_plot_a, label='y_a') >>> plt.plot(x_plot, y_plot_b, label='y_b') >>> plt.legend() >>> plt.xlabel("x") >>> plt.ylabel("y") >>> plt.show() We see that the two solutions have similar shape, but differ in scale significantly. In the second example, we solve a simple Sturm-Liouville problem: y'' + k*2 y = 0 y(0) = y(1) = 0 It is known that a non-trivial solution y = A * sin(k * x) is possible for k = pi * n, where n is an integer. To establish the normalization constant A = 1 we add a boundary condition: y'(0) = k Again, we rewrite our equation as a first-order system and implement its right-hand side evaluation: y1' = y2 y2' = -k*2 y1 >>> def fun(x, y, p): ... k = p[0] ... return np.vstack((y[1], -k*2 y[0])) Note that parameters p are passed as a vector (with one element in our case). Implement the boundary conditions: >>> def bc(ya, yb, p): ... k = p[0] ... return np.array([ya[0], yb[0], ya[1] - k]) Set up the initial mesh and guess for y. We aim to find the solution for k = 2 * pi, to achieve that we set values of y to approximately follow sin(2 * pi * x): >>> x = np.linspace(0, 1, 5) >>> y = np.zeros((2, x.size)) >>> y[0, 1] = 1 >>> y[0, 3] = -1 Run the solver with 6 as an initial guess for k. >>> sol = solve_bvp(fun, bc, x, y, p=[6]) We see that the found k is approximately correct: >>> sol.p[0] 6.28329460046 And, finally, plot the solution to see the anticipated sinusoid: >>> x_plot = np.linspace(0, 1, 100) >>> y_plot = sol.sol(x_plot)[0] >>> plt.plot(x_plot, y_plot) >>> plt.xlabel("x") >>> plt.ylabel("y") >>> plt.show() #### Previous topic scipy.integrate.complex_ode.successful #### Next topic Interpolation (scipy.interpolate)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7394933104515076, "perplexity": 3370.887038649028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991553.4/warc/CC-MAIN-20210510235021-20210511025021-00152.warc.gz"}
https://www.physicsoverflow.org/25631/help-writing-basis-clifford-algebra-%24cl-and-the-quotient-%24cl	# Help in writing the basis for Clifford algebra $Cl^2 (W)$ and the quotient $Cl^3 (W) / Cl^2 (W)$ + 4 like - 0 dislike 111 views Consider a basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W=V \oplus V^$. I need help to write the basis for Clifford algebra $Cl^2 (V \oplus V^)$=$Cl^2 (W)$. By definition, $$Cl^2 (V \oplus V^) := \bigoplus_{l =0 }^2 T^l(W)/I,$$ where $I$ is two sided ideal generated by $(xy+yx)-b(x,y).1$ where $b$ is the canonical bilinear form attached to $W.$ I wrote $$Cl^2 (V \oplus V^) = \mathbb{C}/I \oplus W / I \oplus W \otimes W/I.$$ Is this correct? What is the basis for $Cl^2 (V \oplus V^)$? Also, how can I write basis for quotient space $Cl^3(W)/Cl^2(W)$? For the latter, I am assuming that it is enough to understand the basis for $Cl^3(W)$. Then I can simply consider the coset of each basis element. Right? Added: Here is the context of the question: Let $V$ be a 3-dimensional vector space, $W=V\oplus V^$ and $C(W)$ be the Clifford algebra defined by $W$ and its symmetric form. Take the basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W$. The Lie algebra $\mathfrak{so}(6)$ is embedded as a Lie subalgebra $\mathfrak{s}$ in $C(W)$ by sending a matrix $$\begin{pmatrix} A & B \\ C & -A^t \end{pmatrix}$$ with $B,C$ anti-symmetric to the element $$\sum_{i,j} A_{i,j} (c_i^{\dagger}c_j-c_jc_i ^{\dagger})+B_{i,j}c_i ^{\dagger}c_j^{\dagger} +C_{i,j}c_ic_j.$$ 1) First thing is to explain why $\mathfrak{s}$ is represented on the quotient vector space $C^3 / C^2$. I have explained why $\mathfrak{so}(6)$ acts on $C^3/C^2$. But I am unable to explain why $\mathfrak{s}$ acts on $C^3/C^2.$ 2) Secondly, I need to choose a basis of this quotient such that matrices with $A=Diag(\lambda_1,\lambda_2,\lambda_3)$ and $B=C=0$ are diagonalized in this action. I don't get what I am supposed to do in this part. The reason I asked the original question was to understand how does the basis of $C^2$ and $C^3/C^2$ look like because so far I don't understand this. If you are willing to help with actual questions 1) and 2), that is fine too. This post imported from StackExchange Physics at 2014-12-28 09:00 (UTC), posted by SE-user monomorphic Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7375448346138, "perplexity": 311.5252202573104}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742569.45/warc/CC-MAIN-20181115075207-20181115100622-00007.warc.gz"}
https://www.physicsforums.com/threads/vibration-and-frequency-questions.200254/	# Vibration and Frequency Questions 1. Nov 24, 2007 ### Draco 1. The problem statement, all variables and given/known data A 75 g bungee cord has an equilibrium length of 1.20 m. The cord is stretched to a length of 1.80 m, then vibrated at 20 Hz. This produces a standing wave with two antinodes. 2. Relevant equations Linear Density = m/L 3. The attempt at a solution So far I've only found the linear density to be 0.0626kg/m. I'm stuck after this and have no clue on where to start next. 1. The problem statement, all variables and given/known data Earthquakes are essentially sound waves traveling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse seismic waves. These travel at different speeds. The speed of longitudinal waves, called P waves, is 8000m/s . Transverse waves, called S waves, travel at a slower 4500m/s . A seismograph records the two waves from a distant earthquake. 2. Relevant equations Not sure. 3. The attempt at a solution I'm not exactly sure on how to start this. This topic is very new to me. Can someone get me started? Last edited: Nov 24, 2007 2. Nov 24, 2007 ### strikingleafs01 So what I did for this question is, attempt to plug in values for 'm' so, for example 390 = m v/2L 390 = 1 343/2L and find length that way, it turned out that 390 represents the 3rd modal where (m = 3). Velocity is of course 343 m/s because sound will travel thorugh the tube which is open-open. 3. Nov 24, 2007 ### Draco oh yeah, i got that question just after I posted. Thanks for explaining a different method though. I used a different way to find the length of the tube. 4. Nov 24, 2007 ### Draco um can anyone fill me in on these two questions? Similar Discussions: Vibration and Frequency Questions	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9223892092704773, "perplexity": 997.9607229076798}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190753.92/warc/CC-MAIN-20170322212950-00150-ip-10-233-31-227.ec2.internal.warc.gz"}
https://physics.stackexchange.com/questions/92874/how-does-one-calculate-the-polarization-state-of-random-light-after-total-intern?noredirect=1	# How does one calculate the polarization state of random light after total internal reflection How does one calculate the polarization state of random light after having been totally reflected by a single dielectric interface? Please consider pure specular reflexions from a plane interface between two dielectric mediums of indexes $n_1,\,n_2$ when the angle of incidence $\theta_1$ is greater than the critical angle $\arcsin(n_2/n_1)$. • I think you wrote it quite well. By the way, you are like the optics lord here! – Yossarian Jan 9 '14 at 0:08 ## 1 Answer I'll stick to pure total internal, specular reflexion in this answer. When TIR happens, both linear polarisation components are fully relfected, but the phase change (the Goos-Hänchen shift) is different for the two states. In scalar theory the Goos-Hänchen shift (see my answer here) for the two states, but the full vector theory shows a subtle difference. What this means practically is that the two polarisation states seem to reflect from ever so slightly different depths into the denser medium beyond the totally internally reflecting interface. The Fresnel equations still apply in this situation. Now, of course, we get $\sin\theta_t>1$ so that $\cos\theta_t = \sqrt{1-\sin\theta_t^2}$ is imaginary. We interpret the trigonometric functions exactly as they are interpreted in the derivation of the Fresnel equations (e.g. in Reference [1]), to wit, the sine and cosine are the ratios $k_x/k$ and $k_z/k$ of the wavevector components tangential and normal to the interface, respectively. The cosine is imaginary beyond the interface, so the wave is evanescent and exponentially decaying with depth as in my answer cited above. So, from the Fresnel equations: $$r_s = \frac{n_1 \cos \theta_i - n_2 \cos \theta_t}{n_1 \cos \theta_i + n_2 \cos \theta_t} = \frac{n_1 c_1 - i\, n_2 c_2}{n_1 c_1 + i\,n_2 c_2}$$ $$t_s = \frac{2 n_1 \cos \theta_i}{n_1 \cos \theta_i + n_2 \cos \theta_t}= \frac{2 n_1 c_1}{n_1 c_1 + i\,n_2 c_2}$$ $$r_p = \frac{n_2 \cos \theta_i - n_1 \cos \theta_\text{t}}{n_1 \cos \theta_t + n_2 \cos \theta_i} = \frac{n_2 c_1 -i\, n_1 c_2}{n_1 c_2 + i\, n_2 c_1} = -i$$ $$t_p = \frac{2 n_1\cos \theta_i}{n_1 \cos \theta_t + n_2 \cos \theta_i}= \frac{2 n_1 c_1}{n_1 c_2 + i\,n_2 c_1}\tag{1}$$ where $c_1 = \cos\theta_1$ and $c_2 = \text{Im}(\cos\theta_2)$ are both real numbers. Take heed that the reflexion co-efficients are either a complex numbers of the form $r_s=z^*/z$ or $r_p = -i$ and thus have unity magnitude, so all the power is reflected. The Goos-Hänchen shifts for the two orthogonal linear polarisations are: $$\phi_s = -2\,\arg(n_1 c_1 + i\,n_2 c_2) = -2\,\arctan\left(\frac{n_2 c_2}{n_1 c_1}\right)$$ $$\phi_p = -\frac{\pi}{2}\tag{2}$$ which are almost equal in most cases: they deviate more significantly from one another for highly glancing angles $\theta_1\approx\pi/2$ (which condition invalidates the scalar theory of my other answer). In particular, as the incidence angle approaches $\pi/2$ and the reflexion becomes highly grazing, $\phi_s\to-\pi$ and $\phi_p\to-\pi/2$. That is, the TIR mechanism mimics a quarter wave plate. So, for your random polarisation, you are going to either represent it as a known, pure polarisation with a $2\times1$ Jones vector $X = \left(\begin{array}{c}x_p\\x_s\end{array}\right)$ and transform it by: $$X\mapsto U\,X;\;\text{where}\;U = \left(\begin{array}{cc}e^{i\,\phi_p}&0\\0&e^{i\,\phi_s}\end{array}\right)$$ or if the light is depolarised (a mixed state) you will use the Mueller calculus / density matrix formalism: $$\rho\mapsto U\,\rho\,U^\dagger;\;\text{where}\;\rho = \sum\limits_{j=0}^3 s_j \sigma_j$$ $\sigma_0 = {\rm id}$ is the $2\times 2$ identity matrix, $\sigma_j$ the Pauli spin matrice and the co-efficients $s_j$ are the four Stokes parameters as described in my answer on dealing with calculations with depolarised light. Reference: [1] §1.5 "Reflexion and Refraction of a Plane Wave" in the seventh edition of Born and Wolf, "Principles of Optics".	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8930518627166748, "perplexity": 338.79871901537837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487626008.14/warc/CC-MAIN-20210616190205-20210616220205-00186.warc.gz"}
https://chemistry.stackexchange.com/questions/146075/is-cyclohexane-1-3-dicarboxylic-acid-a-correct-iupac-name?noredirect=1	# Is cyclohexane‐1,3‐dicarboxylic acid a correct IUPAC name? I did the numbering by taking all the functional groups in principal chain (scheme A), but my teacher did it differently and proposed the name cyclohexane‐1,3‐dicarboxylic acid (scheme B): The principal chain in my teacher's answer is shorter. Also, he did not put any $$\ce{-COOH}$$ in the principal chain which he could have done. Where am I wrong? • The principal "chain" is the ring of six atoms. Feb 9 '21 at 16:41 • The truth is naming organic compounds is screwy. There is a massive set of rules for naming which is still growing as inventive organic chemists synthesize weirder and weirder compounds. So it takes practice to learn to name properly. – MaxW Feb 9 '21 at 20:01 • There are two problems with your attempt at naming. First the carboxylate is a functional group. So it might or might not be part of the primary carbon chain. Second if the carboxylate group had been a methyl group then the molecule would be 1,3-dimethylcyclohexane (which doesn't completely specify the particular isomer). So there is no one word name for that particular carbon structure, eg toluene. – MaxW Feb 9 '21 at 20:14 The answer to this question is actually pretty interesting, I think. @user55119 's comment that the principal chain is the cyclohexane ring is correct -- but it seems like you understand that that's what the teacher is saying, and you're asking, "why?" The best short answer, too, is imho in your comments: @MaxW 's "naming organic compounds is just screwy" is the bottom-line answer, because there are so many crazy special cases to deal with that it sometimes seems like sorcery to come up with them. But if one is willing to dig into it a little, 99% of them do actually have consistent reasoning behind them -- even if coming up with that reasoning can feel pretty byzantine right up to the point where you get it. This can be particularly true for cyclic compounds, which can in some cases even have more than one valid IUPAC name. In this particular case, the reason your answer is incorrect is because what you defined with your numbering isn't a chain. Per IUPAC, a chain is a sequence of linked units that is bounded by precisely two "boundary units" -- either a branch point or an "end unit", which is itself defined by IUPAC as a unit* of the macromolecule that is bound to only one other unit. Now, this clearly causes problems with cyclic compounds, so the IUPAC has some caveats included within the definition of a chain -- one being that a cyclic macromolecule with no end groups is also a chain. So, a chain is either linear and has two boundary groups terminating it, or it's strictly cyclical and includes no end units. That means a cycloalkane -- including the cyclohexane in your example -- is a valid chain. Your numbering, however, contains one boundary group. The unit at position 1 is a valid boundary group, as it is connected to only one other group. The unit at position 7, however, is not a boundary group; it is connected to a total of two other units (#6 and #2), and worse, it's a non-boundary that you can't continue counting from either since you've already counted the other molecule it attaches to! "Maybe it solves the problem if I start from the unit I called #3, and leave the hanging methyl group to be the last part of the chain instead of starting from it -- so the chain would start at current #3 and end at current #1, 3-4-5-6-7-2-1." (You may want to be looking at your illustration while you read this; it'll probably make the counting I am referring to clearer than just imagining it.) No, that also doesn't work; it obviously that just puts the problem at the other end -- it still has only one boundary unit, because while the unit at the end of that new "chain" attempt, #1, is only connected to one other, the first unit in the "chain" (#3) is connected to two units, not one, so is not a boundary unit. Still stuck! And of course, you can try counting from your #2 instead of your #1 or #3, but then you go all the way around and never have a way to include that methyl group hanging off of #2 -- you'd have to "step" onto #2 again, effectively counting it twice, to then "step" onto the tail CH. (I wish I could make a video of this "pointing" at each unit in the count; it would come out much clearer, I think!) So in effect, really the only way you have to count the components of the macromolecule and get a coherent chain out of it is to start on #2 and end on the connection between #7 and #2 -- meaning you end up with the most practical solution being exactly the caveat that IUPAC settled on in specifying that a cyclic alkene can itself also be counted as a chain. From there, I imagine you can figure out the rest: the cyclohexane as a valid chain is also the longest chain, you start numbering from the functional group, etc. Hope that helped explain the teacher's answer! Edit/addendum: Some in the comment have asked for authoritative sourcing of the above. Good point, and thank you for bringing to my attention that I only stated what I used for sourcing but neither completely cited nor offered links to said sourcing. First post here in the chemistry StackExchange; please forgive my error. IUPAC's authoritative reference work for nomenclature is their so-called Gold Book -- the Compendium of Chemical Terminology. IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook. I cite various definitions from within the Gold Book, the following two being of particular importance: IUPAC - chain (Note 2). https://goldbook.iupac.org/terms/view/C00946 IUPAC - end-group. https://goldbook.iupac.org/terms/view/E02092 To address one of the comments in particular: it should be noted that nowhere in the Gold Book is there an authoritative definition of the term "ring." That would make claim that "according to IUPAC a ring cannot be a chain," already dubious based on the definition I cited above, even less likely to be the case -- they certainly aren't going to "prohibit" something they don't even define. The complaint about the definition of "chain" referring to "polymers" as if it were a notation of exclusivity and implying that the Gold Book's definition somehow contradicts or does not apply to the Blue Book should be pretty handily refuted with a bit more reading in both the Blue and Gold books. The Blue Book states it doesn't deal with preferred naming of polymers, but does still define the systematic naming of organic structures whether polymeric or not. (Blue Book P-11, p.3) The Gold Book defines "polymer," very simply and with no caveats, as "a substance composed of macromolecules." Since "chain" and "polymer" thus both use the term "macromolecules," one would naturally be led to read that definition as well, where along with the (hopefully) uncontroversial definition of being a high-molecular mass chain of repeated low-molecular mass components, one might stumble onto Note 2: "If a part or the whole of the molecule has a high relative molecular mass and essentially comprises the multiple repetition of units derived, actually or conceptually, from molecules of low relative molecular mass, it may be described as either macromolecular or polymeric, or by polymer used adjectivally." No clearer acknowledgement of the lack of some "hard line" apparently perceived by some between polymers and "regular" organic macromolecules should be necessary. When it comes to systematic naming, especially, polymers aren't a "special class" of substance which are somehow privy to different definitions and procedures -- indeed, the Blue Book tells us that's precisely the opposite of the case. So, for those strictly systematic purposes (which is the entire point of the question, and should be in most introductory academic settings because the point isn't to have students memorize some esoteric rules but instead to teach them something about the structure of molecules and their function within a larger compound) - the definition of "chain" given applies just fine. In fact, the definition of "polymer" is expressly defined in the Gold Book, again IUPAC's authoritative text, such that it could apply to any simple alkane with a side branch for the purposes of that text. Rather than second-guess clearly written prose, I'd rather assume the working group knew what it was doing there by leaving it vague and allowing common usage to define terms that are inherently ambiguous. (Also see: What is the difference between Alkanes and Polymers?) They seem more aware that strict legalism doesn't serve their purposes than most of the practitioners of their work do. And for those of you who skipped this addendum entirely: good for you. As interesting as some of the contentions below are, it seems generally agreed by all that none actually change the correct answer to the original question asked anyway; whatever your take on the semantic issues around the Blue and Gold (and Purple!) books that took up half the comments, they're just that -- semantics. One can think that an answer is based on a definition that is "irrelevant" due to its source, which I find an odd and not particularly useful criticism (but hey, that's just me); but thinking then that said definition is "incorrect" is false logic. If someone who has never seen a math book in their life believes 2+2=4 because some hot dog vendor on a street corner told them so, it doesn't make 2+2 equal anything other than 4. Fortunately for them, just because their source has "no relevance" doesn't mean that they can't have a grasp of basic math. • (to explain the asterisk in the "end unit" definition) * IUPAC states that a "feature group" that provides a macromolecule its name, like a carboxyl group in a carboxylic acid, can count as an end unit as well. Figured somebody might mention that, if I didn't. :) Doesn't directly impact the core problem, though. Feb 10 '21 at 0:21 • @George "A cyclic macromolecule has no end groups but may nevertheless be regarded as a chain." – You quote the Glossary of basic terms in polymer science (IUPAC Recommendations 1996), which is not applicable here since it applies to the polymer chain of a macromolecule. You should have looked into Nomenclature of Organic Chemistry: IUPAC Recommendations and Preferred Names 2013 instead. Feb 14 '21 at 8:25 • There is a misunderstanding here. The Gold Book entry explicitly says "chain (in polymers)". The given compound is not a polymer, and even if it were, the cited definition does not help in choosing a preferred IUPAC name. In this context the "chain" that user55119 mentioned refers to something entirely different, namely the "parent structure" as described in P-12 of the IUPAC Blue Book. I don't mean to be unkind, but I do mean to be blunt, since this "discussion" has gone on so long already: you have based your answer on a definition that is irrelevant, which unfortunately makes it incorrect. Feb 14 '21 at 14:03 • So your claim is that, while the Blue Book assigns a preferred name, you can use polymer terminology from the Gold Book to assign a systematic name to a molecule that is decidedly neither a polymer nor macromolecule? Feb 22 '21 at 23:45 • That is OK, as I have no intention of continuing this discussion. I understand now where the disagreement lies. It is not about systematic or preferred naming (I never brought that up, so you saying that I conflated it with something else is slightly ironic), nor is it about the provenance of your Gold Book citation (note that there's no need to perform a generic search for the text, as the Gold Book tells you where the definition is taken from). The issue is just that you think this thing can be called, or considered, a polymer. 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https://99wevote.org/tag/the-riddler/	# Alice And Bob Fall In Love Welcome back to The Riddler — I hope you had a wonderful Thanksgiving. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-size and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,1 and you may get a shoutout in next week’s column. If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter. ## Riddler Express From Graydon Snider, road race intimidation tactics: Two runners, Alice and Bob, are participating in a footrace. The route is a straight line out some distance and the same straight line back — the starting point and the finish line are the same. As the starting gun is about to go off, Alice hatches a race plan: Her legs feel good and she wants to run fast enough compared to Bob that after the U-turn, they are staring face-to-face for as long as possible. How much faster than Bob should Alice run to spend the maximum amount of time facing Bob before they pass each other going in opposite directions? Assume that, on the advice of their coaches, they’ve each committed to running at a constant speed the whole time, and that the turn-around time at the halfway point is negligible. ## Riddler Classic From Dan Johnston, in which Alice and Bob then fall in love: After their Riddler Express footrace, Alice and Bob fell in love and got married. Now they want lots of kids. However, as you may know, having one child, let alone many, is a lot of work. But Alice and Bob realized children require less of their parents’ time as they grow older. Alice and Bob, romantics that they are, decided to calculate how this relationship worked. They figured out that the work involved in having a child equals one divided by the age of the child in years. (Yes, that means the work is infinite for a child right after they are born. That may be true.) Anyhow, since having a new child is a lot of work, Alice and Bob don’t want to have another child until the total work required by all their other children is 1 or less. Suppose they have their first child at time T=0. When T=1, their only child is turns 1, so the work involved is 1, and so they have their second child. After roughly another 1.61 years, their children are roughly 1.61 and 2.61, the work required has dropped back down to 1, and so they have their third child. And so on. (Feel free to ignore twins, deaths, the real-world inability to decide exactly when you have a child, and so on.) Five questions: Does it make sense for Alice and Bob to have an infinite number of children? Does the time between kids increase as they have more and more kids? What can we say about when they have their Nth child — can we predict it with a formula? Does the size of their brood over time show asymptotic behavior? If so, what are its bounds? ## Solution to the previous Riddler Express Congratulations to 👏 Chris Sears 👏 of Maysville, Kentucky, winner of the previous Riddler Express! The World Chess Championship ended this week after 12 straight draws followed by a series of speedier tie-breaking games. That’s fitting, because two weeks ago we posed this question: What are the chances that the better player wins a 12-game match? Specifically, suppose one of the players is better than his opponent to the degree that he wins 20 percent of all games and loses 15 percent of games; the other 65 percent end in draws.2 What are the chances the better player wins a 12-game match? How many games would a match have to be in order to give the better player a 75 percent chance of winning the match outright? A 90 percent chance? A 99 percent chance? That better player wins a 12-game match about 52 percent of the time. The number of games required for those larger thresholds are, in order, 82, 248 and 773. (Call me crazy, but I’m totally game for a two-year-long World Chess Championship.) So how do we get there? Solver Dan Swenson suggested this tidy approach. Consider the following expression: \begin{equation}\left(0.2x + 0.65 + 0.15x^{-1}\right)^{12}\end{equation} The coefficients 0.2, 0.65 and 0.15 are the probabilities of the three outcomes of an individual chess game, and the whole thing is raised to the 12th power because of the 12 games of the match. Expand that expression, multiplying it all out and grouping its like terms. Then, the coefficient on $$x^r$$, where the exponent $$r$$ is some integer, is the probability that the better player wins the match by $$r$$ games. Finally, add the coefficients on all the positive powers of $$x$$, which gives the probability that the better player wins the match. The result is about 0.5198, or about 52 percent. For the second part of the problem, we can use that same approach, and just change the “12” in the exponent of the expression, and then, the same as before, expand it and sum the coefficients on the positive powers of $$x$$. The smallest exponent that gives a 75 percent chance or better is 82, the smallest that gives a 90 percent chance or better is 248, and the smallest that gives a 99 percent chance or better is 773. Solver Chris Sears illustrated how the probability that the better player wins the match increases as the number of games increases. Somebody get FIDE on the phone! ## Solution to the previous Riddler Classic Congratulations to 👏 Scott Wu 👏 of Baton Rouge, Louisiana, winner of the previous Riddler Classic! Two weeks ago, we presented you with a sort of Riddlerified version of beer pong. You had an infinite supply of ping-pong balls, each labeled with some number 1 through N. There was also a group of N cups, labeled 1 through N, each of which could hold an unlimited number of ping-pong balls. The game was played in rounds that had two phases: throwing and pruning. During the throwing phase, you draw balls randomly, one at a time, from the supply and toss them at the cups. The phase ends when every cup contains at least one ball. Next comes the pruning phase, in which you go through all the cups and remove any ball whose number does not match the number on its cup. Every ball drawn had a uniformly random number, every ball landed in a uniformly random cup, and every throw landed in some cup. The game was over when, after a round was completed, there were no empty cups. How many balls would you expect to need to draw and throw to finish this game? How many rounds would you expect to need before finishing this game? The first question — how many throws would you expect to need — is like a “coupon collector’s problem,” and versions of it have appeared in this column before, for example in a problem about collecting Riddler League football cards. In this case, we’re filling cups instead of collecting coupons.The solution to this problem is well-known and stems from the fact that as we collect coupons — or fill cups — it becomes more and more difficult to collect or fill the ones that remain. The specific solution is quite mathy and involves something called the Euler-Mascheroni constant, but the main takeaway is that the more coupons or cups there are, the more and more time we can expect to spend collecting coupons or throwing balls. Solver Laurent Lessard illustrated how the expected number of throws increases roughly quadratically as the number of cups (and numbers on the balls) increases: The second question — how many rounds would you expect to play — turns out to be much trickier. Lessard also illustrated his mathematical approach when playing with three cups. The cups in his diagram are either empty (white), filled with a correct ball (green), or filled with an incorrect ball (red). The arrows and the numbers next to them represent transition probabilities — the chances a cup goes from empty to correctly filled, for example, are 1/N. Our goal, of course, appears in the bottom right of the diagram, where all three cups are correctly filled and the game is over. From there, Laurent describes how to turn this diagrammatic approach into an answer. He uses a Markov chain, which describes probabilistic sequences of events that depend on the current state of events — such as the balls currently in our cups. And from there he invokes a holy Riddler trinity: absorbing states and limiting distributions and nilpotent matrices. The result of all this is that while the number of throws required grows roughly quadratically, the number of rounds required grows roughly linearly. Of course, this is also a problem that admits programmatic, simulation-based approaches. Solver Robin Chin shared Python code for a Monte Carlo simulation, and Michael Branicky shared his code as well. Finally, solver Tim Book shared the results of his computer simulations, which illustrate how the expected number of rounds increases as the number of cups (and the number of numbers on the balls) increases — roughly linearly: Happy tossing! ## Want more riddles? Well, aren’t you lucky? There’s a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. It’s called “The Riddler,” and it’s in stores now! Consider your holiday shopping done. ## Want to submit a riddle? Email me at [email protected] # So You Want To Tether Your Goat. Now What? 🚨🚨🚨 “The Riddler” book is out now! It’s chock-full of the best puzzles from this column (and, fret not, their answers) and some that have never been seen before. I hope you enjoy it, and thank you for riddling with us these past three years. 🚨🚨🚨 Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-size and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,1 and you may get a shoutout in next week’s column. If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter. ## Riddler Express From Luke Robinson, a serenading stumper: My daughter really likes to hear me sing “The Unbirthday Song” from “Alice in Wonderland” to her. She also likes to sing it to other people. Obviously, the odds of my being able to sing it to her on any random day2 are 364 in 365, because I cannot sing it on her birthday. The question is, though, how many random people would she expect to be able to sing it to on any given day before it became more likely than not that she would encounter someone whose birthday it is? In other words, what is the expected length of her singing streak? ## Riddler Classic From Moritz Hesse, some grazing geometry: A farmer owns a circular field with radius R. If he ties up his goat to the fence that runs along the edge of the field, how long does the goat’s tether need to be so that the goat can graze on exactly half of the field, by area? (The great thing about this puzzle, Moritz notes, is that if you get sick of math, you can find the answer through trial and error with your own circular field and your favorite goat, horse, cow, kangaroo, sheep, unicorn, centaur or sphinx.) ## Solution to last week’s Riddler Express Congratulations to 👏 Stefan Heidekrüger 👏 of Munich, winner of last week’s Riddler Express! Last week’s Express brought a fill-in-the-blank challenge: What’s the next number in this series? 9, 10, 19, 24, 31, 40, 51, 64, 79, 90, ? It’s A9. A9?! What the … ? The trick to filling in this blank was recognizing that these numbers aren’t in our usual base-10, or decimal, number system. Rather they are in the base-16, or hexadecimal, system. Hexadecimal is often used by computer programmers. And since we only have 10 digits that go with our usual base-10 system, base-16 uses the letters A through F to represent the values 10 through 15. OK, back to our series. The pattern is $$(N+2)^2$$, where $$N$$ is the number’s position in the sequence. So the first number is $$3^2$$, or 9. The second number is $$4^2$$, or 16, which is represented as 10 in hexadecimal. The third number is $$5^2$$, or 25, which is 19 in hexadecimal. And so on. Our missing number is $$13^2$$, or 169 — or A9 in hexadecimal. ## Solution to last week’s Riddler Classic Congratulations to 👏 Eric Roshan-Eisner 👏 of San Francisco, winner of last week’s Riddler Classic! Last week you were brought in to solve a serious problem at the Riddler Intelligence Agency, or RIA. Namely, the RIA had been infiltrated by spies, and your job was to root them out. There were N agents total, and K of them were spies. You knew the values of N and K but not, at first, the identities of the spies. You could send any number of agents on a remote island retreat as many times as you wanted. If all of the spies were on the retreat, they would assemble for a secret spy meeting; if any of the spies were not on the retreat, the meeting would not take place. The only thing you learned from each retreat was whether or not this meeting happened. It cost $1,000 per person to send agents on these retreats. What was the least you could spend while still identifying all of the spies? Assume you knew that N = 1,024 agents and K = 17 spies. This week’s winner, Eric Roshan-Eisner, explained that it will take at least 122 retreats to identify all the spies: The total number of possible spy configurations within those 1,024 people is N choose K, or about $$4\cdot 10^{36}$$. That number is very big — but it doesn’t mean we need that many retreats to suss out our spies. Think of each retreat as a moment to learn one bit of information about the arrangement of spies (that is, whether or not the spy meeting occurred). I’m using “bit” there intentionally — the key to solving this puzzle is to translate that $$4\cdot 10^{36}$$ number into binary, a number that needs 122 “bits” — one digit in a binary number — to be expressed. Every retreat you organize returns to you exactly one piece of yes-or-no, 0-or-1 information — that is, whether or not the spy meeting happened. That’s your bit. Therefore, the minimum number of retreats necessary to differentiate between all four undecillion possibilities is 122. The exact strategy you should use to do this, Eric wrote, is left as an exercise to the reader. Others picked up the baton there. Tim Black, a grad student at the University of Chicago (Go Maroons!), found a strategy to identify the spies in 131 retreats, and also proved that it is impossible to do so in fewer than 122. Thomas Swayze also found a 131-retreat strategy, which cost about$93 million. Thomas wrote that his strategy to find the spies was to use a form of binary search to single them out one by one: Start with some subset, S, of the agents that we know contains at least one spy. Then pick a subset, T, of S, and keep the agents in T at home while sending the entire rest of the agency on the retreat. If there’s a meeting, we know that T contains no spy — so leave all the agents in T out of all future retreats. If there is no meeting, there is some spy in T. Either way, we now have a smaller set that we know has a spy. Once we’ve outed one spy, we send him on all future retreats and repeat the process to find all the remaining spies, one by one. The tricky part is deciding how big to make the set T. It depends on the number of agents left, the number of spies we’ve caught, and the size of the set S. Thomas was kind enough to share the Python code he used. Finally, Laurent Lessard described a strategy that used more retreats (191) but cost less (about \$66 million). But hey, if identifying spies were easy — or cheap — everybody would do it. ## Want to submit a riddle? 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https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-2-solving-equations-2-4-solving-equations-with-variables-on-both-sides-practice-and-problem-solving-exercises-page-105/10	## Algebra 1 5x - 1 = x + 15 First, you need to get the variable (x) on one side of the equation, so subtract x from both sides of the equation. This will remove the x from the right side of the equation. 5x - 1 - x = x + 15 - x Simplify to get this: 4x - 1 = 15 Add 1 to each side. 4x - 1 + 1 = 15 + 1 Simplify to get: 4x = 16 Divide each side by 4. $\frac{4x}{4}$ = $\frac{16}{4}$ Simplify to x = 16 $\div$ 4 x = 4	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8056027293205261, "perplexity": 94.59145790259382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741510.42/warc/CC-MAIN-20181113215316-20181114001316-00500.warc.gz"}
https://math.stackexchange.com/questions/925916/how-can-i-find-this-limit-involving-thrice-iterated-logarithm/925936	# How can I find this limit involving thrice-iterated logarithm? $$\lim_{x \to 0}\dfrac{\ln \ln \ln \left[x+(1+x)^{(1+x)^{1/x}/x}\right]+x\left[1-\dfrac{1}{e^{e+1}}\right]}{x^2}$$ How can I find the limit of this question? Any hint. Thank you so much. • Maple outputs $$1/24\,{\frac {{{\rm e}^{-2}}{{\rm e}^{2}} \left( {{\rm e}^{{\rm e}}} \right) ^{2}+24\,{{\rm e}^{-2}}{{\rm e}^{2}}{{\rm e}^{{\rm e}}}+48\,{ {\rm e}^{-2}}{\rm e}{{\rm e}^{{\rm e}}}-12\,{{\rm e}^{-2}}{\rm e}-24\, {{\rm e}^{-2}}}{ \left( {{\rm e}^{{\rm e}}} \right) ^{2}}}.$$ – user64494 Sep 10 '14 at 5:51 • It could be, then, as a somewhat harder than usual, but direct application of asymptotic computations with big $O$. – Jean-Claude Arbaut Sep 13 '14 at 5:46 • This was a real tough one in terms of calculation. I have presented the approach based on LHR (not my favorite though) because Taylor series approach was already given by Claude Leibovici. – Paramanand Singh Sep 13 '14 at 6:44 You can find it with a lot of patience ! I give you what I did (hoping that a simpler solution will be provided) : First, I look at the exponent and, going first to logarithms, obtain $$\frac{(x+1)^{\frac{1}{x}}}{x}=\frac{e}{x}-\frac{e}{2}+\frac{11 e x}{24}-\frac{7 e x^2}{16}+O\left(x^3\right)$$ Then $$(x+1)^{\frac{(x+1)^{\frac{1}{x}}}{x}}=e^e-e^{1+e} x+\frac{1}{24} e^{1+e} (25+12 e) x^2+O\left(x^3\right)$$ So$$A=x+(x+1)^{\frac{(x+1)^{\frac{1}{x}}}{x}}=e^e+\left(1-e^{1+e}\right) x+\frac{1}{24} e^{1+e} (25+12 e) x^2+O\left(x^3\right)$$ Now, let me play with the logarithms $$\log A=e+\left(e^{-e}-e\right) x+\left(\frac{25 e}{24}+e^{1-e}-\frac{e^{-2 e}}{2}\right) x^2+O\left(x^3\right)$$ $$\log\log A=1+\left(e^{-1-e}-1\right) x+\left(\frac{13}{24}-\frac{1}{2} e^{-2-2 e}-\frac{1}{2} e^{-1-2 e}+e^{-1-e}+e^{-e}\right) x^2+O\left(x^3\right)$$ $$\log\log\log A=\left(e^{-1-e}-1\right) x+\frac{1}{24} e^{-2-2 e} \left(-24-12 e+48 e^{1+e}+24 e^{2+e}+e^{2+2 e}\right) x^2+O\left(x^3\right)$$ where you can notice that the first term is $$-x\left[1-\frac{1}{e^{e+1}}\right]$$ So, the limit is $$\frac{1}{24} e^{-2-2 e} \left(-24-12 e+48 e^{1+e}+24 e^{2+e}+e^{2+2 e}\right)=\frac{1}{24}+\frac{1}{2} e^{-2 (1+e)} (2+e) \left(2 e^{1+e}-1\right)$$ All of the above used successively the development (Taylor series) of $\log(1+y)$ close to $y=0$. Let's give it a try using elementary techniques. First we take care of $f(x) = (1 + x)^{(1 + x)^{1/x}/x}$. Clearly $$\log f(x) = (1 + x)^{1/x}\frac{\log(1 + x)}{x} \to e\tag{1}$$ so that $f(x) \to e^{e}$. We have to deal with the expression $$F(x) = \dfrac{\log \log \log(x + f(x)) + x(1 - e^{-e - 1})}{x^{2}}\tag{2}$$ It can be seen that both the numerator and denominator of $F(x)$ tend to $0$. I am somehow forced now to use L'Hospital Rule. This gives us another complicated expression $$G(x) = \frac{1}{2x}\left\{\left(1 - e^{-e - 1}\right) + \frac{1}{\log \log (x + f(x))}\cdot\frac{1}{\log(x + f(x))}\cdot\frac{1 + f'(x)}{x + f(x)}\right\}\tag{3}$$ The challenge now is to calculate $f'(x)$ and show that it tends to $-e^{e + 1}$ as $x \to 0$. This will ensure that we can apply LHR on $G(x)$ also. Assuming that we have done so we can see that the next application of LHR on $G(x)$ will give rise to the expression $$H(x) = \frac{1}{2}\left\{\frac{g(x)f''(x) - (1 + f'(x))g'(x)}{\{g(x)\}^{2}}\right\}\tag{4}$$ where $g(x) = (x + f(x))\cdot\log(x + f(x))\cdot\log\log(x + f(x))$. Note that $g(x) \to e^{e + 1}$ so I hope that we don't need further application of LHR on $H(x)$ and the final limit would be $$\dfrac{A}{2e^{2e + 2}}\tag{5}$$ where $$A = \lim_{x \to 0}g(x)f''(x) - (1 + f'(x))g'(x)\tag{6}$$ We can now see that the real challenge is to evaluate $f'(x), f''(x), g'(x)$ and it would require a reasonable amount of calculation. I will have to leave my keyboard and go for a pen-paper calculation to handle this. Will post the calculation if I succeed. The easiest part is $g'(x)$ given by \begin{aligned}g'(x) &= (1 + f'(x))\log(x + f(x))\log\log(x + f(x))\\ &\,\,\,\,+\,\, (1 + f'(x))\log\log(x + f(x))\\ &\,\,\,\,+\,\, (1 + f'(x))\\ &= ( 1 + f'(x))\{1 + \log\log(x + f(x)) + \log(x + f(x))\log\log(x + f(x))\}\end{aligned} Based on the assumption made in bold we can see that $(1 + f'(x))g'(x)$ tends to $$(1 - e^{e + 1})^{2}(e + 2)\tag{7}$$ From $(1)$ we can see that $\log f(x) = p(x)\log p(x)$ where $p(x) = (1 + x)^{1/x}$. Hence \begin{align}\frac{f'(x)}{f(x)} &= p'(x)(1 + \log p(x))\notag\\ &= p(x)(1 + \log p(x))(\log p(x))'\notag\\ &= p(x)(1 + \log p(x))\left(\frac{\log (1 + x)}{x}\right)'\notag\\ &= p(x)(1 + \log p(x))\left(\frac{x - (1 + x)\log(1 + x)}{x^{2}(1 + x)}\right)\tag{8}\end{align} Now we can see that $p(x) \to e$ so that $f'(x)/f(x)$ tends to $(2e)(-1/2) = -e$. Since $f(x) \to e^{e}$ we see that $f'(x) \to -e^{e + 1}$ and our assumption (mentioned in bold earlier) is verified (which at the same time verifies $(7)$). Now the hardest part is to differentiate $(8)$ and obtain $f''(x)$. We have \begin{aligned}f''(x) &= f'(x)p(x)\{1 + \log p(x)\}\left(\frac{x - (1 + x)\log(1 + x)}{x^{2}(1 + x)}\right)\\ &\,\,+\,\,f(x)\{1 + \log p(x)\}p(x)\left(\frac{x - (1 + x)\log(1 + x)}{x^{2}(1 + x)}\right)^{2}\\ &\,\,+\,\,f(x)p(x)\left(\frac{x - (1 + x)\log(1 + x)}{x^{2}(1 + x)}\right)^{2}\\ &\,\,+\,\,f(x)\{1 + \log p(x)\}p(x)\left(\frac{x - (1 + x)\log(1 + x)}{x^{2}(1 + x)}\right)'\end{aligned} Hence we can see that $f''(x)$ tends to $$e^{e + 2} + \frac{e^{e + 1}}{2} + \frac{e^{e + 1}}{4} + \frac{4e^{e + 1}}{3} = \frac{12e^{e + 2} + 25e^{e + 1}}{12}$$ The desired limit $A$ is thus \begin{aligned}A &= \frac{12e^{2e + 3} + 25e^{2e + 2}}{12} - (e + 2)(1 - e^{e + 1})^{2}\\ &= \frac{12e^{2e + 3} + 25e^{2e + 2} - 12(e + 2)(1 - 2e^{e + 1} + e^{2e + 2})}{12}\\ &= \frac{12e^{2e + 3} + 25e^{2e + 2} - 12(e - 2e^{e + 2} + e^{2e + 3} + 2 - 4e^{e + 1} + 2e^{2e + 2})}{12}\\ &= \frac{e^{2e + 2} + 24e^{e + 2} + 48e^{e + 1} - 12e - 24}{12}\end{aligned} The final limit is $A/2e^{2e + 2}$ so that the final limit is given by $$\frac{e^{2e + 2} + 24e^{e + 2} + 48e^{e + 1} - 12e - 24}{24e^{2e + 2}} = \frac{1}{24} + \frac{2e^{e + 2} + 4e^{e + 1} - e - 2}{2e^{2e + 2}}$$ Note: The limit of $$a(x) = \frac{x - (1 + x)\log(1 + x)}{x^{2}(1 + x)} = \dfrac{\dfrac{x}{1 + x} - \log(1 + x)}{x^{2}} = -\frac{1}{2} + \frac{2}{3}x + \cdots$$ and its derivative were calculated using the Taylor expression given above leading to $a(x) \to -1/2$ and $a'(x) \to 2/3$ as $x \to 0$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9979504942893982, "perplexity": 392.8922068832101}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358798.23/warc/CC-MAIN-20210227084805-20210227114805-00591.warc.gz"}
https://www.physicsforums.com/threads/positron-electron-boom.726813/	# Positron Electron boom 1. ### PsychonautQQ 476 so if a positron and an electron boom each other and create 2 photons, how do I tell the total energy of the two photons? does a positron have anti mass equal to the magnitude of the electron? so the two photons total energy would be 2(m_e)*c^2 ### Staff: Mentor Positrons and electrons have equal mass. Total energy of the photons would be equal to the sum of the two particles' masses plus the kinetic energy they had at the time of the annihilation event.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9014483094215393, "perplexity": 494.85676171377116}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207926924.76/warc/CC-MAIN-20150521113206-00142-ip-10-180-206-219.ec2.internal.warc.gz"}
https://documen.tv/question/here-is-the-situation-a-puck-is-resting-on-the-floor-of-a-large-moving-van-assume-that-the-floor-15157661-61/	## Here is the situation: A puck is resting on the floor of a large moving van. Assume that the floor of the van is frictionless. The van and t Question Here is the situation: A puck is resting on the floor of a large moving van. Assume that the floor of the van is frictionless. The van and the puck are initially at rest, then the van undergoes a positive uniform acceleration. Determine whether each statement below is:a. True b. False. in progress 0 6 months 2021-08-23T20:04:58+00:00 1 Answers 21 views 0 Statement 1: Someone outside the van sees the puck remain stationary with respect to the ground. TRUE. This is because the puck relative to the ground initially was stationary. When the van starts to move forward or accelerate, the puck will tend to maintain its stationary state with the ground by moving in the opposite direction. This movement is only relative to the van and not the ground because the puck and the ground are at rest as the puck has no net force acting on it. Someone in the van sees the puck move toward the front of the van. FALSE. The puck has no net force acting in it so by newton’s first law it should have no relative movement to the ground. In order to achieve this, the puck by newton’s third law of motion must move in the opposite direction (backwards) as the van to maintain its stationary state with the ground. The acceleration of the van with respect to the ground is > the acceleration of the puck with respect to the ground. If the van turns to the right, the puck will hit the left wall. WHAT DOES THIS HAVE TO DO WITH THE STATEMENTS? I’ll assume the truck continues in a straight line and this statement if irrelevant. It does say “If” The magnitude of the acceleration of the van with respect to the ground is = the magnitude of the puck’s acceleration with respect to the van. The puck is stationary with respect to the van during this acceleration. FALSE Explanation:	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8161715865135193, "perplexity": 380.9857308475933}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710710.91/warc/CC-MAIN-20221129164449-20221129194449-00475.warc.gz"}
http://mathhelpforum.com/geometry/205774-help-about-right-triangles.html	# Math Help - help about right triangles! 1. ## help about right triangles! hello! i have question about right triangles and perimeter: "If the perimeter of a right triangle is 12 and it's hypotenuse is 5, what is the area of the triangle?" formula for area: a=bh/2 perimeter = b+h+hypotenuse = 12 this is solved by substitution? b+h+5=12 b+h=7 b=7/h a=bh/2 a=[(7/h)h]/2 a=7/2 a=3.5 2. ## Re: help about right triangles! Let $a$ and $b$ be the two legs of the right triangle. Since the perimeter is 12 and the hypotenuse is 5, we know: $a+b=7$ By the Pythagorean theorem, we also know: $a^2+b^2=5^2$ I recommend squaring the first equation to find $ab$, since the second equation gives you the value of the sum of the squares of the legs.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9053477048873901, "perplexity": 458.8524512585485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430459200931.63/warc/CC-MAIN-20150501054640-00059-ip-10-235-10-82.ec2.internal.warc.gz"}
https://dsp.stackexchange.com/questions/14907/is-there-a-mathematical-relation-between-percent-overlap-and-window-length-or-am	# is there a mathematical relation between percent overlap and window length or amplitude I am trying to plot short time energy of an utterance with percent overlap as an input variable. As I referred to the short time energy equation it can be represented as square summation of $x(n)*w(n-m)$ where $w(n)$ seems to be window length. But I couldn't figure out how to write it as percent overlap.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9362505078315735, "perplexity": 453.33401027920206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540527620.19/warc/CC-MAIN-20191210123634-20191210151634-00289.warc.gz"}
https://socratic.org/questions/59b8c0adb72cff30bb1083eb	Chemistry Topics # What are the oxidation states exhibited in the SF_4 molecule? Sep 13, 2017 Do you want the $\text{excited state or oxidation states..........??}$ #### Explanation: Now we cannot really assess the excited state; it depends on the form of spectroscopy or form of excitation you use. Certainly we can assess the oxidation state of each species, which is the conceptual charge left on an atom of interest in a molecule when all the bonding pairs of electrons are BROKEN with the charge assigned to the MOST electronegative atom..... If we gots $S {F}_{4}$, the fluorine is definitely more electronegative than sulfur, and we thus assign oxidation numbers of $\stackrel{+ I V}{S}$, and $\stackrel{- I}{F}$. As always the weighted sum of the oxidation numbers is equal to the charge on the ion or molecule. For $S {F}_{4}$, we gots a neutral molecule, and $4 \times - I + I V = 0$ AS REQUIRED. For $S {F}_{6}$ we gots $\stackrel{+ V I}{S}$, and $\stackrel{- I}{F}$. And for the interhalogen we gots $\stackrel{+ V I I}{I}$, and $\stackrel{- I}{F}$. ##### Impact of this question 473 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9325805306434631, "perplexity": 1315.985195228947}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664567.4/warc/CC-MAIN-20191112024224-20191112052224-00199.warc.gz"}
https://www.edaboard.com/threads/question-regarding-execution-of-statement-in-verilog.336518/	# question regarding execution of statement in verilog!! Status Not open for further replies. #### Indrajit Ghosh ##### Junior Member level 2 Joined Feb 17, 2015 Messages 21 Helped 0 Reputation 0 Reaction score 0 Trophy points 1 Activity points 170 suppose I have instantiated to ipcores c1,c2 in a module suppose called top module.now a control signal activates core c1 and a control signal from core c1 activates c2. Now suppose i include an always block for positive clock cycles and inside it I write some conditional logic where the signal activating core c1 turns off.Now my question is that will the core keep executing itself over and over again till the control signal is activated or will it run only once and the always block will run over and over again? PS:- I am new in verilog #### TrickyDicky Joined Jun 7, 2010 Messages 7,110 Helped 2,079 Reputation 4,177 Reaction score 2,043 Trophy points 1,393 Activity points 39,625 If you set the enable bit low, it will then halt at whatever state it's in. - - - Updated - - - PS. This is not a verilog question - this is a digital logic question, which is nothing to do with verilog. You need to go and do a bit more learning about how digital systems work before you write ANY verilog. Verilog is an HDL (hardware description language), so you need to understand the circuit before you write any code. The usualy approach is to draw a diagram of the circuit before you write the HDL. #### Indrajit Ghosh ##### Junior Member level 2 Joined Feb 17, 2015 Messages 21 Helped 0 Reputation 0 Reaction score 0 Trophy points 1 Activity points 170 well I am trying to implement the "DIGITAL LOGIC QUESTION" using verilog...:smile: #### andre_luis ##### Super Moderator Staff member Joined Nov 7, 2006 Messages 9,373 Helped 1,170 Reputation 2,359 Reaction score 1,164 Trophy points 1,403 Location Brazil Activity points 54,577 I´m not too familiar with HDL designs, but sounds like related to FSM scope, due each state represents an enable condition to execute each core. #### TrickyDicky Joined Jun 7, 2010 Messages 7,110 Helped 2,079 Reputation 4,177 Reaction score 2,043 Trophy points 1,393 Activity points 39,625 well I am trying to implement the "DIGITAL LOGIC QUESTION" using verilog...:smile: Yes - but before you write ANY verilog, you need to know what the digital logic is. Have you drawn a diagram of the circuit? (you dont need verilog for this). ##### Super Moderator Staff member Joined Sep 10, 2013 Messages 7,906 Helped 1,821 Reputation 3,652 Reaction score 1,799 Trophy points 1,393 Location USA Activity points 59,783 Now suppose i include an always block for positive clock cycles and inside it I write some conditional logic where the signal activating core c1 turns off. This looks like a Verilog question on how an edge triggered always block behaves. Now my question is that will the core keep executing itself over and over again till the control signal is activated or will it run only once and the always block will run over and over again? but then you come back with this confusing question. which core are you talking about? the one that sends or the one that receives the turn off signal. The always block will continually be activated (scheduled) each time the positive edge of the clock arrives. Code Verilog - [expand]1 always @ (posedge clock) begin /...some code to execute.../ end so whatever outputs that always block generates will continue to be updated every clock edge. Reactions: Indrajit Ghosh Points: 2	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2654903531074524, "perplexity": 4417.089902016163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103850139.45/warc/CC-MAIN-20220630153307-20220630183307-00022.warc.gz"}
https://www.machinecurve.com/index.php/2020/12/07/introducing-pca-with-python-and-scikit-learn-for-machine-learning/	Introducing PCA with Python and Scikit-learn for Machine Learning Last Updated on 11 January 2021 Training a Supervised Machine Learning model – whether that is a traditional one or a Deep Learning model – involves a few steps. The first is feeding forward the data through the model, generating predictions. The second is comparing those predictions with the actual values, which are also called ground truth. The third, then, is to optimize the model based on the minimization of some objective function. In this iterative process, the model gets better and better, and sometimes it even gets really good. But what data will you feed forward? Sometimes, your input sample will contain many columns, also known as features. It is common knowledge (especially with traditional models) that using every column in your Machine Learning model will mean trouble, the so-called curse of dimensionality. In this case, you’ll have to selectively handle the features you are working with. In this article, we’ll cover Principal Component Analysis (PCA), which is one such way. It provides a gentle but extensive introduction to feature extraction for your Machine Learning model with PCA. It is structured as follows. First of all, we’ll take a look at what PCA is. We do this through the lens of the Curse of Dimensionality, which explains why we need to reduce dimensionality especially with traditional Machine Learning algorithms. This also involves the explanation of the differences between Feature Selection and Feature Extraction technique, which have a different goal. PCA, which is part of the Feature Extraction branch of techniques, is then introduced. When we know sufficiently about PCA conceptually, we’ll take a look at it from a Python point of view. For a sample dataset, we’re going to perform PCA in a step-by-step fashion. We’ll take a look at all the individual components. Firstly, we’ll compute the covariance matrix for the variables. Then, we compute the eigenvectors and eigenvalues, and select which ones are best. Subsequently, we compose the PCA projection matrix for mapping the data onto the axes of the principal components. This allows us to create entirely new dimensions which capture most of the variance from the original dataset, at a fraction of the dimensions. Note that SVD can also be used instead of eigenvector decomposition; we’ll also take a look at that. Once we clearly understand how PCA happens by means of the Python example, we’ll show you how you don’t have to reinvent the wheel if you’re using PCA. If you understand what’s going on, it’s often better to use a well-established library for computing the PCA. Using Scikit-learn’s sklearn.decomposition.PCA API, we will finally show you how to compute principal components and apply them to perform dimensionality reduction for your dataset. All right. Enough introduction for now. Let’s get to work! 😎 Update 11/Jan/2021: added quick code example to start using PCA straight away. Also corrected a few spelling issues. Code example: using PCA with Python This quick code example allows you to start using Principal Component Analysis with Python immediately. If you want to understand the concepts and code in more detail, make sure to read the rest of this article 🙂 .wp-block-code{border:0;padding:0;}.wp-block-code > div{overflow:auto;}.shcb-language{border:0;clip:rect(1px,1px,1px,1px);-webkit-clip-path:inset(50%);clip-path:inset(50%);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;word-wrap:normal;word-break:normal;}.hljs{box-sizing:border-box;}.hljs.shcb-code-table{display:table;width:100%;}.hljs.shcb-code-table > .shcb-loc{color:inherit;display:table-row;width:100%;}.hljs.shcb-code-table .shcb-loc > span{display:table-cell;}.wp-block-code code.hljs:not(.shcb-wrap-lines){white-space:pre;}.wp-block-code code.hljs.shcb-wrap-lines{white-space:pre-wrap;}.hljs.shcb-line-numbers{border-spacing:0;counter-reset:line;}.hljs.shcb-line-numbers > .shcb-loc{counter-increment:line;}.hljs.shcb-line-numbers .shcb-loc > span{padding-left:.75em;}.hljs.shcb-line-numbers .shcb-loc::before{border-right:1px solid #ddd;content:counter(line);display:table-cell;padding:0 .75em;text-align:right;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;white-space:nowrap;width:1%;}from sklearn import datasets from sklearn.decomposition import PCA from sklearn.preprocessing import StandardScaler X = iris.data y = iris.target # Standardize scaler = StandardScaler() scaler.fit(X) X = scaler.transform(X) # PCA pca = PCA(n_components=2) pca.fit(X) print(pca.explained_variance_ratio_) print(pca.components_) X = pca.transform(X)Code language: PHP (php) What is Principal Component Analysis? Before we dive in to the specifics of PCA, I think we should first take a look at why it can be really useful for Machine Learning projects. For this reason, we will first take a look at Machine Learning projects and the Curse of Dimensionality, which is especially present when using older Machine Learning algorithms (Support Vector Machines, Logistic Regression, …). Then, we’ll discuss what can be done against it – dimensionality reduction – and explain the difference between Feature Selection and Feature Extraction. Finally, we’ll get to PCA – and provide a high-level introduction. Machine Learning and the Curse of Dimensionality If you are training a Supervised Machine Learning model, at a high level, you are following a three-step, iterative process: Let's pause for a second! 👩‍💻 Blogs at MachineCurve teach Machine Learning for Developers. Sign up to MachineCurve's free Machine Learning update today! You will learn new things and better understand concepts you already know. We send emails at least every Friday. Welcome! By signing up, you consent that any information you receive can include services and special offers by email. Since Supervised Learning means that you have a dataset at your disposal, the first step in training a model is feeding the samples to the model. For every sample, a prediction is generated. Note that at the first iteration, the model has just been initialized. The predictions therefore likely make no sense at all. This becomes especially evident from what happens in the second step, where predictions and ground truth (= actual targets) are compared. This comparison produces an error or loss value which illustrates how bad the model performs. The third step is then really simple: you improve the model. Depending on the Machine Learning algorithm, optimization happens in different ways. In the case of Neural networks, gradients are computed with backpropagation, and subsequently optimizers are used for changing the model internals. Weights can also be changed by minimizing one function only; it just depends on the algorithm. You then start again. Likely, because you have optimized the model, the predictions are a little bit better now. You simply keep iterating until you are satisfied with the results, and then you stop the training process. Underfitting and overfitting a model When you are performing this iterative process, you are effectively moving from a model that is underfit to a model that demonstrates a good fit. If you want to understand these concepts in more detail, this article can help, but let’s briefly take a look at them here as well. In the first stages of the training process, your model is likely not able to capture the patterns in your dataset. This is visible in the left part of the figure below. The solution is simple: just keep training until you achieve the right fit for the dataset (that’s the right part). Now, you can’t keep training forever. If you do, the model will learn to focus too much on patterns hidden within your training dataset – patterns that may not be present in other real-world data at all; patterns truly specific to the sample with which you are training. The result: a model tailored to your specific dataset, visible in the middle part of the figure. In other words, training a Machine Learning model involves finding a good balance between a model that is underfit and a model that is overfit. Fortunately, many techniques are available to help you with this, but it’s one of the most common problems in Supervised ML today. Having a high-dimensional feature vector I think the odds are that I can read your mind at this point. Overfitting, underfitting, and training a Machine Learning model – how are they related to Principal Component Analysis? That’s a fair question. What I want to do is to illustrate why a large dataset – in terms of the number of columns – can significantly increase the odds that your model will overfit. Suppose that you have the following feature vector: $$\textbf{x} = [1.23, -3.00, 45.2, 9.3, 0.1, 12.3, 8.999, 1.02, -2.45, -0.26, 1.24]$$ This feature vector is 11-dimensional. Never miss new Machine Learning articles ✅ Blogs at MachineCurve teach Machine Learning for Developers. Sign up to MachineCurve's free Machine Learning update today! You will learn new things and better understand concepts you already know. We send emails at least every Friday. Welcome! By signing up, you consent that any information you receive can include services and special offers by email. Now suppose that you have 200 samples. Will a Machine Learning model be able to generalize across all eleven dimensions? In other words, do we have sufficient samples to cover large parts of the domains for all features in the vector (i.e., all the axes in the 11-dimensional space)? Or does it look like a cheese with (massive) holes? I think it’s the latter. Welcome to the Curse of Dimensionality. The Curse of Dimensionality Quoted from Wikipedia: In machine learning problems that involve learning a “state-of-nature” from a finite number of data samples in a high-dimensional feature space with each feature having a range of possible values, typically an enormous amount of training data is required to ensure that there are several samples with each combination of values. Wikipedia (n.d.) In other words, that’s what we just described. The point with “[ensuring that there are several samples with each combination of values” is that when this is performed well, you will likely be able to train a model that (1) performs well and (2) generalizes well across many settings. With 200 samples, however, it’s 100% certain that you don’t meet this requirement. The effect is simple: your model will overfit to the data at hand, and it will become worthless if it is used with data from the real world. Since increasing dimensionality equals an increasingly growing need for more data, the only way out of this curse is to reduce the number of dimensions in our dataset. This is called Dimensionality Reduction, and we’ll now take a look at two approaches – Feature Selection and Feature Extraction. Dimensionality Reduction: Feature Selection vs Feature Extraction We saw that if we want to decrease the odds of overfitting, we must reduce the dimensionality of our data. While this can easily be done in theory (we can simply cut off a few dimensions, who cares?), this gets slightly difficult in practice (which dimension to cut… because, how do I know which one contributes most?). And what if each dimension contributes an equal emount to the predictive power of the model? What then? In the field of Dimensionality Reduction, there are two main approaches that you can use: Feature Selection and Feature Extraction. • Feature Selection involves “the process of selecting a subset of relevant features (variables, predictors) for use in model construction” (Wikipedia, 2004). In other words, Feature Selection approaches attempt to measure the contribution of each feature, so that you can keep the ones that contribute most. What must be clear is that your model will be trained with the original variables; however, with only a few of them. • Feature Selection can be a good idea if you already think that most variance within your dataset can be explained by a few variables. If the others are truly non-important, then you can easily discard them without losing too much information. • Feature Extraction, on the other hand, “starts from an initial set of measured data and builds derived values (features) intended to be informative and non-redundant” (Wikipedia, 2003). In other words, a derived dataset will be built that can be used for training your Machine Learning model. It is lower-dimensional compared to the original dataset. It will be as informative as possible (i.e., as much information from the original dataset is pushed into the new variables) while non-redundant (i.e., we want to avoid that information present in one variable in the new dataset is also present in another variable in the new dataset). In other words, we get a lower-dimensional dataset that explains most variance in the dataset, while keeping things relatively simple. • Especially in the case where each dimension contributes an equal emount, Feature Extraction can be preferred over Feature Selection. The same is true if you have no clue about the contribution of each variable to the model’s predictive power. Introducing Principal Component Analysis (PCA) Now that we are aware of the two approaches, it’s time to get to the point. We’ll now introduce PCA, a Feature Extraction technique, for dimensionality reduction. Principal Component Analysis is defined as follows: Principal component analysis (PCA) is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest. Wikipedia (2002) Well, that’s quite a technical description, isn’t it. And what are “principal components”? Join hundreds of other learners! 😎 Blogs at MachineCurve teach Machine Learning for Developers. Sign up to MachineCurve's free Machine Learning update today! You will learn new things and better understand concepts you already know. We send emails at least every Friday. Welcome! By signing up, you consent that any information you receive can include services and special offers by email. The principal components of a collection of points in a real p-space are a sequence of $$p$$ direction vectors, where the $$i^{th}$$ vector is the direction of a line that best fits the data while being orthogonal to the first $$i – 1$$ vectors. Wikipedia (2002) I can perfectly get it when you still have no idea what PCA is after reading those two quotes. I had the same. For this reason, let’s break down stuff step-by-step. The goal of PCA: finding a set of vectors (principal components) that best describe the spread and direction of your data across its many dimensions, allowing you to subsequently pick the top-$$n$$ best-describing ones for reducing the dimensionality of your feature space. The steps of PCA: 1. If you have a dataset, its spread can be expressed in orthonormal vectors – the principal directions of the dataset. Orthonormal, here, means that the vectors are orthogonal to each other (i.e. they have an angle of 90°) and are of size 1. 2. By sorting these vectors in order of importance (by looking at their relative contribution to the spread of the data as a whole), we can find the dimensions of the data which explain most variance. 3. We can then reduce the number of dimensions to the most important ones only. 4. And finally, we can project our dataset onto these new dimensions, called the principal components, performing dimensionality reduction without losing much of the information present in the dataset. The how: Although we will explain how later in this article, we’ll now visually walk through performing PCA at a high level. This allows you to understand what happens first, before we dive into how it happens. Another important note is that for step (1), decomposing your dataset into vectors can be done in two different ways – by means of (a) eigenvector decomposition of the covariance matrix, or (b) Singular Value Decomposition. Later in this article, we’ll walk through both approaches step-by-step. Suppose that we generate a dataset based on two overlapping blobs which we consider to be part of just one dataset: from sklearn.datasets import make_blobs import matplotlib.pyplot as plt import numpy as np # Configuration options num_samples_total = 1000 cluster_centers = [(1,1), (1.25,1.5)] num_classes = len(cluster_centers) # Generate data X, y = make_blobs(n_samples = num_samples_total, centers = cluster_centers, n_features = num_classes, center_box=(0, 1), cluster_std = 0.15) e) # Make plot plt.scatter(X[:, 0], X[:, 1]) axes = plt.gca() axes.set_xlim([0, 2]) axes.set_ylim([0, 2]) plt.show()Code language: PHP (php) …which looks as follows: If you look closely at the dataset, you can see that it primarily spreads into two directions. These directions are from the upper right corner to the lower left corner and from the lower right middle to the upper left middle. Those directions are different from the axis directions, which are orthogonal to each other: the x and y axes have an angle of 90 degrees. No other set of directions will explain as much as the variance than the one we mentioned above. After standardization, we can visualize the directions as a pair of two vectors. These vectors are called the principal directions of the data (StackExchange, n.d.). There are as many principal directions as the number of dimensions; in our case, there are two. We call these vectors eigenvectors. Their length is represented by what is known as an eigenvalue. They play a big role in PCA because of the following reason: [The eigenvectors and related] eigenvalues explain the variance of the data along the new feature axes. Raschka (2015) In other words, they allow us to capture both the (1) direction and (2) magnitude of the spread in your dataset. Blogs at MachineCurve teach Machine Learning for Developers. Sign up to MachineCurve's free Machine Learning update today! You will learn new things and better understand concepts you already know. We send emails at least every Friday. Welcome! By signing up, you consent that any information you receive can include services and special offers by email. Notice that the vectors are orthogonal to each other. Also recall that our axes are orthogonal to each other. You can perhaps now imagine that it becomes possible to perform a transformation to your dataset, so that the directions of the axes are equal to the directions of the eigenvectors. In other words, we change the “viewpoint” of our data, so that the axes and vectors have equal directions. This is the core of PCA: projecting the data to our principal directions, which are then called principal components. The benefit here is that while the eigenvectors tell us something about the directions of our projection, the corresponding eigenvalues tell us something about the importance of that particular principal direction in explaining the variance of the dataset. It allows us to easily discard the directions that don’t contribute sufficiently enough. That’s why before projecting the dataset onto the principal components, we must first sort the vectors and reduce the number of dimensions. Sorting the vectors in order of importance Once we know the eigenvectors and eigenvalues that explain the spread of our dataset, we must sort them in order of descending importance. This allows us to perform dimensionality reduction, as we can keep the principal directions which contribute most significantly to the spread in our dataset. Sorting is simple: we sort the list with eigenvalues in descending order and ensure that our list with eigenvectors is sorted in the same way. In other words, the pairs of eigenvectors and eigenvalues are jointly sorted in a descending order based on the eigenvalue. As the largest eigenvalues indicate the biggest explanation for spread in your dataset, they must be on top of the list. For the example above, we can see that the eigenvalue for the downward-oriented eigenvector exceeds the one for the upward-oriented vector. If we draw a line through the dataset that overlaps with the vector, we can also see that variance for that line as a whole (where variance is defined as the squared distance of each point to the mean value for the line) is biggest. We can simply draw no line where variance is larger. In fact, the total (relative) contribution of the eigenvectors to the spread for our example is as follows: [0.76318124 0.23681876]Code language: JSON / JSON with Comments (json) (We’ll look at how we can determine this later.) So, for our example above, we now have a sorted list with eigenpairs. Reducing the number of dimensions As we saw above, the first eigenpair explains 76.3% of the spread in our dataset, whereas the second one explains only 23.7%. Jointly, they explain 100% of the spread, which makes sense. Using PCA for dimensionality reduction now allows us to take the biggest-contributing vectors (if your original feature space was say 10-dimensional, it is likely that you can find a smaller set of vectors which explains most of the variance) and only move forward with them. If our goal was to reduce dimensionality to one, we would now move forward and take the 0.763 contributing eigenvector for data projection. Note that this implies that we will lose 0.237 worth of information about our spread, but in return get a lower number of dimensions. Clearly, this example with only two dimensions makes no rational sense as two dimensions can easily be handled by Machine Learning algorithms, but this is incredibly useful if you have many dimensions to work with. Projecting the dataset Once we have chosen the number of eigenvectors that we will use for dimensionality reduction (i.e. our target number of dimensions), we can project the data onto the principal components – or component, in our case. This means that we will be changing the axes so that they are now equal to the eigenvectors. In the example below, we can project our data to one eigenvector. We can see that only the $$x$$ axis has values after projecting, and that hence our feature space has been reduced to one dimension. We have thus used PCA for dimensionality reduction. The how of generating eigenpairs: Eigenvector Decomposition or Singular Value Decomposition Above, we covered the general steps of performing Principal Component Analysis. Recall that they are as follows: 1. Decomposing the dataset into a set of eigenpairs. 2. Sorting the eigenpairs in descending order of importance. 3. Selecting $$n$$ most important eigenpairs, where $$n$$ is the desired number of dimensions. 4. Projecting the data to the $$n$$ eigenpairs so that their directions equal the ones of our axes. In step (1), we simply mentioned that we can express the spread of our data by means of eigenpairs. On purpose, we didn’t explain how this can be done, for the sake of simplicity. In fact, there are two methods that are being used for this purpose today: Eigenvector Decomposition (often called “EIG”) and Singular Value Decomposition (“SVD”). Using different approaches, they can be used to obtain the same end result: expressing the spread of your dataset in eigenpairs, the principal directions of your data, which can subsequently be used to reduce the number of dimensions by projecting your dataset to the most important ones, the principal components. While mathematically and hence formally you can obtain the same result with both, in practice PCA-SVD is numerically more stable (StackExchange, n.d.). For this reason, you will find that most libraries and frameworks favor a PCA-SVD implementation over a PCA-EIG one. Nevertheless, you can still achieve the same result with both approaches! In the next sections, we will take a look at clear and step-by-step examples of PCA with EIG and PCA with SVD, allowing you to understand the differences intuitively. We will then look at sklearn.decomposition.PCA, Scikit-learn’s implementation of Principal Component Analysis based on PCA-SVD. There is no need to perform PCA manually if there are great tools out there, after all! 😉 PCA-EIG: Eigenvector Decomposition with Python Step-by-Step One of the ways in which PCA can be performed is by means of Eigenvector Decomposition (EIG). More specifically, we can use the covariance matrix of our $$N$$-dimensional dataset and decompose it into $$N$$ eigenpairs. We can do this as follows: 1. Standardizing the dataset: EIG based PCA only works well if the dataset is centered and has a mean of zero (i.e. $$\mu = 0.0$$). We will use standardization for this purpose, which also scales the data to a standard deviation of one ($$\sigma = 1.0$$). 2. Computing the covariance matrix of the variables: a covariance matrix indicates how much variance each individual variable has, and how much they ‘covary’ – in other words, how much certain variables move together. 3. Decomposing the covariance matrix into eigenpairs: mathematically, we can rewrite the covariance matrix so that we can get a set of eigenvectors and eigenvalues, or eigenpairs. 4. Sorting the eigenpairs in decreasing order of importance, to find the principal directions in your dataset which contribute to the spread most significantly. 5. Selecting the variance contribution of your principal directions and selecting $$n$$ principal components: if we know the relative contributions to the spread for each principal direction, we can perform dimensionality reduction by selecting only the $$n$$ most contributing principal components. 6. Building the projection matrix for projecting our original dataset onto the principal components. We can see that steps (1), (4), (5) and (6) are general – we also saw them above. Steps (2) and (3) are specific to PCA-EIG and represent the core of what makes eigenvector decomposition based PCA unique. We will now cover each step in more detail, including step-by-step examples with Python. Note that the example in this section makes use of native / vanilla Python deliberately, and that Scikit-learn based implementations of e.g. standardization and PCA will be used in another section. Using the multidimensional Iris dataset If we want to show how PCA works, we must use a dataset where the number of dimensions $$> 2$$. Fortunately, Scikit-learn provides the Iris dataset, which can be used to classify three groups of Iris flowers based on four characteristics (and hence features or dimensions): petal length, petal width, sepal length and sepal width. This code can be used for visualizing two dimensions every time: from sklearn import datasets import matplotlib.pyplot as plt import numpy as np # Configuration options dimension_one = 1 dimension_two = 3 X = iris.data y = iris.target # Shape print(X.shape) print(y.shape) # Dimension definitions dimensions = { 0: 'Sepal Length', 1: 'Sepal Width', 2: 'Petal Length', 3: 'Petal Width' } # Color definitions colors = { 0: '#b40426', 1: '#3b4cc0', 2: '#f2da0a', } # Legend definition legend = ['Iris Setosa', 'Iris Versicolour', 'Iris Virginica'] # Make plot colors = list(map(lambda x: colors[x], y)) plt.scatter(X[:, dimension_one], X[:, dimension_two], c=colors) plt.title(f'Visualizing dimensions {dimension_one} and {dimension_two}') plt.xlabel(dimensions[dimension_one]) plt.ylabel(dimensions[dimension_two]) plt.show()Code language: PHP (php) This yields the following plots, if we play with the dimensions: The images illustrate that two of the Iris flowers cannot be linearly separated, but that this group can be separated from the other Iris flower. Printing the shape yields the following: (150, 4) (150,) …indicating that we have only 150 samples, but that our feature space is four-dimensional. Clearly a case where feature extraction could be beneficial for training our Machine Learning model. Performing standardization We first add Python code for standardization, which brings our data to $$\mu = 0.0, \sigma = 1.0$$ by performing $$x = \frac{x – \mu}{\sigma}$$ for each dimension (MachineCurve, 2020). # Perform standardization for dim in range(0, X.shape[1]): print(f'Old mean/std for dim={dim}: {np.average(X[:, dim])}/{np.std(X[:, dim])}') X[:, dim] = (X[:, dim] - np.average(X[:, dim])) / np.std(X[:, dim]) print(f'New mean/std for dim={dim}: {np.abs(np.round(np.average(X[:, dim])))}/{np.std(X[:, dim])}') # Make plot colors = list(map(lambda x: colors[x], y)) plt.scatter(X[:, dimension_one], X[:, dimension_two], c=colors) plt.title(f'Visualizing dimensions {dimension_one} and {dimension_two}') plt.xlabel(dimensions[dimension_one]) plt.ylabel(dimensions[dimension_two]) plt.show()Code language: PHP (php) And indeed: Old mean/std for dim=0: 5.843333333333334/0.8253012917851409 New mean/std for dim=0: 0.0/1.0 Old mean/std for dim=1: 3.0573333333333337/0.4344109677354946 New mean/std for dim=1: 0.0/0.9999999999999999 Old mean/std for dim=2: 3.7580000000000005/1.759404065775303 New mean/std for dim=2: 0.0/1.0 Old mean/std for dim=3: 1.1993333333333336/0.7596926279021594 New mean/std for dim=3: 0.0/1.0Code language: PHP (php) Computing the covariance matrix of your variables The next step is computing the covariance matrix for our dataset. In probability theory and statistics, a covariance matrix (…) is a square matrix giving the covariance between each pair of elements of a given random vector. Wikipedia (2003) If you’re not into mathematics, I can understand that you don’t know what this is yet. Let’s therefore briefly take a look at a few aspects related to a covariance matrix before we move on, based on Lambers (n.d.). A variable: such as $$X$$. A mathematical representation of one dimension of the data set. For example, if $$X$$ represents $$\text{petal width}$$, numbers such as $$1.19, 1.20, 1.21, 1.18, 1.16, …$$ which represent the petal width for one flower can all be described by variable $$X$$. Variable mean: the average value for the variable. Computed as the sum of all available values divided by the number of values summed together. As petal width represents dim=3 in the visualization above, with a mean of $$\approx 1.1993$$, we can see how the numbers above fit. Variance: describing the “spread” of data around the variable. Computed as the sum of squared differences between each number and the mean, i.e. the sum of $$(x – \mu)^2$$ for each number. Covariance: describing the joint variability (or joint spread) of two variables. For each pair of numbers from both variables, covariance is computed as $$Cov(x, y) = (x – \mu_x)(y – \mu_y)$$. Covariance matrix for $$n$$ variables: a matrix representing covariances for each pair of variables from some set of variables (dimensions) $$V = [X, Y, Z, ….]$$. A covariance matrix for two dimensions $$X$$ and $$Y$$ looks as follows: $$\begin{pmatrix}Cov(X, X) & Cov(X, Y)\\ Cov(Y, X) & Cov(Y, Y)\end{pmatrix}$$ Fortunately, there are some properties which make covariance matrices interesting for PCA (Lambers, n.d.): • $$Cov(X, X) = Var(X)$$ • $$Cov(X, Y) = Cov(Y, X)$$. By consequence, our covariance matrix is a symmetrical and square, $$n \times n$$ matrix and can hence also be written as follows: $$\begin{pmatrix}Var(X) & Cov(X, Y)\\ Cov(Y, X) & Var(Y)\end{pmatrix}$$ We can compute the covariance matrix by generating a $$n \times n$$ matrix and then filling it by iterating over its rows and columns, setting the value to the average covariance for each respective number from both variables: # Compute covariance matrix cov_matrix = np.empty((X.shape[1], X.shape[1])) # 4 x 4 matrix for row in range(0, X.shape[1]): for col in range(0, X.shape[1]): cov_matrix[row][col] = np.round(np.average([(X[i, row] - np.average(X[:, row]))*(X[i, col]\ - np.average(X[:, col])) for i in range(0, X.shape[0])]), 2)Code language: PHP (php) If we compare our self-computed covariance matrix with one generated with NumPy’s np.cov, we can see the similarities: # Compare the matrices print('Self-computed:') print(cov_matrix) print('NumPy-computed:') print(np.round(np.cov(X.T), 2)) > Self-computed: > [[ 1. -0.12 0.87 0.82] > [-0.12 1. -0.43 -0.37] > [ 0.87 -0.43 1. 0.96] > [ 0.82 -0.37 0.96 1. ]] > NumPy-computed: > [[ 1.01 -0.12 0.88 0.82] > [-0.12 1.01 -0.43 -0.37] > [ 0.88 -0.43 1.01 0.97] > [ 0.82 -0.37 0.97 1.01]]Code language: PHP (php) Decomposing the covariance matrix into eigenvectors and eigenvalues Above, we have expressed the spread of our dataset across the dimensions in our covariance matrix. Recall that PCA works by expressing this spread in terms of vectors, called eigenvectors, which together with their corresponding eigenvalues tell us something about the direction and magnitude of the spread. The great thing of EIG-PCA is that we can decompose the covariance matrix into eigenvectors and eigenvalues. We can do this as follows: $$\mathbf C = \mathbf V \mathbf L \mathbf V^\top$$ Here, $$\mathbf V$$ is a matrix of eigenvectors where each column is an eigenvector, $$\mathbf L$$ is a diagonal matrix with eigenvalues and $$\mathbf V^\top$$ is the transpose of $$\mathbf V$$. We can use NumPy’s numpy.linalg.eig to compute the eigenvectors for this square array: # Compute the eigenpairs eig_vals, eig_vect = np.linalg.eig(cov_matrix) print(eig_vect) print(eig_vals)Code language: PHP (php) This yields the following: [[ 0.52103086 -0.37921152 -0.71988993 0.25784482] [-0.27132907 -0.92251432 0.24581197 -0.12216523] [ 0.57953987 -0.02547068 0.14583347 -0.80138466] [ 0.56483707 -0.06721014 0.63250894 0.52571316]] [2.91912926 0.91184362 0.144265 0.02476212]Code language: JSON / JSON with Comments (json) If we compute how much each principal dimension contributes to variance explanation, we get the following: # Compute variance contribution of each vector contrib_func = np.vectorize(lambda x: x / np.sum(eig_vals)) var_contrib = contrib_func(eig_vals) print(var_contrib) print(np.sum(var_contrib)) > [0.72978232 0.2279609 0.03606625 0.00619053] > 1.0Code language: PHP (php) In other words, the first principal dimension contributes for 73%; the second one for 23%. If we therefore reduce the dimensionality to two, we get to keep approximately $$73 + 23 = 96%$$ of the variance explanation. Sorting the eigenpairs in decreasing order of importance Even though the eigenpairs above have already been sorted, it’s a thing we must definitely do – especially when you perform the decomposition in eigenpairs in a different way. Sorting the eigenpairs happens by eigenvalue: the eigenvalues must be sorted in a descending way; the corresponding eigenvectors must therefore also be sorted equally. # Sort eigenpairs eigenpairs = [(np.abs(eig_vals[x]), eig_vect[:,x]) for x in range(0, len(eig_vals))] eig_vals = [eigenpairs[x][0] for x in range(0, len(eigenpairs))] eig_vect = [eigenpairs[x][1] for x in range(0, len(eigenpairs))] print(eig_vals)Code language: PHP (php) This yields sorted eigenpairs, as we can see from the eigenvalues: [2.919129264835876, 0.9118436180017795, 0.14426499504958146, 0.024762122112763244]Code language: JSON / JSON with Comments (json) Selecting n principal components Above, we saw that 96% of the variance can be explained by only two of the dimensions. We can therefore reduce the dimensionality of our feature space from $$n = 4$$ to $$n = 2$$ without losing much of the information. Building the projection matrix The final thing we must do is generate the projection matrix and project our original data onto the (two) principal components (Raschka, 2015): # Build the projection matrix proj_matrix = np.hstack((eig_vect[0].reshape(4,1), eig_vect[1].reshape(4,1))) print(proj_matrix) # Project onto the principal components X_proj = X.dot(proj_matrix)Code language: PHP (php) Voilà, you’ve performed PCA If we now plot the projected data, we get the following plot: # Make plot of projection plt.scatter(X_proj[:, 0], X_proj[:, 1], c=colors) plt.title(f'Visualizing the principal components') plt.xlabel('Principal component 1') plt.ylabel('Principal component 2') plt.show()Code language: PHP (php) That’s it! You just performed Principal Component Analysis using Eigenvector Decomposition and have reduced dimensionality to two without losing much of the information in the dataset. PCA-SVD: Singular Value Decomposition with Python Step-by-Step Above, we covered performing Principal Component Analysis with Eigenvector Decomposition of the dataset’s covariance matrix. A more numerically stable method is using Singular Value Decomposition on the data matrix itself instead of Eigenvector Decomposition on its covariance matrix. In this section, we’ll cover the SVD approach in a step-by-step fashion, using Python. Note that here as well, we’ll use a vanilla / native Python approach to performing PCA, since it brings more clarity. In the next section, we’ll use the framework provided tools (i.e. sklearn.decomposition.PCA) instead of the native ones. Starting with the standardized Iris dataset In the PCA-SVD approach, we also use the Iris dataset as an example. Using the code below, we’ll load the Iris data and perform standardization, which means that your mean will become $$\mu = 0.0$$ and your standard deviation will become $$\sigma = 1.0$$. from sklearn import datasets import matplotlib.pyplot as plt import numpy as np X = iris.data y = iris.target # Shape print(X.shape) print(y.shape) # Color definitions colors = { 0: '#b40426', 1: '#3b4cc0', 2: '#f2da0a', } # Legend definition legend = ['Iris Setosa', 'Iris Versicolour', 'Iris Virginica'] # Perform standardization for dim in range(0, X.shape[1]): print(f'Old mean/std for dim={dim}: {np.average(X[:, dim])}/{np.std(X[:, dim])}') X[:, dim] = (X[:, dim] - np.average(X[:, dim])) / np.std(X[:, dim]) print(f'New mean/std for dim={dim}: {np.abs(np.round(np.average(X[:, dim])))}/{np.std(X[:, dim])}')Code language: PHP (php) Performing SVD on the data matrix In the EIG variant of PCA, we computed the covariance matrix of our dataset, and then performed Eigenvector Decomposition on this matrix to find the eigenvectors and eigenvalues. We could then use these to sort the most important ones and project our dataset onto the most important ones, i.e. the principal components. In the SVD variant, we compute the singular values of the data matrix instead. It is a generalization of the Eigenvector Decomposition, meaning that it can also be used on non-square and non-symmetric matrices (which in the EIG case required us to use the covariance matrix, which satisfies both criteria). # Compute SVD u, s, vh = np.linalg.svd(X.T, full_matrices=True)Code language: PHP (php) In SVD, we decompose a matrix into three components: • Unitary arrays $$U$$ • Vectors with the singular values $$s$$ • Unitary arrays $$vh$$ Here, the columns of the unitary arrays give results equal to the eigenvectors of the covariance matrix in the PCA-EIG approach, and the singular value vectors are equal to the square roots of the eigenvalues of the covariance matrix (StackExchange, n.d.). Translating SVD outputs to usable vectors and values In other words, by performing SVD on the data matrix, we can create the same results as with the PCA-EIG approach. With that approach, the eigenvectors of the covariance matrix were as follows: [[ 0.52103086 -0.37921152 -0.71988993 0.25784482] [-0.27132907 -0.92251432 0.24581197 -0.12216523] [ 0.57953987 -0.02547068 0.14583347 -0.80138466] [ 0.56483707 -0.06721014 0.63250894 0.52571316]] Code language: JSON / JSON with Comments (json) Now compare them to the output of vh: print(vh) > [[ 0.52106591 -0.26934744 0.5804131 0.56485654] > [-0.37741762 -0.92329566 -0.02449161 -0.06694199] > [ 0.71956635 -0.24438178 -0.14212637 -0.63427274] > [ 0.26128628 -0.12350962 -0.80144925 0.52359713]]Code language: CSS (css) Except for the sign, the columns of vh equal the rows of the EIG-based eigenvectors. Sorting eigenvalues and eigenvectors In the PCA-EIG scenario, you had to sort eigenpairs in descending order of the eigenvalues. np.linalg.svd already sorts in descending order, so this is no longer necessary. Selecting n components Here, too, we can simply select $$n$$ components. As with the PCA-EIG scenario, here we also take $$n = 2$$ and hence reduce our dimensionality from 4 to 2. Building the projection matrix We can now easily build the projection matrix as we did in the PCA-EIG case, project our data onto the principal components, and make a plot of the projection. # Build the projection matrix proj_matrix = np.hstack((vh[0].reshape(4,1), vh[1].reshape(4,1))) print(proj_matrix) # Project onto the principal components X_proj = X.dot(proj_matrix) # Make plot of projection colors = list(map(lambda x: colors[x], y)) plt.scatter(X_proj[:, 0], X_proj[:, 1], c=colors) plt.title(f'Visualizing the principal components') plt.xlabel('Principal component 1') plt.ylabel('Principal component 2') plt.show()Code language: PHP (php) The end result: It’s the same! Easy PCA with Scikit-learn for real datasets In the previous two sections, we manually computed the principal components and manually projected our dataset onto these components – for the sake of showing you how stuff works. Fortunately, this task is not necessary when using modern Machine Learning libraries such as Scikit-learn. Instead, it provides the functionality for PCA out of the box, through sklearn.decomposition.PCA. Really easy! To be more precise, Scikit-learn utilizes PCA-SVD for computing the Principal Components of your dataset. Let’s now take a look at how Scikit’s approach works, so that you can finish this article both knowing how (1) PCA-EIG and PCA-SVD work (previous sections) and (2) how you can implement PCA pragmatically (this section). Restarting with the Iris dataset from sklearn import datasets import matplotlib.pyplot as plt import numpy as np from sklearn.decomposition import PCA from sklearn.preprocessing import StandardScaler X = iris.data y = iris.targetCode language: PHP (php) Performing Scikit-learn based standardization As we could read in another article, Scikit-learn provides standardization out of the box through the StandardScaler, so we also implement it here: # Standardize scaler = StandardScaler() scaler.fit(X) X = scaler.transform(X)Code language: PHP (php) Performing sklearn.decomposition.PCA We can then easily implement PCA as follows. First, we initialize sklearn.decomposition.PCA and instruct it to extract two principal components (just like we did before) based on the Iris dataset (recall that X = iris.data): # PCA pca = PCA(n_components=2) pca.fit(X)Code language: PHP (php) print(pca.explained_variance_ratio_) print(pca.components_)Code language: CSS (css) > [0.72962445 0.22850762] > [[ 0.52106591 -0.26934744 0.5804131 0.56485654] > [ 0.37741762 0.92329566 0.02449161 0.06694199]]Code language: CSS (css) We can see that our explained variance ratio is equal to the ones we found manually; that the same is true for the PCA components. We can now easily project the data onto the principal components with .transform(X): X = pca.transform(X) Visualizing the data… # Color definitions colors = { 0: '#b40426', 1: '#3b4cc0', 2: '#f2da0a', } # Make plot of projection colors = list(map(lambda x: colors[x], y)) plt.scatter(X[:, 0], X[:, 1], c=colors) plt.title(f'Visualizing the principal components with Scikit-learn based PCA') plt.xlabel('Principal component 1') plt.ylabel('Principal component 2') plt.show()Code language: PHP (php) …gives the following result: Voila, precisely as we have seen before! Full PCA code The full code for performing the PCA with Scikit-learn on the Iris dataset is as follows: from sklearn import datasets import matplotlib.pyplot as plt import numpy as np from sklearn.decomposition import PCA from sklearn.preprocessing import StandardScaler X = iris.data y = iris.target # Standardize scaler = StandardScaler() scaler.fit(X) X = scaler.transform(X) # PCA pca = PCA(n_components=2) pca.fit(X) print(pca.explained_variance_ratio_) print(pca.components_) X = pca.transform(X) # Color definitions colors = { 0: '#b40426', 1: '#3b4cc0', 2: '#f2da0a', } # Make plot of projection colors = list(map(lambda x: colors[x], y)) plt.scatter(X[:, 0], X[:, 1], c=colors) plt.title(f'Visualizing the principal components with Scikit-learn based PCA') plt.xlabel('Principal component 1') plt.ylabel('Principal component 2') plt.show()Code language: PHP (php) Summary In this article, we read about performing Principal Component Analysis on the dimensions of your dataset for the purpose of dimensionality reduction. Some datasets have many features and few samples, meaning that many Machine Learning algorithms will be struck by the curse of dimensionality. Feature extraction approaches like PCA, which attempt to construct a lower-dimensional feature space based on the original dataset, can help reduce this curse. Using PCA, we can attempt to recreate our feature space with fewer dimensions and with minimum information loss. After defining the context for applying PCA, we looked at it from a high-level perspective. We saw that we can compute eigenvectors and eigenvalues and sort those to find the principal directions in your dataset. After generating a projection matrix for these directions, we can map our dataset onto these directions, which are then called the principal components. But how these eigenvectors can be derived was explained later, because there are two methods for doing so: using Eigenvector Decomposition (EIG) and the more generalized Singular Value Decomposition (SVD). In two step-by-step examples, we saw how we can apply both PCA-EIG and PCA-SVD for performing a Principal Component Analysis. In the first case, we saw that we can compute a covariance matrix for the standardized dataset which illustrates the variances and covariances of its variables. This matrix can then be decomposed into eigenvectors and eigenvalues, which illustrate the direction and magnitude of the spread expressed by the covariance matrix. Sorting the eigenpairs, we can select the principal directions that contribute most to variance, generate the projection matrix and project our data. While PCA-EIG works well with symmetric and square matrices (and hence with our covariance matrix), it can be numerically unstable. That’s why PCA-SVD is very common in today’s Machine Learning libraries. In another step-by-step example, we looked at how the SVD can be used directly on the standardized data matrix for deriving the eigenvectors we also found with PCA-EIG. They can be used for generating a projection matrix which allowed us to arrive at the same end result as when performing PCA-EIG. Finally, knowing how PCA-EIG and PCA-SVD work, we moved to a Scikit-learn based implementation of Principal Component Analysis. Because why reinvent the wheel if good implementations are already available? Using the Scikit StandardScaler and PCA implementations, we performed the standardization and (SVD-based) PCA that we also performed manually, once again finding the same results. It’s been a thorough read, that’s for sure. Still, I hope that you have learned something. Please share this article or drop a comment in the comments section below if you find it useful 💬 Please do the same when you have additional questions, remarks or suggestions for improvement. Where possible, I’ll respond as quickly as I can. Thank you for reading MachineCurve today and happy engineering! 😎 References Wikipedia. (n.d.). Curse of dimensionality. Wikipedia, the free encyclopedia. Retrieved December 3, 2020, from https://en.wikipedia.org/wiki/Curse_of_dimensionality Wikipedia. (2004, November 17). Feature selection. Wikipedia, the free encyclopedia. Retrieved December 3, 2020, from https://en.wikipedia.org/wiki/Feature_selection Wikipedia. (2003, June 8). Feature extraction. Wikipedia, the free encyclopedia. Retrieved December 3, 2020, from https://en.wikipedia.org/wiki/Feature_extraction Wikipedia. (2002, August 26). Principal component analysis. Wikipedia, the free encyclopedia. Retrieved December 7, 2020, from https://en.wikipedia.org/wiki/Principal_component_analysis StackExchange. (n.d.). Relationship between SVD and PCA. How to use SVD to perform PCA? Cross Validated. https://stats.stackexchange.com/questions/134282/relationship-between-svd-and-pca-how-to-use-svd-to-perform-pca Wikipedia. (n.d.). Eigenvalues and eigenvectors. Wikipedia, the free encyclopedia. Retrieved December 7, 2020, from https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Lambers, J. V. (n.d.). PCA – Mathematical Backgroundhttps://www.math.usm.edu/lambers/cos702/cos702_files/docs/PCA.pdf Raschka, S. (2015, January 27). Principal component analysis. Dr. Sebastian Raschka. https://sebastianraschka.com/Articles/2015_pca_in_3_steps.html StackExchange. (n.d.). Why does Andrew Ng prefer to use SVD and not EIG of covariance matrix to do PCA? Cross Validated. https://stats.stackexchange.com/questions/314046/why-does-andrew-ng-prefer-to-use-svd-and-not-eig-of-covariance-matrix-to-do-pca MachineCurve. (2020, November 19). How to normalize or standardize a dataset in Python? – MachineCurvehttps://www.machinecurve.com/index.php/2020/11/19/how-to-normalize-or-standardize-a-dataset-in-python/ Wikipedia. (2003, March 4). Covariance matrix. Wikipedia, the free encyclopedia. Retrieved December 7, 2020, from https://en.wikipedia.org/wiki/Covariance_matrix NumPy. (n.d.). Numpy.linalg.svd — NumPy v1.19 manualhttps://numpy.org/doc/stable/reference/generated/numpy.linalg.svd.html StackExchange. (n.d.). Understanding the output of SVD when used for PCA. Cross Validated. https://stats.stackexchange.com/questions/96482/understanding-the-output-of-svd-when-used-for-pca Do you want to start learning ML from a developer perspective? 👩‍💻 Blogs at MachineCurve teach Machine Learning for Developers. Sign up to learn new things and better understand concepts you already know. We send emails every Friday. By signing up, you consent that any information you receive can include services and special offers by email.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5852287411689758, "perplexity": 1133.097943805134}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362899.14/warc/CC-MAIN-20210301182445-20210301212445-00171.warc.gz"}
https://tug.org/pipermail/luatex/2009-August/001003.html	# [luatex] trunk fixes Hartmut Henkel hartmut_henkel at gmx.de Sat Aug 8 17:51:48 CEST 2009 On Sat, 8 Aug 2009, Vafa Khalighi wrote: > "with luatex having RTL tables is quite easy because I could have > \pagedir TRT and automatically tables becomes RTL. but sometimes we > actually are writing LTR and we need to have a section which takes > about say 2/3 of the page and it is RTL. So having \pagedir TRT does > not make sense in this case. Is there any primitive avaliable in > LuaTeX that makes tabular RTL? I can do it with macros like what I > have done in XeTeX but I think having a primitive that makes tabular > RTL would help a lot. What do you think?" there are also \pardir and \textdir, and you can give a box an internal direction, like: \pagewidth=210mm \pageheight=297mm \nopagenumbers \pagedir TLT \bodydir TLT \pardir TLT \textdir TLT \input tufte \line{\hfill\vbox dir TRT{\halign{#\unskip\hfill&#\unskip\hfill\cr aasfdkljal fjklsdjl & 1 \cr xxx & asdflk \cr }}\hfill} \input tufte \bye Regards, Hartmut	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9783815145492554, "perplexity": 20543.758120463342}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00024.warc.gz"}
https://dolfin-adjoint-doc.readthedocs.io/en/latest/features/index.html	## Generality¶ dolfin-adjoint works for both steady and time-dependent models, and for both linear and nonlinear models. The user interface is exactly the same in both cases. For an example of adjoining a nonlinear time-dependent model, see the tutorial. ## Ease of use¶ dolfin-adjoint has been carefully designed to try to make its use as easy as possible. In many cases the only change to the forward model is to add from dolfin_adjoint import * at the top of the model. For example, deriving the adjoint of the tutorial example requires adding precisely two lines to the forward model; implementing a checkpointing scheme requires adding another two. dolfin-adjoint also makes it extremely easy to verify the correctness of the adjoint model. It offers a powerful syntax for expressing general functionals. ## Efficiency¶ Efficiency of the resulting model is absolutely crucial for real applications. The efficiency of an adjoint model is measured as (time for forward and adjoint run)/(time for forward run). Naumann (2011) states that a typical value for this ratio when using algorithmic differentiation tools is in the range 3–30. By contrast, dolfin-adjoint is extremely efficient; consider the following examples from the papers: PDE Theoretical optimum Achieved efficiency Cahn-Hilliard 1.2 1.22 Stokes 2.0 1.86 Viscoelasticity 2.0 2.029 Gross-Pitaevskii 1.5 1.54 Gray-Scott 2.0 2.04 Navier-Stokes 1.33 1.41 Mathematical programming with equilibrium constraints 1.125 1.126 Shallow water 1.125 1.125 Wetting and drying 1.5 1.55 ## Parallelism¶ Parallelism is ubiquitous in modern computational science. However, applying algorithmic differentiation to parallel codes is still a major research challenge. Algorithmic differentiation tools must be specially modified to understand MPI and OpenMP directives, and translate them into their parallel equivalents. By contrast, because of the high-level abstraction taken in libadjoint, the problem of parallelism simply disappears. In fact, there is no code whatsoever in either dolfin-adjoint or libadjoint to handle parallelism; by deriving the adjoint at the right level of abstraction, the problem no longer exists. If the forward model runs in parallel, the adjoint model also runs in parallel, with no modification. For more details, see the manual section on parallelism and the dolfin-adjoint paper. ## Checkpointing¶ The adjoint model is a linearisation of the forward model. If the forward model is nonlinear, then the solution of that forward model must be available to linearise the forward model. By default, dolfin-adjoint stores every variable computed in memory, as this is the fastest and most straightforward option; however, this may not be feasible for large runs, or for runs with very many timesteps. The solution to this problem is to employ a checkpointing scheme. Rather than store every variable during the forward run, checkpoints are stored at strategically chosen intervals, from which the model may recompute the missing solutions. During the adjoint run, if a forward variable is necessary and unavailable, the forward model is restarted from the nearest available checkpoint to compute the missing solutions; once these are available, the adjoint run continues. Thus, to employ a checkpointing scheme, the control flow of the adjoint run must seamlessly jump between assembling and solving the adjoint equations, and assembling and solving parts of the forward run. Coding a checkpointing scheme is quite complicated, and so most hand-coded adjoint models do not use them. However, the libadjoint library underlying dolfin-adjoint embeds the excellent revolve library of Griewank and Walther, and can automatically employ optimal checkpointing schemes for almost no marginal user effort. For more details, see the manual section on checkpointing and the dolfin-adjoint paper. ## Optimisation with PDE constraints¶ Many computational problems in engineering and science can be formulated as optimisation problems in which a system of partial differential equation occur as a constraint. To solve these problems efficiently, the use of gradient based optimisation algorithms is essential. The fact that dolfin-adjoint has direct access to the gradient information made it possible to directly interface dolfin-adjoint to a range of powerful optimisation algorithms. That means, that an existing FEniCS forward model can be easily used in the context of an optimisation problem with minimal development effort. For more details, see the manual section on optimisation. ## Generalised stability analysis¶ Generalised stability analysis is an extension of linear stability analysis with two important features: it allows for the stability analysis of non-normal systems that permit transient perturbations that grow in magnitude before decaying, and it allows for the analysis of the stability of time-dependent base solutions. The core computation involved in conducting a generalised stability analysis is the singular value decomposition of the propagator: each action of the propagator requires the solution of the tangent linear system, and each adjoint action (for the singular value decomposition) requires the solution of the adjoint system. Since dolfin-adjoint automates the solution of these systems, it can also automate generalised stability analysis by embedding these computations in a matrix-free Krylov–Schur algorithm for computing the SVD of the propagator. For more details, see the manual section on generalised stability analysis.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8257046937942505, "perplexity": 1180.353700815177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141182794.28/warc/CC-MAIN-20201125125427-20201125155427-00245.warc.gz"}
https://socratic.org/questions/how-do-you-simplify-4-5-2-45	Algebra Topics # How do you simplify 4√5 - 2√45? Oct 9, 2015 $- 2 \sqrt{5}$ #### Explanation: Your starting expression looks like this $4 \sqrt{5} - 2 \sqrt{45}$ Notice that you can write $45$ as $45 = 3 \cdot 15 = 3 \cdot 3 \cdot 5 = {3}^{2} \cdot 5$ This means that you have $4 \sqrt{5} - 2 \sqrt{{3}^{2} \cdot 5} = 4 \sqrt{5} - 2 \cdot \sqrt{{3}^{2}} \cdot \sqrt{5}$ $= 4 \sqrt{5} - 2 \cdot 3 \sqrt{5}$ $= 4 \sqrt{5} - 6 \sqrt{5}$ $= \sqrt{5} \cdot \left(4 - 6\right)$ $= \textcolor{g r e e n}{- 2 \sqrt{5}}$ ##### Impact of this question 695 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6054804921150208, "perplexity": 4261.448230625614}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655912255.54/warc/CC-MAIN-20200710210528-20200711000528-00313.warc.gz"}
http://uncyclopedia.wikia.com/wiki/The_World's_Hardest_Maths_Problems	# The World's Hardest Maths Problems For the religious among us who choose to believe lies, the so-called experts at Wikipedia have an article about The World's Hardest Maths Problems. Over the years, in many forms or another, people have been set Maths problems, either by other people or by God. Some of these problems however are infinitely harder than the rest, and The World's Hardest Maths Problems is a collection of these for your enjoyment, or rather the lack of it.[citation needed] ## History These individual questions were first created by Euclidean Protectorate in order to pose a test for the officers to see if they were fit to join the ranks in the fight against the Non-Euclidean Witchcraft created during World War π. The idea was that there could only be one of three outcomes, each with different decisions made concerning the individual to have provided those answers: • The officer attempted to answer the questions in the time allowed, and failed on account of their impossibility. If they did not complain, and merely took it on themselves as failure, then they were counted as having enough faith in the Order to be allowed to fight. • However, if the officer complained that the questions were impossible, then whilst their faith was still demonstrated, they were shot for insubordination. In the order there can be no room for such insolence! • The third outcome was that the officer would answer all the questions, and worst of all get them right. As this was only possible by practicing Non-Euclidean Magicks, the officer has deemed a Heretic and burned at the stake for witchcraft ## Objections! Many (i.e. most) historians argue that the above section is utter nonsense, particularly as the whole World War π thing never happened anyway. Instead they hold that the questions are merely the product of a deranged mind that has been spending too much time with Puff the Magic Dragon, if you catch my meaning. ## The Questions ### Information required to complete the questions You are given a triangle that has sides of 66cm, 73cm, and 94cm. One of the angles is right-angled (meaning that it is possible by trial and error to calculate what each of the angles are). Inside this triangle is a square, so that three corners are in contact with the lines bounding the triangle. One of the sides or the square, which we shall now dub z, is also tangent to a circle, with a radius such that the centre of the circle lies along the side of the triangle with length 73cm. You are also given a regular octagon, which you are told is the same area as the total are of the circle and triangle if they are taken together (i.e. the overlapping area is not counted twice), and one side of this octagon forms another side of equal length belonging to a second square. The area of this square is dubbed x. ### Question 1 Give the value, to three significant figures, of x. [15 marks] ### Question 2 An isosceles triangle is drawn so that it has the same area as the above square (i.e. x), and with two sides that are equal to the square root of x (henceforth dubbed y). What is the length of the third side? [100+90-7685*58/92 marks] ### Question 3 Prove that the triangle above exists. [25 marks] ### Question 4 What is the area of a octagon of side length y, in cubic inches. (Note that this question uses non-euclidean goemetry) [2πr marks] ### Question 5 Through cunning use of Pythaogoras' Theorem, prove that aliens do not exist. [-0 marks] ### Question 6 If $48y^2+{\pi}y-6778=x$, then what does y smell like? [-10 marks] ### Question 7 What is the answer of this question?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6476126313209534, "perplexity": 836.3261001466971}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207930895.96/warc/CC-MAIN-20150521113210-00010-ip-10-180-206-219.ec2.internal.warc.gz"}
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_4064-fm230-1-1	PL EN Preferencje Język Widoczny [Schowaj] Abstrakt Liczba wyników Czasopismo ## Fundamenta Mathematicae 2015 \| 230 \| 1 \| 1-13 Tytuł artykułu ### Decomposing Borel functions using the Shore-Slaman join theorem Autorzy Treść / Zawartość Warianty tytułu Języki publikacji EN Abstrakty EN Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{σ}$ set under that function is again $F_{σ}$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals α and β with α ≤ β < α·2, every function between Polish spaces having small transfinite inductive dimension is decomposable into countably many Baire class γ functions with $Δ⁰_{β+1}$ domains such that γ + α ≤ β if and only if the preimage of each $Σ^{0}_{α+1}$ set under that function is $Σ^{0}_{β+1}$, and the transformation of a $Σ^{0}_{α+1}$ set into the $Σ^{0}_{β+1}$ preimage is continuous. Słowa kluczowe Kategorie tematyczne Czasopismo Rocznik Tom Numer Strony 1-13 Opis fizyczny Daty wydano 2015 Twórcy autor • School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292 Japan Bibliografia Typ dokumentu Bibliografia Identyfikatory	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.696819007396698, "perplexity": 851.1420868138958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125948426.82/warc/CC-MAIN-20180426164149-20180426184149-00180.warc.gz"}
https://www.shaalaa.com/question-bank-solutions?type=courses&c=maharashtra-state-board-ssc-hsc-hsc-science-general-12th-board-exam_573&subjects=756	HSC Science (General) 12th Board ExamMaharashtra State Board Account It's free! User Share Books Shortlist All Solutions for HSC Science (General) 12th Board Exam - Maharashtra State Board - Mathematics and Statistics Subjects Chapters Subjects Popular subjects Chapters Mathematics and Statistics < prev 1 to 20 of 452 next > The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices. [2] Matrices Chapter: [2] Matrices Concept: Elementary Operation (Transformation) of a Matrix Appears in 2 question papers The equation of tangent to the curve y=y=x^2+4x+1 at (-1,-2) is............... (a) 2x -y = 0 (b) 2x+y-5 = 0 (c) 2x-y-1=0 (d) x+y-1=0 [6] Conics Chapter: [6] Conics Concept: Conics - Tangents and normals - equations of tangent and normal at a point Appears in 1 question paper Given that X ~ B(n= 10, p). If E(X) = 8 then the value of p is ........... (a) 0.6 (b) 0.7 (c) 0.8 (d) 0.4 [20] Bernoulli Trials and Binomial Distribution Chapter: [20] Bernoulli Trials and Binomial Distribution Concept: Bernoulli Trials and Binomial Distribution Appears in 1 question paper Find the area bounded by the curve y2 = 4axx-axis and the lines x = 0 and x = a. [16] Applications of Definite Integral Chapter: [16] Applications of Definite Integral Concept: Area of the Region Bounded by a Curve and a Line Appears in 2 question papers Evaluate : int (sinx)/sqrt(36-cos^2x)dx [15] Integration Chapter: [15] Integration Concept: Methods of Integration - Integration by Substitution Appears in 1 question paper Evaluate : int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx [15] Integration Chapter: [15] Integration Concept: Methods of Integration - Integration by Substitution Appears in 1 question paper The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method. [2] Matrices Chapter: [2] Matrices Concept: Elementary Operation (Transformation) of a Matrix Appears in 1 question paper A fair coin is tossed five times. Find the probability that it shows exactly three times head. [19] Probability Distribution Chapter: [19] Probability Distribution Concept: Conditional Probability Appears in 1 question paper Find the inverse of the matrix, A=[[1,3,3],[1,4,3],[1,3,4]]by using column transformations. [2] Matrices Chapter: [2] Matrices Concept: Elementary Operation (Transformation) of a Matrix Appears in 1 question paper Find the area of the region bounded by the parabola y2 = 4ax and its latus rectum. [16] Applications of Definite Integral Chapter: [16] Applications of Definite Integral Concept: Area of the Region Bounded by a Curve and a Line Show that: int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c [15] Integration Chapter: [15] Integration Concept: Methods of Integration - Integration by Substitution Appears in 1 question paper Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0 [4] Pair of Straight Lines Chapter: [4] Pair of Straight Lines Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation Appears in 1 question paper Solve the following equations by the method of reduction : 2x-y + z=1, x + 2y +3z = 8, 3x + y-4z=1. [2] Matrices Chapter: [2] Matrices Concept: Elementary Operation (Transformation) of a Matrix Appears in 1 question paper Evaluate :intxlogxdx [15] Integration Chapter: [15] Integration Concept: Methods of Integration - Integration by Substitution Appears in 1 question paper The probability that a certain kind of component will survive a check test is 0.5. Find the probability that exactly two of the next four components tested will survive. [19] Probability Distribution Chapter: [19] Probability Distribution Concept: Conditional Probability Appears in 1 question paper Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis [16] Applications of Definite Integral Chapter: [16] Applications of Definite Integral Concept: Area of the Region Bounded by a Curve and a Line Appears in 1 question paper Show that the function defined by f(x) =\|cosx\| is continuous function. [12] Continuity Chapter: [12] Continuity Concept: Introduction of Continuity Appears in 1 question paper A bakerman sells 5 types of cakes. Profits due to the sale of each type of cake is respectively Rs. 3, Rs. 2.5, Rs. 2, Rs. 1.5, Rs. 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake? [18] Statistics Chapter: [18] Statistics Concept: Statistics - Bivariate Frequency Distribution Appears in 1 question paper Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3] [14] Applications of Derivative Chapter: [14] Applications of Derivative Concept: Mean Value Theorem Appears in 1 question paper Prove that int_a^bf(x)dx=f(a+b-x)dx. Hence evaluate : int_a^bf(x)/(f(x)+f(a-b-x))dx [15] Integration Chapter: [15] Integration Concept: Methods of Integration - Integration by Substitution Appears in 1 question paper < prev 1 to 20 of 452 next > All Solutions for HSC Science (General) 12th Board Exam Maharashtra State Board Mathematics and Statistics. You can further filter Question Answers by subjects and chapters. Solutions for most of the questions for Maharashtra State Board can be found here on shaalaa.com. You can use these solutions to prepare for your studies and ace in exams. Solving questions is a great way to practice and with shaalaa.com, you can answer a question and then also check your answer with the solutions provided. S	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6698651313781738, "perplexity": 3604.498527312326}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583867214.54/warc/CC-MAIN-20190122182019-20190122204019-00168.warc.gz"}
https://crypto.stackexchange.com/questions/53066/zero-knowledge-proof-for-complete-new-user-authentication/53222	# zero knowledge proof for complete new user authentication Consider an environment, an user A who is completely new to the environment is entering. An user B who is member of the environment need to verify A as genuine not liar. No information about user A is known to B. Is it possible using the zero knowledge proof. • If there's no information about A known how would you distinguish A from C or D or E? – SEJPM Nov 12 '17 at 16:36 • Or, another way to put it 'verify A as genuine not liar'; liar about what? – poncho Nov 12 '17 at 17:37 • it is that he/she is A not B – Venkatesan S. Venkatesan Nov 13 '17 at 7:14 • How do you define "being A" without any external information? Besides, B will easily convince himself that the new player is not B... – Geoffroy Couteau Nov 13 '17 at 9:41 • Solution is need to detect that. Is it possible with zero knowledge – Venkatesan S. Venkatesan Nov 13 '17 at 11:14 No, you can not. You assume: • $B$ does not know anything about $A$. • You need to know if $A$ is actually called Alice. Well, that does not work. There is no possible proof that $A$ is actually called Alice. That is unless you verify some other credentials, e.g. an ID card. But trust can not be generated out of nothing. That is simply impossible. For example, if there is no way to verify a persons name, then no one in the world can decide if he is actually talking to the real Alice or to Eve who is just calling herself Alice. The question has nothing to do with zero knowledge - that is just impossible in general. You can generate a random keys pair for each user, and store the public key of each user somewhere on the net (blockchain, Usenet, ZeroDB, a website...) Then, each of your users could share a link to their public key for their friends. Friends could then verify signature of challenge to be sure it's the real friend and not a liar. • The question states, the user is "completely new to the environment", so even if there are friends who could verify the person, they are not part of the environment (and thus not trustworthy either). And some random key on the web does not prove it is actually Alice and not Eve calling herself Alice. – tylo Nov 17 '17 at 15:03 • Well, it's not the random key that prove it, it's the public key of the user stored somewhere on the net. For example, this could be a hash ID in the public twitter user's profile, where this hash refer to a public key. So when A come in the environment and claim to be "Alice", a user B can retrieve the public key of "Alice" and verify the signature of user A to detect if it's a liar or not, without having any information on A inside the environment. – lakano Nov 17 '17 at 15:17 • What if there is not already an Alice? Do you just trust every new user who's name is not already known? Say I am the first to claim to be Alice but I am really Eve and that is accepted, not along comes the real Alice—oops. – zaph Nov 17 '17 at 15:41 • @zaph If Alice doesn't exists, so the question to known if it's a liar doesn't exists, because he can't impersonate someone that doesn't exists. – lakano Nov 17 '17 at 15:49 • @zaph When the evil Alice register but can't have any proof, the system could add a flag « unverifed » to inform other users. When the real Alice sign-up, she can add a proof (eg: Twitter hash on his profile), the system can now warn every contacts of evil Alice was a liar, and replace it with the real Alice. – lakano Nov 17 '17 at 16:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.449205607175827, "perplexity": 1023.6063174028359}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541294513.54/warc/CC-MAIN-20191214202754-20191214230754-00290.warc.gz"}
https://www.sciencenews.org/article/mathematicians-progress-riemann-hypothesis-proof?mode=blog&context=101	# Mathematicians report possible progress on proving the Riemann hypothesis ## A new study of Jensen polynomials revives an old approach Researchers have made what might be new headway toward a proof of the Riemann hypothesis, one of the most impenetrable problems in mathematics. The hypothesis, proposed 160 years ago, could help unravel the mysteries of prime numbers. Mathematicians made the advance by tackling a related question about a group of expressions known as Jensen polynomials, they report May 21 in Proceedings of the National Academy of Sciences. But the conjecture is so difficult to verify that even this progress is not necessarily a sign that a solution is near (SN Online: 9/25/18). At the heart of the Riemann hypothesis is an enigmatic mathematical entity known as the Riemann zeta function. It’s intimately connected to prime numbers — whole numbers that can’t be formed by multiplying two smaller numbers — and how they are distributed along the number line. The Riemann hypothesis suggests that the function’s value equals zero only at points that fall on a single line when the function is graphed, with the exception of certain obvious points. But, as the function has infinitely many of these “zeros,” this is not easy to confirm. The puzzle is considered so important and so difficult that there is a \$1 million prize for a solution, offered up by the Clay Mathematics Institute. But Jensen polynomials might be a key to unlocking the Riemann hypothesis. Mathematicians have previously shown that the Riemann hypothesis is true if all the Jensen polynomials associated with the Riemann zeta function have only zeros that are real, meaning the values for which the polynomial equals zero are not imaginary numbers — they don’t involve the square root of negative 1. But there are infinitely many of these Jensen polynomials. Studying Jensen polynomials is one of a variety of strategies for attacking the Riemann hypothesis. The idea is more than 90 years old, and previous studies have proved that a small subset of the Jensen polynomials have real roots. But progress was slow, and efforts had stalled. Now, mathematician Ken Ono and colleagues have shown that many of these polynomials indeed have real roots, satisfying a large chunk of what’s needed to prove the Riemann hypothesis. “Any progress in any direction related to the Riemann hypothesis is fascinating,” says mathematician Dimitar Dimitrov of the State University of São Paulo. Dimitrov thought “it would be impossible that anyone will make any progress in this direction,” he says, “but they did.” It’s hard to say whether this progress could eventually lead to a proof. “I am very reluctant to predict anything,” says mathematician George Andrews of Penn State, who was not involved with the study. Many strides have been made on the Riemann hypothesis in the past, but each advance has fallen short. However, with other major mathematical problems that were solved in recent decades, such as Fermat’s last theorem (SN: 11/5/94, p. 295), it wasn’t clear that the solution was imminent until it was in hand. “You never know when something is going to break.” The result supports the prevailing viewpoint among mathematicians that the Riemann hypothesis is correct. “We’ve made a lot of progress that offers new evidence that the Riemann hypothesis should be true,” says Ono, of Emory University in Atlanta. If the Riemann hypothesis is ultimately proved correct, it would not only illuminate the prime numbers, but would also immediately confirm many mathematical ideas that have been shown to be correct assuming the Riemann hypothesis is true. In addition to its Riemann hypothesis implications, the new result also unveils some details of what’s known as the partition function, which counts the number of possible ways to create a number from the sum of positive whole numbers (SN: 6/17/00, p. 396). For example, the number 4 can be made in five different ways: 3+1, 2+2, 2+1+1, 1+1+1+1, or just the number 4 itself. The result confirms an earlier proposition about the details of how that partition function grows with larger numbers. “That was an open question … for a long time,” Andrews says. The real prize would be proving the Riemann hypothesis, he notes. That will have to wait.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8412078022956848, "perplexity": 420.33578826772464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514577478.95/warc/CC-MAIN-20190923172009-20190923194009-00216.warc.gz"}
http://soft-matter.seas.harvard.edu/index.php?title=Hydrodynamic_metamaterials:_Microfabricated_arrays_to_steer,_refract_and_focus_streams_of_biomaterials&diff=prev&oldid=12572	# Difference between revisions of "Hydrodynamic metamaterials: Microfabricated arrays to steer, refract and focus streams of biomaterials" Original entry: Warren Lloyd Ung, APPHY 225, Fall 2009 "Hydrodynamic metamaterials: Microfabricated arrays to steer, refractt, and focus streams of biomaterials" Keith J. Morton, Kevin Loutherback, David W. Inglis, Ophelia K. Tsui, James C. Sturm, Stephen Y. Chou, and Robert H. Austin. Proceedings of the National Academy of Science. ## Soft Matter Keywords Hydrodynamic metamaterials, microfluidics, biomaterials, separation Figure 1: Hydrodynamic metamaterial: an asymmetric array of posts (A) schematic and (B) fluoresecne image of a particle larger than the critical size (red) and smaller than the critical size (green). Figure 2: Focusing of flowing particles: (A) The analogous case for light is an axicon lens, which focuses light to a line, (B) Schematic of the different regions of distinct metamaterials, (C) Micrograph of the distinct metamaterial regions, (D) fluorescence image showing particle focusing by this geometry of posts. Figure 3: Steering of cells through a microfluidic channel ## Summary Hydrodynamic metamaterials are microfabricated structures that have can be used to manipulate particles to flow along particular paths. In many ways, they are analogous to traditional optical materials, except rather than modifying the propagation of electromagnetic light waves, these metamaterials modify the propagation of particles through a microfluidic channel. The hydrodynamic metamaterial discussed in here is an asymmetric array of posts. The asymmetry arises, because each subsequent column of posts is vertically offset from the previous column by a small distance. As a result, the rows of posts are at an angle, $\alpha$, relative to the direction of flow (Figure 1). Small particles and molecules follow the streamlines of fluid through the channel, and as such, they simply follow the direction of bulk fluid flow through the channel, while moving around any posts in their way. On the other hand, particles larger than some critical size cannot do the same, they cannot fully move around the posts, so each time they encounter a new post, they are deflected it. The large particles, thus, follow the asymmetry of the channel, and propagate at the angle $\alpha$ relative to the flow direction. This ability to segregate particles according to their sizes can be compared with the ability of birefringent crystals to separate different polarizations of light in space. These metamaterials offer exquisite control over the flow direction of particles in solution. By putting metamaterials with different parameters next to one another, it is possible to create devices to control the direction, along which particles of different sizes move. It is also possible to place several different arrays within a single channel to achieve complex devices. The authors showcase a range of possible applications for these simple microstructures, each time demonstrating its analogy in optics. ## Soft Matter Discussion The parameters of the array uniquely define the angle, at which large particles are deflected, and also the critical size of particles, at which particles transition from passing around the posts in the array to being deflected along the rows. The authors define four parameters, the horizontal spacing $\lambda$, the vertical gap between posts, G, the vertical offset between rows, $\delta$, and the obstacle size, D. Qualitatively, for a given G, the critical particle size decreases as the angle is increased $\alpha = tan^{-1}\frac{\delta}{\lambda}$ For a fixed particle size and angle, the critical particle size decreases with G. These behave more or less what as we would expect. The likely cause of the observed results is that the hydraulic object is much larger than the cross-sectional gap between the posts. Objects with large hydraulic diameters are less likely to fit through the space between the post. We would also expect the critical size to change as a function of flow rate. The inertia of particles in the flow is a parameter that we expect to contribute to which mode the particle finds itself in. This dependence is not addressed in the paper. ## Applications Using these asymmetric arrays of posts as building blocks, it is possible to create devices which refract, focus or disperse particles selectively based on size. This size can be tuned to select for the distinct particles of interest in a system, whether they be surfactant-stabilized droplets, vesicles, cells or beads. As a result, this technique is applicable to many relevant systems of interest. Although these ideas are applicable, in general, to nearly any system where micro-scale particles should be manipulated based on size, because this is intrinsically a microfluidic technique, many of the most interesting applications are in the field of microfluidics. Because these posts are already within a microfluidic channel and they are compatible with the standard soft lithography methds, this is a powerful tool, which can already be integrated directly with many of the other complex microfluidic systems. For instance, one can imagine a system for studying mitotic cells, in which we separate mitotic cells based on their size from interphase cells in a label-free manner. This could be combined with methods for optical interrogation and on-chip cell culture techniques to achieve a highly integrated system for studying cells. Another potential microfluidics application would be in separating a polydisperse suspension of particles into a set of monodisperse flows. These flows could then be split and routed to other parts of the microfluidic device for highly parallel experiments, in which particle size is a parameter of interest.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5409762859344482, "perplexity": 1272.4041419776597}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304798.1/warc/CC-MAIN-20220125070039-20220125100039-00123.warc.gz"}
https://infogalactic.com/info/Star	# Star For other uses, see Star (disambiguation). False-color imagery of the Sun, a G-type main-sequence star, the closest to Earth A star is a luminous sphere of plasma held together by its own gravity. The nearest star to Earth is the Sun. Other stars are visible from Earth during the night, appearing as a multitude of fixed luminous points in the sky due to their immense distance from Earth. Historically, the most prominent stars were grouped into constellations and asterisms, and the brightest stars gained proper names. Extensive catalogues of stars have been assembled by astronomers, which provide standardized star designations. For at least a portion of its life, a star shines due to thermonuclear fusion of hydrogen into helium in its core, releasing energy that traverses the star's interior and then radiates into outer space. Once the hydrogen in the core of a star is nearly exhausted, almost all naturally occurring elements heavier than helium are created by stellar nucleosynthesis during the star's lifetime and, for some stars, by supernova nucleosynthesis when it explodes. Near the end of its life, a star can also contain degenerate matter. Astronomers can determine the mass, age, metallicity (chemical composition), and many other properties of a star by observing its motion through space, luminosity, and spectrum respectively. The total mass of a star is the principal determinant of its evolution and eventual fate. Other characteristics of a star, including diameter and temperature, change over its life, while the star's environment affects its rotation and movement. A plot of the temperature of many stars against their luminosities, known as a Hertzsprung–Russell diagram (H–R diagram), allows the age and evolutionary state of a star to be determined. A star's life begins with the gravitational collapse of a gaseous nebula of material composed primarily of hydrogen, along with helium and trace amounts of heavier elements. Once the stellar core is sufficiently dense, hydrogen becomes steadily converted into helium through nuclear fusion, releasing energy in the process.[1] The remainder of the star's interior carries energy away from the core through a combination of radiative and convective processes. The star's internal pressure prevents it from collapsing further under its own gravity. Once the hydrogen fuel at the core is exhausted, a star with at least 0.4 times the mass of the Sun[2] expands to become a red giant, in some cases fusing heavier elements at the core or in shells around the core. The star then evolves into a degenerate form, recycling a portion of its matter into the interstellar environment, where it will contribute to the formation of a new generation of stars with a higher proportion of heavy elements.[3] Meanwhile, the core becomes a stellar remnant: a white dwarf, a neutron star, or (if it is sufficiently massive) a black hole. Binary and multi-star systems consist of two or more stars that are gravitationally bound, and generally move around each other in stable orbits. When two such stars have a relatively close orbit, their gravitational interaction can have a significant impact on their evolution.[4] Stars can form part of a much larger gravitationally bound structure, such as a star cluster or a galaxy. ## Observation history People have seen patterns in the stars since ancient times.[5] This 1690 depiction of the constellation of Leo, the lion, is by Johannes Hevelius.[6] The constellation of Leo as it can be seen by the naked eye. Lines have been added. Historically, stars have been important to civilizations throughout the world. They have been part of religious practices and used for celestial navigation and orientation. Many ancient astronomers believed that stars were permanently affixed to a heavenly sphere, and that they were immutable. By convention, astronomers grouped stars into constellations and used them to track the motions of the planets and the inferred position of the Sun.[5] The motion of the Sun against the background stars (and the horizon) was used to create calendars, which could be used to regulate agricultural practices.[7] The Gregorian calendar, currently used nearly everywhere in the world, is a solar calendar based on the angle of the Earth's rotational axis relative to its local star, the Sun. The oldest accurately dated star chart appeared in ancient Egyptian astronomy in 1534 BC.[8] The earliest known star catalogues were compiled by the ancient Babylonian astronomers of Mesopotamia in the late 2nd millennium BC, during the Kassite Period (ca. 1531–1155 BC).[9] The first star catalogue in Greek astronomy was created by Aristillus in approximately 300 BC, with the help of Timocharis.[10] The star catalog of Hipparchus (2nd century BC) included 1020 stars, and was used to assemble Ptolemy's star catalogue.[11] Hipparchus is known for the discovery of the first recorded nova (new star).[12] Many of the constellations and star names in use today derive from Greek astronomy. In spite of the apparent immutability of the heavens, Chinese astronomers were aware that new stars could appear.[13] In 185 AD, they were the first to observe and write about a supernova, now known as the SN 185.[14] The brightest stellar event in recorded history was the SN 1006 supernova, which was observed in 1006 and written about by the Egyptian astronomer Ali ibn Ridwan and several Chinese astronomers.[15] The SN 1054 supernova, which gave birth to the Crab Nebula, was also observed by Chinese and Islamic astronomers.[16][17][18] Medieval Islamic astronomers gave Arabic names to many stars that are still used today, and they invented numerous astronomical instruments that could compute the positions of the stars. They built the first large observatory research institutes, mainly for the purpose of producing Zij star catalogues.[19] Among these, the Book of Fixed Stars (964) was written by the Persian astronomer Abd al-Rahman al-Sufi, who observed a number of stars, star clusters (including the Omicron Velorum and Brocchi's Clusters) and galaxies (including the Andromeda Galaxy).[20] According to A. Zahoor, in the 11th century, the Persian polymath scholar Abu Rayhan Biruni described the Milky Way galaxy as a multitude of fragments having the properties of nebulous stars, and also gave the latitudes of various stars during a lunar eclipse in 1019.[21] According to Josep Puig, the Andalusian astronomer Ibn Bajjah proposed that the Milky Way was made up of many stars which almost touched one another and appeared to be a continuous image due to the effect of refraction from sublunary material, citing his observation of the conjunction of Jupiter and Mars on 500 AH (1106/1107 AD) as evidence.[22] Early European astronomers such as Tycho Brahe identified new stars in the night sky (later termed novae), suggesting that the heavens were not immutable. In 1584 Giordano Bruno suggested that the stars were like the Sun, and may have other planets, possibly even Earth-like, in orbit around them,[23] an idea that had been suggested earlier by the ancient Greek philosophers, Democritus and Epicurus,[24] and by medieval Islamic cosmologists[25] such as Fakhr al-Din al-Razi.[26] By the following century, the idea of the stars being the same as the Sun was reaching a consensus among astronomers. To explain why these stars exerted no net gravitational pull on the Solar System, Isaac Newton suggested that the stars were equally distributed in every direction, an idea prompted by the theologian Richard Bentley.[27] The Italian astronomer Geminiano Montanari recorded observing variations in luminosity of the star Algol in 1667. Edmond Halley published the first measurements of the proper motion of a pair of nearby "fixed" stars, demonstrating that they had changed positions from the time of the ancient Greek astronomers Ptolemy and Hipparchus.[23] William Herschel was the first astronomer to attempt to determine the distribution of stars in the sky. During the 1780s, he performed a series of gauges in 600 directions, and counted the stars observed along each line of sight. From this he deduced that the number of stars steadily increased toward one side of the sky, in the direction of the Milky Way core. His son John Herschel repeated this study in the southern hemisphere and found a corresponding increase in the same direction.[28] In addition to his other accomplishments, William Herschel is also noted for his discovery that some stars do not merely lie along the same line of sight, but are also physical companions that form binary star systems. The science of stellar spectroscopy was pioneered by Joseph von Fraunhofer and Angelo Secchi. By comparing the spectra of stars such as Sirius to the Sun, they found differences in the strength and number of their absorption lines—the dark lines in a stellar spectra due to the absorption of specific frequencies by the atmosphere. In 1865 Secchi began classifying stars into spectral types.[29] However, the modern version of the stellar classification scheme was developed by Annie J. Cannon during the 1900s. Alpha Centauri A and B over limb of Saturn The first direct measurement of the distance to a star (61 Cygni at 11.4 light-years) was made in 1838 by Friedrich Bessel using the parallax technique. Parallax measurements demonstrated the vast separation of the stars in the heavens.[23] Observation of double stars gained increasing importance during the 19th century. In 1834, Friedrich Bessel observed changes in the proper motion of the star Sirius, and inferred a hidden companion. Edward Pickering discovered the first spectroscopic binary in 1899 when he observed the periodic splitting of the spectral lines of the star Mizar in a 104-day period. Detailed observations of many binary star systems were collected by astronomers such as William Struve and S. W. Burnham, allowing the masses of stars to be determined from computation of the orbital elements. The first solution to the problem of deriving an orbit of binary stars from telescope observations was made by Felix Savary in 1827.[30] The twentieth century saw increasingly rapid advances in the scientific study of stars. The photograph became a valuable astronomical tool. Karl Schwarzschild discovered that the color of a star and, hence, its temperature, could be determined by comparing the visual magnitude against the photographic magnitude. The development of the photoelectric photometer allowed very precise measurements of magnitude at multiple wavelength intervals. In 1921 Albert A. Michelson made the first measurements of a stellar diameter using an interferometer on the Hooker telescope at Mount Wilson Observatory.[31] Important theoretical work on the physical structure of stars occurred during the first decades of the twentieth century. In 1913, the Hertzsprung-Russell diagram was developed, propelling the astrophysical study of stars. Successful models were developed to explain the interiors of stars and stellar evolution. Cecilia Payne-Gaposchkin first proposed that stars were made primarily of hydrogen and helium in her 1925 PhD thesis.[32] The spectra of stars were further understood through advances in quantum physics. This allowed the chemical composition of the stellar atmosphere to be determined.[33] With the exception of supernovae, individual stars have primarily been observed in the Local Group,[34] and especially in the visible part of the Milky Way (as demonstrated by the detailed star catalogues available for our galaxy).[35] But some stars have been observed in the M100 galaxy of the Virgo Cluster, about 100 million light years from the Earth.[36] In the Local Supercluster it is possible to see star clusters, and current telescopes could in principle observe faint individual stars in the Local Group[37] (see Cepheids). However, outside the Local Supercluster of galaxies, neither individual stars nor clusters of stars have been observed. The only exception is a faint image of a large star cluster containing hundreds of thousands of stars located at a distance of one billion light years[38]—ten times further than the most distant star cluster previously observed. ## Designations This view contains blue stars known as "Blue stragglers", for their apparent location on the Hertzsprung–Russell diagram The concept of the constellation was known to exist during the Babylonian period. Ancient sky watchers imagined that prominent arrangements of stars formed patterns, and they associated these with particular aspects of nature or their myths. Twelve of these formations lay along the band of the ecliptic and these became the basis of astrology.[39] Many of the more prominent individual stars were also given names, particularly with Arabic or Latin designations. As well as certain constellations and the Sun itself, individual stars have their own myths.[40] To the Ancient Greeks, some "stars", known as planets (Greek πλανήτης (planētēs), meaning "wanderer"), represented various important deities, from which the names of the planets Mercury, Venus, Mars, Jupiter and Saturn were taken.[40] (Uranus and Neptune were also Greek and Roman gods, but neither planet was known in Antiquity because of their low brightness. Their names were assigned by later astronomers.) Circa 1600, the names of the constellations were used to name the stars in the corresponding regions of the sky. The German astronomer Johann Bayer created a series of star maps and applied Greek letters as designations to the stars in each constellation. Later a numbering system based on the star's right ascension was invented and added to John Flamsteed's star catalogue in his book "Historia coelestis Britannica" (the 1712 edition), whereby this numbering system came to be called Flamsteed designation or Flamsteed numbering.[41][42] The only internationally recognized authority for naming celestial bodies is the International Astronomical Union (IAU).[43] A number of private companies sell names of stars, which the British Library calls an unregulated commercial enterprise.[44][45] However, the IAU has disassociated itself from this commercial practice, and these names are neither recognized by the IAU nor used by them.[46] One such star naming company is the International Star Registry, which, during the 1980s, was accused of deceptive practice for making it appear that the assigned name was official. This now-discontinued ISR practice was informally labeled a scam and a fraud,[47][48][49][50] and the New York City Department of Consumer Affairs issued a violation against ISR for engaging in a deceptive trade practice.[51][52] ## Units of measurement Although stellar parameters can be expressed in SI units or CGS units, it is often most convenient to express mass, luminosity, and radii in solar units, based on the characteristics of the Sun: solar mass: M☉ = 1.9891 × 1030 kg[53] solar luminosity: L⊙ = 3.827 × 1026 W[53] solar radius R⊙ = 6.960 × 108 m[54] Large lengths, such as the radius of a giant star or the semi-major axis of a binary star system, are often expressed in terms of the astronomical unit —approximately equal to the mean distance between the Earth and the Sun (150 million km or 93 million miles). ## Formation and evolution Stellar evolution of low-mass (left cycle) and high-mass (right cycle) stars, with examples in italics Main article: Stellar evolution Stars form within extended regions of higher density in the interstellar medium, although the density is still lower than the inside of a vacuum chamber. These regions - known as molecular clouds - consist mostly of hydrogen, with about 23 to 28 percent helium and a few percent heavier elements. One example of such a star-forming region is the Orion Nebula.[55] Most stars form in groups of dozens to hundreds of thousands of stars.[56] Massive stars in these groups may powerfully illuminate those clouds, ionizing the hydrogen, and creating H II regions. Such feedback effects from star formation may ultimately disrupt the cloud and prevent further star formation. All stars spend the majority of their existence as main sequence stars, fueled primarily by the nuclear fusion of hydrogen into helium within their cores. However, stars of different masses have markedly different properties at various stages of their development. The ultimate fate of more massive stars differs from that of less massive stars, as do their luminosity and the impact they have on their environment. Accordingly, astronomers often group stars by their mass:[57] • Very low mass stars with masses below 0.5 M are fully convective and distribute helium evenly throughout the whole star while on the main sequence. Therefore, they never undergo shell burning, never become red giants, and are theorized to become helium white dwarfs which simply cool off after exhausting their hydrogen.[58] However, as the lifetime of 0.5 M stars is longer than the age of the universe, no such star has yet reached the white dwarf stage. • Low mass stars (including the Sun), with a main sequence mass above about 0.5 M and below 1.8–2.5 M depending on composition, do become red giants when their core hydrogen is depleted, then ignite a degenerate helium core in a helium flash, develop a degenerate carbon-oxygen core on the asymptotic giant branch, and finally produce a planetary nebula to become a white dwarf. • Intermediate-mass stars, between 1.8–2.5 M and 5–10 M, pass through similar evolutionary stages to the low mass stars, but after a relatively short period on the RGB they ignite helium without a flash and spend an extended period in the red clump before forming a degenerate carbon-oxygen core. • Massive stars generally have a minimum mass of 7–10 M, but this may be as low as 5–6 M. After exhausting the hydrogen at the core these stars become supergiants and go on to fuse elements heavier than helium. They end their lives when their cores collapse and they explode as supernovae. ### Star formation Main article: Star formation The formation of a star begins with gravitational instability within a molecular cloud, caused by regions of higher density - often triggered by compression of clouds by radiation from massive stars, expanding bubbles in the interstellar medium, the collision of different molecular clouds, or the collision of galaxies (as in a starburst galaxy).[59][60] Once a region reaches a sufficient density of matter to satisfy the criteria for Jeans instability, it begins to collapse under its own gravitational force.[61] Artist's conception of the birth of a star within a dense molecular cloud. As the cloud collapses, individual conglomerations of dense dust and gas form "Bok globules". As a globule collapses and the density increases, the gravitational energy converts into heat and the temperature rises. When the protostellar cloud has approximately reached the stable condition of hydrostatic equilibrium, a protostar forms at the core.[62] These pre–main sequence stars are often surrounded by a protoplanetary disk and powered mainly by the conversion of gravitational energy. The period of gravitational contraction lasts about 10 to 15 million years. A cluster of approximately 500 young stars lies within the nearby W40 stellar nursery. Early stars of less than 2 M are called T Tauri stars, while those with greater mass are Herbig Ae/Be stars. These newly formed stars emit jets of gas along their axis of rotation, which may reduce the angular momentum of the collapsing star and result in small patches of nebulosity known as Herbig–Haro objects.[63][64] These jets, in combination with radiation from nearby massive stars, may help to drive away the surrounding cloud from which the star was formed.[65] Early in their development, T Tauri stars follow the Hayashi track—they contract and decrease in luminosity while remaining at roughly the same temperature. Less massive T Tauri stars follow this track to the main sequence, while more massive stars turn onto the Henyey track. Most stars are observed to be members of binary star systems, and the properties of these binaries are the result of the conditions in which they formed.[66] A gas cloud must lose its angular momentum in order to collapse and form a star, and fragmentation of the cloud into multiple stars uses up some of the angular momentum. The primordial binaries will get processed by gravitational interactions during close encounters with other stars in young stellar clusters. These interactions tend to split apart wider (soft) binaries while causing closer (hard) binaries to become more tightly bound, producing the distribution of binary separations seen in the field. ### Main sequence Main article: Main sequence Stars spend about 90% of their existence fusing hydrogen into helium in high-temperature and high-pressure reactions near the core. Such stars are said to be on the main sequence, and are called dwarf stars. Starting at zero-age main sequence, the proportion of helium in a star's core will steadily increase, the rate of nuclear fusion at the core will slowly increase, as will the star's temperature and luminosity.[67] The Sun, for example, is estimated to have increased in luminosity by about 40% since it reached the main sequence 4.6 billion (4.6 × 109) years ago.[68] Every star generates a stellar wind of particles that causes a continual outflow of gas into space. For most stars, the mass lost is negligible. The Sun loses 10−14 M every year,[69] or about 0.01% of its total mass over its entire lifespan. However, very massive stars can lose 10−7 to 10−5 M each year, significantly affecting their evolution.[70] Stars that begin with more than 50 M can lose over half their total mass while on the main sequence.[71] An example of a Hertzsprung–Russell diagram for a set of stars that includes the Sun (center). (See "Classification" below.) The duration that a star spends on the main sequence depends primarily on the amount of fuel it has to fuse and the rate at which it fuses that fuel, i.e. its initial mass and its luminosity. For the Sun, its life is estimated to be about 10 billion (1010) years. Massive stars consume their fuel very rapidly and are short-lived. Low mass stars consume their fuel very slowly. Stars less massive than 0.25 M, called red dwarfs, are able to fuse nearly all of their mass as fuel while stars of about 1 M can only use about 10% of their mass as fuel. The combination of their slow fuel-consumption and relatively large usable fuel supply allows about 0.25 M stars to last for about one trillion (1012) years according to stellar-evolution calculations, while the least-massive hydrogen-fusing stars (0.08 M) will last for about 12 trillion years. Red dwarfs become hotter and more luminous as they accumulate helium. When they eventually run out of hydrogen, they contract into a white dwarf and start to cool.[58] However, since the lifespan of such stars is greater than the current age of the universe (13.8 billion years), no stars under about 0.85 M[72] are expected to have moved off the main sequence. Besides mass, the elements heavier than helium can play a significant role in the evolution of stars. Astronomers consider all elements heavier than helium "metals", and call the chemical concentration of these elements the metallicity. The metallicity can influence the duration that a star will burn its fuel, control the formation of magnetic fields,[73] and modify the strength of the stellar wind.[74] Older, population II stars have substantially less metallicity than the younger, population I stars due to the composition of the molecular clouds from which they formed. Over time these clouds become increasingly enriched in heavier elements as older stars die and shed portions of their atmospheres. ### Post–main sequence Main article: Red giant As stars of at least 0.4 M[2] exhaust their supply of hydrogen at their core, they start to fuse hydrogen in a shell outside the helium core. Their outer layers expand and cool greatly to form a red giant. In about 5 billion years, when the Sun enters this phase, it will expand to a maximum radius of roughly 1 astronomical unit (150 million kilometres), 250 times its present size. As a giant, the Sun will lose roughly 30% of its current mass.[68][75] As hydrogen shell burning produces more helium, the core increases in mass and temperature. In a red giant of up to 2.25 M, the helium core becomes degenerate before it is compressed enough to start helium fusion. When the temperature increases sufficiently helium fusion begins explosively in the helium flash, and the star rapidly shrinks in radius, increases its surface temperature, and moves to the horizontal branch. For more massive stars, the helium core fusion starts before the core becomes degenerate, and the star spends some time in the red clump before the outer convective envelope collapses and the star moves to the horizontal branch.[4] After the star has consumed the helium at the core, fusion continues in a shell around a hot core of carbon and oxygen. The star then follows an evolutionary path (the asymptotic giant branch or AGB) that parallels the original red giant phase at a higher luminosity. The more massive AGB stars may undergo a brief period of carbon fusion before the core becomes degenerate. #### Massive stars Main articles: Supergiant and Hypergiant During their helium-burning phase, very high-mass stars with more than nine solar masses expand to form red supergiants. Once this fuel is exhausted at the core, they continue to fuse elements heavier than helium. The core contracts until the temperature and pressure suffice to fuse carbon (see Carbon burning process). This process continues, with the successive stages being fueled by neon (see neon burning process), oxygen (see oxygen burning process), and silicon (see silicon burning process). Near the end of the star's life, fusion continues along a series of onion-layer shells within the star. Each shell fuses a different element, with the outermost shell fusing hydrogen; the next shell fusing helium, and so forth.[76] The final stage occurs when a massive star begins producing iron. Since iron nuclei are more tightly bound than any heavier nuclei, any fusion beyond iron does not produce a net release of energy—the process would, on the contrary, consume energy. Likewise, since they are more tightly bound than all lighter nuclei, energy cannot be released by fission.[77] In relatively old, very massive stars, a large core of inert iron will accumulate in the center of the star. The heavier elements in these stars can work their way to the surface, forming evolved objects known as Wolf-Rayet stars that have a dense stellar wind which sheds the outer atmosphere. #### Collapse As a star's core shrinks, the intensity of radiation from that surface increases, creating such radiation pressure on the outer shell of gas that it will push those layers away, forming a planetary nebula. If what remains after the outer atmosphere has been shed is less than 1.4 M, it shrinks to a relatively tiny object about the size of Earth, known as a white dwarf. White dwarfs lack the mass for further gravitational compression to take place.[78] The electron-degenerate matter inside a white dwarf is no longer a plasma, even though stars are generally referred to as being spheres of plasma. Eventually, white dwarfs fade into black dwarfs over a very long period of time. The Crab Nebula, remnants of a supernova that was first observed around 1050 AD In larger stars, fusion continues until the iron core has grown so large (more than 1.4 M) that it can no longer support its own mass. This core will suddenly collapse as its electrons are driven into its protons, forming neutrons, neutrinos, and gamma rays in a burst of electron capture and inverse beta decay. The shockwave formed by this sudden collapse causes the rest of the star to explode in a supernova. Supernovae become so bright that they may briefly outshine the star's entire home galaxy. When they occur within the Milky Way, supernovae have historically been observed by naked-eye observers as "new stars" where none seemingly existed before.[79] Supernova explosions blow away the star's outer layers, leaving remnants such as the Crab Nebula.[79] There remains a neutron star (which sometimes manifests itself as a pulsar or X-ray burster) or, in the case of the largest stars (large enough to leave a remnant greater than roughly 4 M), a black hole.[80] In a neutron star the matter is in a state known as neutron-degenerate matter, with a more exotic form of degenerate matter, QCD matter, possibly present in the core. Within a black hole the matter is in a state that is not currently understood. The blown-off outer layers of dying stars include heavy elements, which may be recycled during the formation of new stars. These heavy elements allow the formation of rocky planets. The outflow from supernovae and the stellar wind of large stars play an important part in shaping the interstellar medium.[79] #### Binary stars The post–main-sequence evolution of binary stars may be significantly different from the evolution of single stars of the same mass. If stars in a binary system are sufficiently close, when one of the stars expands to become a red giant it may overflow its Roche lobe, the region around a star where material is gravitationally bound to that star, leading to transfer of material to the other star. A variety of phenomena can result from these systems, including contact binaries, common-envelope binaries, cataclysmic variables, and type Ia supernovae. ## Distribution A white dwarf star in orbit around Sirius (artist's impression). In addition to isolated stars, a multi-star system can consist of two or more gravitationally bound stars that orbit each other. The simplest and most common multi-star system is a binary star, but systems of three or more stars are also found. For reasons of orbital stability, such multi-star systems are often organized into hierarchical sets of binary stars.[81] Larger groups called star clusters also exist. These range from loose stellar associations with only a few stars, up to enormous globular clusters with hundreds of thousands of stars. It has been a long-held assumption that the majority of stars occur in gravitationally bound, multiple-star systems. This is particularly true for very massive O and B class stars, where 80% of the stars are believed to be part of multiple-star systems. However the proportion of single star systems increases for smaller stars, so that only 25% of red dwarfs are known to have stellar companions. As 85% of all stars are red dwarfs, most stars in the Milky Way are likely single from birth.[82] Stars are not spread uniformly across the universe, but are normally grouped into galaxies along with interstellar gas and dust. A typical galaxy contains hundreds of billions of stars, and there are more than 100 billion (1011) galaxies in the observable universe.[83] In 2010, one estimate of the number of stars in the observable universe was 300 sextillion (3 × 1023) in the observable universe.[84] While it is often believed that stars only exist within galaxies, intergalactic stars have been discovered.[85] The nearest star to the Earth, apart from the Sun, is Proxima Centauri, which is 39.9 trillion kilometres, or 4.2 light-years away. Travelling at the orbital speed of the Space Shuttle (8 kilometres per second—almost 30,000 kilometres per hour), it would take about 150,000 years to get there.[86] Distances like this are typical inside galactic discs, including in the vicinity of the solar system.[87] Stars can be much closer to each other in the centres of galaxies and in globular clusters, or much farther apart in galactic halos. Due to the relatively vast distances between stars outside the galactic nucleus, collisions between stars are thought to be rare. In denser regions such as the core of globular clusters or the galactic center, collisions can be more common.[88] Such collisions can produce what are known as blue stragglers. These abnormal stars have a higher surface temperature than the other main sequence stars with the same luminosity in the cluster.[89] ## Characteristics Some of the well-known stars with their apparent colors and relative sizes. Almost everything about a star is determined by its initial mass, including essential characteristics such as luminosity and size, as well as its evolution, lifespan, and eventual fate. ### Age Most stars are between 1 billion and 10 billion years old. Some stars may even be close to 13.8 billion years old—the observed age of the universe. The oldest star yet discovered, HD 140283, nicknamed Methuselah star, is an estimated 14.46 ± 0.8 billion years old.[90] (Due to the uncertainty in the value, this age for the star does not conflict with the age of the Universe, determined by the Planck satellite as 13.799 ± 0.021).[90][91] The more massive the star, the shorter its lifespan, primarily because massive stars have greater pressure on their cores, causing them to burn hydrogen more rapidly. The most massive stars last an average of a few million years, while stars of minimum mass (red dwarfs) burn their fuel very slowly and can last tens to hundreds of billions of years.[92][93] ### Chemical composition When stars form in the present Milky Way galaxy they are composed of about 71% hydrogen and 27% helium,[94] as measured by mass, with a small fraction of heavier elements. Typically the portion of heavy elements is measured in terms of the iron content of the stellar atmosphere, as iron is a common element and its absorption lines are relatively easy to measure. The portion of heavier elements may be an indicator of the likelihood that the star has a planetary system.[95] The star with the lowest iron content ever measured is the dwarf HE1327-2326, with only 1/200,000th the iron content of the Sun.[96] By contrast, the super-metal-rich star μ Leonis has nearly double the abundance of iron as the Sun, while the planet-bearing star 14 Herculis has nearly triple the iron.[97] There also exist chemically peculiar stars that show unusual abundances of certain elements in their spectrum; especially chromium and rare earth elements.[98] Stars with cooler outer atmospheres, including the Sun, can form various diatomic and polyatomic molecules.[99] ### Diameter Stars vary widely in size. In each image in the sequence, the right-most object appears as the left-most object in the next panel. The Earth appears at right in panel 1 and the Sun is second from the right in panel 3. The rightmost star at panel 6 is UY Scuti, the largest known star. Due to their great distance from the Earth, all stars except the Sun appear to the unaided eye as shining points in the night sky that twinkle because of the effect of the Earth's atmosphere. The Sun is also a star, but it is close enough to the Earth to appear as a disk instead, and to provide daylight. Other than the Sun, the star with the largest apparent size is R Doradus, with an angular diameter of only 0.057 arcseconds.[100] The disks of most stars are much too small in angular size to be observed with current ground-based optical telescopes, and so interferometer telescopes are required to produce images of these objects. Another technique for measuring the angular size of stars is through occultation. By precisely measuring the drop in brightness of a star as it is occulted by the Moon (or the rise in brightness when it reappears), the star's angular diameter can be computed.[101] Stars range in size from neutron stars, which vary anywhere from 20 to 40 km (25 mi) in diameter, to supergiants like Betelgeuse in the Orion constellation, which has a diameter approximately 1,070 times that of the Sun—about 1,490,171,880 km (925,949,878 mi). Betelgeuse, however, has a much lower density than the Sun.[102] ### Kinematics Main article: Stellar kinematics The Pleiades, an open cluster of stars in the constellation of Taurus. These stars share a common motion through space.[103] The motion of a star relative to the Sun can provide useful information about the origin and age of a star, as well as the structure and evolution of the surrounding galaxy. The components of motion of a star consist of the radial velocity toward or away from the Sun, and the traverse angular movement, which is called its proper motion. Radial velocity is measured by the doppler shift of the star's spectral lines, and is given in units of km/s. The proper motion of a star is determined by precise astrometric measurements in units of milli-arc seconds (mas) per year. By determining the parallax of a star, the proper motion can then be converted into units of velocity. Stars with high rates of proper motion are likely to be relatively close to the Sun, making them good candidates for parallax measurements.[104] Once both rates of movement are known, the space velocity of the star relative to the Sun or the galaxy can be computed. Among nearby stars, it has been found that younger population I stars have generally lower velocities than older, population II stars. The latter have elliptical orbits that are inclined to the plane of the galaxy.[105] A comparison of the kinematics of nearby stars has also led to the identification of stellar associations. These are most likely groups of stars that share a common point of origin in giant molecular clouds.[106] ### Magnetic field Surface magnetic field of SU Aur (a young star of T Tauri type), reconstructed by means of Zeeman-Doppler imaging The magnetic field of a star is generated within regions of the interior where convective circulation occurs. This movement of conductive plasma functions like a dynamo, generating magnetic fields that extend throughout the star. The strength of the magnetic field varies with the mass and composition of the star, and the amount of magnetic surface activity depends upon the star's rate of rotation. This surface activity produces starspots, which are regions of strong magnetic fields and lower than normal surface temperatures. Coronal loops are arching magnetic fields that reach out into the corona from active regions. Stellar flares are bursts of high-energy particles that are emitted due to the same magnetic activity.[107] Young, rapidly rotating stars tend to have high levels of surface activity because of their magnetic field. The magnetic field can act upon a star's stellar wind, functioning as a brake to gradually slow the rate of rotation with time. Thus, older stars such as the Sun have a much slower rate of rotation and a lower level of surface activity. The activity levels of slowly rotating stars tend to vary in a cyclical manner and can shut down altogether for periods of time.[108] During the Maunder minimum, for example, the Sun underwent a 70-year period with almost no sunspot activity. ### Mass Main article: Stellar mass One of the most massive stars known is Eta Carinae,[109] which, with 100–150 times as much mass as the Sun, will have a lifespan of only several million years. Studies of the most massive open clusters suggests 150 M as an upper limit for stars in the current era of the universe.[110] This represents an empirical value for the theoretical limit on the formation of massive stars due to increasing radiation pressure on the accreting gas cloud. Several stars in the R136 cluster in the Large Magellanic Cloud have been measured with larger masses,[111] but it has been determined that they could have been created through the collision and merger of massive stars in close binary systems, sidestepping the 150 M limit on massive star formation.[112] The reflection nebula NGC 1999 is brilliantly illuminated by V380 Orionis (center), a variable star with about 3.5 times the mass of the Sun. The black patch of sky is a vast hole of empty space and not a dark nebula as previously thought. The first stars to form after the Big Bang may have been larger, up to 300 M or more,[113] due to the complete absence of elements heavier than lithium in their composition. This generation of supermassive population III stars is likely to have existed in the very early universe (i.e., at high redshift), and may have started the production of chemical elements heavier than hydrogen that are needed for the later formation of planets and life. In June 2015, astronomers reported evidence for Population III stars in the Cosmos Redshift 7 galaxy at z = 6.60.[114][115] With a mass only 80 times that of Jupiter (MJ), 2MASS J0523-1403 is the smallest known star undergoing nuclear fusion in its core.[116] For stars with similar metallicity to the Sun, the theoretical minimum mass the star can have, and still undergo fusion at the core, is estimated to be about 75 MJ.[117][118] When the metallicity is very low, however, a recent study of the faintest stars found that the minimum star size seems to be about 8.3% of the solar mass, or about 87 MJ.[118][119] Smaller bodies are called brown dwarfs, which occupy a poorly defined grey area between stars and gas giants. The combination of the radius and the mass of a star determines the surface gravity. Giant stars have a much lower surface gravity than main sequence stars, while the opposite is the case for degenerate, compact stars such as white dwarfs. The surface gravity can influence the appearance of a star's spectrum, with higher gravity causing a broadening of the absorption lines.[33] ### Rotation Main article: Stellar rotation The rotation rate of stars can be determined through spectroscopic measurement, or more exactly determined by tracking the rotation rate of starspots. Young stars can have a rapid rate of rotation greater than 100 km/s at the equator. The B-class star Achernar, for example, has an equatorial rotation velocity of about 225 km/s or greater, causing its equator to be slung outward and giving it an equatorial diameter that is more than 50% larger than the distance between the poles. This rate of rotation is just below the critical velocity of 300 km/s where the star would break apart.[120] By contrast, the Sun only rotates once every 25 – 35 days, with an equatorial velocity of 1.994 km/s. The star's magnetic field and the stellar wind serve to slow a main sequence star's rate of rotation by a significant amount as it evolves on the main sequence.[121] Degenerate stars have contracted into a compact mass, resulting in a rapid rate of rotation. However they have relatively low rates of rotation compared to what would be expected by conservation of angular momentum—the tendency of a rotating body to compensate for a contraction in size by increasing its rate of spin. A large portion of the star's angular momentum is dissipated as a result of mass loss through the stellar wind.[122] In spite of this, the rate of rotation for a pulsar can be very rapid. The pulsar at the heart of the Crab nebula, for example, rotates 30 times per second.[123] The rotation rate of the pulsar will gradually slow due to the emission of radiation. ### Temperature The surface temperature of a main sequence star is determined by the rate of energy production at the core and by its radius, and is often estimated from the star's color index.[124] The temperature is normally given as the effective temperature, which is the temperature of an idealized black body that radiates its energy at the same luminosity per surface area as the star. Note that the effective temperature is only a representative value, as the temperature increases toward the core.[125] The temperature in the core region of a star is several million kelvins.[126] The stellar temperature will determine the rate of ionization of various elements, resulting in characteristic absorption lines in the spectrum. The surface temperature of a star, along with its visual absolute magnitude and absorption features, is used to classify a star (see classification below).[33] Massive main sequence stars can have surface temperatures of 50,000 K. Smaller stars such as the Sun have surface temperatures of a few thousand K. Red giants have relatively low surface temperatures of about 3,600 K; but they also have a high luminosity due to their large exterior surface area.[127] The energy produced by stars, as a product of nuclear fusion, radiates into space as both electromagnetic radiation and particle radiation. The particle radiation emitted by a star is manifested as the stellar wind,[128] which streams from the outer layers as electrically charged protons and alpha and beta particles. Although almost massless there also exists a steady stream of neutrinos emanating from the star's core. The production of energy at the core is the reason stars shine so brightly: every time two or more atomic nuclei fuse together to form a single atomic nucleus of a new heavier element, gamma ray photons are released from the nuclear fusion product. This energy is converted to other forms of electromagnetic energy of lower frequency, such as visible light, by the time it reaches the star's outer layers. The color of a star, as determined by the most intense frequency of the visible light, depends on the temperature of the star's outer layers, including its photosphere.[129] Besides visible light, stars also emit forms of electromagnetic radiation that are invisible to the human eye. In fact, stellar electromagnetic radiation spans the entire electromagnetic spectrum, from the longest wavelengths of radio waves through infrared, visible light, ultraviolet, to the shortest of X-rays, and gamma rays. From the standpoint of total energy emitted by a star, not all components of stellar electromagnetic radiation are significant, but all frequencies provide insight into the star's physics. Using the stellar spectrum, astronomers can also determine the surface temperature, surface gravity, metallicity and rotational velocity of a star. If the distance of the star is known, such as by measuring the parallax, then the luminosity of the star can be derived. The mass, radius, surface gravity, and rotation period can then be estimated based on stellar models. (Mass can be calculated for stars in binary systems by measuring their orbital velocities and distances. Gravitational microlensing has been used to measure the mass of a single star.[130]) With these parameters, astronomers can also estimate the age of the star.[131] ### Luminosity The luminosity of a star is the amount of light and other forms of radiant energy it radiates per unit of time. It has units of power. The luminosity of a star is determined by the radius and the surface temperature. However, many stars do not radiate a uniform flux (the amount of energy radiated per unit area) across their entire surface. The rapidly rotating star Vega, for example, has a higher energy flux at its poles than along its equator.[132] Surface patches with a lower temperature and luminosity than average are known as starspots. Small, dwarf stars such as our Sun generally have essentially featureless disks with only small starspots. Larger, giant stars have much larger, more obvious starspots,[133] and they also exhibit strong stellar limb darkening. That is, the brightness decreases towards the edge of the stellar disk.[134] Red dwarf flare stars such as UV Ceti may also possess prominent starspot features.[135] ### Magnitude The apparent brightness of a star is expressed in terms of its apparent magnitude, which is the brightness of a star and is a function of the star's luminosity, distance from Earth, and the altering of the star's light as it passes through Earth's atmosphere. Intrinsic or absolute magnitude is directly related to a star's luminosity, and is what the apparent magnitude a star would be if the distance between the Earth and the star were 10 parsecs (32.6 light-years). Number of stars brighter than magnitude Apparent magnitude Number of stars[136] 0 4 1 15 2 48 3 171 4 513 5 1,602 6 4,800 7 14,000 Both the apparent and absolute magnitude scales are logarithmic units: one whole number difference in magnitude is equal to a brightness variation of about 2.5 times[137] (the 5th root of 100 or approximately 2.512). This means that a first magnitude star (+1.00) is about 2.5 times brighter than a second magnitude (+2.00) star, and approximately 100 times brighter than a sixth magnitude star (+6.00). The faintest stars visible to the naked eye under good seeing conditions are about magnitude +6. On both apparent and absolute magnitude scales, the smaller the magnitude number, the brighter the star; the larger the magnitude number, the fainter. The brightest stars, on either scale, have negative magnitude numbers. The variation in brightness (ΔL) between two stars is calculated by subtracting the magnitude number of the brighter star (mb) from the magnitude number of the fainter star (mf), then using the difference as an exponent for the base number 2.512; that is to say: $\Delta{m} = m_\mathrm{f} - m_\mathrm{b}$ $2.512^{\Delta{m}} = \Delta{L}$ Relative to both luminosity and distance from Earth, a star's absolute magnitude (M) and apparent magnitude (m) are not equivalent;[137] for example, the bright star Sirius has an apparent magnitude of −1.44, but it has an absolute magnitude of +1.41. The Sun has an apparent magnitude of −26.7, but its absolute magnitude is only +4.83. Sirius, the brightest star in the night sky as seen from Earth, is approximately 23 times more luminous than the Sun, while Canopus, the second brightest star in the night sky with an absolute magnitude of −5.53, is approximately 14,000 times more luminous than the Sun. Despite Canopus being vastly more luminous than Sirius, however, Sirius appears brighter than Canopus. This is because Sirius is merely 8.6 light-years from the Earth, while Canopus is much farther away at a distance of 310 light-years. As of 2006, the star with the highest known absolute magnitude is LBV 1806-20, with a magnitude of −14.2. This star is at least 5,000,000 times more luminous than the Sun.[138] The least luminous stars that are currently known are located in the NGC 6397 cluster. The faintest red dwarfs in the cluster were magnitude 26, while a 28th magnitude white dwarf was also discovered. These faint stars are so dim that their light is as bright as a birthday candle on the Moon when viewed from the Earth.[139] ## Classification Surface temperature ranges for different stellar classes[140] Class Temperature Sample star O 33,000 K or more Zeta Ophiuchi B 10,500–30,000 K Rigel A 7,500–10,000 K Altair F 6,000–7,200 K Procyon A G 5,500–6,000 K Sun K 4,000–5,250 K Epsilon Indi M 2,600–3,850 K Proxima Centauri The current stellar classification system originated in the early 20th century, when stars were classified from A to Q based on the strength of the hydrogen line.[141] It was not known at the time that the major influence on the line strength was temperature; the hydrogen line strength reaches a peak at over 9000 K, and is weaker at both hotter and cooler temperatures. When the classifications were reordered by temperature, it more closely resembled the modern scheme.[142] Stars are given a single-letter classification according to their spectra, ranging from type O, which are very hot, to M, which are so cool that molecules may form in their atmospheres. The main classifications in order of decreasing surface temperature are: O, B, A, F, G, K, and M. A variety of rare spectral types have special classifications. The most common of these are types L and T, which classify the coldest low-mass stars and brown dwarfs. Each letter has 10 sub-divisions, numbered from 0 to 9, in order of decreasing temperature. However, this system breaks down at extreme high temperatures: class O0 and O1 stars may not exist.[143] In addition, stars may be classified by the luminosity effects found in their spectral lines, which correspond to their spatial size and is determined by the surface gravity. These range from 0 (hypergiants) through III (giants) to V (main sequence dwarfs); some authors add VII (white dwarfs). Most stars belong to the main sequence, which consists of ordinary hydrogen-burning stars. These fall along a narrow, diagonal band when graphed according to their absolute magnitude and spectral type.[143] The Sun is a main sequence G2V yellow dwarf of intermediate temperature and ordinary size. Additional nomenclature, in the form of lower-case letters, can follow the spectral type to indicate peculiar features of the spectrum. For example, an "e" can indicate the presence of emission lines; "m" represents unusually strong levels of metals, and "var" can mean variations in the spectral type.[143] White dwarf stars have their own class that begins with the letter D. This is further sub-divided into the classes DA, DB, DC, DO, DZ, and DQ, depending on the types of prominent lines found in the spectrum. This is followed by a numerical value that indicates the temperature index.[144] ## Variable stars Main article: Variable star The asymmetrical appearance of Mira, an oscillating variable star. Variable stars have periodic or random changes in luminosity because of intrinsic or extrinsic properties. Of the intrinsically variable stars, the primary types can be subdivided into three principal groups. During their stellar evolution, some stars pass through phases where they can become pulsating variables. Pulsating variable stars vary in radius and luminosity over time, expanding and contracting with periods ranging from minutes to years, depending on the size of the star. This category includes Cepheid and Cepheid-like stars, and long-period variables such as Mira.[145] Eruptive variables are stars that experience sudden increases in luminosity because of flares or mass ejection events.[145] This group includes protostars, Wolf-Rayet stars, and flare stars, as well as giant and supergiant stars. Cataclysmic or explosive variable stars are those that undergo a dramatic change in their properties. This group includes novae and supernovae. A binary star system that includes a nearby white dwarf can produce certain types of these spectacular stellar explosions, including the nova and a Type 1a supernova.[4] The explosion is created when the white dwarf accretes hydrogen from the companion star, building up mass until the hydrogen undergoes fusion.[146] Some novae are also recurrent, having periodic outbursts of moderate amplitude.[145] Stars can also vary in luminosity because of extrinsic factors, such as eclipsing binaries, as well as rotating stars that produce extreme starspots.[145] A notable example of an eclipsing binary is Algol, which regularly varies in magnitude from 2.3 to 3.5 over a period of 2.87 days. ## Structure Main article: Stellar structure Internal structures of main sequence stars, convection zones with arrowed cycles and radiative zones with red flashes. To the left a low-mass red dwarf, in the center a mid-sized yellow dwarf, and, at the right, a massive blue-white main sequence star. The interior of a stable star is in a state of hydrostatic equilibrium: the forces on any small volume almost exactly counterbalance each other. The balanced forces are inward gravitational force and an outward force due to the pressure gradient within the star. The pressure gradient is established by the temperature gradient of the plasma; the outer part of the star is cooler than the core. The temperature at the core of a main sequence or giant star is at least on the order of 107 K. The resulting temperature and pressure at the hydrogen-burning core of a main sequence star are sufficient for nuclear fusion to occur and for sufficient energy to be produced to prevent further collapse of the star.[147][148] As atomic nuclei are fused in the core, they emit energy in the form of gamma rays. These photons interact with the surrounding plasma, adding to the thermal energy at the core. Stars on the main sequence convert hydrogen into helium, creating a slowly but steadily increasing proportion of helium in the core. Eventually the helium content becomes predominant, and energy production ceases at the core. Instead, for stars of more than 0.4 M, fusion occurs in a slowly expanding shell around the degenerate helium core.[149] In addition to hydrostatic equilibrium, the interior of a stable star will also maintain an energy balance of thermal equilibrium. There is a radial temperature gradient throughout the interior that results in a flux of energy flowing toward the exterior. The outgoing flux of energy leaving any layer within the star will exactly match the incoming flux from below. The radiation zone is the region within the stellar interior where radiative transfer is sufficiently efficient to maintain the flux of energy. In this region the plasma will not be perturbed, and any mass motions will die out. If this is not the case, however, then the plasma becomes unstable and convection will occur, forming a convection zone. This can occur, for example, in regions where very high energy fluxes occur, such as near the core or in areas with high opacity as in the outer envelope.[148] The occurrence of convection in the outer envelope of a main sequence star depends on the mass. Stars with several times the mass of the Sun have a convection zone deep within the interior and a radiative zone in the outer layers. Smaller stars such as the Sun are just the opposite, with the convective zone located in the outer layers.[150] Red dwarf stars with less than 0.4 M are convective throughout, which prevents the accumulation of a helium core.[2] For most stars the convective zones will also vary over time as the star ages and the constitution of the interior is modified.[148] This diagram shows a cross-section of the Sun. The portion of a star that is visible to an observer is called the photosphere. This is the layer at which the plasma of the star becomes transparent to photons of light. From here, the energy generated at the core becomes free to propagate out into space. It is within the photosphere that sun spots, or regions of lower than average temperature, appear. Above the level of the photosphere is the stellar atmosphere. In a main sequence star such as the Sun, the lowest level of the atmosphere is the thin chromosphere region, where spicules appear and stellar flares begin. This is surrounded by a transition region, where the temperature rapidly increases within a distance of only 100 km (62 mi). Beyond this is the corona, a volume of super-heated plasma that can extend outward to several million kilometres.[151] The existence of a corona appears to be dependent on a convective zone in the outer layers of the star.[150] Despite its high temperature, the corona emits very little light. The corona region of the Sun is normally only visible during a solar eclipse. From the corona, a stellar wind of plasma particles expands outward from the star, propagating until it interacts with the interstellar medium. For the Sun, the influence of its solar wind extends throughout the bubble-shaped region of the heliosphere.[152] ## Nuclear fusion reaction pathways Overview of the proton-proton chain The carbon-nitrogen-oxygen cycle A variety of different nuclear fusion reactions take place inside the cores of stars, depending upon their mass and composition, as part of stellar nucleosynthesis. The net mass of the fused atomic nuclei is smaller than the sum of the constituents. This lost mass is released as electromagnetic energy, according to the mass-energy equivalence relationship E = mc2.[1] The hydrogen fusion process is temperature-sensitive, so a moderate increase in the core temperature will result in a significant increase in the fusion rate. As a result, the core temperature of main sequence stars only varies from 4 million kelvin for a small M-class star to 40 million kelvin for a massive O-class star.[126] In the Sun, with a 10-million-kelvin core, hydrogen fuses to form helium in the proton-proton chain reaction:[153] 41H → 22H + 2e+ + 2νe(2 x 0.4 MeV) 2e+ + 2e- → 2γ (2 x 1.0 MeV) 21H + 22H → 23He + 2γ (2 x 5.5 MeV) 23He → 4He + 21H (12.9 MeV) These reactions result in the overall reaction: 41H → 4He + 2e+ + 2γ + 2νe (26.7 MeV) where e+ is a positron, γ is a gamma ray photon, νe is a neutrino, and H and He are isotopes of hydrogen and helium, respectively. The energy released by this reaction is in millions of electron volts, which is actually only a tiny amount of energy. However enormous numbers of these reactions occur constantly, producing all the energy necessary to sustain the star's radiation output. Minimum stellar mass required for fusion Element Solar masses Hydrogen 0.01 Helium 0.4 Carbon 5[154] Neon 8 In more massive stars, helium is produced in a cycle of reactions catalyzed by carbon—the carbon-nitrogen-oxygen cycle.[153] In evolved stars with cores at 100 million kelvin and masses between 0.5 and 10 M, helium can be transformed into carbon in the triple-alpha process that uses the intermediate element beryllium:[153] 4He + 4He + 92 keV → 8Be 4He + 8Be + 67 keV → 12C 12C → 12C + γ + 7.4 MeV For an overall reaction of: 34He → 12C + γ + 7.2 MeV In massive stars, heavier elements can also be burned in a contracting core through the neon burning process and oxygen burning process. The final stage in the stellar nucleosynthesis process is the silicon burning process that results in the production of the stable isotope iron-56. Fusion can not proceed any further except through an endothermic process, and so further energy can only be produced through gravitational collapse.[153] The example below shows the amount of time required for a star of 20 M to consume all of its nuclear fuel. As an O-class main sequence star, it would be 8 times the solar radius and 62,000 times the Sun's luminosity.[155] Fuel material Temperature (million kelvins) Density (kg/cm3) Burn duration (τ in years) H 37 0.0045 8.1 million He 188 0.97 1.2 million C 870 170 976 Ne 1,570 3,100 0.6 O 1,980 5,550 1.25 S/Si 3,340 33,400 0.0315[156] ## References 1. Bahcall, John N. (June 29, 2000). "How the Sun Shines". Nobel Foundation. Retrieved 2006-08-30. 2. Richmond, Michael. "Late stages of evolution for low-mass stars". Rochester Institute of Technology. Retrieved 2006-08-04. 3. "Stellar Evolution & Death". NASA Observatorium. Archived from the original on 2008-02-10. Retrieved 2006-06-08. 4. Iben, Icko, Jr. (1991). "Single and binary star evolution". Astrophysical Journal Supplement Series. 76: 55–114. Bibcode:1991ApJS...76...55I. doi:10.1086/191565. 5. Forbes, George (1909). History of Astronomy. London: Watts & Co. ISBN 1-153-62774-4. 6. Hevelius, Johannis (1690). Firmamentum Sobiescianum, sive Uranographia. Gdansk. 7. Tøndering, Claus. "Other ancient calendars". WebExhibits. Retrieved 2006-12-10. 8. von Spaeth, Ove (2000). "Dating the Oldest Egyptian Star Map". Centaurus International Magazine of the History of Mathematics, Science and Technology. 42 (3): 159–179. Bibcode:2000Cent...42..159V. doi:10.1034/j.1600-0498.2000.420301.x. Retrieved 2007-10-21. 9. North, John (1995). The Norton History of Astronomy and Cosmology. New York and London: W.W. Norton & Company. pp. 30–31. ISBN 0-393-03656-1. 10. Murdin, P. (November 2000). "Aristillus (c. 200 BC)". Encyclopedia of Astronomy and Astrophysics. Bibcode:2000eaa..bookE3440 Check \|bibcode= length (help). ISBN 0-333-75088-8. doi:10.1888/0333750888/3440. 11. Grasshoff, Gerd (1990). The history of Ptolemy's star catalogue. Springer. pp. 1–5. ISBN 0-387-97181-5. 12. Pinotsis, Antonios D. "Astronomy in Ancient Rhodes". Section of Astrophysics, Astronomy and Mechanics, Department of Physics, University of Athens. Retrieved 2009-06-02. 13. Clark, D. H.; Stephenson, F. R. (June 29, 1981). "The Historical Supernovae". Supernovae: A survey of current research; Proceedings of the Advanced Study Institute. Cambridge, England: Dordrecht, D. Reidel Publishing Co. pp. 355–370. Bibcode:1982sscr.conf..355C. 14. Zhao, Fu-Yuan; Strom, R. G.; Jiang, Shi-Yang (2006). "The Guest Star of AD185 Must Have Been a Supernova". Chinese Journal of Astronomy and Astrophysics. 6 (5): 635–640. Bibcode:2006ChJAA...6..635Z. doi:10.1088/1009-9271/6/5/17. 15. "Astronomers Peg Brightness of History's Brightest Star". NAOA News. March 5, 2003. Retrieved 2006-06-08. 16. Frommert, Hartmut; Kronberg, Christine (August 30, 2006). "Supernova 1054 – Creation of the Crab Nebula". SEDS. University of Arizona. 17. Duyvendak, J. J. L. (April 1942). "Further Data Bearing on the Identification of the Crab Nebula with the Supernova of 1054 A.D. Part I. The Ancient Oriental Chronicles". Publications of the Astronomical Society of the Pacific. 54 (318): 91–94. Bibcode:1942PASP...54...91D. doi:10.1086/125409. Mayall, N. U.; Oort, Jan Hendrik (April 1942). "Further Data Bearing on the Identification of the Crab Nebula with the Supernova of 1054 A.D. Part II. The Astronomical Aspects". Publications of the Astronomical Society of the Pacific. 54 (318): 95–104. Bibcode:1942PASP...54...95M. doi:10.1086/125410. 18. Brecher, K.; et al. (1983). "Ancient records and the Crab Nebula supernova". The Observatory. 103: 106–113. Bibcode:1983Obs...103..106B. 19. Kennedy, Edward S. (1962). "Review: The Observatory in Islam and Its Place in the General History of the Observatory by Aydin Sayili". Isis. 53 (2): 237–239. doi:10.1086/349558. 20. Jones, Kenneth Glyn (1991). Messier's nebulae and star clusters. Cambridge University Press. p. 1. ISBN 0-521-37079-5. 21. Zahoor, A. (1997). "Al-Biruni". Hasanuddin University. Archived from the original on 2008-06-26. Retrieved 2007-10-21. 22. Montada, Josep Puig (September 28, 2007). "Ibn Bajja". Stanford Encyclopedia of Philosophy. Retrieved 2008-07-11. 23. Drake, Stephen A. (August 17, 2006). "A Brief History of High-Energy (X-ray & Gamma-Ray) Astronomy". NASA HEASARC. Retrieved 2006-08-24. 24. Greskovic, Peter; Rudy, Peter (July 24, 2006). "Exoplanets". ESO. Retrieved 2012-06-15. 25. Ahmad, I. A. (1995). "The impact of the Qur'anic conception of astronomical phenomena on Islamic civilization". Vistas in Astronomy. 39 (4): 395–403 [402]. Bibcode:1995VA.....39..395A. doi:10.1016/0083-6656(95)00033-X. 26. Setia, Adi (2004). "Fakhr Al-Din Al-Razi on Physics and the Nature of the Physical World: A Preliminary Survey". Islam & Science. 2 (2) – via Questia. 27. Hoskin, Michael (1998). "The Value of Archives in Writing the History of Astronomy". Space Telescope Science Institute. Retrieved 2006-08-24. 28. Proctor, Richard A. (1870). "Are any of the nebulæ star-systems?". Nature. 1 (13): 331–333. Bibcode:1870Natur...1..331P. doi:10.1038/001331a0. 29. MacDonnell, Joseph. "Angelo Secchi, S.J. (1818–1878) the Father of Astrophysics". Fairfield University. Archived from the original on 2011-07-21. Retrieved 2006-10-02. 30. Aitken, Robert G. (1964). The Binary Stars. New York: Dover Publications Inc. p. 66. ISBN 0-486-61102-7. 31. Michelson, A. A.; Pease, F. G. (1921). "Measurement of the diameter of Alpha Orionis with the interferometer". Astrophysical Journal. 53: 249–259. Bibcode:1921ApJ....53..249M. doi:10.1086/142603. 32. "" Payne-Gaposchkin, Cecilia Helena." CWP". University of California. Retrieved 2013-02-21. 33. Unsöld, Albrecht (2001). The New Cosmos (5th ed.). New York: Springer. pp. 180–185, 215–216. ISBN 3-540-67877-8. 34. e. g. Battinelli, Paolo; Demers, Serge; Letarte, Bruno (2003). "Carbon Star Survey in the Local Group. V. The Outer Disk of M31". The Astronomical Journal. 125 (3): 1298–1308. Bibcode:2003AJ....125.1298B. doi:10.1086/346274. 35. "Millennium Star Atlas marks the completion of ESA's Hipparcos Mission". ESA. December 8, 1997. Retrieved 2007-08-05. 36. Villard, Ray; Freedman, Wendy L. (October 26, 1994). "Hubble Space Telescope Measures Precise Distance to the Most Remote Galaxy Yet". Hubble Site. Retrieved 2007-08-05. 37. "Hubble Completes Eight-Year Effort to Measure Expanding Universe". Hubble Site. May 25, 1999. Retrieved 2007-08-02. 38. "UBC Prof., alumnus discover most distant star clusters: a billion light-years away.". UBC Public Affairs. January 8, 2007. Retrieved 2015-06-28. 39. Koch-Westenholz, Ulla; Koch, Ulla Susanne (1995). Mesopotamian astrology: an introduction to Babylonian and Assyrian celestial divination. Carsten Niebuhr Institute Publications. 19. Museum Tusculanum Press. p. 163. ISBN 87-7289-287-0. 40. Coleman, Leslie S. "Myths, Legends and Lore". Frosty Drew Observatory. Retrieved 2012-06-15. 41. "Naming Astronomical Objects". International Astronomical Union (IAU). Retrieved 2009-01-30. 42. "Naming Stars". Students for the Exploration and Development of Space (SEDS). Retrieved 2009-01-30. 43. Lyall, Francis; Larsen, Paul B. (2009). "Chapter 7: The Moon and Other Celestial Bodies". Space Law: A Treatise. Ashgate Publishing, Ltd. p. 176. ISBN 0-7546-4390-5. 44. "Star naming". Scientia Astrophysical Organization. 2005. Retrieved 2010-06-29. 45. "Disclaimer: Name a star, name a rose and other, similar enterprises". British Library. The British Library Board. Archived from the original on 2010-01-19. Retrieved 2010-06-29. 46. Andersen, Johannes. "Buying Stars and Star Names". International Astronomical Union. Retrieved 2010-06-24. 47. Pliat, Phil (September–October 2006). "Name Dropping: Want to Be a Star?". Skeptical Inquirer. 30.5. Retrieved 2010-06-29. 48. Adams, Cecil (April 1, 1998). "Can you pay \$35 to get a star named after you?". The Straight Dope. Retrieved 2006-08-13. 49. Golden, Frederick; Faflick, Philip (January 11, 1982). "Science: Stellar Idea or Cosmic Scam?". Times Magazine. Time Inc. Retrieved 2010-06-24. 50. Di Justo, Patrick (December 26, 2001). "Buy a Star, But It's Not Yours". Wired. Condé Nast Digital. Retrieved 2010-06-29. 51. Plait, Philip C. (2002). Bad astronomy: misconceptions and misuses revealed, from astrology to the moon landing "hoax". John Wiley and Sons. pp. 237–240. ISBN 0-471-40976-6. 52. Sclafani, Tom (May 8, 1998). "Consumer Affairs Commissioner Polonetsky Warns Consumers: "Buying A Star Won't Make You One"". National Astronomy and Ionosphere Center, Aricebo Observatory. Retrieved 2010-06-24. 53. Sackmann, I.-J.; Boothroyd, A. I. (2003). "Our Sun. V. A Bright Young Sun Consistent with Helioseismology and Warm Temperatures on Ancient Earth and Mars". The Astrophysical Journal. 583 (2): 1024–1039. Bibcode:2003ApJ...583.1024S. arXiv:. doi:10.1086/345408. 54. Tripathy, S. C.; Antia, H. M. (1999). "Influence of surface layers on the seismic estimate of the solar radius". Solar Physics. 186 (1/2): 1–11. Bibcode:1999SoPh..186....1T. doi:10.1023/A:1005116830445. 55. Woodward, P. R. (1978). "Theoretical models of star formation". Annual review of astronomy and astrophysics. 16 (1): 555–584. Bibcode:1978ARA&A..16..555W. doi:10.1146/annurev.aa.16.090178.003011. 56. Lada, C. J.; Lada, E. A. (2003). "Embedded Clusters in Molecular Clouds". Annual review of astronomy and astrophysics. 41 (1): 57–115. Bibcode:2003ARA&A..41...57L. arXiv:. doi:10.1146/annurev.astro.41.011802.094844. 57. Kwok, Sun (2000). The origin and evolution of planetary nebulae. Cambridge astrophysics series. 33. Cambridge University Press. pp. 103–104. ISBN 0-521-62313-8. 58. Adams, Fred C.; Laughlin, Gregory; Graves, Genevieve J. M. "Red Dwarfs and the End of the Main Sequence" (PDF). Gravitational Collapse: From Massive Stars to Planets. Revista Mexicana de Astronomía y Astrofísica. pp. 46–49. Bibcode:2004RMxAC..22...46A. Retrieved 2008-06-24. 59. Elmegreen, B. G.; Lada, C. J. (1977). "Sequential formation of subgroups in OB associations". Astrophysical Journal, Part 1. 214: 725–741. Bibcode:1977ApJ...214..725E. doi:10.1086/155302. 60. Getman, K. V.; et al. (2012). "The Elephant Trunk Nebula and the Trumpler 37 cluster: contribution of triggered star formation to the total population of an H II region". Monthly Notices of the Royal Astronomical Society. 426 (4): 2917–2943. Bibcode:2012MNRAS.426.2917G. arXiv:. doi:10.1111/j.1365-2966.2012.21879.x. 61. Smith, Michael David (2004). The Origin of Stars. Imperial College Press. pp. 57–68. ISBN 1-86094-501-5. 62. Seligman, Courtney. "Slow Contraction of Protostellar Cloud". Self-published. Archived from the original on 2008-06-23. Retrieved 2006-09-05. 63. Bally, J.; Morse, J.; Reipurth, B. (1996). "The Birth of Stars: Herbig-Haro Jets, Accretion and Proto-Planetary Disks". In Benvenuti, Piero; Macchetto, F. D.; Schreier, Ethan J. Science with the Hubble Space Telescope – II. Proceedings of a workshop held in Paris, France, December 4–8, 1995. Space Telescope Science Institute. p. 491. Bibcode:1996swhs.conf..491B. 64. Smith, Michael David (2004). The origin of stars. Imperial College Press. p. 176. ISBN 1-86094-501-5. 65. Megeath, Tom (May 11, 2010). "Herschel finds a hole in space". ESA. Retrieved 2010-05-17. 66. Duquennoy, A.; Mayor, M. (1991). "Multiplicity among solar-type stars in the solar neighbourhood. II - Distribution of the orbital elements in an unbiased sample". Astronomy & Astrophysics. 248 (2): 485–524. Bibcode:1991A&A...248..485D. 67. Mengel, J. G.; et al. (1979). "Stellar evolution from the zero-age main sequence". Astrophysical Journal Supplement Series. 40: 733–791. Bibcode:1979ApJS...40..733M. doi:10.1086/190603. 68. Sackmann, I. J.; Boothroyd, A. I.; Kraemer, K. E. (1993). "Our Sun. III. Present and Future". Astrophysical Journal. 418: 457. Bibcode:1993ApJ...418..457S. doi:10.1086/173407. 69. Wood, B. E.; et al. (2002). "Measured Mass-Loss Rates of Solar-like Stars as a Function of Age and Activity". The Astrophysical Journal. 574 (1): 412–425. Bibcode:2002ApJ...574..412W. arXiv:. doi:10.1086/340797. 70. de Loore, C.; de Greve, J. P.; Lamers, H. J. G. L. M. (1977). "Evolution of massive stars with mass loss by stellar wind". Astronomy and Astrophysics. 61 (2): 251–259. Bibcode:1977A&A....61..251D. 71. "The evolution of stars between 50 and 100 times the mass of the Sun". Royal Greenwich Observatory. Retrieved 2015-11-17. 72. "Main Sequence Lifetime". Swinburne Astronomy Online Encyclopedia of Astronomy. Swinburne University of Technology. 73. Pizzolato, N.; et al. (2001). "Subphotospheric convection and magnetic activity dependence on metallicity and age: Models and tests". Astronomy & Astrophysics. 373 (2): 597–607. Bibcode:2001A&A...373..597P. doi:10.1051/0004-6361:20010626. 74. "Mass loss and Evolution". UCL Astrophysics Group. June 18, 2004. Archived from the original on 2004-11-22. Retrieved 2006-08-26. 75. Schröder, K.-P.; Smith, Robert Connon (2008). "Distant future of the Sun and Earth revisited". Monthly Notices of the Royal Astronomical Society. 386 (1): 155–163. Bibcode:2008MNRAS.386..155S. arXiv:. doi:10.1111/j.1365-2966.2008.13022.x. See also Palmer, Jason (February 22, 2008). "Hope dims that Earth will survive Sun's death". NewScientist.com news service. Retrieved 2008-03-24. 76. "The Evolution of Massive Stars and Type II Supernovae". Penn Stats College of Science. Retrieved 2016-01-05. 77. Sneden, Christopher (February 8, 2001). "Astronomy: The age of the Universe". Nature. 409 (6821): 673–675. ISSN 0028-0836. doi:10.1038/35055646. 78. Liebert, J. (1980). "White dwarf stars". Annual review of astronomy and astrophysics. 18 (2): 363–398. Bibcode:1980ARA&A..18..363L. doi:10.1146/annurev.aa.18.090180.002051. 79. "Introduction to Supernova Remnants". Goddard Space Flight Center. April 6, 2006. Retrieved 2006-07-16. 80. Fryer, C. L. (2003). "Black-hole formation from stellar collapse". Classical and Quantum Gravity. 20 (10): S73–S80. Bibcode:2003CQGra..20S..73F. doi:10.1088/0264-9381/20/10/309. 81. Szebehely, Victor G.; Curran, Richard B. (1985). Stability of the Solar System and Its Minor Natural and Artificial Bodies. Springer. ISBN 90-277-2046-0. 82. "Most Milky Way Stars Are Single" (Press release). Harvard-Smithsonian Center for Astrophysics. January 30, 2006. Retrieved 2006-07-16. 83. "What is a galaxy? How many stars in a galaxy / the Universe?". Royal Greenwich Observatory. Retrieved 2006-07-18. 84. Borenstein, Seth (December 1, 2010). "Universe's Star Count Could Triple". CBS News. Retrieved 2011-07-14. 85. "Hubble Finds Intergalactic Stars". Hubble News Desk. January 14, 1997. Retrieved 2006-11-06. 86. 3.99 × 1013 km / (3 × 104 km/h × 24 × 365.25) = 1.5 × 105 years. 87. Holmberg, J.; Flynn, C. (2000). "The local density of matter mapped by Hipparcos". Monthly Notices of the Royal Astronomical Society. 313 (2): 209–216. Bibcode:2000MNRAS.313..209H. arXiv:. doi:10.1046/j.1365-8711.2000.02905.x. 88. "Astronomers: Star collisions are rampant, catastrophic". CNN News. June 2, 2000. Archived from the original on 2007-01-07. Retrieved 2014-01-21. 89. Lombardi, Jr., J. C.; et al. (2002). "Stellar Collisions and the Interior Structure of Blue Stragglers". The Astrophysical Journal. 568 (2): 939–953. Bibcode:2002ApJ...568..939L. arXiv:. doi:10.1086/339060. 90. H. E. Bond; E. P. Nelan; D. A. VandenBerg; G. H. Schaefer; D. Harmer (2013). "HD 140283: A Star in the Solar Neighborhood that Formed Shortly After the Big Bang". The Astrophysical Journal Letters. 765 (1): L12. Bibcode:2013ApJ...765L..12B. arXiv:. doi:10.1088/2041-8205/765/1/L12. 91. Planck Collaboration (2015). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd).". Bibcode:2015arXiv150201589P. arXiv:. 92. Naftilan, S. A.; Stetson, P. B. (July 13, 2006). "How do scientists determine the ages of stars? Is the technique really accurate enough to use it to verify the age of the universe?". Scientific American. Retrieved 2007-05-11. 93. Laughlin, G.; Bodenheimer, P.; Adams, F. C. (1997). "The End of the Main Sequence". The Astrophysical Journal. 482 (1): 420–432. Bibcode:1997ApJ...482..420L. doi:10.1086/304125. 94. Irwin, Judith A. (2007). Astrophysics: Decoding the Cosmos. John Wiley and Sons. p. 78. ISBN 0-470-01306-0. 95. Fischer, D. A.; Valenti, J. (2005). "The Planet-Metallicity Correlation". The Astrophysical Journal. 622 (2): 1102–1117. Bibcode:2005ApJ...622.1102F. doi:10.1086/428383. 96. "Signatures Of The First Stars". ScienceDaily. April 17, 2005. Retrieved 2006-10-10. 97. Feltzing, S.; Gonzalez, G. (2000). "The nature of super-metal-rich stars: Detailed abundance analysis of 8 super-metal-rich star candidates". Astronomy & Astrophysics. 367 (1): 253–265. Bibcode:2001A&A...367..253F. doi:10.1051/0004-6361:20000477. 98. Gray, David F. (1992). The Observation and Analysis of Stellar Photospheres. Cambridge University Press. pp. 413–414. ISBN 0-521-40868-7. 99. Jørgensen, Uffe G. (1997), "Cool Star Models", in van Dishoeck, Ewine F., Molecules in Astrophysics: Probes and Processes, International Astronomical Union Symposia. Molecules in Astrophysics: Probes and Processes, 178, Springer Science & Business Media, p. 446, ISBN 079234538X 100. "The Biggest Star in the Sky". ESO. March 11, 1997. Retrieved 2006-07-10. 101. Ragland, S.; Chandrasekhar, T.; Ashok, N. M. (1995). "Angular Diameter of Carbon Star Tx-Piscium from Lunar Occultation Observations in the Near Infrared". Journal of Astrophysics and Astronomy. 16: 332. Bibcode:1995JApAS..16..332R. 102. Davis, Kate (December 1, 2000). "Variable Star of the Month—December, 2000: Alpha Orionis". AAVSO. Archived from the original on 2006-07-12. Retrieved 2006-08-13. 103. Loktin, A. V. (September 2006). "Kinematics of stars in the Pleiades open cluster". Astronomy Reports. 50 (9): 714–721. Bibcode:2006ARep...50..714L. doi:10.1134/S1063772906090058. 104. "Hipparcos: High Proper Motion Stars". ESA. September 10, 1999. Retrieved 2006-10-10. 105. Johnson, Hugh M. (1957). "The Kinematics and Evolution of Population I Stars". Publications of the Astronomical Society of the Pacific. 69 (406): 54. Bibcode:1957PASP...69...54J. doi:10.1086/127012. 106. Elmegreen, B.; Efremov, Y. N. (1999). "The Formation of Star Clusters". American Scientist. 86 (3): 264. Bibcode:1998AmSci..86..264E. doi:10.1511/1998.3.264. Archived from the original on March 23, 2005. Retrieved 2006-08-23. 107. Brainerd, Jerome James (July 6, 2005). "X-rays from Stellar Coronas". The Astrophysics Spectator. Retrieved 2007-06-21. 108. Berdyugina, Svetlana V. (2005). "Starspots: A Key to the Stellar Dynamo". Living Reviews. Retrieved 2007-06-21. 109. Smith, Nathan (1998). "The Behemoth Eta Carinae: A Repeat Offender". Mercury Magazine. Astronomical Society of the Pacific. 27: 20. Retrieved 2006-08-13. 110. Weidner, C.; Kroupa, P. (February 11, 2004). "Evidence for a fundamental stellar upper mass limit from clustered star formation" (PDF). Monthly Notices of the Royal Astronomical Society. 348 (1): 187–191. Bibcode:2004MNRAS.348..187W. ISSN 0035-8711. arXiv:. doi:10.1111/j.1365-2966.2004.07340.x. 111. Hainich, R.; Rühling, U.; Todt, H.; Oskinova, L. M.; Liermann, A.; Gräfener, G.; Foellmi, C.; Schnurr, O.; Hamann, W. -R. (2014). "The Wolf-Rayet stars in the Large Magellanic Cloud". Astronomy & Astrophysics. 565: A27. Bibcode:2014A&A...565A..27H. arXiv:. doi:10.1051/0004-6361/201322696. 112. Banerjee, Sambaran; Kroupa, Pavel; Oh, Seungkyung (October 21, 2012). "The emergence of super-canonical stars in R136-type starburst clusters". Monthly Notices of the Royal Astronomical Society. 426 (2): 1416–1426. Bibcode:2012MNRAS.426.1416B. ISSN 0035-8711. arXiv:. doi:10.1111/j.1365-2966.2012.21672.x. 113. "Ferreting Out The First Stars". Harvard-Smithsonian Center for Astrophysics. September 22, 2005. Retrieved 2006-09-05. 114. Sobral, David; Matthee, Jorryt; Darvish, Behnam; Schaerer, Daniel; Mobasher, Bahram; Röttgering, Huub J. A.; Santos, Sérgio; Hemmati, Shoubaneh (4 June 2015). "Evidence For POPIII-Like Stellar Populations In The Most Luminous LYMAN-α Emitters At The Epoch Of Re-Ionisation: Spectroscopic Confirmation" (PDF). The Astrophysical Journal. Bibcode:2015ApJ...808..139S. arXiv:. doi:10.1088/0004-637x/808/2/139. Retrieved 17 June 2015. 115. Overbye, Dennis (17 June 2015). "Astronomers Report Finding Earliest Stars That Enriched Cosmos". New York Times. Retrieved 17 June 2015. 116. "2MASS J05233822-1403022". SIMBAD - Centre de Données astronomiques de Strasbourg. Retrieved 14 December 2013. 117. Boss, Alan (April 3, 2001). "Are They Planets or What?". Carnegie Institution of Washington. Archived from the original on 2006-09-28. Retrieved 2006-06-08. 118. Shiga, David (August 17, 2006). "Mass cut-off between stars and brown dwarfs revealed". New Scientist. Archived from the original on 2006-11-14. Retrieved 2006-08-23. 119. Leadbeater, Elli (August 18, 2006). "Hubble glimpses faintest stars". BBC. Retrieved 2006-08-22. 120. "Flattest Star Ever Seen". ESO. June 11, 2003. Retrieved 2006-10-03. 121. Fitzpatrick, Richard (February 13, 2006). "Introduction to Plasma Physics: A graduate course". The University of Texas at Austin. Archived from the original on 2010-01-04. Retrieved 2006-10-04. 122. Villata, Massimo (1992). "Angular momentum loss by a stellar wind and rotational velocities of white dwarfs". Monthly Notices of the Royal Astronomical Society. 257 (3): 450–454. Bibcode:1992MNRAS.257..450V. doi:10.1093/mnras/257.3.450. 123. "A History of the Crab Nebula". ESO. May 30, 1996. Retrieved 2006-10-03. 124. Strobel, Nick (August 20, 2007). "Properties of Stars: Color and Temperature". Astronomy Notes. Primis/McGraw-Hill, Inc. Archived from the original on 2007-06-26. Retrieved 2007-10-09. 125. Seligman, Courtney. "Review of Heat Flow Inside Stars". Self-published. Retrieved 2007-07-05. 126. "Main Sequence Stars". The Astrophysics Spectator. February 16, 2005. Retrieved 2006-10-10. 127. Zeilik, Michael A.; Gregory, Stephan A. (1998). Introductory Astronomy & Astrophysics (4th ed.). Saunders College Publishing. p. 321. ISBN 0-03-006228-4. 128. Koppes, Steve (June 20, 2003). "University of Chicago physicist receives Kyoto Prize for lifetime achievements in science". The University of Chicago News Office. Retrieved 2012-06-15. 129. "The Colour of Stars". Australian Telescope Outreach and Education. Retrieved 2006-08-13. 130. "Astronomers Measure Mass of a Single Star—First Since the Sun". Hubble News Desk. July 15, 2004. Retrieved 2006-05-24. 131. Garnett, D. R.; Kobulnicky, H. A. (2000). "Distance Dependence in the Solar Neighborhood Age-Metallicity Relation". The Astrophysical Journal. 532 (2): 1192–1196. Bibcode:2000ApJ...532.1192G. arXiv:. doi:10.1086/308617. 132. Staff (January 10, 2006). "Rapidly Spinning Star Vega has Cool Dark Equator". National Optical Astronomy Observatory. Retrieved 2007-11-18. 133. Michelson, A. A.; Pease, F. G. (2005). "Starspots: A Key to the Stellar Dynamo". Living Reviews in Solar Physics. Max Planck Society. 134. Manduca, A.; Bell, R. A.; Gustafsson, B. (1977). "Limb darkening coefficients for late-type giant model atmospheres". Astronomy and Astrophysics. 61 (6): 809–813. Bibcode:1977A&A....61..809M. 135. Chugainov, P. F. (1971). "On the Cause of Periodic Light Variations of Some Red Dwarf Stars". Information Bulletin on Variable Stars. 520: 1–3. Bibcode:1971IBVS..520....1C. 136. "Magnitude". National Solar Observatory—Sacramento Peak. Archived from the original on 2008-02-06. Retrieved 2006-08-23. 137. "Luminosity of Stars". Australian Telescope Outreach and Education. Retrieved 2006-08-13. 138. Hoover, Aaron (January 15, 2004). "Star may be biggest, brightest yet observed". HubbleSite. Archived from the original on 2007-08-07. Retrieved 2006-06-08. 139. "Faintest Stars in Globular Cluster NGC 6397". HubbleSite. August 17, 2006. Retrieved 2006-06-08. 140. Smith, Gene (April 16, 1999). "Stellar Spectra". University of California, San Diego. Retrieved 2006-10-12. 141. Fowler, A. (February 1891). "The Draper Catalogue of Stellar Spectra". Nature. 45: 427–8. Bibcode:1892Natur..45..427F. doi:10.1038/045427a0. 142. Jaschek, Carlos; Jaschek, Mercedes (1990). The Classification of Stars. Cambridge University Press. pp. 31–48. ISBN 0-521-38996-8. 143. MacRobert, Alan M. "The Spectral Types of Stars". Sky and Telescope. Retrieved 2006-07-19. 144. "White Dwarf (wd) Stars". White Dwarf Research Corporation. Archived from the original on 2009-10-08. Retrieved 2006-07-19. 145. "Types of Variable". AAVSO. May 11, 2010. Retrieved 2010-08-20. 146. "Cataclysmic Variables". NASA Goddard Space Flight Center. 2004-11-01. Retrieved 2006-06-08. 147. Hansen, Carl J.; Kawaler, Steven D.; Trimble, Virginia (2004). Stellar Interiors. Springer. pp. 32–33. ISBN 0-387-20089-4. 148. Schwarzschild, Martin (1958). Structure and Evolution of the Stars. Princeton University Press. ISBN 0-691-08044-5. 149. "Formation of the High Mass Elements". Smoot Group. Retrieved 2006-07-11. 150. "What is a Star?". NASA. 2006-09-01. Retrieved 2006-07-11. 151. "The Glory of a Nearby Star: Optical Light from a Hot Stellar Corona Detected with the VLT" (Press release). ESO. August 1, 2001. Retrieved 2006-07-10. 152. Burlaga, L. F.; et al. (2005). "Crossing the Termination Shock into the Heliosheath: Magnetic Fields". Science. 309 (5743): 2027–2029. Bibcode:2005Sci...309.2027B. PMID 16179471. doi:10.1126/science.1117542. 153. Wallerstein, G.; et al. (1999). "Synthesis of the elements in stars: forty years of progress" (PDF). Reviews of Modern Physics. 69 (4): 995–1084. Bibcode:1997RvMP...69..995W. doi:10.1103/RevModPhys.69.995. Retrieved 2006-08-04. 154. Girardi, L.; Bressan, A.; Bertelli, G.; Chiosi, C. (2000). "Evolutionary tracks and isochrones for low- and intermediate-mass stars: From 0.15 to 7 Msun, and from Z=0.0004 to 0.03". Astronomy and Astrophysics Supplement. 141 (3): 371–383. Bibcode:2000A&AS..141..371G. arXiv:. doi:10.1051/aas:2000126. 155. Woosley, S. E.; Heger, A.; Weaver, T. A. (2002). "The evolution and explosion of massive stars". Reviews of Modern Physics. 74 (4): 1015–1071. Bibcode:2002RvMP...74.1015W. doi:10.1103/RevModPhys.74.1015. 156. 11.5 days is 0.0315 years.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.893088161945343, "perplexity": 2590.57337894379}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676595531.70/warc/CC-MAIN-20180723071245-20180723091245-00335.warc.gz"}
https://forum.math.toronto.edu/index.php?PHPSESSID=1rvpd3nee4mqr5qb3qi3r03n41&topic=873.msg3120	### Author Topic: Julia Set (Read 3463 times) #### Laura Campbell • Newbie • Posts: 2 • Karma: 0 ##### Julia Set « on: March 14, 2017, 03:18:47 PM » #### Thierry Serafin Nadeau • Newbie • Posts: 3 • Karma: 0 ##### Julia Set « Reply #1 on: March 15, 2017, 06:07:38 PM » Turns out that there are 4D analogues to the 2D Julia sets which use quaternions ($i^2 = j^2 = k^2 = ijk = -1$) instead of regular complex numbers. Seeing as they're 4D they can only be visualised as 3D slices of the whole set, which end up looking quite a bit different than the regular Julia sets depending on the chosen slice. Here's a link to a video which passes through multiple 3D slices of a quaternion Julia set: https://www.youtube.com/watch?v=VkmqT6MQoDE And bellow is an image of such a slice. « Last Edit: March 15, 2017, 06:12:37 PM by Thierry Serafin Nadeau » #### Leonora Boci • Newbie • Posts: 3 • Karma: 0 ##### Re: Julia Set « Reply #2 on: March 27, 2017, 11:19:20 PM » Julia set with the parameter µ taken from the center of the circle on top of the cardioid. #### kojak • Guest ##### Re: Julia Set « Reply #3 on: April 05, 2017, 12:56:26 AM » Julia Sets: zn+1 = c sin(zn) zn+1 = c exp(zn) zn+1 = c i cos(zn) zn+1 = c zn (1 - zn) A property of the the Julia Set is that if the domain of c is real numbers the the Julia Set it mirrored about the Real axis. If c is a complex number with an imaginary component then then the symmetry is rotational at 180 degrees.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6670787334442139, "perplexity": 4301.456698083274}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046150266.65/warc/CC-MAIN-20210724125655-20210724155655-00543.warc.gz"}
http://cogsci.stackexchange.com/questions/3478/do-people-estimate-combined-probabilities-differently-to-uncombined-ones?answertab=active	# Do people estimate combined probabilities differently to uncombined ones? Suppose, somebody has to estimate the likelihood of one of the following events (or has to estimate which event is more likely): 1. A coin is tossed six times and each time the result is heads. (combined) 2. A 64-sided die is rolled and the result is 1. (uncombined) Since 2⁶ = 64, both events are obviously equiprobable. However, somebody with less statistical prowess might think otherwise. Also, in a more complex scenario (e.g., in a board game) at least one of the probabilities may be impossible to calculate for almost anybody (at least in a short time). I am interested in studies that investigate, whether combined probabilities (1) are estimated to be higher, lower or equal than uncombined probabilities (2), even if the actual probabilities are identical. I am also interested in studies, which investigate the influence of the presentation of the scenario on the estimation of probabilities: Is the same scenario perceived differently, if presented in a way that emphasises combinedness? For example, one could simply ask somebody to estimate the probability that one of his close relatives will die within a year (uncombined) or one could ask him to list all his close relatives, their age and state of health first (combined). After all, a lot of probabilities with real-life applications are combined. Note that I am not interested in which of the estimates is more accurate, only which one is higher. - Welcome to CogSci. This sounds like an interesting question, but after the example in the last paragraph, I'm not sure I understand what you mean by cumulative and non cumulative. The term cumulative as I know it refers to questions like "what is the probability of at least 4 heads in tosses", while non cumulative would be "what is the probability of exactly 4 heads in 6 tosses". Is that what you mean? If not, can you be more specific and define what you mean by the term? – Ofri Raviv May 4 at 17:28 Thanks, I forgot that cumulated is a predefined term in statistics and I changed it to combined, which should be the correct term. I denote an event as combined if it occurs if and only if a collection of other events occur. – Wrzlprmft May 4 at 18:20 The probability of conjunctive events (all six tosses are heads) are overestimated, relative to a single event of similar overall probability. This result has been shown by Paul Slovic, in an experiment that is described in its abstract as follows: This study examined the effects on the attractiveness of a gamble, of manipulating the number and structure of the independent events on which the gamble's outcomes were contingent, while holding the expected value constant. 3 manipulations were studied. The most effective was the situation where a gamble offering a payoff with probability equal to p was changed into a gamble whose payoff was contingent on the joint occurrence of 4 independent events, each having a probability equal to p^(1/4). Although individual differences were substantial, the majority of Ss behaved as if the probability of the compound event was much greater than p, its true value. Similar results were obtained by Maya Bar-Hillel, in several experiments. In one of them she presented subjects with displays similar to this one: A path in this array is defined as any line which originates at any element of the first row, connects it with any element in the second row, proceeds to pass through any element in the third row, and so on to the last row. The subjects were asked to estimate the proportion of paths in such displays that cross only X's, relative to all paths. Results show a robust (and quite large) overestimation of this proportion. References • Slovic, P. (1969). Manipulating the attractiveness of a gamble without changing its expected value. Journal of Experimental Psychology, 79(1p1), 139. • Bar-Hillel, M. (1973). On the subjective probability of compound events. Organizational behavior and human performance, 9(3), 396-406. -	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8069168329238892, "perplexity": 668.2584555905107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164034245/warc/CC-MAIN-20131204133354-00002-ip-10-33-133-15.ec2.internal.warc.gz"}
https://www.ideals.illinois.edu/handle/2142/96994	## Files in this item FilesDescriptionFormat application/pdf 2627.pdf (15kB) (no description provided)PDF ## Description Title: THE ATMOSPHERIC CHEMISTRY EXPERIMENT (ACE): LATEST RESULTS Author(s): Bernath, Peter F. Subject(s): Atmospheric science Abstract: ACE (also known as SCISAT) is making a comprehensive set of simultaneous measurements of numerous trace gases, thin clouds, aerosols and temperature by solar occultation from a satellite in low earth orbit. A high inclination orbit gives ACE coverage of tropical, mid-latitudes and polar regions. The primary instrument is a high-resolution (0.02 cm$^{-1}$) infrared Fourier Transform Spectrometer (FTS) operating in the 750--4400 cm$^{-1}$ region, which provides the vertical distribution of trace gases, and the meteorological variables of temperature and pressure. Aerosols and clouds are being monitored through the extinction of solar radiation using two filtered imagers as well as by their infrared spectra. After 14 years in orbit, the ACE-FTS is still operating well. A short introduction and overview of the ACE mission will be presented (see http://www.ace.uwaterloo.ca for more information). This talk will focus on recent ACE results and comparisons with chemical transport models. Issue Date: 6/19/2017 Publisher: International Symposium on Molecular Spectroscopy Citation Info: APS Genre: CONFERENCE PAPER/PRESENTATION Type: Text Language: English URI: http://hdl.handle.net/2142/96994 DOI: 10.15278/isms.2017.MJ08 Date Available in IDEALS: 2017-07-27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7355533242225647, "perplexity": 8772.385339798206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084888041.33/warc/CC-MAIN-20180119144931-20180119164931-00345.warc.gz"}
https://squ.pure.elsevier.com/en/publications/magnetocaloric-properties-of-nisub50submnsub28subgasub22sub-melt-	# Magnetocaloric properties of Ni50Mn28Ga22 melt-spun ribbons Deepak Kumar Satapathy, I. A. Al-Omari, Shampa Aich* *Corresponding author for this work Research output: Contribution to journalArticlepeer-review 1 Citation (Scopus) ## Abstract NiMnGa-based Heusler alloys are known for their shape-memory effect. However, in this study, the focus will be on the magneto-thermal (magnetocaloric) properties of rapidly solidified Ni50Mn28Ga22 ribbons melt spun at 1300 and 1600 RPM as well as annealed bulk specimens. The ribbons, both as-spun and annealed, and the annealed bulk specimens were tested in a SQUID to determine the magnetic properties at a field of 50 kOe with a temperature step of 3 K in the temperature range 355–385 K. The magnetic data from the isotherms were used to achieve Arrott plots. Second-order transitions were observed in the materials at the TC temperature. The values of magnetic entropy ΔSm were calculated from the magnetic data and these were further used to calculate the values of refrigeration capacity RC. The highest RC value of 273 J/kg was obtained for 1300NMG5800 where 1300, 800 and 5, represent the melt spinning rate in RPM, the annealing temperature in Celsius and the duration of annealing in hours, respectively. The values of ΔSm were similar for all the ribbons. To confirm the order of the magnetic transition, universal curves were plotted which led to the conclusion that the transformation at the Curie temperature is a second-order ferromagnetic to paramagnetic transition. Original language English Philosophical Magazine Letters https://doi.org/10.1080/09500839.2021.1962015 Accepted/In press - 2021 Yes ## Keywords • Heusler alloys • magnetic entropy • magnetic properties • magnetocaloric • Refrigeration capacity ## ASJC Scopus subject areas • Condensed Matter Physics	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9026092886924744, "perplexity": 5627.837019256563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103620968.33/warc/CC-MAIN-20220629024217-20220629054217-00642.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-derivative-of-f-x-sinx-1-cosx-2	Calculus Topics # How do you find the derivative of F(x) = (sinx/(1+cosx))^2? Aug 2, 2016 $\frac{2 \sin x}{1 + \cos x} ^ 2$ #### Explanation: The derivative of a power is stated as follows: $\left({\left(u\right)}^{n}\right) ' = n \cdot u \cdot u '$ Applying this property the derivative of the given power is : $\textcolor{b l u e}{\left(\sin \frac{x}{1 + \cos x} ^ 2\right) ' = 2 \left(\sin \frac{x}{1 + \cos x}\right) \left(\sin \frac{x}{1 + \cos x}\right) '}$ Let's find$\left(\sin \frac{x}{1 + \cos x}\right) '$: The derivative of the quotient $\left(\frac{u}{v}\right) ' = \frac{u ' v - v ' u}{v} ^ 2$ To apply the derivative of a quotient on (sinx)/(1+cosx) we've to find : $\left(\sin x\right) ' = \cos x$ $\left(1 + \cos x\right) ' = - \sin x$ So, $\textcolor{b l u e}{\left(\sin \frac{x}{1 + \cos x}\right) '}$ $= \frac{\left(\sin x\right) ' \left(1 + \cos x\right) - \left(1 + \cos x\right) ' \sin x}{1 + \cos x} ^ 2$ $= \frac{\cos x \left(1 + \cos x\right) - \left(- \sin x\right) \left(\sin x\right)}{1 + \cos x} ^ 2$ $= \frac{\cos x + {\left(\cos x\right)}^{2} + {\left(\sin x\right)}^{2}}{1 + \cos x} ^ 2$ Knowing that${\left(\cos x\right)}^{2} + {\left(\sin x\right)}^{2} = 1$ $= \frac{\cos x + 1}{1 + \cos x} ^ 2$ Simplifying by $1 + \cos x$ Therefore, $\left(\sin \frac{x}{1 + \cos x}\right) ' = \frac{1}{1 + \cos x}$ So ,the derivative of the given power: $\left({\left(\sin \frac{x}{1 + \cos x}\right)}^{2}\right) '$ $= 2 \left(\sin \frac{x}{1 + \cos x}\right) \left(\sin \frac{x}{1 + \cos x}\right) '$ $= 2 \left(\sin \frac{x}{1 + \cos x}\right) \left(\frac{1}{1 + \cos x}\right)$ $= \frac{2 \sin x}{1 + \cos x} ^ 2$ ##### Impact of this question 2259 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 19, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9222739934921265, "perplexity": 2668.873707764736}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153854.42/warc/CC-MAIN-20210729074313-20210729104313-00627.warc.gz"}
https://gmatclub.com/forum/which-of-the-following-could-not-be-the-lengths-of-the-sides-of-a-righ-218532.html	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 27 Jun 2019, 01:27 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Which of the following could not be the lengths of the sides of a righ Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 55804 Which of the following could not be the lengths of the sides of a righ [#permalink] ### Show Tags 14 May 2016, 03:51 00:00 Difficulty: 5% (low) Question Stats: 85% (00:44) correct 15% (00:57) wrong based on 152 sessions ### HideShow timer Statistics Which of the following could not be the lengths of the sides of a right angled triangle? A. 3, 4, 5 B. 5, 12, 13 C. 8, 15, 17 D. 12, 15, 18 E. 9, 12, 15 _________________ Intern Joined: 02 Mar 2015 Posts: 2 Re: Which of the following could not be the lengths of the sides of a righ [#permalink] ### Show Tags 14 May 2016, 06:42 1 D. 12,15,18 Except this all are Pythagorean triplet or their multiples 3,4,5 5,12,13 8,15,17 9,12,15 : 3x3,3x4,3x5 (3,4,5 is a triplet) Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4505 Location: India GPA: 3.5 Re: Which of the following could not be the lengths of the sides of a righ [#permalink] ### Show Tags 14 May 2016, 07:29 Bunuel wrote: Which of the following could not be the lengths of the sides of a right angled triangle? A. 3, 4, 5 B. 5, 12, 13 C. 8, 15, 17 D. 12, 15, 18 E. 9, 12, 15 A. $$5^2$$ = $$3^2$$ + $$4^2$$ B. $$13^2$$ = $$12^2$$ + $$5^2$$ C. $$17^2$$ = $$15^2$$ + $$8^2$$ D. $$18^2$$ $$not$$ = $$12^2$$ + $$15^2$$ E. $$15^2$$ = $$12^2$$ + $$9^2$$ _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club \| Rules for Posting in QA forum \| Writing Mathematical Formulas \|Rules for Posting in VA forum \| Request Expert's Reply ( VA Forum Only ) Senior Manager Joined: 24 Nov 2015 Posts: 496 Location: United States (LA) Re: Which of the following could not be the lengths of the sides of a righ [#permalink] ### Show Tags 18 Sep 2016, 03:52 option A, B , C and E satisfy the pythagoras theorem which is $$one side^2$$ + $$another side ^2$$ = $$hypotenuse^2$$ option D doesn't satisfy this theorem Non-Human User Joined: 09 Sep 2013 Posts: 11448 Re: Which of the following could not be the lengths of the sides of a righ [#permalink] ### Show Tags 15 Sep 2018, 06:41 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Re: Which of the following could not be the lengths of the sides of a righ [#permalink] 15 Sep 2018, 06:41 Display posts from previous: Sort by	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5460987091064453, "perplexity": 2651.4421008856584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628001014.85/warc/CC-MAIN-20190627075525-20190627101525-00053.warc.gz"}
https://lasseschultebraucks.com/ai/machine%20learning/a-short-history-of-artificial-intelligence.html	A Short History of Artificial Intelligence What is Artificial Intelligence? What has Artificial Intelligence reached in the past? What are appreciable milestones of Artificial Intelligence in the last years? What will Artificial Intelligence solve in 5 years, 20 years, 50 years or 100 years from now? And how will look Artificial Intelligence the future? There are many question around Artificial Intelligence. We already can answer some of them and talk about what we have solved with Artificial Intelligence in the past. On the other hand talking about future scenarios, especially if we talk about superintelligence, turns out difficult. But there are some scenarios we can think and evaluate about. Definition of Artificial Intelligence Artificial Intelligence is a term, which consists of two words. Artificial Artificial is something that is not real and which is kind of fake because it is simulated. The simplest thing what I can think of which is artificial is artificial grass. Artificial grass is not real grass, so it is kind of fake. It is used to substitute real grass for various reason. Artificial grass is often used for sports, because it is more resistant and therefore can be used longer than real grass. It is also easier to care than real grass. I am sure there are many more reason for artificial grass and against real grass. But that is not the point I want to make. The point is, that there are reasons why some things are artificial and substitute real things. Intelligence Intelligence is are very complex term. It can be defined in many different ways like logic, understanding, self-awareness, learning, emotional knowledge, planning, creativity and of course problem solving. We call us, humans, intelligent, because we all do this mentioned things. We perceive our environment, learn from it and take action based on what we discovered. The same applies to animals. The interesting point about intelligence on animals is, that there are many different species and because of that we can compare intelligence on between species. In both cases (human intelligence and animal intelligence) we talk about natural intelligence (NI). Next to humans and animals there has been argued about plant intelligence. Intelligence in plants shows off kind of different from humans or animals. The main reason is here because plants are not having a brain or neuronal network, but they react to their environment. Plant intelligence is a very interesting topic on its own, because plant intelligence is not instantly visible through reactions through movement or lute. If we talk about Artificial Intelligence (AI) we refer to a subfield of Computer Science. Artificial Intelligence is acted by machines, computers and mainly software. Machines mimic, here we see why it is called artificial, some kind of cognitive function based on environment, observations, rewards and learning process. To understand more about Artificial Intelligence we look at the history of Artificial Intelligence to see what Artificial Intelligence is capable of and how his status quo is related to the present. History and Milestone of Artificial Intelligence The history of Artificial Intelligence is quite interesting and started around 100 years ago. Rossum’s Universal Robots (R.U.R.) In 1920 the Czech writer Karel Čapek published a science fiction play named Rossumovi Univerzální Roboti (Rossum’s Universal Robots), also better known as R.U.R. The play introduced the word robot. R.U.R. deals about a factory, which creates artificial people named as robots. They differentiate from today’s term of robot. In R.U.R. robots are living creatures, who are more similar to the term of clones. The robots in R.U.R. first worked for the humans, but then there comes are robot rebellion which leads to the extinction of the human race. The play is quite interesting, because of different reason. First it is introducing the term robot, even if represents not exactly the modern idea of robots. Next it is also telling the story of the creation of robots, so some kind of artificial intelligence, which first seems to be a positive effect to the humans, but later on the is the robot rebellion which threat the whole human race. Artificial Intelligence in literature and movies is a big topic for its own. The example of R.U.R. should have shown the importance and influence for Artificial Intelligence on researches and society. Alan Turing Alan Turing was born on 23th June 1912 in London. He is widely known, because the encrypted the code of the enigma, which were used from Nazi Germany to communicate. Alan Turing’s study also led to his theory of computation, which deals about how efficient problems can be solved. His presented his idea in the model of the Turing machine, which is today still a popular term in Computer Science. The Turing machine is an abstract machine, which can ,despite the model’s simplicity, construct any algorithm’s logic. Because of discoveries in neurology, information theory and cybernetics in the same time researches and with them Alan Turing created the idea that it is possible to build an electronic brain. Some years after the end of World War 2, Turing introduced his widely known Turing Test, which was an attempt to define machines intelligent. The idea behind the test was that are machine (e.g. a computer) is then called intelligent, if a machine (A) and a person (B) communicate through natural language and a second person (C), a so-called elevator, can not detect which of the communicators (A or B) is the machine. The Dartmouth conference In 1956 there was probably the first workshop of Artificial Intelligence and with it the field of AI research was born. Researcher from Carnegie Mellon University (CMU), Massachusetts Institute of Technology (MIT) and employee from IBM met together and founded the AI research. In the following years they made huge process. Nearly everybody was very optimistic. Machines will be capable, within twenty years, of doing any work what man can do.” – Herbert A. Simon (CMU) “Within a generation … the problem of creating ‘artificer intelligence’ will substantially be solved” – Marvin Minsky (MIT) That was in the 1960s. The progress slowed down in the following years. Because of the failing recognizing of the difficulty of the tasks promises were broken. The first AI Winter Because of the over optimistic settings and the not occurred breakthroughs U.S. and British government cut of exploratory research in AI. The following years were called (first) AI Winter. The enthusiasm was lost, nobody wanted to fund AI research. The interest of publicity on Artificial Intelligence decreased. This was around 1974. Expert Systems After the first AI Winter, Artificial Intelligence came back in a form of so-called “expert systems”. Expert systems are programs that answer question and solve problems in a specific domain. They emulate an expert in a specific branch and solve problems by rules. There are two types of engines in expert systems: First, there is the knowledge engine, which represents facts and rules about a specific topic. Second, there is the inference engine, which applies the rules and facts from the knowledge engine to new facts. Synthesis of Integral Design In 1981 an expert system named SID (Synthesis of Integral Design) designed 93% of the VAX 9000 CPU logic gates. The SID system was existing out of 1,000 hand-written-rules. The final design of the CPU took 3 hours to calculate and outperformed in many ways the human experts. As an example, the SID produced a faster 64-bit adder than the manually designed one. Also the bug per gate rate, which where around 1 bug per 200 gates from human experts, was much lower at around 1 bug per 20,000 gates at the final result of the SID system. The second AI Winter The second AI Winter came in the later 80s and early 90s after a series of financial setbacks. The fall of expert systems and hardware companies who suffered through desktop computers built by Apple and IBM led again to decreasing AI interest, on the one hand side from publicity and on the other side from investors. Deep Blue After many ups and downs Deep Blue became the first chess computer to beat a world chess champions, Garry Kasparov. On 11 May 1997 IBM’s chess computer defeated Garry Kasparov after six games with 3½–2½. Deep Blue used tree search to calculate up to a maximum of 20 possible moves. It evaluated positions by a value function mainly written by hand, which was later optimized by analyzing thousand of games. Deep Blue also contained an opening and endgame library of many grandmaster games. In 1997 DeepBlue was the 259th most powerful supercomputer with 11.38 GFLOPS. In comprising: The most powerful supercomputer in 1997 had 1,068 GFLOPS and today (December 2017) the most powerful supercomputer has 93,015 GFLOPS. FLOPS stand for floating-point operations per second and the ‘G’ in GFLOPS stands for Giga. So the equivalent of 1 GFLOPS are 10⁹ FLOPS. 21st Century: Deep learning, Big Data and Artificial General Intelligence In the last two decades, Artificial intelligence grow heavily. The AI market (hardware and software) has reached $8 billion in 2017 and the research firm IDC (International Data Corporation) predicts that the market will be$47 billion by 2020. This all is possible through Big data, faster computers and advancements in machine learning techniques in the last years. With the usage of Neural Networks complicated tasks like video processing, text analysis and speech recognition can be tackled now and the solutions which are already existing will become better in the next years. Atari Games In 2013 DeepMind, one of the world’s foremost AI research, introduced an AI which could play a couple of Atari games on top of a level of human players. This first seems not very expressive, but they just used reinforcement learning and neural networks to let the AI self learn these games. Also they just used the pixels as an input to the agent, so there was no direct reward score given to the agent depending on the moves he did. In 2015 they further introduced a smarter agent, who successfully played 49 classic Atari games by itself. Next to classic games from old retro consoles DeepMind is developing an AI for more complex game, like e.g. Starcraft 2. Starcraft 2 is a Real Time Strategy (RTS) game, which is the most popular 1 vs.1 E-Sport title. Starcraft 2 is very popular in South Korea and the best Starcraft 2 pro player come from South Korea. Nevertheless there are many European and North American pro player who play for living. Starcraft 2 is a much more complex game than classic video games: There are much more possible actions you can do, you do not know everything about your opponent and you have to scout him to explore what he is doing. In Starcraft 2 there are also dozens of strategy decision to choose from every minute and in general much more to care about comparing to classic video games. The current AI is not very good at the moment and it only can play mini games like building units. About the Starcraft 2 AI I am very exciting about, because I am a big Starcraft 2 fan and I am exciting about how the AI will change the Starcraft 2 meta game and what new tactics it will explore. AlphaGo Next to classic Atari games, DeepMind also managed to defeat the world best human Go player with his AI AlphaGo. In October 2015 they first defeated the European Go champion Fan Hui five to zero. After the match there was a lot of skeptics in the Go scene about AlphaGo, because Fan Hui is ‘only’ an 2-dan (out of 9-dan, which is best) European Champion. Therefore the DeepMind team flew to South Korean to face Lee Sedol, a 9-dan Go Player. Lee Sedol is known as one of the best Go players in the world. After DeepMind managed to win the first 3 matches Lee Sedol seemed very desperate. But in the fourth game AlphaGo lost after it made an obvious mistake. In the last match AlphaGo could win again. In the end AlphaGo managed to win with 4-1 against Lee Sedol. If you are more interested in the story about AlphaGo I recommend the movie about it. In my opinion the movie is great and shows, next to the technical impact of the AI, the impact on the Go community. In 2017 DeepMind published the next generation of AlphaGo. AlphaGo Zero is build up on reduced hardware and just learned Go to play against itself. After three days of training AlphaGo Zero was stronger than the version of AlphaGo who defeated Lee Sedol and won against his younger version with 100-0. After 40 days of training it also defeated his former version of AlphaGo Zero. What is coming next? Well nobody knows and I also do not know. This answer is probably very unsatisfying, therefore let’s talk about possible scenarios. Artificial General Intelligence, Superintelligence – AI takeover? Until now there has been also some Artificial Intelligence systems who were specialists, e.g. AlphaGo (Zero), who has mastered Go and could outperform human Grandmaster player. But there is no Artificial General Intelligence (AGI) yet. AGI refers to an Artificial Intelligence, who could successfully perform any intellectual task that a human being can. The Turing test is one example to test if an AI is an AGI. Superintelligence takes AGI to the next stage. superintelligence refers to an AGI, who is smarter than a human and can outperform a human in any intellectual task that a human being can. The question, when there will be AGI and when there will be superintelligence, is of course difficult to answer (as nearly all answers who refers to future prediction). Nick Bostrom, who is a AI research on Oxford University, evaluates this question in a paper. The AI takeover refers to the scenario, that a superintelligence will take over the world and will fight against the human race. It is a popular science fiction theme and it is started to use 100 years ago in R.U.R. and still being used in more modern movies like Terminator or Matrix. So is it only fiction or is it possible, that AI can take over the world one day? I am not 100% sure about it. I favor more the opinion about that there will no AI takeover. But there are also many people like Elon Musk, Stephen Hawking and Bill Gates who are concerned about possible the possibilities of AI. Therefore I think we should not call AI takeover as a pure fiction scenario, we should be aware of it and act responsible. If you are interested in the topic of AGI and Superintelligence, I recommend you the read the book by Nick Bostrom “Superintelligence – Path, Dangers, Strategies”. Conclusion There are already Artificial Intelligence systems who can outperform humans in specifics areas, like e.g. playing GO or data analysis. Today, if we talk about Artificial Intelligence systems in production we refer to specialists. But there in no Artificial General Intelligence (AGI) yet, who can perform like a human, and neither there is a superintelligence, who is smarter than a human being.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23565258085727692, "perplexity": 1911.142532136039}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358064.34/warc/CC-MAIN-20210227024823-20210227054823-00196.warc.gz"}
http://mathoverflow.net/questions/132842/spectral-decomposition-action-on-the-unitary-group	# “Spectral decomposition” action on the unitary group Consider a matrix $U$ from the unitary group $U_N(\mathbb{C})$ and consider the map $f:U_N(\mathbb{C})\rightarrow U_N(\mathbb{C})$ where $f(U)$ is the matrix of the eigenvectors of $U$. What is known for the orbits of this map, i.e. $(f^n(U):\;n\in\mathbb N)$ ? And what can we say about the eigenvalues of each $f^n(U)$ ? One observation is that if you endow $U_N(\mathbb{C})$ with its Haar measure, then its image by $f$ is still Haar distributed : it kind of preserves disorder. An other observation is that if you embed $U_{N-1}(\mathbb{C})$ in an obvious way in $U_N(\mathbb{C})$, then $f$ preserves that subgroup. I guess this has been studied somewhere, but I'm not able to find any reference (but maybe it is just that I don't know a proper name for it). - Your map is not well-defined. Even if $U$ has distinct eigenvalues we are still free to multiply the eigenvectors by complex numbers of absolute value one, or to permute them. There is even more freedom if eigenvalues are repeated. Different choices of $f(U)$ will give completely unrelated answers for $f^2(U)$. – Neil Strickland Jun 5 '13 at 21:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9767270088195801, "perplexity": 134.38891181275605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535924131.19/warc/CC-MAIN-20140901014524-00122-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/special-relativity-problem-with-solving-about-finding-velocity-in-a-reference-frame.662247/	# Special Relativity: problem with solving about finding velocity in a reference frame 1. Jan 3, 2013 1. The problem statement, all variables and given/known data A pion in its rest frame decays into a muon and a neutrino. Find the velocity of the muon and its mean-lifetime in the pion rest frame. (I've done this part). The muon decays into an electron and two neutrinos. If the two neutrinos happen to travel in the same direction in the rest frame of the muon, find the velocity of the resulting electron in both the rest frame of the muon (done this as well) and in the rest frame of the original pion. Hence show that the maximum velocity of the electron in the pion's rest-frame satisfies: $\frac{U}{c}=\frac{\mu^{2}-m^{2}}{\mu^{2}+m^{2}}$ where $\mu$ is the pion's rest-mass, M is the muon's rest-mass, and m is the electron's rest-mass neutrino's are massless Hint: You will need to appreciate that the relative orientation of the muon velocity to the electron velocity is an issue. 2. Relevant equations relative mass: ie m(v) = γ*m(0) 3. The attempt at a solution For the parts I've done: I used the conversation of mass and momentum to get two equations for the first split (pion -> muon+neutrino), while the I assign (-p) for the momentum of the neutrino and so (p/c) for its mass. So I get (eventually after some algebra..): $\frac{V}{c}=\frac{\mu^{2}-M^{2}}{\mu^{2}+M^{2}}$ which is the first bit done. I also get the velocity of the electron in the muon's rest-frame: $\frac{V}{c}=\frac{M^{2}-m^{2}}{M^{2}+m^{2}}$ But when I attempt to use the velocity addition rule, to find this velocity in the rest frame of the pion, I don't quite get the answer..it's messy.. And I don't understand where the "maximum" velocity of electron comes in! I also don't get the meaning of the hint. Thankyou for taking time to read.. Last edited: Jan 3, 2013 2. Jan 3, 2013 ### BruceW Re: Special Relativity: problem with solving about finding velocity in a reference fr These are correct. Darn straight it is complicated. This is the part where you must use the hint. What answer are you getting? Also, what do you mean you don't quite get the answer? They don't provide you with the answer to this bit. Showing the maximum velocity of the electron in the pion's rest frame is actually easier than the previous part of the problem. Again, you must make use of the hint, and think of the case in which the electron has most energy (relative to the inertial frame in which the pion is at rest). Also, welcome to physicsforums :) 3. Jan 3, 2013 Re: Special Relativity: problem with solving about finding velocity in a reference fr Thank you for welcoming me! .. and helping.. So I've managed to simplify the mess: I get the velocity of the electron in the pion's rest frame, U to be: $\frac{U}{c}=\frac{\mu^{2}(M^{2}-m^{2})}{M^{2}(\mu^{2}+m^{2})}$ Do you think this is correct? I'm thinking on what you said about the energy.. ..although the above expression equals the required one in the question when $\mu=M$ 4. Jan 3, 2013 Re: Special Relativity: problem with solving about finding velocity in a reference fr I'm really not sure whether there should be an angle involved, since the hint mentions "orientation"? 5. Jan 3, 2013 ### BruceW Re: Special Relativity: problem with solving about finding velocity in a reference fr No, that's not right. You need to take into consideration "relative orientation of the muon velocity to the electron velocity." They want us to find the velocity of the electron according to the (original) pion rest frame, right? So think through what happens according to this frame. The pion decays into a muon going in one direction (let's call that the z axis), and a neutrino in the opposite direction. Then that muon decays into an electron and a couple of neutrinos. Now is there any reason why the electron would come out from the decay going in the z direction as well? And from this answer, will an angle be involved? Edit: This is in reply to post #3, not post 4 6. Jan 3, 2013 Re: Special Relativity: problem with solving about finding velocity in a reference fr Ah yes it will be Thanks, I'll get back to you after trying to maths it out.. 7. Jan 3, 2013 Re: Special Relativity: problem with solving about finding velocity in a reference fr Thanks man! You're a life saver! I assumed an angle of alpha to the Z axis in pions frame. So when adding the velocities (using the rule...) take the component of velocity in muon's frame parallel to the Z axis. So when I got an "unsimplifyable" expression, I put cos(alpha)=1 and it worked out exactly as expected. Once again, thank you. 8. Jan 4, 2013 ### BruceW Re: Special Relativity: problem with solving about finding velocity in a reference fr Similar Discussions: Special Relativity: problem with solving about finding velocity in a reference frame	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8443760871887207, "perplexity": 749.3911589179456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815318.53/warc/CC-MAIN-20180224033332-20180224053332-00715.warc.gz"}
http://mathhelpforum.com/statistics/65008-standard-deviation.html	1. ## Standard Deviation A study of the weights of the brains of Swedish men found that the weight X was a random variable with mean 1400 grams and standard deviation 20 grams. Find positive numbers a and b such that Y = a + bX has mean 0 and a standard deviation 1. 2. Originally Posted by DINOCALC09 A study of the weights of the brains of Swedish men found that the weight X was a random variable with mean 1400 grams and standard deviation 20 grams. Find positive numbers a and b such that Y = a + bX has mean 0 and a standard deviation 1. Use the following formulae: E(Y) = a + b E(X) Var(Y) = b^2 Var(X)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.892662525177002, "perplexity": 404.4282827440009}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281649.59/warc/CC-MAIN-20170116095121-00135-ip-10-171-10-70.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/934160/proving-a-limit-with-epsilon-delta-definition	# Proving a limit with epsilon delta definition I am in honors Calculus I and my teacher is really stressing this limit proof. I understand the examples she goes over in class but she gave us a problem for home work and i just dont know how to start it. I appreciate any help! $$\lim_{x\to 3} \frac{2}{x+1} =\frac12$$ • That is supposed to say lim as x approaches 3 – Jacob Culleny Sep 16 '14 at 20:36 • Isn't there any example that looks familiar? How about obtaining an expression for $\frac{2}{x+1} - \frac{1}{2}$? – imranfat Sep 16 '14 at 20:37 • Its how my teacher gave it to us. It just means the limit value is 3. Heres the step im stuck at: f(x)=L+- ε 2=(1/2+-ε)x + 1 x= 3+- 2/ε I think i have to define values for x(0) and x(1) – Jacob Culleny Sep 16 '14 at 20:44 • check this out: math.stackexchange.com/questions/65667/… – imranfat Sep 16 '14 at 20:54 • You also might find this helpful: math.stackexchange.com/questions/930576/… – user84413 Sep 16 '14 at 21:20 Hint: To get started, you need to work with the inequality $\displaystyle\left\|\frac{2}{x+1}-\frac{1}{2}\right\|<\epsilon$. We can rewrite this as $\displaystyle\left\|\frac{4-(x+1)}{2(x+1)}\right\|<\epsilon$, which gives $\displaystyle\left\|\frac{3-x}{2(x+1)}\right\|<\epsilon$ or, equivalently, $\displaystyle\frac{\left\|x-3\right\|}{2\|x+1\|}<\epsilon$. Now we need to get an upper bound for the factor $\frac{1}{\|x+1\|}$, so one way to do this is to assume that our value of $\delta \le 1$. Under this assumption, $0<\|x-3\|<\delta\implies\|x-3\|<1\implies 2<x<4\implies3<x+1<5\implies$ $\;\;\;\displaystyle\frac{1}{3}>\frac{1}{x+1}>\frac{1}{5}\implies \frac{1}{\left\|x+1\right\|}<\frac{1}{3}$. Now you need to find a $\delta>0$ which satisfies $\delta\le1$, and $\displaystyle0<\|x-3\|<\delta\implies\frac{\|x-3\|}{2\|x+1\|}=\frac{\|x-3\|}{2}\frac{1}{\|x+1\|}<\epsilon$. • Thank you so much for taking the time to help me!! – Jacob Culleny Sep 17 '14 at 1:54 • You're welcome. (The actual proof would start something like this: Let $\epsilon>0$ be given, and let $\delta=\cdots$.) – user84413 Sep 17 '14 at 20:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8383021354675293, "perplexity": 363.4444269563965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986673250.23/warc/CC-MAIN-20191017073050-20191017100550-00027.warc.gz"}
https://quant.stackexchange.com/questions/10763/use-of-girsanovs-theorem-in-bond-pricing/10789	# Use of Girsanov's theorem in bond pricing Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma dB_t=b(r_t)dt+\sigma dB_t$. I was shown that one could write the price as $B(0,T) = E_{\hat{P}}[\exp(-\int_0^TB^{}_sds)\exp(\int_0^Tb(B^{}_s)dB^{}s-\frac{1}{2}\int_0^Tb^2(B^{}_s)ds)]$ where $B_t^{}$ is the "new" Brownian motion from the Girsanov theorem. However, when I try to implement it, it results in prices that are too low. Here is what I did: Since $dr_t=b(r_t)dt+\sigma dB_t$, Girsanov's theorem gives a new Brownian motion with dynamics $dB_t^{}=\frac{1}{\sigma}b(r_t)dt+dB_t$, and the dynamics of $r_t$ becomes $dr_t=\sigma dB_t^{}$. So $r_t=r_0+\sigma B_t^{}$ and $dB_t^{}= \frac{1}{\sigma}b(r_0+\sigma B_t^{})dt+dB_t$. With this last expression I tried to use Euler discretization to find $B_t^{}$, then finally I approximated the three integrals as sums. What am I doing wrong? Secondly, I also wonder what the starting point of $B_t^{}$ should be, i.e. $B_0^{}$. • it could help if you worked with the explicit solution of $r_t$ - do you know it or shall I post it here ? – Probilitator Mar 31 '14 at 20:12 • @Probilitator: Thanks. I'm not 100% sure how to do it, so I would really appreciate it if you could post it! – DSilva21 Apr 1 '14 at 12:03 • how do you know that your prices are too low ?- what is your benchmark ? – Probilitator Apr 1 '14 at 12:37 • As a start I am only using $b(r_t)=k(\theta-r_t)$ i.e. Vasicek, which I am comparing to the ZCB bond prices computed the usual way. The idea is to extend $b(r_t)$ to incorporate regime-switching, but first I want to make sure I have simulated $B_t^{}$ correctly. Now that I am running the simulations again, I see that the results are very unstable, resulting in both higher and lower prices than the results I am comparing them to. – DSilva21 Apr 1 '14 at 12:52 • by the way do you have a source for the bond dynamics ? Also note that in Vasicek Model one rarely uses monte carlo to price anything - in most cases pricing is done on the tree. – Probilitator Apr 2 '14 at 11:29 ## Bond Price Dynamics I do not know the source of the bond dynamics you show above but seeing how we are dealing with an affine model there is a very elegant way to derive those. Due to the model being affine the bond price is given by $$P(t,T)=A(t,T)e^{-r(t)B(t,T)}$$ you can find the exact formulas for $A(t,T)$ and $B(t,T)$ in this document (or just read the relvant chapter in Brigo Mercurio. If you are curious to know how one arrives at above pricing formula I suggest this paper. Now that we know above closed form formua, getting the dynamics of $P(t,T)$ is just a matter of applying Itô and using some algebra ## Getting a solution for $r_t$ The generalized Vasicek model is given by $$dr_t=\kappa(t)(\theta(t)-r_t)d t+\sigma(t)dW_t$$ The unique solution is $$r_t=A^{-1}(t)\left[r_0+\int_0^tA(s)\kappa(s)\theta(s)ds+\int_0^t A(s)\sigma(s)dW_s\right]$$ with $A(t)=\exp\left(\int_0^t\kappa(s)ds\right)$ Seeing how you have constant and not time dependant parameters the above simplifies to $$r_t=e^{-\kappa t}\left[r_0+\int_0^t\kappa\theta e^{\kappa s} ds+\int_0^t \sigma e^{\kappa s} dW_s\right]$$ P.S.: To arrive at the above solution one simply applies Ito's Formula to $d(r_t \cdot e^{\int_0^t\kappa(s)ds})$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8769450783729553, "perplexity": 309.9841477179068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540502120.37/warc/CC-MAIN-20191207210620-20191207234620-00084.warc.gz"}
https://mathoverflow.net/questions/20789/approximate-a-probability-distribution-by-moment-matching	# approximate a probability distribution by moment matching Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ up to the $m^{\rm th}$ order: (1) $\mathbb{E}[X^k] = \mathbb{E}[Z^k]$ for $k = 1, \ldots m$. Here is a positive result, which is a simple consequence of convex analysis (Caratheodory's theorem): there exists $Z$ with at most $m+1$ atoms such that (1) holds. Here are my questions: 1) Is there a converse result about this? Say $X$ has an absolutely continuous distribution supported on $\mathbb{R}$ (e.g. Gaussian). When $m$ is large, given that $Z$ has only $m$ atoms, can we conclude that we cannot approximate all $2m$ moments of $X$ well, i.e., can we lower bound the error $\max_{1 \leq k \leq 2m}\|\mathbb{E}[X^k] - \mathbb{E}[Z^k]\|$? My intuition is the following: for a Gaussian $X$, $\mathbb{E}[X^k]$ grows like $k^{\frac{k}{2}}$ superexponentially. When we find a $Z$ who matches all moments of $X$ up to $m$, it cannot catch up with higher-order moments $X$; if $Z$ matches all moments from $m+1$ up to $2m$, then its low-order moments will be quite different from $X$. 2) Is there an efficient algorithm to compute the location and weights of the approximating discrete distribution? Does there exist a table to record these for approximating common distribution (e.g. Gaussian) for each fixed $m$? It could be very handy... 3) I heard from folklore that when (1) holds, the total variation distance between their distributions can be upper bounded by, say, $e^{-m}$ or $1/m!$. Of course, this won't be true for a discrete $Z$. But let's say $X$ and $Z$ both has smooth and bounded density on $\mathbb{R}$. Could this be true? Now two characteristic functions matches at $0$ up to $m^{\rm th}$ derivatives. They should be pretty close? - For (1) and (2) just forget about probability and recall everything you ever learned about orthogonal polynomials and the Gauss quadrature formulae. 3) is false as stated: there are plenty of Schwartz functions orthogonal to all polynomials, so you can have all moments coincide and still have a large distance (in any sense). Something like that may be true but I cannot think of any good formulation right away. - thanks fedja! I understand (2) now: just place the atoms at the roots of the $m^{\rm th}$-order orthogonal polynomial. For the converse in (1), can you elaborate a bit more please? I haven't been able to see how to lower bound the approximate error for any choice of locations and weights. – mr.gondolier Apr 10 '10 at 0:04 Sure. If the nodes are $x_j$, integrate $\prod_j(x-x_j)^2$ which is a polynomial of degree $2m$. You get something strictly positive showing that the quadrature formula cannot hold exactly. You can also estimate the integral if you know the weight thus getting some lower bound. – fedja Apr 10 '10 at 3:44 hi fedja, I recalled a theorem about Gauss quadrature states that placing the nodes at the roots of the orth. poly. $p_m$ approximates the integral of all poly. up to order $2m-1$ exactly but there exists poly. of order $2m$ (e.g. the one you gave, $p_m^2$) that cannot be. Therefore the error of approximating $x^{2m}$ is the same as approximating $p_m^2$. However, this gave the approximation error of one particular method. My original question is about the worst case error of approximating $x^i$ for $i=1,\ldots, 2m$ regardless of the node locations. Is it optimal to put the nodes at the roots? – mr.gondolier Apr 16 '10 at 17:58 Depends on what you mean by optimal. How exactly do you measure "the worst case error" size? – fedja Apr 16 '10 at 19:17 The error is measured either by $e_{2m} = \max_{k = 1, \ldots, 2m} \|\mathbb{E}[X^k]-\mathbb{E}[Z^k]\|$ (worst case) or $\sum_{k = 1}^{2m} \|\mathbb{E}[X^k]-\mathbb{E}[Z^k]\|^2$ (square error). Since I am only interested in the asymptotics when $m$ is large, these two do not differ much. If only $m$ nodes (in $Z$) are allowed, is it optimal to place them at the roots to minimize $e_{2m}$? Let's say $X$ is standard normal, then $\mathbb{E}[\|X\|^k]$ grows very fast like $k^{\frac{k}{2}}$. Moments of a discrete $Z$ cannot follow this growth. This is my intuition of lower bounding $e_{2m}$. – mr.gondolier Apr 17 '10 at 8:43 In the context of 3), what I have heard from folklore is that when (1) holds, the Kolmogorov distance (not total variation) is bounded by something like $1/\sqrt{m}$. This bound follows if (1) holds only approximately, and exact equality in (1) suggests that a much stronger bound holds but does not formally imply it, even under the assumption of smooth bounded densities. Under exactly what conditions? I gave an example of two different probability distributions with densities in $S$ and exactly the same moments of all order. You need some fairly strong decay of tails to get any bound at all. Are you talking about compactly supported distributions? – fedja Apr 10 '10 at 3:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9700586199760437, "perplexity": 246.22897074773738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394020703260/warc/CC-MAIN-20140305115823-00087-ip-10-183-142-35.ec2.internal.warc.gz"}
http://math.stackexchange.com/users/4204/auguste-hoang-duc	# Auguste Hoang Duc less info reputation 15 bio website location age member for 3 years, 3 months seen Jan 17 at 17:53 profile views 68 5 Multiplicative norm on $\mathbb{R}[X]$. 3 sequential convergence and continuity 2 If every continuous $f:X\to X$ has $\text{Fix}(f)\subseteq X$ closed, must $X$ be Hausdorff? 1 Continuous surjective functions from the unit disk to itself that agree nowhere 1 Why do we use 'this' Gamma Function. # 353 Reputation +5 How to calculate the asymptotic expansion of $\sum \sqrt{k}$? +10 Continuous surjective functions from the unit disk to itself that agree nowhere +10 Multiplicative norm on $\mathbb{R}[X]$. +10 sequential convergence and continuity # 2 Questions 8 How to extend Galois character? 3 How to calculate the asymptotic expansion of $\sum \sqrt{k}$? # 16 Tags 6 general-topology × 3 5 inner-product-space 6 linear-algebra × 2 3 homework 5 norm 2 examples-counterexamples 5 polynomials 1 representation-theory 5 normed-spaces 1 gamma-function # 2 Accounts Mathematics 353 rep 15 MathOverflow 320 rep 1211	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8722691535949707, "perplexity": 2836.684203056676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999657009/warc/CC-MAIN-20140305060737-00045-ip-10-183-142-35.ec2.internal.warc.gz"}
https://arxiv.org/list/gr-qc/1409	# General Relativity and Quantum Cosmology ## Authors and titles for Sep 2014 [ total of 356 entries: 1-25 \| 26-50 \| 51-75 \| 76-100 \| ... \| 351-356 ] [ showing 25 entries per page: fewer \| more \| all ] [1] Title: Exploring New Physics Frontiers Through Numerical Relativity Comments: 156 pages, 21 figures. Published in Living Reviews in Relativity Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Astrophysical Phenomena (astro-ph.HE); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th) [2] Title: The Conformal Flow of Metrics and the General Penrose Inequality Subjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Differential Geometry (math.DG) [3] Title: A Shape Dynamics Tutorial Authors: Flavio Mercati Comments: v1.0, 71 pages, two columns in landscape format, for optimal reading on a computer screen or on a short-sided binding printout Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th) [4] Title: Hawking Evaporation is Inconsistent with a Classical Event Horizon at $r=2M$ Authors: Borun D. Chowdhury (1), Lawrence M. Krauss (1,2) ((1) Arizona State University, (2) Australian National University) Comments: 4 pages, 2 figs, submitted to Phys. Rev. Lett; revised in response to referees reports Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th) [5] Title: Dark matter in ghost-free bigravity theory: From a galaxy scale to the universe Journal-ref: Phys. Rev. D 90, 124089 (2014) Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Physics - Theory (hep-th) [6] Title: Non-embeddable relational configurations Authors: William Edwards Subjects: General Relativity and Quantum Cosmology (gr-qc) [7] Title: Constraining the Cardoso-Pani-Rico metric with future observations of SgrA$^*$ Authors: Cosimo Bambi Comments: 8 pages, 3 figures. v2: refereed version Journal-ref: Class.Quant.Grav.32:065005,2015 Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Astrophysical Phenomena (astro-ph.HE) [8] Title: Discrete Newtonian Cosmology: Perturbations Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO) [9] Title: Electrically charged matter in rigid rotation around magnetized black hole Journal-ref: Physical Review D 90, 044029 (2014) Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Astrophysical Phenomena (astro-ph.HE) [10] Title: On tidal capture of primordial black holes by neutron stars Journal-ref: Phys. Rev. D 90, 103522 (2014) Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO) [11] Comments: MSc. thesis submitted to ODTU. Table of contents entries referring to subsections of the index are added (The only addition to the library version). v2: more comprehensive, visually attractive v3: Published as a book by LAP LAMBERT Academic Publishing (December 20, 2016) Subjects: General Relativity and Quantum Cosmology (gr-qc) [12] Title: Genericity aspects of black hole formation in the collapse of spherically symmetric slightly inhomogeneous perfect fluids Comments: 8 pages, no figures, published version Journal-ref: Int. J. Mod. Phys. D 25, 1650023 (2016) Subjects: General Relativity and Quantum Cosmology (gr-qc) [13] Title: The Loudest Gravitational Wave Events Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Astrophysical Phenomena (astro-ph.HE) [14] Title: Dynamic conformal spherically symmetric solutions in an accelerated background Comments: Accepted by Rom. Rep. Phys Journal-ref: Romanian Reports in Physics, Vol. 68, No. 4, P. 1382--1396 (2016) Subjects: General Relativity and Quantum Cosmology (gr-qc) [15] Title: On the nonlinear instability of confined geometries Comments: 14 pages, 9 figures; v2:Some improvements in convergence results, accepted for publication in Physical Review D Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th) [16] Title: Prospects for studies of the free fall and gravitational quantum states of antimatter Comments: This work reviews contributions made at the GRANIT 2014 workshop on prospects for the observation of the free fall and gravitational quantum states of antimatter Journal-ref: Advances in High Energy Physics, vol. 2015, article ID 379642 (2015) Subjects: General Relativity and Quantum Cosmology (gr-qc); Atomic Physics (physics.atom-ph); Quantum Physics (quant-ph) [17] Title: $σ$CDM coupled to radiation. Dark energy and Universe acceleration Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO) [18] Title: Breaking the cosmological background degeneracy by two-fluid perturbations in f(R) gravity Authors: Amare Abebe Journal-ref: Int. J. Mod. Phys. D Vol. 24, No. 7 (2015) 1550053 Subjects: General Relativity and Quantum Cosmology (gr-qc) [19] Title: An improved derivation of minimum information quantum gravity Comments: 21 pages. v2: more details in the proof of the space-time structure; text improved; some parts transfered to appendix, added by summaries of some previously published concepts, a few formulations more comprehensive. Prepared for submission to Annals of Physics. v3: Minor revision (boundary law: "non-thin-layers", clean notation; Legendre transf.: tetrad bath; proof local trivializ. on subsystems) Subjects: General Relativity and Quantum Cosmology (gr-qc) [20] Title: The black hole challenge in Randall-Sundrum II model Subjects: General Relativity and Quantum Cosmology (gr-qc) [21] Title: A note on the secondary simplicity constraints in loop quantum gravity Comments: 20 pages. v2 many improvements: slightly different Hamiltonian, more details on the canonical analysis and on the extension to arbitrary graphs, new section added. Main conclusion unchanged, but made more precise Subjects: General Relativity and Quantum Cosmology (gr-qc) [22] Title: Stability and Binding Energy of Small Asymptotically Randall-Sundrum Black Holes Comments: 12 pages, 7 figures, REVTeX 4.1; added new section VII; matches published version. This paper draws on arXiv:1408.4425 Journal-ref: Phys. Rev. D 92, 024032 (2015) Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th) [23] Title: Identification of a gravitational arrow of time Comments: To appear in Physical Review Letters Journal-ref: Phys.Rev.Lett.113:181101,2014 Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th) [24] Title: The Entropy of Higher Dimensional Nonrotating Isolated Horizons from Loop Quantum Gravity Subjects: General Relativity and Quantum Cosmology (gr-qc) [25] Title: Limits on a Gravitational Field Dependence of the Proton--Electron Mass Ratio from H$_2$ in White Dwarf Stars Authors: Julija Bagdonaite (1), Edcel J. Salumbides (1,2), Simon P. Preval (3), Martin A. Barstow (3), John D. Barrow (4), Michael T. Murphy (5), Wim Ubachs (1) ((1) VU University Amsterdam, Netherlands (2) University of San Carlos, Philippines (3) University of Leicester, UK (4) DAMTP, UK (5) Swinburne University of Technology, Australia) Comments: 5 pages, 3 figures. Accepted by PRL Journal-ref: Phys. Rev. Lett. 113, 123002 (2014) Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO) [ total of 356 entries: 1-25 \| 26-50 \| 51-75 \| 76-100 \| ... \| 351-356 ] [ showing 25 entries per page: fewer \| more \| all ] Disable MathJax (What is MathJax?)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17336632311344147, "perplexity": 6111.638014321525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123549.87/warc/CC-MAIN-20170423031203-00301-ip-10-145-167-34.ec2.internal.warc.gz"}
https://infoscience.epfl.ch/record/188981?ln=en	## Electrical conductivity of multi-walled carbon nanotubes-SU8 epoxy composites We have characterized the electrical conductivity of the composite which consists of multi-walled carbon nanotubes dispersed in SU8 epoxy resin. Depending on the processing conditions of the epoxy (ranging from non-polymerized to cross-linked), we obtained tunneling and percolating-like regimes of the electrical conductivity of the composites. We interpret the observed qualitative change of the conductivity behavior in terms of reduced separation between the nanotubes induced by polymerization of the epoxy matrix. (C) 2013 AIP Publishing LLC. Published in: Applied Physics Letters, 102, 22 Year: 2013 Publisher: Melville, Amer Inst Physics ISSN: 0003-6951 Laboratories:	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8742358684539795, "perplexity": 7982.5821286511655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347399820.9/warc/CC-MAIN-20200528135528-20200528165528-00591.warc.gz"}
https://www.physicsforums.com/threads/continuously-differentiable.282643/	# Continuously differentiable 1. Jan 3, 2009 ### johnson12 I'm having trouble with this inequality: let f be (real valued) continuously differentiable on [0,1] with f(0)=0, prove that sup$$_{x\in[0,1]}$$ $$\left\|f(x)\right\|$$ $$\leq$$ $$\int^{1}_{0}\left\|f\acute{}(x)\right\| dx$$ Thanks for any help. 2. Jan 3, 2009 ### jgens Clarifying questions: Do you intend to find the supremum of the function or the interval? Furthermore, is the second f(x) term an f(x) or f'(x)? (It looks like there is a little tick next to it but I cannot tell for sure) 3. Jan 3, 2009 ### johnson12 Sorry for the discrepancy, the problem is to show that the supremum of \|f(x)\| over [0,1] is less than or equal to the integral from 0 to 1 of \|f ' (x)\|, where f ' (x) is the derivative of f. 4. Jan 4, 2009 ### dirk_mec1 Look at this: $$\|f(t)\| = \left\|\int_0^t f'(x) \mbox{ d}x \right\| \leq \int_0^t \|f'(x)\|\ \mbox{ d}x \leq \int_0^1 \|f'(x)\|\ \mbox{ d}x$$ Can you finish this?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9735912680625916, "perplexity": 1200.4281370599851}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281069.89/warc/CC-MAIN-20170116095121-00257-ip-10-171-10-70.ec2.internal.warc.gz"}
https://infoscience.epfl.ch/record/32827	Infoscience Thesis Ozone and water vapor measurements by Raman lidar in the planetary boundary layer The temporal and spatial retrieve of ozone (O3) concentration and water vapor (H2O) mixing ratio in the troposphere is of essential interest. Contrary to the stratospheric case, the tropospheric ozone can have a harmful impact, with its toxic effect, on humans and vegetation, accelerating the degradation of the minerals and participating in the green-house problem. Concerning the water vapor, knowledge of its highly variable concentration is essential to both the chemistry of the troposphere (O(1D) + H2O —> 2 OH) where it participates, among others, in the generation of the hydroxyl radical (OH) and to the meteorology. Water vapor is the dominant green-house gas, it plays an important role in the atmospheric chemistry. The conversion and transport of water in the atmosphere is the essential point in the earth's radiation budget. Due to the complexity and the non-linearity of the air pollution system including emissions, chemistry, thermal radiation, transport and deposition, pollution abatement strategies can only be designed rightly by the use of a three-dimensional mesoscale Eulerian photochemical transport model. To check such models, measurement campaigns are undertaken, in which many physical (wind, temperature, H2O, etc.) and chemical parameters (emissions and imissions) are measured at different parts of the atmosphere. LIDAR (LIght Detection And Ranging), which is a real-time method for measuring air pollutants in situ, is one of the best tools to make 3-D measurements of gases concentrations like O3, H2O and others. Contrary to the ground based measurements that are highly sensitive to the very local conditions, lidar sensitivity and resolution in space and time is optimal to obtain measurements and to compare or give some input data for the models. During the last thirty years, (elastic backscatter) Differential Absorption Lidar (DIAL) has been established as a convenient tool for the monitoring of the three dimensional real time concentrations of air pollutants [Measures, 1992], [Schoulepnikoff et al., 1998]. But the DIAL apparatus has shown limitations: — the operation in layers with high aerosol loading like in the Planetary Boundary Layer (PBL) where they are highly variable — the simultaneous detection of several atmospheric components or pollutants is impossible [Bösenberg, 1996] — the detection at short range is difficult due to the high dynamics. Furthermore, due to its spectrum and the strong influence from other elements, the water vapor can not be easily measured in the UV with classical DIAL systems. The goal of this work was to develop a method to simultaneously measure the ozone absolute concentration and the water vapor mixing ratio in the PBL. Experiments with Nd : YAG and KrF lasers were made and utilization of both analog and photon counting techniques, increasing the dynamic range, were investigated. To retrieve the ozone concentration profile, we take advantage of the simultaneous spontaneous Raman backscattering on the molecules of nitrogen (N2) and oxygen (O2) that have different ozone absorption cross-sections. Thus with a modified DIAL technique, the ozone concentration can be measured without most of the interference from poorly known backscatter by particles. Water vapor mixing ratio profile can also be obtained with a set of three Raman backscattered signals, simultaneously detected, from the molecules of H2O, N2 and O2. The main advantage of this Raman system is its essential independence to the wavelength dependent backscatter problems as induced by aerosols, and the fact that the N2 and O2 concentrations are well known as well as the Raman cross-sections of interest. Although the Raman cross-sections are two or three orders of magnitude lower than the elastic backscattering cross-sections, they are compensated by the proportionally much higher concentrations of O2, N2 and H2O compared to trace gases like O3. The development of the Raman — DIAL method for atmospheric measurements in the PBL presents several challenges. One is the development of highly-sensitive lidar systems, in particular the optical receiver, the spectrometer and the signal acquisition for the Raman part of the Raman-DIAL system. Also the data processing procedure for simultaneous evaluation of the ozone and the water vapor profiles. Both of these challenges present a number of issues, theoretical and practical, that are investigated in the frame of this work. Thèse École polytechnique fédérale de Lausanne EPFL, n° 2351 (2001) Faculté de l'environnement naturel, architectural et construit Institut de génie de l'environnement Jury: Iouri Archinov, François Golay, René Monot, Valentin Simeonov, Hubert van den Bergh, Bernard Vittoz Public defense: 2001-3-16 Reference Record created on 2005-03-16, modified on 2016-08-08	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8376134037971497, "perplexity": 1943.38253987087}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463613738.67/warc/CC-MAIN-20170530011338-20170530031338-00373.warc.gz"}
https://kar.kent.ac.uk/view/divisions/13701=2F1.html	Skip to main content # Items where division is "Faculties > Social Sciences > School of Social Policy Sociology and Social Research > Personal Social Services Research Unit" Up a level Export as ASCII CitationBibTeXDublin CoreEP3 XMLEndNoteHTML CitationJSONMETSMultiline CSVMultiline CSV (Staff)Object IDsOpenURL ContextObjectRDF+N-TriplesRDF+N3RDF+XMLReferReference Manager Group by: Creator's name \| Item Type \| Date \| No Grouping Jump to: A \| B \| C \| D \| E \| F \| G \| H \| I \| J \| K \| L \| M \| N \| O \| P \| Q \| R \| S \| T \| V \| W \| Y Number of items at this level: 1636. ## A Acquilla, S., Challis, David J., Darton, Robin, Johnson, Lynne (1987) Report of the Census of Elderly Patients in Long-Stay Wards in Durham, May 1986. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27168) Acquilla, S., Challis, David J., Darton, Robin, Stone, M. (1987) Report of the Darlington Day Census, 1986. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27167) Acquilla, S., Challis, David J., Darton, Robin, Stone, M. (1987) Report of the Darlington Day Census, 1987. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27169) Adams, Catherine, Bauld, Linda, Judge, Ken F. (2000) Smoking Cessation Services: Early Experiences from Health Action Zones. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27170) Allan, Stephen (2019) Future simulation modelling: Workforce, Developing Adult Social Care Modelling Capability. In: ASCRU/Department of Health and Social Care Workshop, 24th September 2019, London. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78519) Allan, Stephen (2015) Implications of the Care Act 2014 on social care markets for older people. In: Curtis, Lesley A. and Burns, Amanda, eds. Unit Costs of Health and Social Care 2015. PSSRU, University of Kent, Canterbury, UK, pp. 6-10. ISBN 978-1-902671-96-3. (KAR id:77236) Preview Allan, Stephen (2018) Price and quality across English care homes: Evidence from a secret shopper survey, Report commissioned by the Live-in Care Hub, PSSRU Discussion Paper 2945, PSSRU: Canterbury. Discussion paper. PSSRU, Kent, UK (KAR id:76662) Preview Allan, Stephen (2017) Staffing and care home closures. In: Care Market Research Workshop, Department of Health and Social Care, 11 October 2017, London. (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:78526) Allan, Stephen (2018) The impact of staffing on care home closures. In: 5th International Conference on Evidence-based Policy in Long-term Care, 10-12 September 2018, Vienna, Austria. (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:78523) Allan, Stephen (2018) The impact of staffing on care home survival. N/A, 26 pp. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:77984) Allan, Stephen (2018) The relationship between domiciliary care provision and care home markets. In: Bringing Data to Life for Policy and Practice: The BLGDRC Conference 2018, 28 November 2018, London. (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:78528) Allan, Stephen, Forder, Julien (2015) The determinants of care home closure. Health Economics, 24 (S1). pp. 132-145. ISSN 1057-9230. E-ISSN 1099-1050. (doi:10.1002/hec.3149) (KAR id:46279) Preview Allan, Stephen and Forder, Julien E. (2012) Care Markets in England: Lessons from Research. Discussion paper. PSSRU, Canterbury (KAR id:76667) Preview Allan, Stephen, Forder, Julien E. (2012) Competition in the English nursing homes market. In: International Conference on Evidence-based Policy in Long-term Care 2012, 5-8 September 2012, London School of Economics and Political Science, London. (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:78648) Allan, Stephen, Forder, Julien E. (2012) Discussion of Dormont, B and Martin, C., Quality of service and cost-efficiency of French nursing homes. In: 3rd joint CES-HESG meeting on health economics, 11-13 January 2012, Inserm-IRD & Aix-Marseille Université, Aix-en Provence, France. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:78647) Allan, Stephen, Forder, Julien E. (2013) Estimating Self-funder prices by Local Authorities. N/A (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:77237) Allan, Stephen, Forder, Julien E. (2013) The determinants of closure in the English care homes market. In: Health Economists’ Study Group Summer 2013 Conference, 26-28 June 2013, University of Warwick, UK. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78646) Allan, Stephen, Gousia, Katerina, Forder, Julien (2020) Exploring differences between private and public prices in the English care homes market. Health Economics, Policy and Law, . ISSN 1744-1331. E-ISSN 1744-134X. (doi:10.1017/S1744133120000018) (KAR id:79532) Preview Allan, Stephen, Gousia, Katerina, Forder, Julien E. (2016) Assessing the impact of Care Act 2014 funding reforms on the English care homes market. N/A (Submitted) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:77235) Allan, Stephen, Gousia, Katerina, Forder, Julien E. (2014) Assessing the impact of Care Act 2014 funding reforms on the English care homes market. In: Care Act Market Impact Event, Department of Health and Social Care, 12 November 2014, London. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78555) Allan, Stephen and Gousia, Katerina and Forder, Julien E. (2017) Explaining the fees gap between funding types in the English care homes market. Working paper. University of Kent (KAR id:63689) Preview Allan, Stephen, Gousia, Katerina, Forder, Julien E. (2014) Explaining the fees gap between funding types in the English care homes market. In: International Conference on Evidence-based Policy in Long-term Care 2014, 31 August - 3 September 2014, London School of Economics and Political Science, London. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78556) Allan, Stephen, Gridley, Kate, Roland, Daniel, Malisauskaite, Gintare, Jones, Karen C., Birks, Yvonne (2018) The influence of social care on delayed transfer of care (DTOC) among older people. In: Adult Social Care: What's the Evidence 2018? RiPfA/ADASS Seminar, 2 February 2018, London. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78525) Allan, Stephen, Vadean, Florin (2019) The Association Between Staff Retention and English Care Home Quality. Journal of Aging and Social Policy, . ISSN 0895-9420. (In press) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:75795) Allan, Stephen and Vadean, Florin (2017) The impact of workforce composition and characteristics on English care home quality. Working paper. Personal Social Services Research Unit, Canterbury, Kent, UK (Unpublished) (KAR id:63688) Preview Allan, Stephen and Vadean, Florin (2017) The impact of workforce composition and characteristics on English care home quality. Discussion paper. PSSRU, Canterbury (KAR id:76664) Preview Allan, Stephen, Vadean, Florin (2016) The impact of workforce composition and characteristics on English care home quality. In: International Conference on Evidence-based Policy in Long-term Care 2016, 4-7 September 2016, London School of Economics and Political Science, London. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78553) Allen, Caroline (1990) The costs of electrotherapy. Physiotherapy, 76 (11). pp. 680-682. ISSN 0031-9406. (doi:10.1016/S0031-9406(10)63114-8) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26659) Allen, Caroline and Beecham, Jennifer (1993) Costing services: ideals and reality. In: Netten, Ann and Beecham, Jennifer, eds. Costing Community Care:Theory and Practice. Avebury, Aldershot, pp. 25-42. ISBN hbk: 1857420985 / pbk: 1857421027. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26395) Allen, Caroline, Kitchen, Sheila S. (1993) The costs of electrotherapy II. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27171) Almond, Stephen (1999) Behaviour at local elections in England in 1998: Evidence from the British Social Attitudes Survey. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27172) Almond, Stephen and Bebbington, Andrew and Judge, Ken F. and Mangalore, Roshni and O'Donnell, Owen (1999) Poverty, disability and the use of long-term care services. In: Commission on Long Term Care, Royal, ed. A New Era for Older People. The Stationery Office, London, I-36. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26396) Almond, Stephen, Kendall, Jeremy (2000) Taking the employees' perspective seriously: An initial United Kingdom cross-sectoral comparison. Nonprofit and Voluntary Sector Quarterly, 29 (2). pp. 205-231. ISSN 0899-7640. (doi:10.1177/0899764000292001) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:16045) Amaddeo, Francesco, Beecham, Jennifer, Bonizzato, Paola, Fenyo, Andrew J., Knapp, Martin R J., Tansella, Michele (1997) The use of a case register for evaluating the costs of psychiatric care. Acta Psychiatrica Scandinavica, 95 . pp. 195-198. ISSN 0001-690X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26661) Amaddeo, Francesco, Beecham, Jennifer, Bonizzato, Paola, Fenyo, Andrew J., Tansella, Michele, Knapp, Martin R J. (1998) The costs of community-based care for first-ever patients: a case register study. Psychological Medicine, 28 . pp. 173-183. ISSN 0033-2917 (Print) 1469-8978 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26662) Amaddeo, Francesco, Bonizzato, Paola, Beecham, Jennifer, Knapp, Martin R J., Tansella, Michele (1996) ICAP. Un'intervista per la raccolta dei dati necessari per la valutazione dei costi dell'assistenza psichiatrica. Epidemiologia e Psichiatria Sociale, 5 (3). pp. 201-213. ISSN 1121-189X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26663) Amaddeo, Francesco, Bonizzato, Paola, Rossi, Francois, Beecham, Jennifer, Knapp, Martin R J., Tansella, Michele (1995) La valutazione dei costi delle malattie mentali in Italia. Sviluppo di una metodologia e possibili applicazione [Evaluating costs of mental illness in Italy. The development of a methodology and possible applications]. Epidemiologia e Psichiatria Sociale, 4 (2). pp. 145-162. ISSN 1121-189X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26664) Anheier, Helmut K. and Knapp, Martin R J. and Salamon, Lester M. (1993) No numbers, no policy - can Eurostat count the nonprofit sector? In: Saxon-Harrold, Susan and Kendall, Jeremy, eds. Researching the Voluntary Sector. Charities Aid Foundation, Tonbridge and London, pp. 195-205. ISBN 13506447. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26397) Anheier, Helmut K., Knapp, Martin R J., Salamon, Lester M. (1993) Pas de chiffres, pas de politique: est-ce qu'Eurostat peut mesurer le secteur sans but lucratif? Revue des Etudes Coopératives Mutualistes et Associatives, 46 (248). pp. 87-96. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26665) Apps, Joanna, Crowther, Tanya, Forder, Julien E. (2013) Personal outcome measures and postal surveys of social care. Quality and Outcomes of Person-Centred Care Policy Research Unit (KAR id:41644) Preview Ashwin, Chris, Batchelder, Laurie, Brosnan, Mark (2014) The relationship between greater empathy and psychosis traits is mediated by executive functions. In: 17th General Meeting of the European Association of Social Psychology, 09-12 Jul 2014, Amsterdam, Netherlands. (Submitted) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64661) Aspinal, Fiona, Stevens, Martin, Manthorpe, Jill, Woolham, John, Samsi, Kritika, Baxter, Kate, Hussein, Shereen, Ismail, Mohamed (2019) Safeguarding and personal budgets: the experiences of adults at risk. The Journal of Adult Protection, 21 (3). pp. 157-168. ISSN 1466-8203. (doi:10.1108/JAP-12-2018-0030) (KAR id:73801) Preview Atkinson, Teresa, Evans, Simon, Darton, Robin, Cameron, Ailsa, Porteus, Jeremy, Smith, Randall (2014) Creating the asset base – a review of literature and policy on housing with care. Housing, Care and Support, 17 (1). pp. 16-25. ISSN 1460-8790. E-ISSN 2042-8375. (doi:10.1108/HCS-09-2013-0017) (KAR id:41630) Preview ## B Bachmann, Christian J., Beecham, Jennifer, O'Connor, Thomas G., Scott, Adam, Briskman, Jackie, Scott, Stephen (2019) The cost of love: financial consequences of insecure attachment in antisocial youth. Journal of Child Psychology and Psychiatry, 60 (12). pp. 1343-1350. ISSN 0021-9630. (doi:10.1111/jcpp.13103) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:78718) Baines, Barry and Davies, Bleddyn P. (1994) Attitudes to residential care and the subsequent probability of admission: experience of a cohort of new users of community-based social services. In: Challis, David J. and Davies, Bleddyn P. and Traske, Karen, eds. Community Care in the UK and Overseas: New Agendas and Challenges. Ashgate, Aldershot, pp. 59-72. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26398) Baines, Barry, Davies, Bleddyn P. (1991) On the calculation of discounted present values of costs to social services departments through time of home care and residential care of persons admitted from home care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27173) Baines, Barry, Davies, Bleddyn P. (1990) A comparison of the provision of home help services in France and Britain: a research note. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27174) Balarajan, Meera and Blake, Margaret and Darton, Robin and Gray, Michelle and Hancock, Ruth and Pickard, Linda and Wittenberg, Raphael (2015) Survey questions on older people's receipt of, and payment for, formal and unpaid care in the community. In: Curtis, Lesley A. and Burns, Amanda, eds. Unit Costs of Health and Social Care 2015. Personal Social Services Research Unit, Canterbury, UK, pp. 11-15. ISBN 978-1-902671-96-3. E-ISBN 1368230X. (doi:3) (KAR id:70239) Preview Baldock, John C., Hadlow, Jan (2002) Self-talk versus Needs-talk: an exploration of the priorities of housebound older people. Quality in Ageing: Policy, Practice and Research, 3 (1). pp. 42-48. ISSN 1471-7794. (KAR id:7446) Preview Baltruks, D. and Hussein, Shereen and Lara Montero, A. (2017) Investing in the social services workforce: A study on how local public social services are planning, managing and training the social services workforce of the future. Project report. European Social Network, Brighton, UK (KAR id:68301) Preview Barnes, Marian, Bauld, Linda, Benzeval, Michaela, Judge, Ken F., Killoran, Amanda, Robinson, Ray, Wigglesworth, Rachel, Zeilig, Hannah (1999) Health Action Zones: learning to make a difference. Funded/commissioned by: Report submitted to the Department of Health June 1999 PA from mid July 1999 @ £20. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27175) Barnes, Sarah, Torrington, Judith, Darton, Robin, Holder, Jacquetta, Lewis, Alan, McKee, Kevin, Netten, Ann, Orrell, Alison (2012) Does the design of extra-care housing meet the needs of the residents? A focus group study. Ageing and Society, 32 (7). pp. 1193-1214. ISSN 0144-686X. (doi:10.1017/S0144686X11000791) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:34690) Batchelder, Laurie (2017) EXCELC ISPOR Update. . EXCELC Project Blog. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64686) Batchelder, Laurie (2017) EXCELC Team Meeting and Advisory Group in Vienna 2017. . EXCELC Project Blog. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64687) Batchelder, Laurie (2017) EXCELC Team Meeting in Helsinki 2017. . EXCELC Project Blog. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64688) Batchelder, Laurie (2016) Living with long-term conditions: Validation of the Long-Term Conditions Questionnaire (LTCQ). . PSSRU Blog. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64684) Batchelder, Laurie, Brosnan, Mark, Ashwin, Chris (2017) The Development and Validation of the Empathy Components Questionnaire (ECQ). PLoS ONE, 12 (1). ISSN 1932-6203. (doi:10.1371/journal.pone.0169185) (KAR id:62124) Preview Batchelder, Laurie, Brosnan, Mark, Ashwin, Chris (2014) Differences in emotional intelligence associated with Theory of Mind performance between males and females. In: Consortium of European Research on Emotion Conference (CERE), 27-28 Mar 2014, Berlin, Germany. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64680) Batchelder, Laurie, Brosnan, Mark, Ashwin, Chris (2014) Distinguishing abilities and drives in feeling for others: Examining further components of empathy through a self-report questionnaire. In: ‘Changing lives, changing worlds’- the Faculty of Humanities and Social Sciences Postgraduate Conference, 23 Jun 2014, Bath, UK. (Submitted) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64663) Batchelder, Laurie, Brosnan, Mark, Ashwin, Chris (2014) Distinguishing abilities and drives in feeling for others: Examining further components of empathy through a self-report questionnaire. In: The 1st Annual Psychology Department PhD Conference, 14 May 2014, Bath, UK. (Submitted) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64678) Batchelder, Laurie, Brosnan, Mark, Ashwin, Chris (2014) The development of the Empathy Components Questionnaire (ECQ): measuring abilities and drives in empathy through a self-report measure. In: The Association for Psychological Science 26th Annual Convention, 22-25 May 2014, San Francisco, CA. (Submitted) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64664) Batchelder, Laurie, Fox, Diane (2018) Measuring health & well-being: Reaching across disciplinary boundaries. . Blog. Workshop Blog. (doi:Workshop Blog) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:68528) Batchelder, Laurie, Fox, Diane, Jones, Karen C., Forder, Julien E., Potter, Caroline, Geneen, Louise, Fitzpatrick, Ray, Peters, Michele (2017) Development of a Long-Term Conditions Questionnaire (LTCQ) social care subscale (LTCQ-S). In: Advances in Patient Reported Outcomes Research Conference 2017, 08 Jun 2017, Oxford, UK. (Unpublished) (KAR id:64279) Preview Batchelder, Laurie, Fox, Diane, Potter, Caroline, Fitzpatrick, Ray, Forder, Julien E., Jones, Karen C., Kelly, Laura, Peters, Michele (2018) Further structural validation of the Long-Term Conditions Questionnaire (LTCQ): Formation of the Rasch 8-item LTCQ short-form (LTCQ-8). In: Advances in Patient Reported Outcomes Research Conference 2018, 20 Jun 2018, Birmingham, UK. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:68546) Batchelder, Laurie, Fox, Diane, Potter, Caroline, Fitzpatrick, Ray, Forder, Julien E., Jones, Karen C., Peters, Michele (2018) Further structural validation of the Long-Term Conditions Questionnaire (LTCQ): Formation of the Rasch 8-item LTCQ short-form (LTCQ-8). In: ILPN Conference, 10-12 Sept 2018, Vienna, Austria. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:69142) Batchelder, Laurie, Hajji, Assma, Trukeschitz, Birgit (2017) Eliciting preferences to measure social care-related quality of life outcomes: a discussion on methodology and best practice. In: Research Institute for Economics of Aging Seminar, 12 Jul 2017, Vienna, Austria. (Submitted) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:64710) Batchelder, Laurie, Malley, Juliette, Burge, Peter, Lu, Hui, Saloniki, Eirini-Christina, Linnosmaa, Ismo, Trukeschitz, Birgit, Forder, Julien E. (2019) Carer social care-related quality of life outcomes: estimating English preference weights for the Adult Social Care Outcomes Toolkit for Carers (ASCOT-Carer). Value in Health, 22 (12). pp. 1427-1440. ISSN 1098-3015. (doi:10.1016/j.jval.2019.07.014) (KAR id:75662) Preview Batchelder, Laurie, Malley, Juliette, Razik, Kamilla, Forder, Julien E. (2016) How do people make decisions about ‘best’ and ‘worst’ quality of life states? A qualitative exploration of best-worst scaling responses to the ASCOT measure of care-related quality of life. In: ILPN Conference 2016: Abstracts. . p. 8. International Long Term Care Policy Network (KAR id:59079) Preview Batchelder, Laurie, Malley, Juliette, Razik, Kamilla, Forder, Julien E. (2016) How do people make decisions about ‘best’ and ‘worst’ quality of life states? A qualitative exploration of best-worst scaling responses to the ASCOT measure of care-related quality of life. In: WSF Thematic Workshop on "Health Politics, Health Policy, and Health Inequalities", 4-6 October 2016, Mannheim, Germany. (Unpublished) (KAR id:61628) Preview Batchelder, Laurie, Malley, Juliette, Razik, Kamilla, Jokimaki, Hanna, Kieninger, Judith, Trukeschitz, Birgit, Hajji, Assma, Linnosmaa, Ismo (2016) “It’s hard to imagine situations that you’ve not experienced before properly:” Reflections on what aspects of quality of life people value using the Best-Worst Scaling (BWS) Task. . EXCELC Project Blog and research note. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64685) Batchelder, Laurie, Saloniki, Eirini-Christina, Malley, Juliette, Burge, Peter, Lu, Hui, Forder, Julien E. (2017) Carer social care-related quality of life outcomes: establishing preference weights for the Adult Social Care Outcomes Toolkit for carers. In: Value in Health. 20 (5). Elsevier (doi:10.1016/j.jval.2017.05.005) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:62292) Batchelder, Laurie, Saloniki, Eirini-Christina, Malley, Juliette, Burge, Peter, Lu, Hui, Linnosmaa, Ismo, Trukeschitz, Birgit, Forder, Julien E. (2018) Comparing Preferences for Social Care-Related Quality of Life Using the Adult Social Care Outcomes Toolkit (ASCOT-SCT4) Service User Measure in Austria, England and Finland. In: ILPN Conference, 10-12 Sept 2018, Vienna, Austria. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:69143) Bauld, Linda (1998) Care package costs of elderly people. In: Netten, Ann and Dennett, Jane and Knight, John R. P., eds. Unit Costs of Health and Social Care 1998. PSSRU, University of Kent, Canterbury, pp. 135-142. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26400) Bauld, Linda (1997) Older patient participation in multi-disciplinary decision-making: planning in Scotland and British Columbia. In: Scotia Centre on Aging, Nova, ed. Research into Healthy Aging: Challenges in Changing Times. Nova Scotia Centre on Aging, Dalhousie University, Halifax, pp. 110-127. ISBN none. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26399) Bauld, Linda, Chesterman, John, Davies, Bleddyn P., Judge, Ken F., Mangalore, Roshni (2000) Caring for Older People: An Assessment of Community Care in the 1990s. Ashgate, Aldershot, 428 pp. ISBN 754612805. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27425) Bauld, Linda, Findlater, J., Adams, Catherine, Judge, Ken F. (2000) Compendium of Local Evaluation Arrangements in Health Action Zones. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27176) Bauld, Linda and Judge, Ken F. Adapting health systems to tackle public health challenges and inequalities. In: Creating a Better Future for Health in Europe. Conference Proceedings. . ISBN 3 95000989 1 7. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26402) Bauld, Linda and Judge, Ken F. (2000) Evaluating policies to tackle inequalities in health: the contribution of Health Action Zones. In: Creating a Better Future for Health in Europe. Conference Proceedings. European Health Forum, Gastein, Austria, pp. 140-158. ISBN ISBN: 3 95000989 1 7. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26401) Bauld, Linda, Judge, Ken F. Health action zones. ESRC Health Variations Newsletter, 2 . pp. 10-11. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26667) Bauld, Linda, Mangalore, Roshni (1998) Costing intensive home care packages for older people. Funded/commissioned by: Social Work Services Inspectorate for Scotland. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27121) Bauld, Linda, Zeilig, Hannah (1999) Health Action Zones: improving the health of older people? Generations Review, 9 (3). pp. 18-19. ISSN 0965-2000. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26668) Baumann, Matt, Evans, Sherrill, Perkins, Margaret, Curtis, Lesley A., Netten, Ann, Fernández, José-Luis, Huxley, Peter (2008) Implementing the reinbursement scheme - views of health and social care staff in six high performing sites. Research, Policy and Planning, 26 (2). pp. 101-112. ISSN 0264 519X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:14654) Baumann, Matt, Evans, Sherrill, Perkins, Margaret, Curtis, Lesley A., Netten, Ann, Fernández, José-Luis, Huxley, Peter (2007) Organisation and features of hospital, intermediate care and social services in English sites with low rates of delayed discharge. Health & Social Care in the Community, 15 (4). pp. 293-305. ISSN 0966-0410. (doi:10.1111/j.1365-2524.2007.00697.x) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:2078) Beadle-Brown, Julie, Ryan, Sara, Windle, Karen, Holder, Jacquetta, Turnpenny, Agnes, Smith, Nick, Richardson, Lisa, Whelton, Beckie (2012) Engagement of People with Long-Term Conditions in Health and Social Care Research: Barriers and Facilitators to Capturing the Views of Seldom Heard Population. Quality and Outcomes of Person-Centred Care Policy Research Unit, 58 pp. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:44864) Beadle-Brown, Julie and Towers, Ann-Marie and Netten, Ann and Smith, Nick and Trukeschitz, Birgit and Welch, Elizabeth (2011) ASCOT Adult Social Care Outcomes Toolkit: Additional Care Home Guidance v2.1 PSSRU Discussion Paper 2716/2_1, University of Kent. Manual. Personal Social Services Research Unit, University of Kent, Canterbury (KAR id:35172) Preview Bebbington, Andrew (1993) Calculating unit costs of a centre for people with AIDS/HIV. In: Netten, Ann and Beecham, Jennifer, eds. Costing Community Care:Theory and Practice. Avebury, Aldershot, pp. 127-142. ISBN hbk: 1857420985 / pbk: 1857421027. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26403) Bebbington, Andrew (1979) Changes in the provision of social services to the elderly in the community over fourteen years. Social Policy & Administration, 13 (2). pp. 111-123. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1979.tb00649.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26672) Bebbington, Andrew (1995) Children in Need: survey design for SSA purposes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27191) Bebbington, Andrew (1986) Development of needs indicators for the social services. Part II: Exemplification. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27212) Bebbington, Andrew (1991) Expectation of life without disability in England and Wales: 1976-1988. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27200) Bebbington, Andrew (1992) Expectation of life without disability measure from the OPCS disability surveys. Studies on Medical and Population Subjects, 54 . pp. 23-32. ISSN 0072-6400. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26683) Bebbington, Andrew (1995) Health expectancies and health policies in the UK. In: van se Water, H.P.A. and Perenboom, R.J.M., eds. Proceedings of the First Meeting of EuroREVES: Policy Relevance and Conceptual Harmonisation. TNO Prevention and Health, Leiden, Netherlands, pp. 20-27. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26407) Bebbington, Andrew (1996) Health expectancy and long term care costs. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27184) Bebbington, Andrew (1985) Need indicators and target efficiency: a review. In: The Production of Welfare, 1985-12-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27120) Bebbington, Andrew (1990) Predicting length of stay at a drop-in day centre. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27205) Bebbington, Andrew (2001) Predicting the course of care for entrants to care homes. Gerontology, 47 (Supple). 53-. ISSN 0304-324X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:763) Bebbington, Andrew (1993) Regional and social variations in disability-free life expectancy in Great Britain. In: Robine, Jean-Marie and Mathers, Colin D. and Bone, Margaret R. and Romieu, Isabelle, eds. Calculation of Health Expectancies: Harmonization, Consensuses Achieved and Future Perspectives, Vol. 226. Colloque, INSERM/John Libbey Eurotext Ltd, Montrouge, France, pp. 175-191. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26406) Bebbington, Andrew (1977) Scaling indices of disablement. British Journal of Preventive and Social Medicine, 31 (2). pp. 122-126. ISSN 0007-1242. (doi:PMC479007PMC479007) (KAR id:26670) Preview Bebbington, Andrew (1996) Synthetic estimation methods for resource allocation formulae. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27186) Bebbington, Andrew (1991) Volunteers: their length of commitment and value of input. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27198) Bebbington, Andrew (1988) The expectation of life without disability in England and Wales. Social Science and Medicine, 27 . ISSN 0277-9536. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26678) Bebbington, Andrew (1977) The incidence of need among children, measured on data from the NCDS. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27214) Bebbington, Andrew (1978) A method of bivariate trimming for robust estimation of the correlation coefficient. Journal of the Royal Statistical Society: Series C (Applied Statistics), 27 (3). pp. 221-226. ISSN 0035-9254. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26674) Bebbington, Andrew (1977) A needs indicator based on client typography. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27215) Bebbington, Andrew (1992) The statutory/voluntary HIV partnership. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27195) Bebbington, Andrew and Bajekal, M. (2003) Sub-national Variations in Health Expectancy. In: Robine, Jean-Marie and Jagger, Carol and Mathers, Colin D. and Crimmins, Eileen M. and Suzman, Richard M., eds. Determining Health Expectancies. John Wiley, London, pp. 127-147. ISBN 978-0-470-84397-0. (doi:10.1002/0470858885.ch6) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:755) Bebbington, Andrew, Beecham, Jennifer (2003) Children in Need 2001: Seven Collected Reports. Funded/commissioned by: Department for Education and Skills. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27124) Bebbington, Andrew, Beecham, Jennifer (1989) Hospital services for people with AIDS: a basis for costing. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27206) Bebbington, Andrew, Beecham, Jennifer (2007) Social services support and expenditure for children with autism. Autism, 11 (1). pp. 43-61. ISSN 1362-3613. (doi:10.1177/1362361307070911) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2408) Bebbington, Andrew, Bone, Margaret R. (1998) Healthy life expectancy and long-term care. Presentation to the Royal Commission on Long-Term Care. In: UNSPECIFIED, 1998-03-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27086) Bebbington, Andrew, Bone, Margaret R., Jagger, Carol, Morgan, K., Nicolaas, Geraldine (1995) Health Expectancy and its Uses. HMSO, London, 94 pp. ISBN 117020052. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27432) Bebbington, Andrew, Brown, Pamela, Darton, Robin, Miles, Kathryn, Netten, Ann (1999) Survey of Admissions to Residential and Nursing Home Care. 30 Month Follow-Up. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27180) Bebbington, Andrew, Brown, Pamela, Darton, Robin, Miles, Kathryn, Netten, Ann (1998) Survey of Admissions to Residential and Nursing Home Care: 18 Month Follow-Up. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27181) Bebbington, Andrew, Brown, Pamela, Darton, Robin, Netten, Ann (1996) Survey of Admissions to Residential Care: SSA Analysis Report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27189) Bebbington, Andrew, Brown, Pamela, Darton, Robin, Netten, Ann (1996) Survey of Admissions to Residential and Nursing Homes for Elderly People. In: British Congress of Gerontology, 1996-07-01T00:00:00, Manchester. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27094) Bebbington, Andrew, Charnley, Helen (1985) Domiciliary Care Project Entry into Care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27213) Bebbington, Andrew, Charnley, Helen (1990) Joint health and social services care for elderly people - rhetoric and reality. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27201) Bebbington, Andrew, Charnley, Helen, Fitzpatrick, A. (1990) Balance and allocation of services to elderly people in Oxfordshire. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27202) Bebbington, Andrew, Darton, Robin (1995) Alternatives to long-term hospital care for elderly people in London. Funded/commissioned by: King's Fund Fair Shares Initiative. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27125) Bebbington, Andrew, Darton, Robin (1996) Healthy life expectancy in England and Wales: recent evidence. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27187) Bebbington, Andrew, Darton, Robin, Bartholomew, Royston, Netten, Ann (2000) Survey of Admissions to Residential and Nursing Home Care: Final Report of the 42 Month follow-up. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27179) Bebbington, Andrew, Darton, Robin, Bartholomew, Royston, Netten, Ann, Brown, Pamela (2000) Survey of Admissions to Residential and Nursing Home Care. Final Report of the 30 Month Follow-Up. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27178) Bebbington, Andrew, Darton, Robin, Netten, Ann (2001) Care homes for older people: Volume 2 Admissions, needs and outcomes. The 1995/96 National Longitudinal Survey of Publicly-Funded Admissions. Personal Social Services Research Unit, University of Kent, Canterbury, 102 pp. ISBN 1-902671-25-2. (KAR id:762) Preview Bebbington, Andrew, Darton, Robin, Netten, Ann (1997) Lifetime risk of entering residential or nursing home care in England. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27183) Bebbington, Andrew, Davies, Bleddyn P. (1993) Efficient targeting of community care - the case of the home help service. Journal of Social Policy, 22 . pp. 373-391. ISSN 0047-2794. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:20792) Bebbington, Andrew, Davies, Bleddyn P. (1986) Needs indicators for the social services: an overview, a report to the Scottish Development Department. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27211) Bebbington, Andrew and Davies, Bleddyn P. (1982) Patterns of social service provision for the elderly: variations between local authorities of England in 1975/76, 1977/78 and 1979/80. In: Warner, A., ed. Geographical Perspectives on the Elderly. John Wiley, Chichester, pp. 355-374. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26404) Bebbington, Andrew, Davies, Bleddyn P. (1989) Target efficiency of the home help service in 1985. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27207) Bebbington, Andrew, Davies, Bleddyn P. (1980) Territorial need indicators: a new approach. Journal of Social Policy, 9 (2 & 4). 145-68 & 433. ISSN 0047-2794. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26675) Bebbington, Andrew, Davies, Bleddyn P. (1983) Territorial need indicators: a reply. Journal of Social Policy, 12 (2). pp. 246-250. ISSN 0047-2794. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26676) Bebbington, Andrew, Davies, Bleddyn P. (1977) What differences have the cuts made? Social Work Today, 9 (5). pp. 6-7. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26671) Bebbington, Andrew, Davies, Bleddyn P., Coles, Oliver (1979) Social workers and client numbers. British Journal of Social Work, 9 (1). pp. 93-100. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26673) Bebbington, Andrew, Feldman, Rayah, Gatter, P.N., Warren, Pat (1992) Evaluation of the Landmark: final report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27197) Bebbington, Andrew, Gatter, P.N. (1994) AIDS volunteers. AIDS Care, 6 . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26685) Bebbington, Andrew, Kelly, Aidan (1992) Reliability unit cost estimation for personal social services. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27196) Bebbington, Andrew, Kelly, Aidan (1992) Unit costs of personal social services in Inner London and other inner cities: stage 2 proposal. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27194) Bebbington, Andrew, Kelly, Aidan (1991) Unit costs of personal social services in Inner London. Stage one report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27199) Bebbington, Andrew, Kesby, S., Challis, David J., Clarkson, Paul C., Stewart, Karen, Weiner, Kate, Worden, Angela (2001) Promoting Continuity of Care for Older People across Health and Social Care: short report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27177) Bebbington, Andrew, Lawson, Robyn, Warren, Pat (1990) An HIV/AIDS strategy for Lewisham social services department. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27204) Bebbington, Andrew, Miles, John (1988) Children's GRE research report of the survey of foster families. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27209) Bebbington, Andrew, Miles, John (1988) Children's GRE research: children entering care. A need indicator for in-care services for children. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27210) Bebbington, Andrew, Miles, John (1989) The background of children who enter local authority care. British Journal of Social Work, 19 (5). pp. 349-368. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26679) Bebbington, Andrew, Miles, John (1989) The length of stay in care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27208) Bebbington, Andrew, Miles, John (1990) The supply of foster families for children in care. British Journal of Social Work, 20 (4). ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26682) Bebbington, Andrew, Netten, Ann, Darton, Robin, Davies, Bleddyn P. (1995) Elderly People in Residential Care. Survey design for SSA and other purposes. A paper for information and discussion with local government representative bodies. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27190) Bebbington, Andrew, Quine, Lyn (1987) A comment on Hirst's 'Evaluating the Malaise Inventory'. Social Psychiatry, 22 (1). pp. 5-7. (doi:10.1007/BF00583613) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26677) Bebbington, Andrew, Rickard, Wendy (1999) Needs-Based Planning for Community Care: Matching Theory to Practice. PSSRU, Canterbury, 64 pp. ISBN 904938905. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27424) Bebbington, Andrew, Rickard, Wendy, Warren, Pat (1993) Community care for people with AIDS: a review of plans in London. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27193) Bebbington, Andrew, Rickard, Wendy, Warren, Pat (1997) Community care for people with HIV/AIDS. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27182) Bebbington, Andrew, Shapiro, Judith C. (2005) Ageing, Health Status and Determinants of Health Expenditure (AHEAD). Work package III: incidence of poor health and long-term care. Health transitions in Europe: results from the European Community Household Panel Survey and Institutional Data. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27123) Bebbington, Andrew and Tong, M.S. (1986) Trends and changes in old people's homes: provision over twenty years. In: Judge, Ken F. and Sinclair, Ian, eds. Residential Care for Elderly People. HMSO, London, pp. 67-81. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26405) Bebbington, Andrew, Turvey, Karen, Janzon, Karin (1996) Needs based planning for community care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27188) Bebbington, Andrew, Unell, Judith (2003) Care direct: an integrated route to help for older people. Generations Review, 13 (3). pp. 18-21. ISSN 0965-2000. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:760) Bebbington, Andrew, Warren, Pat (1989) Accommodation and HIV infection: the research agenda. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27126) Bebbington, Andrew, Warren, Pat (1989) Services for people with AIDS: what role for local authorities? Social Services Insight, 4 (11). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26681) Bebbington, Andrew, Warren, Pat (1990) Voluntary sector service centres for AIDS/HIV+: strategic problems. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27203) Bebbington, Andrew, Warren, Pat, Rickard, Wendy (1994) Addressing health and social care needs in the community for people with HIV/AIDS. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27192) Beck, Eduard J., Mandalia, Sundhiya, Griffith, Rebecca, Beecham, Jennifer, Walters, M.D. Sam, Boulton, Mary, Miller, David L. (2000) Use and cost of hospital and community service provision for children with HIV infection at an English HIV referral centre. PharmacoEconomics, 17 (1). pp. 53-69. ISSN 1170-7690. (doi:10.2165/00019053-200017010-00004) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26687) Beck, Eduard J., Tolley, Keith, Power, Amanda, Mandalia, Sundhiya, Rutter, Philippa, Izumi, Junichi, Beecham, Jennifer, Gray, Alistair, Barlow, David, Easterbrook, Philippa, and others. (1998) The use and cost of HIV service provision in England in 1996. PharmacoEconomics, 14 (6). pp. 639-52. ISSN 1170-7690. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32412) Becker, Thomas, Knapp, Martin R J., Knudsen, Helle Charlotte, Schene, Aart H., Tansella, Michele, Vázquez-Barquero, Jose Luis (2000) Aims, outcome measures, study sites and patient sample. EPSILON Study 1. British Journal of Psychiatry, 177 (S39). pp. 1-7. ISSN 0007-1250. (doi:10.1192/bjp.177.39.s1) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26688) Becker, Thomas, Knapp, Martin R J., Knudsen, Helle Charlotte, Schene, Aart H., Tansella, Michele, Vázquez-Barquero, Jose Luis, the EPSILON Study Group (1999) The EPSILON study of schizophrenia in five European countries: design and methodology for standardising outcome measures and comparing patterns of care and service costs. British Journal of Psychiatry, 175 (6). pp. 514-521. ISSN 0007-1250 (Print) 1472-1465 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26689) Becker, Thomas, Leese, M.R., McCrone, Paul, Clarkson, Paul C., Szmukler, George, Thornicroft, Graham (1998) Impact of community mental health services on users’ social networks, PriSM psychosis study 7. British Journal of Psychiatry, 173 . pp. 404-408. ISSN 0007-1250 (Print) 1472-1465 (Online). (doi:10.1192/bjp.173.5.404) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26690) Beecham, Jennifer (1995) 150 years of mental health services. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27219) Beecham, Jennifer (2005) Access to mental health supports in England: crisis resolution teams and day services. International Journal of Law and Psychiatry, 28 (5). pp. 574-587. ISSN 0160-2527. (doi:10.1016/j.ijlp.2005.08.009) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:765) Beecham, Jennifer (2014) Annual Research Review: Child and adolescent mental health interventions: a review of progress in economic studies across different disorders. Journal of Child Psychology and Psychiatry, 55 (6). pp. 714-732. ISSN 0021-9630. E-ISSN 1469-7610. (doi:10.1111/jcpp.12216) (KAR id:41631) Preview Beecham, Jennifer (1995) Collecting and estimating costs. In: Knapp, Martin R J., ed. The Economic Evaluation of Mental Health Care. Arena, Aldershot, pp. 157-174. ISBN 1-85742-211-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26425) Beecham, Jennifer (1992) Costing services: an update. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27221) Beecham, Jennifer (2001) Disabled children and their families: the research evidence. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27217) Beecham, Jennifer (1998) Economic evaluation and child psychiatric in-patient services. In: Green, Jonathan and Jacobs, Brian, eds. Inpatient Child Psychiatry: Modern Practice, Research and the Future. Routledge, London, pp. 363-373. ISBN hbk: 0415194393 / pbk: 0415145252. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26414) Beecham, Jennifer (1996) Health economics and children's mental health. Mental Health Research Review, 3 . pp. 37-39. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27083) Beecham, Jennifer (1995) International costs research. Mental Health Research Review, 2 . pp. 35-37. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27064) Beecham, Jennifer (2010) Measures of Success. Community Care, . p. 23. ISSN 0307-5508. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26704) Beecham, Jennifer (1994) Reduced list costing: some implications for research costs and for costs research. In: ARCAP Third Workshop on Costs and Assessment in Psychiatry, The Economics of Schizophrenia, Depression, Anxiety, Dementias: New Research, Methods, Health Policies, 1994-10-01T00:00:00, Venice. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27102) Beecham, Jennifer (2000) Unit Costs: Not Exactly Child’s Play. Project report. Department of Health, Personal Social Services Research Unit and Dartington Social Care Research Unit (KAR id:32449) Preview Beecham, Jennifer (2006) Why costs vary in children's care services. Journal of Children's Services, 1 (3). pp. 50-62. ISSN 1746-6660. (doi:10.1108/17466660200600023) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26703) Beecham, Jennifer (1995) The costs of in-patient and out-patient treatment for children with severe emotional and behavioural problems: a pilot study. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27220) Beecham, Jennifer, Astin, Jack, Mummery, Kathryn (2001) Linking datasets, people and resources. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27216) Beecham, Jennifer, Bebbington, Andrew (2002) Child care costs: variations and unit costs. First annual report to the Department of Health. Funded/commissioned by: Department of Health. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27127) Beecham, Jennifer and Beck, Eduard J. and Mandalia, Sundhiya and Griffith, R. and Walters, M. and Boulton, Mary and Miller, David L. (2000) What is the cost of getting the price wrong? In: Netten, Ann and Curtis, Lesley A., eds. Unit Costs of Health and Social Care 2000. PSSRU, University of Kent at Canterbury, pp. 19-22. ISBN ISBN: 1 902671 19 8/ISSN: 0989 4226. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26427) Beecham, Jennifer and Bonin, Eva-Maria and Byford, Sarah and McDaid, David and Mullally, Gerald and Parsonage, Michael (2011) School-based social and emotional learning programmes to prevent conduct problems in childhood. Project report. Department of Health (KAR id:32450) Preview Beecham, Jennifer, Bonin, Eva-Maria, Görlich, Dennis, Baños, Rosa, Beintner, Ina, Bluntrock, Claudia, Bolinski, Felix, Botella, Christina, Ebert, David Daniel, Herrero, Rocio, and others. (2018) Assessing the costs and cost-effectiveness of ICare internet-based interventions (protocol). Internet Interventions, 16 . pp. 12-19. E-ISSN 2214-7829. (doi:10.1016/j.invent.2018.02.009) (KAR id:66627) Preview Beecham, Jennifer, Cambridge, Paul, Hallam, Angela, Knapp, Martin R J. (1993) The costs of domus care. International Journal of Geriatric Psychiatry, 8 (10). pp. 827-831. ISSN 0885-6230. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:20793) Beecham, Jennifer and Cambridge, Paul and Hallam, Angela and Knapp, Martin R J. (1994) The costs of domus care. In: Murphy, Elaine and Lindesay, James and Dean, Rachel, eds. The Domus Report: Long-Stay Care for People with Severe Dementia. Sainsbury Centre, London, pp. 75-84. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26417) Beecham, Jennifer, Chadwick, Oliver, Fidan, Dogan, Barnard, Sarah (2002) Children with severe learning disabilities: needs, services and costs. Children and Society, 16 (3). pp. 168-181. ISSN 0951-0605. (doi:10.1002/chi.690) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:766) Beecham, Jennifer and Chisholm, Daniel (1995) Mental health economics. In: Wing, John, ed. Measurement for Mental Health. Royal College of Psychiatrists, London, pp. 55-70. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26424) Beecham, Jennifer and Chisholm, Daniel and O'Herlihy, Anne (2002) The costs of child and adolescent psychiatric inpatient units. In: Netten, Ann and Curtis, Lesley A., eds. Unit Costs of Health and Social Care 2002. PSSRU, University of Kent at Canterbury, pp. 21-23. ISBN ISSN: 1368-230X/ISBN: 1-902 671-33-3. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26430) Beecham, Jennifer, Chisholm, Daniel, O'Herlihy, Anne, Astin, Jack (2003) Variations in the costs of child and adolescent psychiatric inpatient units. British Journal of Psychiatry, 183 (3). pp. 220-225. ISSN 0007-1250. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:764) Beecham, Jennifer and Glover, Gyles and Barnes, Diana M (2002) Mapping mental health services in England. In: Netten, Ann and Curtis, Lesley A., eds. Unit Costs of Health and Social Care 2002. PSSRU, University of Kent at Canterbury, pp. 29-31. ISBN ISSN: 1368-230X/ISBN: 1-902 671-33-3. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26431) Beecham, Jennifer, Green, Jonathan, Jacobs, Brian, Dunn, Graham (2009) Cost variation in child and adolescent psychiatric inpatient treatment. European Child & Adolescent Psychiatry, 18 (publis). pp. 535-542. ISSN 1435-165X (Online), 1018-8827 (Print). (doi:10.1007/s00787-009-0008-9) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:20515) Beecham, Jennifer and Hallam, Angela and Knapp, Martin R J. and Baines, Barry and Fenyo, Andrew J. and Asbury, M.P.J. (1997) Costing care in hospital and in the community. In: Leff, Julian, ed. Care in the Community: Illusion or Reality. John Wiley & Sons, Chichester, pp. 93-108. ISBN hbk: 0 47196 981 8 / pbk: 0 47196 982 6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26422) Beecham, Jennifer, Hallam, Angela, Knapp, Martin R J., Baines, Barry, Fenyo, Andrew J., Asbury, M.P.J. (1995) The economic evaluation of community psychiatric reprovision: final report to North Thames Regional Health Authority. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27218) Beecham, Jennifer, Hallam, Angela, Knapp, Martin R J., Cambridge, Paul (1992) The costs of care for people with dual sensory impairment. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27222) Beecham, Jennifer, Hallam, Angela, Knapp, Martin R J., Carpenter, John, Cambridge, Paul, Forrester-Jones, Rachel, Tate, Alison, Wooff, David, Coolen-Schrijner, Pauline (2004) Twelve years on: service use and costs for people with mental health problems who left psychiatric hospitals. Journal of Mental Health, 13 (4). pp. 363-377. ISSN 0963-8237. (doi:10.1080/09638230410001729816) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:767) Beecham, Jennifer, Hayes, L., Knapp, Martin R J., Cambridge, Paul (1996) Costes y consecuencias a largo plazo: atención comunitaria a pacientes con retraso mental [Longer-term costs and consequences: community care for people with learning disabilities]. Siglo Cero, 27 (5). pp. 25-32. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26696) Beecham, Jennifer, Hayes, L., Knapp, Martin R J., Cambridge, Paul (1997) Evaluación de costes en salud mental y en las minusvalias psiquicas. Costes y consecuencias a largo plazo: atención comunitaria a pacientes con retraso mental. Monografias de Psiquiatria, 9 (4). pp. 41-48. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26695) Beecham, Jennifer and Jerram, S. and While, A. (2002) A nurse practitioner service for nursing and residential care homes. In: Netten, Ann and Curtis, Lesley A., eds. Unit Costs of Health and Social Care 2002. PSSRU, University of Kent at Canterbury, pp. 17-19. ISBN 1-902671-33-3. (KAR id:26429) Preview Beecham, Jennifer, Johnson, Sally M., EPCAT group (2000) The European Socio-Demographic Schedule (ESDS): rationale, principles and development. Acta Psychiatrica Scandinavica Supplementum, 102 (s405). pp. 33-46. ISSN 0065-1591. (doi:10.1111/j.0902-4441.2000.t01-1-acp28-02.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26700) Beecham, Jennifer, Knapp, Martin R J. (1996) Analisi costo efficacia e schizofrenia [Cost-effectiveness analysis and schizophrenia]. Annali di Freniatria, 2 (2). pp. 153-157. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26692) Beecham, Jennifer and Knapp, Martin R J. (1992) Costing psychiatric interventions. In: Thornicroft, Graham and Brewin, C. and Wing, J., eds. Measuring Mental Health Needs. Gaskell, London, pp. 179-190. ISBN 902241516. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26416) Beecham, Jennifer and Knapp, Martin R J. (2001) Costing psychiatric interventions. In: Thornicroft, Graham, ed. Measuring Mental Health Needs (Second Edition). Royal College of Psychiatrists, London, pp. 200-224. ISBN 1-901242-60-9. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26426) Beecham, Jennifer, Knapp, Martin R J. (1996) Inclusive and special education: issues of cost-effectiveness. Funded/commissioned by: Organisation for Economic Development and Development. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27129) Beecham, Jennifer and Knapp, Martin R J. (1995) Mental health economic evaluations: unfinished business. In: Knapp, Martin R J., ed. The Economic Evaluation of Mental Health Care. Arena, Aldershot, pp. 229-242. ISBN 1-85742-211-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26411) Beecham, Jennifer, Knapp, Martin R J. (1990) Reprovision: clients, costs and care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27130) Beecham, Jennifer and Knapp, Martin R J. (1995) The costs of child care assessments. In: Sinclair, Ruth and Garnett, Louise and Berridge, David, eds. Social Work and Assessment with Adolescents. National Children's Bureau, London, pp. 245-275. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26418) Beecham, Jennifer and Knapp, Martin R J. and Allen, Caroline (1995) Comparative efficiency and equity in community-based care. In: Knapp, Martin R J., ed. The Economic Evaluation of Mental Health Care. Arena, Aldershot, pp. 175-194. ISBN 1-85742-211-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26410) Beecham, Jennifer and Knapp, Martin R J. and Asbury, M.P.J. (1996) Costs and children's mental health services. In: Netten, Ann and Dennett, Jane, eds. Unit Costs of Health and Social Care 1996. PSSRU, Canterbury, pp. 32-35. ISBN 904938956. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26409) Beecham, Jennifer and Knapp, Martin R J. and Asbury, M.P.J. (1994) The cost dimension. In: Kurtz, Zarrina and Thornes, Rosemary and Wolkind, Stephen, eds. Services for the Mental Health of Children and Young People in England: A National Review. SETRHA, London, pp. 40-48. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26421) Beecham, Jennifer and Knapp, Martin R J. and Fenyo, Andrew J. (1993) Costs, needs and outcomes in community mental health care. In: Netten, Ann and Beecham, Jennifer, eds. Costing Community Care:Theory and Practice. Avebury, Aldershot, pp. 162-176. ISBN hbk: 1857420985 / pbk: 1857421027. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26408) Beecham, Jennifer, Knapp, Martin R J., Fenyo, Andrew J. (1991) Costs, needs and outcomes: community care for people with long term mental health problems. Schizophrenia Bulletin, 17 (3). pp. 427-439. ISSN 0586-7614. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26691) Beecham, Jennifer and Knapp, Martin R J. and Fenyo, Andrew J. and Hallam, Angela (1994) The costs of community care: implementing a psychiatric reprovision policy. In: Institution to Community: Friern Lessons Pack. NETRHA, London, pp. 63-81. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26420) Beecham, Jennifer, Knapp, Martin R J., Fernández, José-Luis, Huxley, Peter, Mangalore, Roshni, McCrone, Paul, Snell, Tom, Wittenberg, Raphael, Winter, Beth (2008) Age Discrimination in Mental Health Services. Personal Social Services Research Unit, 68 pp. (KAR id:13344) Preview Beecham, Jennifer, Knapp, Martin R J., McGilloway, S., Donnelly, Maureen A., Kavanagh, Shane M., Fenyo, Andrew J., Mays, Nicholas (1997) The cost-effectiveness of community care for adults with learning disabilities leaving long-stay hospital in Northern Ireland. Journal of Intellectual Disability Research, 41 (7). pp. 30-41. ISSN 0964-2633. (doi:10.1111/j.1365-2788.1997.tb00674.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26694) Beecham, Jennifer, Knapp, Martin R J., McGilloway, S., Kavanagh, Shane M., Fenyo, Andrew J., Donnelly, Maureen A., Mays, Nicholas (1996) Leaving hospital II: the cost-effectiveness of community care for former long-stay psychiatric hospital patients. Journal of Mental Health, 5 (4). pp. 379-394. ISSN 0963-8237. (doi:10.1080/09638239619293) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26693) Beecham, Jennifer and Knapp, Martin R J. and Schneider, Justine (1996) Policy and finance for community care: the new mixed economy. In: Watkins, Mary and Hervey, Nick and Ritter, Susan and Carson, Jeremy, eds. Collaborative Community Mental Health Care. Edward Arnold, Sevenoaks, pp. 41-57. ISBN 340542411. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26419) Beecham, Jennifer, Law, James, Zeng, Biao, Lindsay, Geoff (2012) Costing children’s speech, language and communication interventions. International Journal of Language and Communication Disorders, 47 (5). pp. 477-486. ISSN 1460-6984. (doi:10.1111/j.1460-6984.2012.00157.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32414) Beecham, Jennifer, Lesage, A.D. (1997) Leçons britanniques sur le transfert des ressources: le système de dotation par patient. Santé Mentale au Québec, XXII (2). pp. 170-194. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26697) Beecham, Jennifer and McCrone, Paul (1996) Cost information for commissioners. In: Thornicroft, Graham and Strathdee, Geraldine, eds. Commissioning Mental Health Services. HMSO, London, pp. 236-246. ISBN hbk: 0113219741 / pbk: 0113219740. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26423) Beecham, Jennifer, Munizza, C (2000) Assessing mental health in Europe. Acta Psychiatrica Scandinavica Supplementum, 102 (45). pp. 5-7. ISSN 0065-1591. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26699) Beecham, Jennifer, Munizza, C (2000) Introduction: assessing mental health in Europe. Acta Psychiatrica Scandinavica Supplementum, 405 (102). pp. 5-7. ISSN 0065-1591. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26701) Beecham, Jennifer, Netten, Ann (1993) Community Care in Action: The Role of Costs. PSSRU, Canterbury, 68 pp. ISBN 904938409. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27402) Beecham, Jennifer, O'Neil, Teresa, Goodman, Robert (2001) Supporting young adults with hemiplegia: services and costs. Health & Social Care in the Community, 9 (1). pp. 51-59. ISSN 0966-0410. (doi:10.1046/j.1365-2524.2001.00279.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26698) Beecham, Jennifer, O'Neil, Teresa, Goodman, Robert (1998) The costs of hemiplegic cerebral palsy. Funded/commissioned by: Children’s Department, Maudsley Hospital. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27128) Beecham, Jennifer, Pearce, Peter, Sewell, Ros, Osman, Sarah (2018) Support and costs for students with emotional problems referred to school-based counselling: findings from the ALIGN study. British Journal of Guidance & Counselling, . ISSN 0306-9885. (doi:10.1080/03069885.2018.1552777) (KAR id:72092) Preview Beecham, Jennifer, Perkins, Margaret, Snell, Tom, Knapp, Martin R J. (2009) Treatment paths and costs for young adults with acquired brain injury in the United Kingdom. Brain Injury, 23 (1). pp. 30-38. ISSN 1362–301X. (doi:10.1080/0269905080259033) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:15491) Beecham, Jennifer, Ramsay, Angus, Gordon, Kate, Maltby, Sophie, Walshe, Kieran, Shaw, Ian, Worral, Adrian, King, Sarah (2010) Cost and impact of a quality improvement programme in mental health services. Journal of Health Services Research and Policy, 15 (2). pp. 69-75. ISSN 1355-8196. (doi:10.1258/jhsrp.2009.009005) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:24888) Beecham, Jennifer and Rowlands, J. and Barker, M. and Lyon, J. and Stafford, M. and Lunt, R. (2001) Child care costs in social services. In: Netten, Ann and Reese, T. and Harrison, G., eds. Unit Costs of Health and Social Care 2001. Personal Social Services Research Unit, University of Kent at Canterbury, pp. 13-17. ISBN 1-902671-22-8. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26428) Beecham, Jennifer, Sinclair, Ian (2007) Costs and Outcomes in Children's Social Care. Jessica Kingsley Publishers, London, UK, 144 pp. ISBN 978-1-84310-496-4. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2770) Beecham, Jennifer, Sleed, Michelle, Knapp, Martin R J., Chiesa, Marco, Drahorhad, Carla (2006) The costs and effectiveness of two psychosocial treatment programmes for personality disorder: a controlled study. European Psychiatry, 21 (2). pp. 102-109. (doi:10.1016/j.eurpsy.2005.05.006) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26702) Beecham, Jennifer, Sloper, Patricia, Greco, Veronica, Webb, Rosemary (2007) The costs of key worker support for disabled children and their families. Child Care Health and Development, 33 (5). pp. 611-618. ISSN 0305-1862. (doi:10.1111/j.1365-2214.2007.00740.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2409) Beecham, Jennifer and Snell, Tom and Perkins, Margaret and Knapp, Martin R J. (2008) After Transition: Health and Social Care Needs of Young Adults with Long-Term Neurological Conditions (PSSRU Research Summary 48). Project report. Personal Social Services Research Unit, University of Kent, Canterbury (KAR id:13318) Preview Beecham, Jennifer, Snell, Tom, Perkins, Margaret, Knapp, Martin R J. (2010) Health and social care costs for young adults with epilepsy in the UK. Health & Social Care in the Community, 18 (5). pp. 465-473. ISSN 0966-0410. (doi:10.1111/j.1365-2524.2010.00919.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:34694) Beecham, Jennifer and Thomason, Corinne (1988) Supporting people with long-term care needs in the community. In: Baldwin, Sally and Parker, Gillian and Walker, Robert, eds. Social Security and Community Care. Avebury, Aldershot, pp. 150-164. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26412) Beecham, Jennifer, Topan, Catherine (1997) Costs and treatment for pre-school children with oppositional defiance disorder. Mental Health Research Review, 4 . pp. 26-29. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27063) Beecham, Jennifer and Topan, Catherine (1996) The direct costs of specialist services for children with mental health problems. In: Kurtz, Zarrina and Thornes, Rosemary and Wolkind, Stephen, eds. Services for the Mental Health of Children and Young People: Assessment of Needs and Unmet Need. Department of Health, London, pp. 59-71. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26413) Beecham, Jennifer and Topan, Catherine and Knapp, Martin R J. (1996) Economic evaluation and services for children with mental health problems. In: Kurtz, Zarrina and Thornes, Rosemary and Wolkind, Stephen, eds. Services for the Mental Health of Children and Young People: Assessment of Needs and Unmet Need. Department of Health, London, A1-A17. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26415) Beecham, Jennifer and Wenborn, Jennifer and Charlesworth, Georgina and Ahmed, Shaheen (2014) RYCT & CSP intervention costs. In: Curtis, Lesley A., ed. Unit Costs of Health and Social Care 2014. PSSRU, Canterbury, pp. 24-28. ISBN 978-1-902671-89-5. (KAR id:49033) Preview Beresford, Bryony, Moran, Nicola, Sloper, Patricia, Cusworth, Linda, Mitchell, Wendy, Spiers, Gemma, Weston, Kath, Beecham, Jennifer (2013) Transition to Adult Services and Adulthood for Young People with Autistic Spectrum Conditions (Working paper no: DH 2525, Department of Health Policy Research Programme Project reference no. 016 0108, Final Report). Social Policy Research Unit, University of York ISBN 978-1-907265-20-4. (KAR id:35444) Preview Beresford, Bryony and Stuttard, Lucy and Clarke, Susan and Maddison, Jane and Beecham, Jennifer (2012) Managing behaviour and sleep problems in disabled children: An investigation into the effectiveness and costs of parent-training interventions. Project report. Department for Education, London DFE-RR204. (doi:DFE-RR204) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32451) Berridge, David, Beecham, Jennifer, Brodie, Isabelle, Cole, Ted, Daniels, Harry, Knapp, Martin R J., MacNeill, Virginia (2003) Services for troubled adolescents: exploring user variation. Child and Family Social Work, 8 (4). pp. 269-279. ISSN 1356-7500. (doi:10.1046/j.1365-2206.2003.00295.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26705) Berridge, David, Dance, Cherilyn, Beecham, Jennifer, Field, Stuart (2008) Educating Difficult Adolescents: Effective education for children in public care or with emotional or behavioural difficulties. Quality Matters in Children's Services . Jessica Kingsley Publishers, London, 224 pp. ISBN 978-1-84310-681-4. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:14653) Biehal, Nina and Ellison, Sarah and Sinclair, Ian and Randerson, Catherine and Richards, Andrew and Mallon, Sharon and Kay, Catherine and Green, Jonathan and Bonin, Eva-Maria and Beecham, Jennifer (2010) A report on the Intensive Fostering Pilot Programme. Project report. Youth Justice Board (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32452) Bindman, Jonathan, Beck, Anatole, Glover, Gyles, Thornicroft, Graham, Knapp, Martin R J., Leese, Morven, Szmukler, George (1999) Evaluating mental health policy in England: the care programme approach and supervision registers. British Journal of Psychiatry, 175 . pp. 327-330. ISSN 0007-1250 (Print) 1472-1465 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26706) Bindman, Jonathan, Beck, Andrew, Thornicroft, Graham, Knapp, Martin R J., Szmukler, George (2000) Psychiatric patients at greatest risk and in greatest need. Impact of the Supervision Register policy. British Journal of Psychiatry, 177 (1). pp. 33-37. ISSN 0007-1250 (Print) 1472-1465 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26707) Blewett, James and Hussein, Shereen and Tunstill, Jane and Manthorpe, Jill and Cowley, Sarah (2011) Children’s Centres in 2011: Improving outcomes for the children who use Action for Children Children’s Centres. Project report. Action for Children (KAR id:68365) Preview Bone, Margaret R. and Bebbington, Andrew and Nicolaas, Geraldine (1994) Policy relevance and comparability problems of health expectancy indicators. In: Proceedings of the Seventh REVES Conference. Australian Institute of Health and Welfare, Canberra, pp. 306-321. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26432) Bonin, Eva-Maria, Beecham, Jennifer, Dance, Cherilyn, Farmer, Elaine (2013) Support for adoption placements: the first six months. British Journal of Social Work, 44 (6). pp. 1508-1525. ISSN 0045-3102. E-ISSN 1468-263X. (doi:10.1093/bjsw/bct008) (KAR id:33544) Preview Bonin, Eva-Maria, Beecham, Jennifer, Swift, Naomi, Raikundalia, Shriti, Brown, June S.L. (2014) Psycho-educational CBT-Insomnia workshops in the community. A cost-effectiveness analysis alongside a randomised controlled trial. Behaviour Research and Therapy, 55 . pp. 40-47. ISSN 0005-7967. (doi:10.1016/j.brat.2014.01.005) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:41629) Bonin, Eva-Maria, Stevens, Madeleine, Beecham, Jennifer, Byford, Sarah, Parsonage, Mike (2011) Costs and longer-term savings of parenting programmes for the prevention of persistent conduct disorder: a modelling study. BMC Public Health, 11 (803). ISSN 1471-2458. (doi:10.1186/1471-2458-11-803) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32415) Bonin, Eva-Maria and Stevens, Madeleine and Beecham, Jennifer and Byford, Sarah and Parsonage, Michael (2012) Do parenting programmes reduce conduct disorder and its costs to society? Project report. Economic and Social Data Service (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32454) Bonin, Eva-Maria and Stevens, Madeleine and Beecham, Jennifer and Byford, Sarah and Parsonage, Michael (2011) Parenting interventions for the prevention of persistent conduct disorders. Project report. Department ofHealth, London (KAR id:32453) Preview Bower, Peter, Byford, Sarah, Barber, Julie, Beecham, Jennifer, Simpson, Sharon, Friedi, Karin, Corney, Roslyn, King, Michael, Harvey, Ian (2003) Meta-analysis of data on costs from trials of counselling in primary care: using individual patient data to overcome sample size limitations in economic analyses. British Medical Journal, 326 (7401). pp. 1247-1250. ISSN 0959-8146. (doi:10.1136/bmj.326.7401.1247) (KAR id:26709) Preview Breeze, Beth (2008) The problem of riches: is philanthropy a solution or part of the problem? In: Maltby, Tony and Kennett, Patricia and Rummery, Kirstein, eds. Social Policy Review 20: Analysis and debate in social policy. Oxford University Press. ISBN 978-1-84742-076-3. (doi:10.1332/policypress/9781847420763.003.0009) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:36130) Brookes, Nadia (2016) Implementation of a community-based approach to dementia care in England: Understanding the experiences of staff. Journal of Social Service Research, 43 (3). pp. 336-345. ISSN 0148-8376. E-ISSN 1540-7314. (doi:10.1080/01488376.2016.1242448) (KAR id:58371) Preview Brookes, Nadia (2015) Personalisation and innovation of social services in a cold financial climate. In: Donati, P and Martignani, L, eds. Towards a New Local Welfare: Best Practices and Networks of Social Inclusion. Bononia University Press, pp. 183-205. ISBN 978-88-7395-998-4. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:64626) Brookes, Nadia and Barrett, Barbara and Netten, Ann and Knapp, Emily (2013) Unit Costs in Criminal Justice. Project report. Personal Social Services Research Unit, Canterbury (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:33547) Brookes, Nadia and Callaghan, Lisa (2014) Shared Lives - improving understanding of the costs of family-based support. In: Curtis, Lesley A., ed. Unit Costs of Health and Social Care 2014. University of Kent, pp. 19-23. ISBN 978-1-902671-89-5. (KAR id:64627) Preview Brookes, Nadia, Callaghan, Lisa (2013) What next for Shared Lives? Family-based support as a potential option for older people. Journal of Care Services Management, 7 (3). pp. 87-94. ISSN 1750-1679. E-ISSN 1750-1687. (doi:10.1179/1750168714Y.0000000029) (KAR id:49547) Preview Brookes, Nadia, Callaghan, Lisa, Collins, Grace, Palmer, Sinead (2018) Shared lives: a multi-method study of a community service for people with intellectual disabilities. Journal of Applied Research in Intellectual Disabilities, 31 (4). pp. 569-588. ISSN 1360-2322. (doi:10.1111/jar.12486) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:74064) Brookes, Nadia, Callaghan, Lisa, Netten, Ann, Fox, Diane (2013) Personalisation and innovation in a cold financial climate. British Journal of Social Work, 45 (1). pp. 86-103. ISSN 0045-3102. E-ISSN 1468-263X. (doi:10.1093/bjsw/bct104) (KAR id:34724) Preview Brookes, Nadia and Kendall, Jeremy and Mitton, Lavinia (2016) Birmingham, Priority to Economics, Social Innovation at the Margins. In: Brandsen, T and Cattacin, S and Evers, A. and Zimmer, A, eds. Social Innovations in the Urban Context. Springer, pp. 83-96. ISBN 978-3-319-21550-1. E-ISBN 978-3-319-21551-8. (doi:10.1007/978-3-319-21551-8_5) (KAR id:64623) Preview Brookes, Nadia and Kendall, Jeremy and Mitton, Lavinia (2016) Birmingham: A "Locality Approach" to Combating Worklessness. In: Brandsen, T and Cattacin, S and Evers, A. and Zimmer, A, eds. Social Innovations in the Urban Context. Springer, pp. 257-263. ISBN 978-3-319-21550-1. E-ISBN 978-3-319-21551-8. (doi:10.1007/978-3-319-21551-8_20) (KAR id:64625) Preview Brookes, Nadia and Kendall, Jeremy and Mitton, Lavinia (2015) Birmingham: The Youth Employment and Enterprise Rehearsal Project. In: Brandsen, T and Cattacin, S and Evers, A. and Zimmer, A, eds. Social Innovations in the Urban Context. Springer, pp. 251-255. ISBN 978-3-319-21551-8. E-ISBN 978-3-319-21551-8. (doi:10.1007/978-3-319-21551-8_19) (KAR id:64624) Preview Brookes, Nadia, Rider, Sinead, Callaghan, Lisa (2016) “I live with other people and not alone”: a survey of the views and experiences of older people using Shared Lives (adult placement). Working with Older People, 20 (3). pp. 179-186. ISSN 1366-3666. E-ISSN 2042-8790. (doi:10.1108/WWOP-03-2016-0005) (KAR id:52926) Preview Brown, June, Slade, Mike, Beecham, Jennifer, Sellwood, Katie, Andiappan, Manoharan, Landau, Sabine, Johnson, Tracy, Smith, Roger (2011) Outcome, costs and patient engagement for group and individual CBT for depression: a naturalistic clinical study. Behavioural and Cognitive Psychotherapy, 39 (3). pp. 355-358. ISSN 1352-4658. (doi:10.1017/S135246581000072X) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32416) Brown, Pamela, Challis, David J., von Abendorff, Richard (1996) The work of a community mental health team for the elderly: Referrals, caseloads, contact history and outcomes. International Journal of Geriatric Psychiatry, 11 (1). pp. 29-39. ISSN 0885-6230. (doi:10.1002/(SICI)1099-1166(199601)11:1<29::AID-GPS265>3.0.CO;2-K) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:18846) Brown, Pamela and von Abendorff, Richard and Challis, David J. and Chesterman, John (1998) Extending the limits of care management: an evaluation of the Lewisham project. In: Services Development Centre, Dementia, ed. Conference Proceedings: Dementia Care: can we afford the risk? Dementia Services Development Centre, Sydney, pp. 29-38. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26433) Buckingham, Ken, Bebbington, Andrew, Campbell, Sue, Dennis, Chris, Evans, Graham R., Freeman, Peter, Martin, Neil D. T., Olver, Lynn (1996) Interim Needs Indicators for Community Health Services. PSSRU, Canterbury, 165 pp. ISBN 904938891. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27414) Burge, Peter, Gallo, Frederico, Netten, Ann (2006) Valuing PSS outputs and quality changes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27223) Burge, Peter, Netten, Ann, Gallo, Federico (2010) Estimating the value of social care. Journal of Health Economics, 29 (6). pp. 883-894. ISSN 0167-6296. (doi:10.1016/j.jhealeco.2010.08.006) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:34477) Buszewicz, Marta and Griffin, Mark and Beecham, Jennifer and Bonin, Eva-Maria and Hutson, Madeline (2011) ProCEED: report of a study of proactive care by practice nurses for people with depression and anxiety. Project report. Mind (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32455) Buszewicz, Marta, Griffin, Mark, McMahon, Elaine, Beecham, Jennifer, King, Michael (2010) Evaluation of a system of structured, pro-active care for chronic depression in primary care: A randomised controlled trial. BMC Psychiatry, 10 (61). pp. 1-9. ISSN 1471-244X. (doi:10.1186/1471-244X-10-61) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:34696) Bäumker, Theresia, Callaghan, Lisa, Darton, Robin, Holder, Jacquetta, Netten, Ann, Towers, Ann-Marie (2012) Deciding to move into extra care housing: residents’ views. Ageing and Society, 32 (7). pp. 1215-1245. ISSN 0144-686X. (doi:10.1017/S0144686X11000869) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32409) Bäumker, Theresia, Darton, Robin, Netten, Ann, Holder, Jacquetta (2011) Development costs and funding of extra care housing. Public Money & Management, 31 (6). pp. 411-418. ISSN 0954-0962. (doi:10.1080/09540962.2011.618765) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32410) Bäumker, Theresia and Netten, Ann (2011) Funding: paying for residential care for older people. In: Dening, Tom and Milne, Alisoun, eds. Mental Health & Care Homes. Oxford University Press, Oxford, pp. 89-100. ISBN 978-0-19-959363-7. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32405) Bäumker, Theresia and Netten, Ann (2011) The cost of extra care housing. In: Curtis, Lesley A., ed. Unit Costs of Health and Social Care 2011. Personal Social Services Research Unit, University of Kent, Canterbury, pp. 7-11. ISBN 978-1-902671-74-1. (KAR id:32404) Preview Bäumker, Theresia, Netten, Ann, Darton, Robin (2008) Costs and Outomes of an Extra-Care Housing Scheme in Bradford. Funded/commissioned by: Joseph Rowntree Foundation. Personal Social Services Research Unit ISBN 978185935. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27122) Bäumker, Theresia, Netten, Ann, Darton, Robin (2010) Costs and outcomes of an extra care housing scheme in England. Journal of Housing for the Elderly, 24 (2). pp. 151-170. ISSN 1540-353X. (doi:10.1080/02763891003757098) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:24894) Bäumker, Theresia, Netten, Ann, Darton, Robin (2008) Costs and outcomes of an extra-care housing scheme in Bradford. Joseph Rowntree Foundation, 49 pp. ISBN 978185935. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:15495) Bäumker, Theresia, Netten, Ann, Darton, Robin, Callaghan, Lisa (2011) Evaluating Extra Care Housing in England for older people: A comparative cost and outcome analysis with residential care. Journal of Service Science & Management, 4 . pp. 523-539. ISSN 1940-9893. (doi:10.4236/jssm.2011.44060) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32411) ## C Caiels, James, Forder, Julien E., Malley, Juliette, Netten, Ann, Windle, Karen (2010) Measuring the outcomes of low-level services: final report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27224) Caiels, James and Fox, Diane and McCarthy, Michelle and Smith, Nick and Malley, Juliette and Beadle-Brown, Julie and Netten, Ann and Towers, Ann-Marie (2010) Development studies for the National Adult Social Care User Experience Survey: technical report. Technical report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27225) Caiels, James, Rand, Stacey, Crowther, Tanya, Collins, Grace, Forder, Julien E. (2019) Exploring the views of being a proxy from the perspective of unpaid carers and paid carers: developing a proxy version of the Adult Social Care Outcomes Toolkit (ASCOT). BMC Health Services Research, 19 (1). ISSN 1472-6963. (doi:10.1186/s12913-019-4025-1) (KAR id:73332) Preview Callaghan, Lisa (2008) Social Well-Being in Extra-Care Housing: An Overview of the Literature (PSSRU Discussion Paper 2528). Personal Social Services Research Unit, University of Kent, 35 pp. (KAR id:13334) Preview Callaghan, Lisa, Brookes, Nadia, Palmer, Sinead (2017) Older people receiving family-based support in the community: A survey of quality of life amongst users of ‘Shared Lives’ in England. Health & Social Care in the Community, 25 (5). pp. 1655-1666. ISSN 0966-0410. E-ISSN 1365-2524. (doi:10.1111/hsc.12422) (KAR id:52924) Preview Callaghan, Lisa and Brookes, Nadia and Rider, Sinead (2015) Developing an outcome measurement tool for Shared Lives (PSSRU DP 2895). Discussion paper. PSSRU (Unpublished) (KAR id:52950) Preview Callaghan, Lisa, Netten, Ann, Darton, Robin (2009) The development of social well-being in new extra care housing schemes. Funded/commissioned by: Joseph Rowntree Foundation. Personal Social Services Research Unit ISBN 978-1-85935-723-1. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27131) Callaghan, Lisa, Netten, Ann, Darton, Robin, Bäumker, Theresia, Holder, Jacquetta (2008) Social Well-Being in Extra-Care Housing: Emerging Themes. Interim Report for the Joseph Rowntree Foundation (PSSRU Discussion Paper 2524/2). Personal Social Services Research Unit, University of Kent, 109 pp. (KAR id:13330) Preview Callaghan, Lisa, Towers, Ann-Marie (2014) Feeling in control: comparing older people’s experiences in different care settings. Ageing and Society, 34 (8). pp. 1427-1451. ISSN 0144-686X. (doi:10.1017/S0144686X13000184) (KAR id:35251) Preview Cambridge, Paul (1987) Care in the community: management and costs - some emerging issues. In: Care in the Community: Partnership with the Mentally Handicapped. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27116) Cambridge, Paul (1992) Case management in community care services: organizational responses. British Journal of Social Work, 22 (5). pp. 495-517. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26710) Cambridge, Paul (1988) From micro budgeting to agency pluralism: some lessons from the Care in the Community initiative. In: UNSPECIFIED, 1988-11-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27112) Cambridge, Paul (1988) Griffiths and care in the community. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27228) Cambridge, Paul (1987) Risk-taking and the care in the community projects: a perspective on the different types and levels of risk. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27229) Cambridge, Paul (1988) Staffing care in the community. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27227) Cambridge, Paul (1989) The information consequences of monitoring and evaluating a central policy initiative. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27226) Cambridge, Paul, Carpenter, John, Beecham, Jennifer, Hallam, Angela, Knapp, Martin R J., Forrester-Jones, Rachel, Tate, Alison (2002) Twelve years on: the long-term outcomes and costs of deinstitutionalisation and community care for people with learning disabilities. Tizard Learning Disability Review, 7 (3). pp. 34-42. ISSN 1359-5474. (doi:10.1108/13595474200200027) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26712) Cambridge, Paul, Carpenter, John, Forrester-Jones, Rachel, Tate, Alison, Knapp, Martin R J., Beecham, Jennifer, Hallam, Angela (2005) The state of care management in learning disability and mental health services twelve years into community care. British Journal of Social Work, 37 (7). pp. 1039-1062. ISSN 0045-3102. (doi:10.1093/bjsw/bch219) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26713) Cambridge, Paul, Hayes, L., Knapp, Martin R J. (1994) Care in the Community: Five Years On. Ashgate, Aldershot, 121 pp. ISBN 1-85742-275-9. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27405) Cambridge, Paul, Knapp, Martin R J. (1997) At what cost? Using cost information for purchasing and providing community care for people with learning disabilities. British Journal of Learning Disabilities, 25 (1). pp. 7-12. ISSN 1354-4187. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26711) Cambridge, Paul, Knapp, Martin R J. (1988) Demonstrating Successful Care in the Community. PSSRU, Canterbury ISBN 904938018. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27419) Cameron, Ailsa M, Johnson, Eleanor K, Evans, Simon, Lloyd, Liz, Darton, Robin, Smith, Randall C, Porteus, Jeremy, Atkinson, Teresa (2020) ‘You have got to stick to your times’: Care workers and managers’ experiences of working in extra care housing. Health and Social Care in the Community, 28 (2). pp. 396-403. ISSN 0966-0410. E-ISSN 1365-2524. (doi:10.1111/hsc.12871) (KAR id:76781) Preview Cameron, Ailsa, Johnson, Eleanor K, Lloyd, Liz, Evans, Simon, Smith, Randall, Porteus, Jeremy, Darton, Robin, Atkinson, Teresa (2019) Using longitudinal qualitative research to explore extra care housing. International Journal of Qualitative Studies on Health & Well-being, 14 (1). pp. 1-10. ISSN 1748-2623. E-ISSN 1748-2631. (doi:10.1080/17482631.2019.1593038) (KAR id:72862) Preview Preview Cardi, Valentina, Ambwani, Suman, Robinson, Emily, Albano, Gaia, MacDonald, Pamela, Aya, Viviana, Rowlands, Katie, Todd, Gill, Schmidt, Ulrike, Landau, Sabine, and others. (2017) Transition Care in Anorexia Nervosa Through Guidance Online from Peer and Carer Expertise (TRIANGLE): Study Protocol for a Randomised Controlled Trial. European Eating Disorders Review, 25 (6). pp. 512-523. ISSN 1072-4133. E-ISSN 1099-0968. (doi:10.1002/erv.2542) (KAR id:65291) Preview Castelli, Adriana and Nizalova, Olena (2011) Avoidable Mortality: What It Means And How It Is Measured. Project report. University of York (KAR id:35673) Preview Chadwick, Oliver, Beecham, Jennifer, Piroth, Nicola, Bernard, Sarah, Taylor, Eric (2002) Respite care for children with severe intellectual disability and their families: who needs it? Who receives it? Child and Adolescent Mental Health, 7 (2). pp. 66-72. ISSN 1475-357X. E-ISSN 1475-3588. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26714) Challis, D.J. and Davies, B.P. and Traske, K. (1994) Community Care in the UK and Overseas: New Agendas and Challenges. Ashgate, Aldershot ISBN 1-85742-208-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26643) Challis, David J. (1992) CASE-MANAGEMENT IN SOCIAL-WORK-PRACTICE - ROSE,SM. British Journal of Social Work, 22 (5). pp. 596-597. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:23251) Challis, David J. (1992) PROVIDING ALTERNATIVES TO LONG-STAY HOSPITAL-CARE FOR FRAIL ELDERLY PATIENTS - IS IT COST-EFFECTIVE. International Journal of Geriatric Psychiatry, 7 (11). pp. 773-781. ISSN 0885-6230. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:23250) Challis, David J., Carpenter, G. Iain, Traske, Karen (1996) Assessment in Continuing Care Homes: Towards a National Standard Instrument. PSSRU, Canterbury ISBN 0-904938-93-X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27420) Challis, David J. and Chesterman, John and Darton, Robin and Traske, Karen (1992) Case management in social and health care. In: Via, J.M. and Portella, E., eds. La Sociedad ante el envejecimiento y la minusvalía, vol. 1. Proceedings of SYSTED 91 - 4th International Conference, Barcelona, Spain, 10-14 June 1991. SG Editores, Barcelona, pp. 129-146. ISBN 8.487621082E9. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26443) Challis, David J. and Chesterman, John and Darton, Robin and Traske, Karen (1993) Case management in the care of the aged: the provision of care in different settings. In: Bornat, Joanna and Pereira, Charmaine and Pilgrim, David and Williams, Fiona, eds. Community Care: A Reader. Macmillan, Basingstoke and London, pp. 184-203. ISBN hbk: 0333587146 / pbk: 0333587154. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26444) Challis, David J., Chesterman, John, Darton, Robin, Traske, Karen (1992) Case management in the care of the aged: the provision of care in different settings. Johoshiryoshitsu Dayori [Bulletin of the Tokyo Metropolitan Welfare Information Centre], . pp. 21-26. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26723) Challis, David J., Chesterman, John, Luckett, R., Stewart, Karen, Chessum, R. (2002) Care Management in Social and Primary Health Care: The Gateshead Community Care Scheme. Ashgate, Aldershot, 274 pp. ISBN 1-85742-206-6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27429) Challis, David J. and Darton, Robin (1990) Evaluation Research and Experiment in Social Gerontology. In: Peace, S.M., ed. Researching Social Gerontology: Concepts, Methods and Issues. Sage, London. ISBN hbk: 0803982844 / pbk: 0803982852. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26453) Challis, David J., Darton, Robin, Hughes, Jane, Huxley, Peter, Stewart, Karen (1998) Emerging models of care management for older people and those with mental health problems in the United Kingdom. Journal of Case Management, 7 (4). pp. 153-160. ISSN 1061-3706. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26726) Challis, David J., Darton, Robin, Hughes, Jane, Stewart, Karen, Weiner, Kate (1998) Care Management Study: Report on National Data. Mapping and Evaluation of Care Management Arrangements for Older People and Those with Mental Health Problems. Funded/commissioned by: Department of Health, Social Services Inspectorate Care Management Study. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27132) Challis, David J. and Darton, Robin and Hughes, Jane and Stewart, Karen and Weiner, Kate (1999) Case or Care Management in the UK: Material for the Project Case Management in Various National Elderly Assistance Systems. In: Engel, Heike and Engels, Dietrich, eds. Case Management in Various National Elderly Assistance Systems. ISG Sozialforschung und Gesellschaftspolitik GmbH, Kohlhammer, Cologne. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26450) Challis, David J., Darton, Robin, Hughes, Jane, Stewart, Karen, Weiner, Kate (2001) Intensive Care Management at Home: An Alternative to Institutional Care? Age and Ageing, 30 (5). pp. 409-413. ISSN 0002-0729. (doi:10.1093/ageing/30.5.409) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26728) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, M. (1988) Multi-skilled carers in the community. Social Work Today, 19 (48). pp. 13-15. ISSN ISSN: 00378070. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27022) Challis, David J. and Darton, Robin and Johnson, Lynne and Stone, M. (1988) Services, Resource Management and the Integration of Health and Social Care in the Darlington Project for Elderly People. In: Cambridge, Paul and Knapp, Martin R J., eds. Demonstrating Successful Care in the Community. PSSRU, Canterbury, pp. 41-45. ISBN 904938018. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26440) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, Malcolm, Traske, Karen (1995) Care Management and Health Care of Older People: The Darlington Community Care Project. Ashgate, Aldershot, 380 pp. ISBN 1-85742-190-6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27407) Challis, David J. and Darton, Robin and Johnson, Lynne and Stone, Malcolm and Traske, Karen (1990) Case management in health and social care: an evaluation of an alternative to long-stay hospital care for frail elderly people. In: Puliafito, P.P., ed. Proceedings of SYSTED 90 Third International Conference on Systems Science in Health-Social Services for the Elderly and the Disabled. Dipartimento Sicurezza Sociale, Regione Emilia-Romagna, Bologna, Italy, 17-21 April, pp. 457-463. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26441) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, Malcolm, Traske, Karen (1991) Evaluación de una alternativa al hospital de larga estancia para pacientes ancianos débiles: I. El modelo de asistencia. Revista de Gerontologia, 1 (4th Ma). pp. 182-190. ISSN 1130-6882. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26730) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, Malcolm, Traske, Karen (1991) Evaluación de una alternativa al hospital de larga estancia para pacientes débiles: II. Costes y eficacia. Revista de Gerontologia, 1 (4th Ma). pp. 191-199. ISSN 1130-6882. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26731) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, Malcolm, Traske, Karen (1990) Supporting Frail Elderly People at Home: The Darlington Community Care Project. Funded/commissioned by: Joseph Rowntree Memorial Trust. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27133) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, Malcolm, Traske, Karen (1991) An evaluation of an alternative to long-stay hospital care for frail elderly patients: Part 2. Costs and effectiveness. Age and Ageing, 20 (4). pp. 245-254. ISSN 0002-0729. (doi:10.1093/ageing/20.4.245) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26722) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, Malcolm, Traske, Karen (1991) An evaluation of an alternative to long-stay hospital care for frail elderly patients: Part I. The model of care. Age and Ageing, 20 (4). pp. 236-244. ISSN 0002-0729. (doi:10.1093/ageing/20.4.236) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26721) Challis, David J., Darton, Robin, Johnson, Lynne, Stone, Malcolm, Traske, Karen, Wall, Barbara (1989) The Darlington Community Care Project: Supporting Frail Elderly People at Home. PSSRU, Canterbury, 66 pp. ISBN 904938042. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27431) Challis, David J. and Darton, Robin and Stewart, Karen (1998) The Darlington Study: Findings and Lessons for Care Management, Health Care and Community Care. In: Challis, David J. and Darton, Robin and Stewart, Karen, eds. Community Care, Secondary Health Care and Care Management. Ashgate, Aldershot, pp. 37-56. ISBN 1-84014-581-1. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26448) Challis, David J. and Darton, Robin and Stewart, Karen (1998) Introduction. In: Challis, David J. and Darton, Robin and Stewart, Karen, eds. Community Care, Secondary Health Care and Care Management. Ashgate, Aldershot, pp. 1-10. ISBN 1-84014-581-1. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26447) Challis, David J. and Darton, Robin and Stewart, Karen (1998) Linking Community Care and Health Care: A New Role for Secondary Health Care Services. In: Challis, David J. and Darton, Robin and Stewart, Karen, eds. Community Care, Secondary Health Care and Care Management. Ashgate, Aldershot, pp. 149-170. ISBN 1-84014-581-1. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26449) Challis, David J., Davies, Bleddyn P. (1986) Case Management in Community Care. Ashgate, Aldershot, 97 pp. ISBN hbk: 0566052873 / pbk: 0566058162. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27411) Challis, David J. and Davies, Bleddyn P. (1981) Community care projects: costs and effectiveness. In: Henrard, Jean Claude, ed. Les Colloques de L'INSERM, Vol. 101. INSERM, Paris, pp. 317-326. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26435) Challis, David J. and Davies, Bleddyn P. (1984) Community care schemes: a development in the home care of the frail elderly. In: Grimley Evans, J. and Caird, F.I., eds. Advanced Geriatric Medicine 4. Pitman, London, pp. 135-144. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26436) Challis, David J., Davies, Bleddyn P. (1985) Decentralised budgeting and care for the elderly. Public Money, 5 (3). pp. 21-24. ISSN 0261-1252. (doi:10.1080/09540968509387344) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26717) Challis, David J., Davies, Bleddyn P. (1984) Home care of the frail elderly in the United Kingdom: matching resources to needs. Home Health Care Services Quarterly, 3 (4). pp. 89-108. ISSN 0162-1424. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26715) Challis, David J., Davies, Bleddyn P. (1985) Long term care for the elderly: the community care scheme. British Journal of Social Work, 15 (6). ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26718) Challis, David J. and Davies, Bleddyn P. (1991) Long-term care: service focussed and case management focussed models of care in the UK. In: Wells, L., ed. An Aging Population. University of Toronto, Canada, pp. 43-61. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26442) Challis, David J. and Davies, Bleddyn P. (1981) The community care project. In: Henrard, Jean Claude, ed. Les Colloques de l'INSERM, vol. 101. INSERM, Paris, pp. 301-316. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26434) Challis, David J. and Davies, Bleddyn P. (1985) A more comprehensive approach to care of the elderly: the community care approach. In: Wells, N. and Freer, C., eds. The Aging Population: Burden or Challenge? Macmillan, London, pp. 191-202. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26439) Challis, David J. and Davies, Bleddyn P. (1986) A more comprehensive approach to care of the elderly: the community care scheme. In: Mann, A., ed. Continuous Care for the Elderly in the United Kingdom: Services, Their Coordination and Innovations. Columbia University Press, New York, pp. 2010-10. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26438) Challis, David J., Davies, Bleddyn P. (1980) A new approach to community care for the elderly. British Journal of Social Work, 10 (1). ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26716) Challis, David J., Knapp, Martin R J. (1990) Efficiency, effectiveness and quality of care in old people's homes in Scotland: a discussion of some of the key study issues. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27231) Challis, David J., Knapp, Martin R J. (1980) An examination of the PGC morale scale in an English context. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27233) Challis, David J. and Knapp, Martin R J. and Davies, Bleddyn P. (1984) Cost effectiveness evaluation in social care. In: Lishman, J., ed. Research Highlights No. 8: Evaluation. University of Aberdeen, Aberdeen, pp. 123-139. ISBN 1-85302-006-0. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26437) Challis, David J. and Netten, Ann (2005) Continuing Care: Policy and Context. In: Roe, Brenda and Beech, Roger, eds. Intermediate and Continuing Care: Policy and Practice. Wiley-Blackwell, Oxford, pp. 139-154. ISBN 978-1-4051-2033-3. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26451) Challis, David J., Schneider, Justine (1996) Commentary. Community Care Management and Planning, 4 (2). pp. 58-59. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26724) Challis, David J., Sutcliffe, Caroline, Hughes, Jane, von Abendorff, Richard, Brown, Pamela, Chesterman, John (2009) Supporting People with Dementia at Home. Challenges and Opportunities for the 21st Century. Ashgate, 242 pp. ISBN 978-0-7546-7479-5. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27440) Challis, David J., Tong, M.S., Traske, Karen (1988) Salisbury Health Authority - survey of the elderly 1988. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27232) Challis, David J. and Traske, Karen (1997) Community care. In: Mayer, Peter P. and Dickinson, Edward J. and Sandler, Martin, eds. Quality Care for Elderly People. Chapman and Hall, London, pp. 97-116. ISBN 0-412-61830-3. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26445) Challis, David J., Warburton, R.W. (1996) Performance indicators for community-based social care: from theory to practice. Care Plan, Care Management Practice Series No. 2, 2 (4). pp. 32-34. ISSN 1355-0454. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26725) Challis, David J., Weiner, Kate, Darton, Robin, Hughes, Jane, Stewart, Karen (2001) Emerging Patterns of Care Management: Arrangements for Older People in England. Social Policy & Administration, 35 (6). pp. 672-687. ISSN 0144-5596. (doi:10.1111/1467-9515.00260) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26727) Challis, David J., von Abendorff, Richard (1992) Lewisham Case Management Scheme report to management group. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27230) Challis, David J. and von Abendorff, Richard and Brown, Pamela and Chesterman, John (1997) Care management and dementia: an evaluation of the Lewisham intensive case management scheme. In: Hunter, Susan, ed. Dementia: Challenges and New Directions. Research Highlights in Social Work 31. Jessica Kingsley, London, pp. 139-164. ISBN 1-85302-312-4. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26446) Challis, David J., von Abendorff, Richard, Brown, Pamela, Chesterman, John, Hughes, Jane (2002) Care management, dementia care and specialist mental health services: an evaluation. International Journal of Geriatric Psychiatry, 17 (4). pp. 315-325. ISSN 0885-6230. (doi:10.1002/gps.595) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26729) Charlesworth, Georgina, Burnell, Karen, Beecham, Jennifer, Hoare, Zoë, Hoe, Juanita, Wenborn, Jennifer, Knapp, Martin R J., Russell, Ian, Woods, Bob, Orrell, Martin and others. (2011) Peer support for family carers of people with dementia, alone or in combination with group reminiscence in a factorial design: study protocol for a randomised controlled trial. Trials, 12 (205). ISSN 1745-6215. (doi:10.1186/1745-6215-12-205) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32417) Charnley, Helen (1990) Case managers and consumers in the social production of welfare: expectations, interventions and outcomes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27234) Charnley, Helen, Baines, Barry, Bebbington, Andrew, Davies, Bleddyn P., Lawson, Robyn, Netten, Ann, Whitley, A. (1989) Deciding one's destiny at the drawing board. Social Services Insight, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26732) Charnley, Helen, Davies, Bleddyn P. (1986) Blockages and the performance of the core tasks of case management. In: UNSPECIFIED, 1986. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27118) Chessum, R. (1983) Polly finds a family (Gateshead Community Care Scheme). Social Work Today, 15 (6). pp. 20-21. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26733) Chester Evans, Simon, Atkinson, Teresa, Cameron, Ailsa, Johnson, Eleanor K., Smith, Randall, Darton, Robin, Porteus, Jeremy, Lloyd, Liz (2018) Can Extra Care Housing support the changing needs of older people living with dementia? Dementia, . ISSN 1471-3012. E-ISSN 1741-2684. (doi:10.1177/1471301218801743) (KAR id:69079) Preview Chesterman, John (1985) Software for facilitating direct input of data off sets of forms. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27238) Chesterman, John (1988) A summary of time trends in the Kent Community Care Scheme. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27236) Chesterman, John, Bauld, Linda, Judge, Ken F. (2000) Satisfaction with the care-managed support of older people: an empirical analysis. Health & Social Care in the Community, 9 (1). pp. 31-42. ISSN 0966-0410. (doi:10.1046/j.1365-2524.2001.00280.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26735) Chesterman, John and Challis, David J. and Davies, Bleddyn P. (1994) Budget-devolved care management in two routine programmes. Have they improved outcomes? In: Challis, David J. and Davies, Bleddyn P. and Traske, Karen, eds. Community Care in the UK and Overseas: New Agendas and Challenges. Ashgate, Aldershot, pp. 173-188. ISBN 1-85742-208-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26454) Chesterman, John, Challis, David J., Davies, Bleddyn P. (1988) Long-term care at home for the elderly: a four-year follow-up. British Journal of Social Work, 18 (S). pp. 43-53. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26734) Chesterman, John, Davies, Bleddyn P., Challis, David J. (1992) Costs and welfare outcomes of case managed community-care for the frail elderly in two routine programmes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27235) Chesterman, John, Tong, M.S. (1986) Data entry users manual. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27237) Chisholm, Daniel, Healey, Andrew T., Knapp, Martin R J. (1997) QALYs and mental health care. Social Psychiatry and Psychiatric Epidemiology, 32 (2). pp. 68-75. ISSN 0933-7954. (doi:10.1007/BF00788923) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26736) Chisholm, Daniel and Knapp, Martin R J. and Astin, Jack (1996) Mental health residential care: is there a London differential? In: Netten, Ann and Dennett, Jane, eds. Unit Costs of Health and Social Care 1996. PSSRU, Canterbury, pp. 19-22. ISBN 904938956. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26455) Chisholm, Daniel, Knapp, Martin R J., Astin, Jack, Audini, Bernard, Lelliott, Paul (1997) The mental health residential care study: the 'hidden costs' of provision. Health & Social Care in the Community, 5 . pp. 162-172. ISSN 0966-0410. (doi:10.1111/j.1365-2524.1997.tb00111.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26738) Chisholm, Daniel, Knapp, Martin R J., Astin, Jack, Beecham, Jennifer, Audini, Bernard, Lelliott, Paul The mental health residential care study: the costs of provision. Journal of Mental Health, 6 (1). pp. 85-99. ISSN 0963-8237 (Print) 1360-0567 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26740) Chisholm, Daniel, Knapp, Martin R J., Astin, Jack, Lelliott, Paul, Audini, Bernard (1997) The mental health residential care study: predicting costs from resident characteristics. British Journal of Psychiatry, 170 (1). pp. 37-42. ISSN 0007-1250 (Print) 1472-1465 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26739) Chisholm, Daniel, Knapp, Martin R J., Knudsen, Helle Charlotte, Amaddeo, Francesco, Gaite, Luis, van Wijngaarden, Bob, the EPSILON Study Group (2000) Client Socio-Demographic and Service Receipt Inventory - European Version: development of an instrument for international research. EPSILON Study 5. British Journal of Psychiatry, 177 (Supple). pp. 28-33. ISSN 0007-1250. (doi:10.1192/bjp.177.39.s28) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26741) Chisholm, Daniel and Knapp, Martin R J. and Lowin, Ana (1997) Mental health care in London: costs. In: Johnson, S., ed. London's Mental Health. King's Fund, London, pp. 305-330. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26456) Chisholm, Daniel, Lowin, Ana, Knapp, Martin R J. (1997) Costing London's mental health services. Mental Health Research Review, 4 . pp. 6-9. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27059) Chisholm, Daniel, Stewart, A.S. (1998) Economics and ethics in mental health care: traditions and trade-offs. Journal of Mental Health Policy and Economics, 1 . pp. 55-62. ISSN 1091-4358 (Print) 1099-176X (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26737) Christensen, Karen, Hussein, Shereen, Ismail, Mohamed (2017) Migrants’ decision-process shaping work destination choice: the case of long-term care work in the United Kingdom and Norway. European Journal of Ageing, 14 (3). pp. 219-232. ISSN 1613-9372. (doi:10.1007/s10433-016-0405-0) (KAR id:68305) Preview Clark, D., Sivam, A., Lennon, J., Wingfield, J., Owen, P., Darton, Robin (1988) Survey of Private and Voluntary Homes for the Elderly, 1986. Cornwall Social Services in association with the PSSRU. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27239) Clark, D., Sivam, A., Owen, P., Darton, Robin (1985) Survey of Local Authority Residential Homes for the Elderly, 1985. Cornwall Social Services with the PSSRU. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27240) Clarke, John, Newman, Janet, Smith, Nick, Vidler, Elizabeth, Westmarland, Louise (2007) Creating Citizen-Consumers: Changing publics and changing public services. Sage, London, 192 pp. ISBN 978-1-4129-2134-3. E-ISBN 978-1-84787-864-9. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:44790) Clarke, John and Smith, Nick and Vidler, Elizabeth (2005) Consumerism and the reform of public services: inequalities and instabilities. In: Social Policy Review 17: Analysis and debate in social policy. The Policy Press, Bristol, pp. 167-182. ISBN 1-86134-669-7. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:44825) Clarke, John, Smith, Nick, Vidler, Elizabeth (2004) Consumerism and the reform of public services: inequalities and instabilities. In: Social Policy Association Annual Conference, 13th-15th July 2004, The University of Nottingham. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:45140) Clarke, John, Smith, Nick, Vidler, Elizabeth (2006) The Indeterminacy of Choice: Political, Policy and Organisational Implications. Social Policy and Society, 5 (03). pp. 327-336. ISSN 1474-7464. E-ISSN 1475-3073. (doi:10.1017/S1474746406003010) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:44829) Clarke, Susan, Sloper, Patricia, Moran, Nicola, Cusworth, Linda, Franklin, Anita, Beecham, Jennifer (2011) Multi-agency transition services: greater collaboration needed to meet the priorities of young disabled people with complex needs as they move into adulthood. Journal of Integrated Care, 19 (5). pp. 74-85. ISSN 1476-9018. (doi:10.1108/14769011111176734) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32418) Clarkson, Paul C., McCrone, Paul (1998) Quality of life and service utilisation of psychotic patients in South London - The PRiSM Study. Journal of Mental Health, 7 (1). pp. 71-80. ISSN 0963-8237 (Print) 1360-0567 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26743) Clarkson, Paul C., McCrone, Paul, Sutherby, K., Johnson, Christine, Johnson, Sonia, Thornicroft, Graham (1999) Outcomes and costs of a community support worker service for the severely mentally ill. Acta Psychiatrica Scandinavica, 99 (3). pp. 196-206. ISSN 0001-690X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26742) Collins, Grace, Brookes, Nadia, Callaghan, Lisa, Palmer, Sinead (2019) Shared Lives: Evidence of Effectiveness: a multi-method study of a community service for people with intellectual disabilities. In: School for Social Care Research (SSCR) Conference, 11 Apr 2019, London. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:74062) Collins, Grace, Brookes, Nadia, Palmer, Sinead, Callaghan, Lisa (2018) Shared Lives: Evidence of Effectiveness: a multi-method study of a community service for people with intellectual disabilities. In: Journal of Applied Research in Intellectual Disabilities. 31 (4). Wiley (doi:10.1111/jar.12486) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:77771) Collins, Grace, Towers, Ann-Marie, Smith, Nick, Palmer, Sinead (2016) Conducting research in care homes during times of austerity: lessons learned from two research studies involving care homes for older adults in England. In: BSG Conference, July 2016, Stirling, Scotland. (KAR id:63192) Collins, Grace, Towers, Ann-Marie, Smith, Nick, Palmer, Sinead, Naick, Madeline, Babaian, Jacinta (2018) Engaging with care home managers. In: British Society of Gerontology Conference, 4-6 Jul 2018, Manchester. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:74063) Collins, Grace, Towers, Ann-Marie, Smith, Nick, Palmer, Sinead, Naick, Madeline, Babaian, Jacinta (2018) MOOCH Symposium: Engaging with care home managers. In: BSG Conference 2018, 4-6 Jul 2018, Manchester. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:77772) Comas-Herrera, Adelina, Knapp, Martin R J., Beecham, Jennifer, Pendaries, Claude, Carthew, Richard (2001) Benefit groups and resource groups for adults with intellectual disabilities in residential accommodation. Journal of Applied Research in Intellectual Disabilities, 14 (2). pp. 120-140. ISSN 1360-2322. (doi:10.1046/j.1468-3148.2001.00061.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26744) Comas-Herrera, Adelina, Knapp, Martin R J., Beecham, Jennifer, Pendaries, Claude, Carthew, Richard (2000) Learning disability groups. Development of benefit and resource groups for adults in residential accommodation, Consultation document. Funded/commissioned by: NHS. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27134) Comas-Herrera, Adelina, Pickard, Linda, Wittenberg, Raphael, Davies, Bleddyn P., Darton, Robin (2003) Future Demand for Long-Term Care, 2001 to 2031: Projections of Demand for Long-Term Care for Older People in England. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27241) Comas-Herrera, Adelina, Wittenberg, Raphael, Pickard, Linda, Knapp, Martin R J., Davies, Bleddyn P. (2001) Cognitive impairment: its implications for future demand for services and costs. Mental Health Research Review, 8 . pp. 37-38. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26745) Cornes, Michelle, Manthorpe, Jill, Moriarty, Joanna, Blendi-Mahota, Saidah, Hussein, Shereen (2013) Assessing the effectiveness of policy interventions to reduce the use of agency or temporary social workers in England. Health and Social Care in the Community, 21 (3). pp. 236-244. ISSN 0966-0410. (doi:10.1111/hsc.12011) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:68336) Cornes, Michelle, Manthorpe, Jill, Moriarty, Joanna, Hussein, Shereen (2013) The experiences and perspectives of agency social workers in England: Findings from interviews with those working in adult services. Social Work & Social Sciences Review, 16 (1). pp. 67-83. ISSN 0953-5225. (doi:10.1921/703160106) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:68353) Cornes, Michelle and Moriarty, Jo and Blendi-Mahota, Saidah and Chittleburgh, Tim and Hussein, Shereen and Manthorpe, Jill (2010) Working for the Agency: The Role and Significance of Temporary Employment Agencies in the Adult Social Care Workforce. Project report. King's College London (KAR id:68383) Preview Cottrell, David J., Wright-Hughes, Alexandra, Collinson, Michelle, Boston, Paula, Eisler, Ivan, Fortune, Sarah, Graham, Elizabeth H., Green, Jonathan, House, Allan O., Kerfoot, Michael, and others. (2018) Effectiveness of systemic family therapy versus treatment as usual for young people after self-harm: a pragmatic, phase 3, multicentre, randomised controlled trial. The Lancet Psychiatry, 5 (3). pp. 203-216. ISSN 2215-0366. E-ISSN 2215-0374. (doi:10.1016/S2215-0366(18)30058-0) (KAR id:65837) +1 more... Cottrell, David and Wright-Hughes, Alexandra and Collinson, Michelle and Boston, Paula and Eisler, Ivan and Fortune, Sarah and Graham, Elizabeth and Green, Jonathan and House, Alan and Kerfoot, Michael and Owens, David and Saloniki, Eirini-Christina and Simic, Mima and Tubeuf, Sandy and Farrin, Amanda (2018) A pragmatic randomised controlled trial and economic evaluation of family therapy versus treatment as usual for young people seen after second or subsequent episodes of self-harm: the Self-Harm Intervention - Family Therapy (SHIFT) trial. Project report. National Institute for Health Research 10.3310/hta22120. (doi:10.3310/hta22120) (KAR id:66368) Preview Craig, Rachel and Darton, Robin and Hancock, Ruth and Henderson, Catherine and Morciano, Marcello and Sadler, Katherine and Wittenberg, Raphael (2012) Social Care. In: Craig, Rachel and Mindell, Jennifer, eds. Health Survey for England - 2011, Health, social care and lifestyles. Health and Social Care Information Centre, Leeds. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:41638) Curran, Claire, Knapp, Martin R J., Beecham, Jennifer (2004) Mental health and employment: some economic evidence. Journal of Mental Health Promotion, 3 (1). pp. 13-25. ISSN 1462-3730. (doi:10.1108/17465729200400003) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26746) Curtis, Lesley A., ed. (2008) Unit Costs of Health and Social Care 2008. Unit Costs of Health and Social Care, 16 . Personal Social Services Research Unit, University of Kent, Canterbury, 200 pp. ISBN 978-1-902671-61-1. (KAR id:15492) Preview Curtis, Lesley A., ed. (2009) Unit Costs of Health and Social Care 2009. Unit Costs of Health and Social Care . Personal Social Services Research Unit, University of Kent, Canterbury, 213 pp. ISBN 978-1-902671-64-2. (KAR id:24882) Preview Curtis, Lesley A., ed. (2011) Unit Costs of Health and Social Care 2011. Personal Social Services Research Unit, University of Kent, Canterbury, 230 pp. ISBN 978-1-902671-74-1. (KAR id:32407) Preview Curtis, Lesley A., ed. (2012) Unit Costs of Health and Social Care 2012. Personal Social Services Research Unit, Canterbury, Kent, 243 pp. ISBN 978-1-902671-82-6. (KAR id:32408) Preview Curtis, Lesley A., ed. (2013) Unit Costs of Health and Social Care 2013. Personal Social Services Research Unit, University of Kent, Canterbury ISBN 978-1-902671-87-1. (KAR id:41636) Preview Curtis, Lesley A. (2014) Unit Costs of Health and Social Care 2014. Other. Personal Social Services Research Unit, Canterbury (KAR id:46138) Preview Curtis, Lesley A. (2005) The costs of recuperative care housing. In: Curtis, Lesley A. and Netten, Ann, eds. Unit Costs of Health and Social Care 2005. Personal Social Services Research Unit, University of Kent, pp. 13-16. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26458) Curtis, Lesley A., Beecham, Jennifer (2018) A survey of Local Authorities and Home Improvement Agencies: identifying the hidden costs of providing a home adaptations service. British Journal of Occupational Therapy, 81 (11). pp. 633-640. ISSN 0308-0226. E-ISSN 1477-6006. (doi:10.1177/0308022618771534) (KAR id:66433) Preview Preview Curtis, Lesley A., Burns, Amanda (2015) Unit Costs of Health and Social Care 2015. Unit Costs of Health and Social Care. Report number: 3. Personal Social Services Research Unit, Kent, UK, 274 pp. ISBN 978-1-902671-96-3. (doi:3) (KAR id:60240) Preview Curtis, Lesley A., Burns, Amanda (2016) Unit Costs of Health and Social Care 2016. Unit Costs of Health and Social Care. Personal Social Services Research Unit, University of Kent, Kent, UK, 226 pp. ISBN 978-1-911353-02-7. (KAR id:60243) Preview Curtis, Lesley A., Burns, Amanda (2019) Unit Costs of Health and Social Care 2019. Unit Costs of Health and Social Care . PSSRU, Kent, UK, 176 pp. ISBN 978-1-911353-10-2. (doi:10.22024/UniKent/01.02.79286) (KAR id:81429) Preview Curtis, Lesley A., Moriarty, Jo, Netten, Ann (2012) The costs of qualifying a social worker. British Journal of Social Work, 42 (4). pp. 706-724. ISSN 0045-3102. (doi:10.1093/bjsw/bcr113) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32419) Curtis, Lesley A., Moriarty, Jo, Netten, Ann (2010) The expected working life of a social worker. British Journal of Social Work, 40 (5). pp. 1628-1643. ISSN 0045-3102. (doi:10.1093/bjsw/bcp039) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:20516) Curtis, Lesley A. and Netten, Ann, eds. (2004) Unit Costs of Health and Social Care 2004. Personal Social Services Research Unit, University of Kent ISBN 1902671392 ISSN: 1368230X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26655) Curtis, Lesley A. and Netten, Ann, eds. (2005) Unit Costs of Health and Social Care 2005. Personal Social Services Research Unit, University of Kent at Canterbury ISBN 1902671406 ISSN: 1368230X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26657) Curtis, Lesley A. and Netten, Ann, eds. (2006) Unit Costs of Health and Social Care 2006. Personal Social Services Research Unit, University of Kent, Canterbury ISBN 978-1-902671-45-1. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26658) Curtis, Lesley A., Netten, Ann (2005) Are pharmacists worth the investment in their training and the ongoing costs? Pharmaceutical Journal, 274 . pp. 275-278. ISSN 0961-7671. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26747) Curtis, Lesley A. and Netten, Ann (2004) Editorial: New developments and data sources for unit costs. In: Netten, Ann and Curtis, Lesley A., eds. Unit Costs of Health and Social Care 2004. Personal Social Services Research Unit, University of Kent, Canterbury. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26457) Curtis, Lesley A., Netten, Ann (2007) The costs of training a nurse practitioner in primary care: the importance of allowing for the cost of education and training when making decisions about changing the professional. Journal of Nursing Management, 15 (4). pp. 449-457. ISSN 0966-0429. (doi:10.1111/j.1365-2834.2007.00668.x) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:2809) Curtis, Lesley A., Robinson, Sarah, Netten, Ann (2009) Changing patterns of male and female nurses' participation in the workforce. Journal of Nursing Management, 17 (7). pp. 843-852. ISSN 0966-0429. (doi:10.1111/j.1365-2834.2009.00982.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:20517) Curtis, Lesley, Burns, Amanda (2017) Unit Costs of Health and Social Care 2017. Personal Social Services Research Unit, University of Kent, 247 pp. ISBN 978-1-911353-04-1. (KAR id:65559) Preview Curtis, Lesley, Burns, Amanda (2018) Unit Costs of Health and Social Care 2018. Report number: 10.22024/UniKent/01.02.70995. University of Kent, 201 pp. ISBN 978-1-911353-06-5. (doi:10.22024/UniKent/01.02.70995) (KAR id:70995) Preview Cylus, Jonathan and Roland, Daniel and Corbett, Jennie and Jones, Karen C. and Forder, Julien E. and Sussex, Jon (2018) Identifying options for funding the NHS and social care in the UK: international evidence. Working paper. The Health Foundation, London, United Kingdom (KAR id:69339) Preview ## D Dance, Cherilyn, Ouwejan, Danielle, Beecham, Jennifer, Farmer, Elaine (2010) Linking and matching: a survey of adoption agency practice in England and Wales. British Association of Adoption and Fostering, London, 198 pp. ISBN 978-1-905664-82-5. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:24880) Dansie, A., Beecham, Jennifer, Knapp, Martin R J. (1994) The cost of MOST. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27242) Darton, Robin (2002) Commentary 1 on Karen Glaser and Cecilia Tomassini (2002) Demography. Living arrangements, receipt of care, residential proximity and housing preferences among older people in Britain and Italy in the 1990s: an overview of trends. In: Sumner, Keith, ed. Our Homes, Our Lives: Choice in Later Life Living Arrangements. Centre for Policy on Ageing, London, pp. 99-104. ISBN 1-901097-85-4. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26466) Darton, Robin (2008) Evaluation of the Extra Care Housing Funding Initiative: Summary of Initial Findings. (PSSRU Research Summary 47). Project report. Personal Social Services Research Unit, University of Kent (KAR id:2913) Preview Darton, Robin (2012) Great expectations: feedback from relatives and residents. Nursing and Residential Care, 14 (10). pp. 534-538. ISSN 1465-9301. (doi:10.12968/nrec.2012.14.10.534) (KAR id:32420) Preview Darton, Robin (2008) Housing and Care for Older People Newsletter 2. Documentation. Personal Social Services Research Unit, University of Kent, Canterbury, United Kingdom (KAR id:2914) Preview Darton, Robin (1994) Length of stay of of residents and patients in residential and nursing homes for elderly people. Research, Policy and Planning, 12 (3). pp. 18-24. (KAR id:26753) Preview Darton, Robin (1992) Length of stay of residents and patients in residential and nursing homes. Statistical report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27248) Darton, Robin (1986) Methodological report of the PSSRU Survey of Residential Accommodation for the Elderly, 1981. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27256) Darton, Robin (1986) PSSRU Survey of Residential Accommodation for the Elderly, 1981. Characteristics of the residents. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27259) Darton, Robin (1986) PSSRU Survey of Residential Accommodation for the Elderly, 1981. Design features and facilities provided by the surveyed homes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27258) Darton, Robin (1986) PSSRU Survey of Residential Accommodation for the Elderly, 1981. General characteristics of the surveyed homes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27257) Darton, Robin (2001) Predicting admission to nursing home rather than to residential home care. Gerontology, 47 (S1). pp. 52-53. ISSN 0304-324X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26752) Darton, Robin (1990) Private and Voluntary Residential and Nursing Homes in Canterbury and Thanet. PSSRU, Canterbury, 36 pp. ISBN 904938077. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27398) Darton, Robin (1994) Review of recent research on elderly people in residential care and nursing homes, with specific reference to dependency. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27247) Darton, Robin (1980) Rotation in factor analysis. Statistician, 29 (3). pp. 167-194. ISSN 0039-0526. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26749) Darton, Robin (1989) Survey of Residential and Nursing Homes in Canterbury and Thanet, 1987. Survey Report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27250) Darton, Robin (1984) Trends 1970-81. In: Laming, Herbert, ed. Residential Care for the Elderly: Present Problems and Future Issues. Policy Studies Institute Discussion Paper No. 8, Policy Studies Institute, London, pp. 9-21. ISBN 853742413. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26459) Darton, Robin (2004) What types of home are closing? The characteristics of homes which closed between 1996 and 2001. Health & Social Care in the Community, 12 (3). pp. 254-264. ISSN 0966-0410. (doi:10.1111/j.1365-2524.2004.00495.x) (KAR id:798) Preview Darton, Robin (1984) The use of SIR at the PSSRU, with particular reference to a complex large-scale survey. In: Proceedings of the European SIR Users Group Meeting, London, 20-21 September. , pp. 53-68. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26461) Darton, Robin, Brown, Pamela (1997) Survey of Admissions to Residential Care: Analyses of Six Month Follow-Up. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27245) Darton, Robin, Bäumker, Theresia, Callaghan, Lisa, Holder, Jacquetta, Netten, Ann, Towers, Ann-Marie (2008) Evaluation of the Extra Care Housing Funding Initiative: Initial Report (PSSRU Discussion Paper 2506/2). Personal Social Services Research Unit, University of Kent, 103 pp. (KAR id:13326) Preview Darton, Robin, Bäumker, Theresia, Callaghan, Lisa, Holder, Jacquetta, Netten, Ann, Towers, Ann-Marie (2012) The characteristics of residents in extra care housing and care homes in England. Health & Social Care in the Community, 20 (1). pp. 87-96. ISSN 0966-0410. (doi:10.1111/j.1365-2524.2011.01022.x) (KAR id:32421) Preview Darton, Robin, Bäumker, Theresia, Callaghan, Lisa, Netten, Ann (2011) Improving housing with care choices for older people: The PSSRU evaluation of extra care housing. Housing, Care and Support, 14 (3). pp. 77-82. ISSN 1460-8790. (doi:10.1108/14608791111199741) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32422) Darton, Robin, Callaghan, Lisa (2009) The role of extra care housing in supporting people with dementia: Early findings from the PSSRU evaluation of extra care housing. Journal of Care Services Management, 3 (3 (Spe). pp. 284-294. ISSN 1750-1679. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:20518) Darton, Robin, Forder, Julien E., Bebbington, Andrew, Netten, Ann, Towers, Ann-Marie, Williams, Jacquetta (2005) Analysis to Support the Development of FSS Formulae for Older People: Interim Report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27244) Darton, Robin, Forder, Julien E., Bebbington, Andrew, Netten, Ann, Towers, Ann-Marie, Williams, Jacquetta (2006) Analysis to Support the Development of the Relative Needs Formula for Older People: Final Report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27243) Darton, Robin and Fox, Diane (2012) Admission Risk to Care Homes (ARCH) - Phase 1: Older People. Discussion paper. University of Kent (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:77659) Darton, Robin, Jefferson, S.F., Sutcliffe, E.M., Wright, Kenneth G. (1987) PSSRU/CHE Survey of Residential and Nursing Homes. Descriptive statistical report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27254) Darton, Robin, Jefferson, S.F., Sutcliffe, E.M., Wright, Kenneth G. (1987) PSSRU/CHE Survey of Residential and Nursing Homes. Descriptive statistical report commentary. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27255) Darton, Robin, Jefferson, S.F., Sutcliffe, E.M., Wright, Kenneth G. (1989) PSSRU/CHE Survey of Residential and Nursing Homes. Interview Questionnaire Descriptive Statistical Report Commentary. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27249) Darton, Robin, Jefferson, S.F., Sutcliffe, E.M., Wright, Kenneth G. (1988) PSSRU/CHE Survey of Residential and Nursing Homes. Interview Questionnaire Descriptive Statistics. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27253) Darton, Robin, Jefferson, S.F., Sutcliffe, E.M., Wright, Kenneth G. (1988) PSSRU/CHE Survey of Residential and Nursing Homes. The costs and charges of the surveyed homes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27252) Darton, Robin and Knapp, Martin R J. (1986) Factors associated with variations in the cost of local authority old people's homes. In: Judge, Ken F. and Sinclair, Ian, eds. Residential Care for Elderly People. HMSO, London, pp. 93-105. ISBN 113210604. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26460) Darton, Robin, Knapp, Martin R J. (1984) The cost of residential care for the elderly: the effects of dependency, design and social environment. Ageing and Society, 4 (2). pp. 157-183. ISSN 0144-686X. (doi:10.1017/S0144686X00010631) (KAR id:26750) Preview Darton, Robin, McCoy, P. (1981) Survey of Residential Accommodation for the Elderly. Clearing House for Local Authority Social Services Research (University of Birmingham), (2). pp. 23-44. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26748) Darton, Robin, Miles, Kathryn (1997) Cross-Sectional Survey of Residential and Nursing Homes for Elderly People. Comparisons of residents in residential and nursing homes for elderly people, 1981-1996. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27246) Darton, Robin and Muncer, Ann-Marie (2005) Alternative Housing and Care Arrangements: The Evidence. In: Roe, Brenda and Beech, Roger, eds. Intermediate and Continuing Care: Policy and Practice. Blackwell Publishing, Oxford, pp. 183-203. ISBN 978-1-4051-2033-3. (doi:10.1002/9780470774960.ch12) (KAR id:26467) Preview Darton, Robin, Netten, Ann (2007) The Development of Extra Care Housing in England: An Alternative or a Replacement for Residential and Nursing Homes? Published abstract. Gerontologist, 47 (SI1). p. 56. ISSN 0016-9013. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26754) Darton, Robin, Netten, Ann (2007) The Development of Extra Care Housing in England: An Alternative or a Replacement for Residential and Nursing Homes? In: Gerontologist. 47, Sp. p. 56. Oxford Journals (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2813) Darton, Robin, Netten, Ann, Brown, Pamela (1997) A Longitudinal Study of Admissions to Residential and Nursing Home Care Following the Community Care Reforms. In: British Society of Gerontology 26th Annual Conference, 19-21 September 1997, Bristol. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27090) Darton, Robin and Netten, Ann and Forder, Julien E. (2000) The Cost Implications of the Changing Population and Characteristics of Care Homes. In: Dickinson, A. and Bartlett, H. and Wade, S., eds. Old Age in a New Age: Proceedings of the British Society of Gerontology 29th Annual Conference. Oxford Brookes University, Oxford, UK, pp. 310-314. ISBN 1-902606-08-6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26465) Darton, Robin, Netten, Ann, Forder, Julien E. (2003) The cost implications of the changing population and the characteristics of care homes. International Journal of Geriatric Psychiatry, 18 (3). pp. 236-243. (doi:10.1002/gps.815) (KAR id:801) Preview Darton, Robin and Netten, Ann and Whitfield, G. (2000) Survey of Self-Funded Admissions to Residential and Nursing Homes. In: Fleiss, A., ed. Research Yearbook 1999/2000. Department of Social Security, Leeds: Corporate Document Services, pp. 57-66. ISBN 1-84123-286-6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26464) Darton, Robin, Sutcliffe, E.M., Wright, Kate (1989) PSSRU/CHE Survey of Residential and Nursing Homes. General Report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27251) Darton, Robin and Wright, Kate (1990) The Characteristics of Non-Statutory Residential and Nursing Homes. In: Parry, R., ed. Research Highlights in Social Work, 18. Privatisation. Jessica Kingsley Publishers, London, pp. 55-95. ISBN 1-85302-017-6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26462) Darton, Robin and Wright, Kate (1992) Residential and nursing homes for elderly people: one sector or two? In: Laczko, Frank and Victor, Christina R., eds. Social Policy and Elderly People: The Role of Community Care. Avebury, Aldershot, pp. 216-44. ISBN 1-85628-303-8. (KAR id:26463) Preview Darton, Robin, Wright, Kenneth G. (1993) Changes in the provision of long-stay care, 1970-1990. Health & Social Care in the Community, 1 (1). pp. 11-25. ISSN 0966-0410. (doi:10.1111/j.1365-2524.1993.tb00191.x) (KAR id:26751) Preview Davey, Vanessa and Snell, Tom and Fernández, José-Luis and Knapp, Martin R J. and Tobin, Roseanne and Jolly, Debbie and Perkins, Margaret and Kendall, Jeremy and Pearson, Charlotte and Vick, Nicola and Swift, Paul and Mercer, Geof and Priestley, Mark (2007) Schemes Providing Support to People Using Direct Payments: A UK Survey. Project report. Personal Social Services Research Unit, London (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:38998) Davidson, Jacqueline, Baxter, Kate, Glendinning, Caroline, Jones, Karen C., Forder, Julien E., Caiels, James, Welch, Elizabeth, Windle, Karen, Dolan, Paul, King, Dominic and others. (2012) Personal health budgets: experiences and outcomes for budget holders at nine months. Fifth interim report. Department of Health, 77 pp. (KAR id:77835) Preview Davies, Bleddyn P. (1991) A+B. Evaluating the impact of the new community care policy: the MSCEP project. In: Health, Social Policy and Ageing, 1991-09-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27111) Davies, Bleddyn P. (1987) Allocation of services in England: facts and myths about the equity and efficiency of social care agencies. In: UNSPECIFIED, 1987. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27114) Davies, Bleddyn P. (1986) American experiments to substitute homes for institutional long-term care: policy logic and evaluation methodology. In: Phillipson, Chris, ed. Dependency and Interdependency in Old Age. Croom Helm, London, pp. 180-197. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26480) Davies, Bleddyn P. (1986) American lessons for British policy and research on long-term care of the elderly. Quarterly Journal of Social Affairs, 2 (3). ISSN 0266-8548. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26768) Davies, Bleddyn P. (1997) Are post-reform British patterns of utilization of community care services more defensible? Gerontologist, 37 (SI1). p. 347. ISSN 0016-9013. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26791) Davies, Bleddyn P. (1979) Area management: long on promise, short on achievement: a review of R. Hambledon, Policy Planning in Local Government'. Journal of Social Policy, 8 (2). ISSN 0047-2794. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26760) Davies, Bleddyn P. (1964) Assessing local services. Oxford Magazine, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26793) Davies, Bleddyn P. (1994) Assuring quality in long-term case management in the community. In: Second International Conference on Long-Term Care Case Management. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27101) Davies, Bleddyn P. (1986) The Audit Commission and equity and efficiency research on the elderly. In: Value for Money, 1986-07-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27117) Davies, Bleddyn P. (1988) British community-based care: implications of evidence about the outcome consequences of resource variations. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27288) Davies, Bleddyn P. (1991) British evaluated care management programmes. implications for training. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27276) Davies, Bleddyn P. (1993) British home and community care: research-based critiques and the challenge of the new policy. Social Science and Medicine, 38 (7). pp. 883-903. ISSN 0277-9536. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26777) Davies, Bleddyn P. (1994) Capitated/premium financed risk-bearing managed care models for community and health care: are they of relevance to the UK? In: Davies, Bleddyn P. and Hobman, David and Hollingbery, Richard and Netten, Ann, eds. Building on EPICS Principles. Helen Hamlyn Foundation, London, pp. 19-30. ISBN 0-9524648-1-0. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26505) Davies, Bleddyn P. (1992) Care Management, Equity and Efficiency: The International Experience. PSSRU, Canterbury, 183 pp. ISBN 0-904938-28-X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27401) Davies, Bleddyn P. (1996) Care management in community care reform. In: Case Management in Community and Long-Term Care, 1996-10-01T00:00:00, Hong Kong. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27095) Davies, Bleddyn P. (1993) Caring for the frail elderly. II: Structures enabling trade-offs between ends and means. An international perspective. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27270) Davies, Bleddyn P. (1993) Caring for the frail elderly: an international perspective. Generations, 17 (4). pp. 51-54. ISSN 0897-019X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26782) Davies, Bleddyn P. (1997) Case management and case coordination in the United Kingdom. In: Davies, G.R. and Mykita, L.T. and Andrews, M.M. and Pearson, S.A. and Gregory, A.J. and Hagger, J.C., eds. Ageing Beyond 2000: One World, One Future. World Congress of Gerontology, Adelaide, p. 158. ISBN 0-646-32782-8. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26509) Davies, Bleddyn P. (1992) Case management and the social services: on breeding the best chameleons. Generations Review, 2 (2). pp. 18-22. ISSN 0965-2000. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26778) Davies, Bleddyn P. (1995) Case management for elderly people: the Kent Community Care Project (the KCCP') and its descendants in their international context. Hong Kong Journal of Gerontology, 9 (1). pp. 33-43. ISSN 1608-2346. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26785) Davies, Bleddyn P. (1993) Case management in British community care: could it and will it succeed? In: XVth Congress of the International-Association-of-Gerontology: Recent Advances in Aging Science, 04-09 Jul 1993, Budapest, Hungary. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26501) Davies, Bleddyn P. (1993) Case management in British community care: could it and will it succeed? Hong Kong Journal of Gerontology, 7 (2). pp. 17-21. ISSN 1608-2346. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26781) Davies, Bleddyn P. (1989) Case management in the UK: current debate and future prospects. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27281) Davies, Bleddyn P. (1988) Case management: why we must develop policy argument and a research agenda. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27287) Davies, Bleddyn P. (1975) Causal processes and techniques in the modelling of policy outcomes. In: Young, K., ed. Essays on the Study of Urban Politics. Macmillan, London, pp. 78-105. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26468) Davies, Bleddyn P. (1991) Checking the aim. Social Services Insight, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26779) Davies, Bleddyn P. (1984) Comments on the Audit Commission's draft guide. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27293) Davies, Bleddyn P. (1990) Comments on the Australian situation: a British perspective. In: Howe, Amanda and Ozanne, Elizabeth and Selby-Smith, C., eds. Community Care Policy and Practice: New Directions in Australia. Monash University Press, Melbourne, pp. 107-118. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26487) Davies, Bleddyn P. (1985) Community alternatives to institutional long-term care: some British research. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27292) Davies, Bleddyn P. (1996) Community care for elderly people: Brian Abel-Smith's Legacy Revisited. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27264) Davies, Bleddyn P. (1990) Community care in Australia and elsewhere: the future. In: Howe, Amanda and Ozanne, Elizabeth and Selby-Smith, C., eds. Community Care Policy and Practice: New Directions in Australia. Monash University Press, Melbourne, pp. 260-263. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26488) Davies, Bleddyn P. (1987) Community care: present practice and implications for the future. In: Launch of Matching Resources to Needs in Community Care and Case Management in Community Care, 1987-03-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27115) Davies, Bleddyn P. (1989) Coordination et le soutien à domicile. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27284) Davies, Bleddyn P. (1993) Costs estimation and community care. Why we must run fast to stand still. In: Netten, Ann and Beecham, Jennifer, eds. Costing Community Care: Theory and Practice. Avebury, Aldershot, pp. 197-208. ISBN hbk: 1857420985 / pbk: 1857421027. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26498) Davies, Bleddyn P. (1989) Costs needs and outcomes in residential and community-based care of the elderly: towards the quantification of optimal targeting criteria. In: Colvez, A., ed. Les Institutions Sanitaires Face au Vieillissement. Ecole Nationale de la Santé Publique, Rennes, France, pp. 139-141. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26485) Davies, Bleddyn P. (1975) Determinants of variation in social service expenditures and the measurement of need: introduction and summary. In: Layfield, F., ed. Report of the Committee of Enquiry into Local Government Finance, Cmnd. 6453, Appendix 10. HMSO, London, microfiche. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26476) Davies, Bleddyn P. (1994) Division of labour and homecare in England. In: Hollingsworth, J. Rogers and Hollingsworth, Ellen Jane, eds. Care of the Chronically and Severely Ill: Comparative Social Policies. Aldine de Gruyter, New York, pp. 107-139. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26490) Davies, Bleddyn P. (1986) Epilogue: past achievements and future promise. In: Judge, Ken F. and Sinclair, Ian, eds. Residential Care for Elderly People. HMSO, London, pp. 203-206. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26481) Davies, Bleddyn P. (2008) Equity and efficiency in community care: supply and financing in an age of fiscal austerity. Ageing and Society, 7 (2). pp. 161-174. ISSN 0144-686X. (doi:10.1017/S0144686X0001254X) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26770) Davies, Bleddyn P. (1989) Evaluating and control of research: the short note. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27286) Davies, Bleddyn P. (1990) Evaluating community care. In: of Health, Department, ed. Yearbook of Research and Development. Department of Health, London. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26497) Davies, Bleddyn P. (1988) Financing long-term social care: challenges for the nineties. Social Policy & Administration, 22 (2). pp. 97-114. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1988.tb00295.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26773) Davies, Bleddyn P. (1991) Financing mechanisms and the mixed economy. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27278) Davies, Bleddyn P. (1997) Forecasting long-term care needs and costs in the UK, Sweden and the Netherlands. In: Davies, G.R. and Mykita, L.T. and Andrews, M.M. and Pearson, S.A. and Gregory, A.J. and Hagger, J.C., eds. Ageing Beyond 2000: One World, One Future. World Congress of Gerontology, Adelaide, p. 73. ISBN 646327828. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26508) Davies, Bleddyn P. (1994) Improving the case management process: potential for improving effectiveness and efficiency of care for frail elderly persons. In: Care of the Frail Elderly. OECD, Paris, pp. 111-143. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26500) Davies, Bleddyn P. (1994) Institutional and family support: how inefficiencies can help cope with the dilemmas. In: Attias-Donfut, Claudine, ed. Personnes âgées et solidarité entre les générations. CNAVTS, Paris. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26504) Davies, Bleddyn P. (1994) La dépendance en Europe: assistance institutionnelle et familiale. Retraite et Société, 5 . pp. 39-40. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26783) Davies, Bleddyn P. (1993) La réforme des soins communautaires - création et mise en oeuvre d'une politique nationale: du désordre au modèle et du modèle au ... Gérontologie et Société, 67 . pp. 112-126. ISSN 0151-0193. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26780) Davies, Bleddyn P. (1976) The Layfield Report. Social Work Today, 7 (7). p. 193. ISSN 0037-8070. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26757) Davies, Bleddyn P. (1976) Layfield's split vision. New Society, 36 (712). pp. 473-474. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26756) Davies, Bleddyn P. (2007) Le livre vert de l'Angleterre sur les services sociaux pour les adultes: indépendance, bien-être et choix. Retraite et Société, 47 . pp. 194-200. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26809) Davies, Bleddyn P. (1992) Lessons for case management. In: Onyett, Steve and Cambridge, Paul, eds. Case Management: Issues in Practice. University of Kent, Canterbury, pp. 32-39. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26493) Davies, Bleddyn P. (2004) Local authority management and adaptability. Lancet, . ISSN 0140-6736. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26795) Davies, Bleddyn P. (1969) Local authority size: some associations with standards of performance of services for deprived children and old people. Public Administration, (Summer). pp. 225-248. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26798) Davies, Bleddyn P. (1994) Maintaining the pressure in community care reform. Social Policy & Administration, 28 (3). pp. 197-205. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1994.tb00423.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26786) Davies, Bleddyn P. (1977) Needs and outputs. In: Heisler, H., ed. Fundamentals of Social Administration. Macmillan, London, 129-162 and 237. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26472) Davies, Bleddyn P. (1993) New policies and old logics:costs information and modal choice. In: Netten, Ann and Beecham, Jennifer, eds. Costing Community Care:Theory and Practice. Avebury, Aldershot, pp. 104-126. ISBN hbk: 1857420985 / pbk: 1857421027. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26496) Davies, Bleddyn P. (1990) New priorities in home care principles from the PSSRU experiments. In: Howe, Amanda and Ozanne, Elizabeth and Selby-Smith, C., eds. Community Care Policy and Practice: New Directions in Australia. Monash University Press, Melbourne, pp. 47-72. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26492) Davies, Bleddyn P. Notes of L.E.A provision, Appendix XIV. In: Children and their Primary Schools, The Plowden Report. . ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26519) Davies, Bleddyn P. (1993) On babies and the emptying of bathwater: contract content, incentives, and the nature of research contributions. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27269) Davies, Bleddyn P. (1975) On local expenditure and a standard level of service. In: Layfield, F., ed. Report of the Committee of Enquiry into Local Government Finance, Cmnd. 6453, Appendix 10. HMSO, London, p. 23. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26473) Davies, Bleddyn P. (2000) On the sensitivity of needs estimates to targeting criteria. International Journal of Health Services, 4 (4). pp. 157-167. ISSN 0020-7314. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26776) Davies, Bleddyn P. (1996) On the targeting of care management: some implications of PSSRU research. In: Targeting Care Management, 1996. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27092) Davies, Bleddyn P. (1990) PSSRU research on the production of welfare in the community care of the elderly. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27279) Davies, Bleddyn P. (1966) Planning local health and welfare services: an administrative structure for effective planning. Local Government Chronicle, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26805) Davies, Bleddyn P. (1966) Planning local health and welfare services: inequality and the correlation of standards with needs. Local Government Chronicle, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26804) Davies, Bleddyn P. (1966) Planning local health and welfare services: the relative expansion of substitute services. Local Government Chronicle, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26803) Davies, Bleddyn P. (1987) Plans for the HPSS in Northern Ireland. In: UNSPECIFIED, 1987-02-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27113) Davies, Bleddyn P. (1980) Policy options for charges and means tests. In: Judge, Ken F., ed. Pricing the Social Services. Macmillan, London, pp. 133-153. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26477) Davies, Bleddyn P. (1989) Pow!... Zap!... Crunch!... Screech! Is PSSRU research impacting where it matters? Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27282) Davies, Bleddyn P. (1995) Production of welfare. In: Maddox, G., ed. The Encyclopedia of Aging. Springer Publishing, New York, pp. 761-763. ISBN 826148409. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26515) Davies, Bleddyn P. (1985) Production of welfare approach. In: The Production of Welfare Approach, 1985. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27119) Davies, Bleddyn P. (1995) Production of welfare evidence: PSSRU's budget-devolved case management experiments. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27266) Davies, Bleddyn P. (2007) Public spending levels for social care of older people: why we must call in the debt. Policy and Politics, 35 (4). pp. 719-726. (doi:10.1332/030557307782452976) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26810) Davies, Bleddyn P. (1970) Quantified theory and planning techniques in social welfare services. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27136) Davies, Bleddyn P. (1989) Rational funding policies. In: Butler, E., ed. The Future of Community Care. ASI, London, pp. 25-30. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26489) Davies, Bleddyn P. (1991) Resources needs and outcomes: the messages distilled. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27274) Davies, Bleddyn P. (1991) Resources, needs and outcomes in community services: why academic caution is useful. In: Morgan, Kevin, ed. Gerontology: Responding to an Ageing Society. Jessica Kingsley Publishers, London, pp. 215-48. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26494) Davies, Bleddyn P. (1987) Review article: making a reality of community care. British Journal of Social Work, 18 (S). pp. 173-187. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26769) Davies, Bleddyn P. (2003) Review of private lives in public places: a research-based critique of residential life in local authority old peoples homes. Times Higher Education Supplement, . p. 15. ISSN 0049-3929. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26772) Davies, Bleddyn P. (2007) Securing good care for older people: taking a long-term view. Ageing Horizons, 6 . pp. 12-27. ISSN 1746–1081. (KAR id:12641) Preview Davies, Bleddyn P. (1998) Shelter with care and the community care reforms. In: Jack, Raymond, ed. Residential Versus Community Care: The Role of Institutions in Welfare Provision. Macmillan, London, pp. 71-111. ISBN 0-333-66518-X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26506) Davies, Bleddyn P. (1968) Social Needs and Resources in Local Services: A Study of Variations in Provision of Social Services between Local Authority Areas. Joseph Rowntree, London, 371 pp. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27434) Davies, Bleddyn P. (2007) Social Planning: Classics in Planning I - Edited by Jessie P. H. Poon, Kenneth Button and Peter Nijkamp; Review article. Social Policy & Administration, 41 . pp. 525-529. ISSN 0144-5596. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26811) Davies, Bleddyn P. (1976) Social and economic indicators: the academic's contribution. In: Kirkland, K., ed. Statistics for Local Government: Proceedings of the Statistics User's Conference, 1975. Standing Committee of Statistics Users, London, pp. 41-52. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26471) Davies, Bleddyn P. (1990) Social services in the city: context change, service response and service outcomes. Statistician, 39 (3). pp. 229-245. ISSN 0039-0526. (doi:10.2307/2349040) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26774) Davies, Bleddyn P. (1977) Social-service studies and the explanation of policy outcomes. Policy and Politics, 5 (3). pp. 41-59. ISSN 0305-5736. (doi:10.1332/030557378782842713) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26759) Davies, Bleddyn P. (1971) Some constraints on school meal policy. Social and Economic Administration, 5 (1). pp. 34-52. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1971.tb00715.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26800) Davies, Bleddyn P. (1974) Standards setting and the theory of social welfare planning. Funded/commissioned by: Report of the Expert Group on Standard Setting in Social Welfare (SOA/ESDP/1974 2). Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27135) Davies, Bleddyn P. (1981) Strategic goals and piecemeal innovations: adjusting to the new balance of needs and resources. In: Goldberg, E.M., ed. A New Look at the Personal Social Services. Policy Studies Institute, London, pp. 96-121. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26478) Davies, Bleddyn P. (1976) Territorial injustice. New Society, 36 (710). pp. 352-354. ISSN 0028-6729. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26755) Davies, Bleddyn P. (1975) Territorial injustices in the social services. In: Lambert, Camilla and Wier, David, eds. Cities in Modern Britain. Fontana, London, pp. 344-50. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26470) Davies, Bleddyn P. (1993) Thinking long in community care. In: Deakin, Nicholas and Page, Robert, eds. The Costs of Welfare. Avebury, Aldershot, pp. 200-266. ISBN 1-85628-513-8. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26499) Davies, Bleddyn P. (1997) To what degree have the British community care reforms met the pre-reform criticisms of targeting? In: UNSPECIFIED, 1997-07-01T00:00:00. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27088) Davies, Bleddyn P. (1993) Towards the integration of social and economic rationales in care management. In: Evers, Adalbert and van der Zanden, Gerard H., eds. Better Care for Dependent People Living at Home: Meeting the New Agenda in Services for the Elderly. NIG, Bunnik, The Netherlands, pp. 153-196. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26495) Davies, Bleddyn P. (1996) Two questions for the design and analysis of evaluations. In: Case Management in Community and Long-term Care, 1996-10-01T00:00:00, Hong Kong. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27096) Davies, Bleddyn P. (1975) Values, needs, and the outputs of local services. In: Jerzy Wiatr, I. and Rose, Richard, eds. Comparing Public Policies. Polska Akademia Nauk, Warsaw. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26469) Davies, Bleddyn P. (1969) Welfare departments and territorial justice: some implications for the reform of local government. Social and Economic Administration, 3 (4). pp. 235-252. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1969.tb00700.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26799) Davies, Bleddyn P. (1977) Welfare needs and local autonomy. In: Davies, Ross and Hall, Peter, eds. Issues in the Urban Society. Penguin, London, pp. 216-241. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26475) Davies, Bleddyn P. (1996) Who gets what community care services: reform impacts, dilemmas and context. In: Third International Conference on Long-Term Care Case Management: Bridging the Many Worlds of Case Management, 1996-12-01T00:00:00, San Diego, California. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27098) Davies, Bleddyn P. (1989) Why we must fight the eighth deadly sin: parochialism. Journal of Aging and Social Policy, 1 . ISSN 0895-9420. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26775) Davies, Bleddyn P. (1989) Why we need a metaphor to guide policy: the example of user financing mechanisms and the mixed economy. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27285) Davies, Bleddyn P. (1988) The community care approach and the development of institutions intermediate between formal and informal care. In: Evans, A., ed. Towards Better Links and Balances between Social and Health Services in the Care of the Elderly. European Centre for Social Work Training and Research, Vienna, pp. 25-49. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26484) Davies, Bleddyn P. (1968) The cost-effectiveness of education spending. Social Services for All?, . pp. 49-60. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27033) Davies, Bleddyn P. (1997) The economics of community care case management: arguments from British research. In: Davies, Gary R. and Mykita, L.T. and Andrews, M.M. and Pearson, S.A. and Gregory, A.J. and Hagger, J.C., eds. Ageing Beyond 2000: One World, One Future. World Congress of Gerontology, Adelaide, p. 566. ISBN 646327828. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26513) Davies, Bleddyn P. (1968) The future of local government statistics. Journal of the Royal Statistical Society: Series A (Statistics in Society), 131 (1). ISSN 0964-1998. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26797) Davies, Bleddyn P. (1964) An index of variations in needs of county boroughs for old people's homes. Sociological Review, 12 (1). pp. 5-38. ISSN 0038-0261. (doi:10.1111/j.1467-954X.1964.tb01244.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26794) Davies, Bleddyn P. (1975) The measurement of needs and the allocation of grant. In: Layfield, F., ed. Report of the Committee of Enquiry into Local Government Finance, Cmnd. 6453, Appendix 10. HMSO, London, microfiche. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26474) Davies, Bleddyn P. (1987) The new managerialist argument and the supply and financing of care. In: di Gregario, S., ed. Social Gerontology: New Directions. Croom Helm, Beckenham, pp. 75-89. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26482) Davies, Bleddyn P. (1995) The production of welfare approach: conceptual framework and methodology. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27265) Davies, Bleddyn P. (1995) The reform of community and long-term care of elderly persons: an international perspective. In: Scharf, Thomas and Wenger, Clare, eds. International Perspectives on Community Care for Older People. Avebury, Aldershot, pp. 21-38. ISBN 1-85972-056-0. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26503) Davies, Bleddyn P. (1989) The third age and beyond: insuring financial security mechanisms for independent financing of long-term social care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27283) Davies, Bleddyn P. (1990) The trade and industry policy' metaphor: the SSD in the post- Griffiths world. In: Bytheway, Bill and Johnson, Julia, eds. Welfare and the Ageing Experience: A Multidisciplinary Analysis. Gower, Aldershot, pp. 14-28. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26486) Davies, Bleddyn P. (1991) A universal challenge: making the best use of community-based and residential care modes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27277) Davies, Bleddyn P. (1970) The uptake of school meals after the price increase. New Society, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26807) Davies, Bleddyn P., Baines, Barry (1991) On lifetime costs and targeting: effects of current case management practice on future resource commitments with entropic assumptions about productivities. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27275) Davies, Bleddyn P., Baines, Barry (1992) On the silting up of social service department resources and the stability of need states in a cohort of new recipients of community ¦ based social services. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27273) Davies, Bleddyn P., Baines, Barry (1992) Targeting and the silting-up of resources in the community-based social services: the consequences of alternative policies. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27272) Davies, Bleddyn P. and Baines, Barry and Chesterman, John (1996) The effects of care management on efficiency in long-term care: a new evaluation model applied to British and American data. In: Phillips, Judith and Penhale, Bridget, eds. Reviewing Care Management for Older People. Jessica Kingsley, London, pp. 87-101. ISBN 1-85302-317-5. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26502) Davies, Bleddyn P. and Baldock, John C. (1991) Community care of the elderly: on the evaluation of strategic innovations and change. In: Kraan, Robert J., ed. Care for the Elderly: Significant Innovations in Three European Countries. Campus/Westview, Boulder, Colorado, pp. 203-226. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26491) Davies, Bleddyn P., Barton, A. (1973) The 'silting up' of unadjustable resources and the planning of the personal social services. Policy and Politics, 1 (4). pp. 341-355. ISSN 0305-5736. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26802) Davies, Bleddyn P., Barton, A., MacMillan, I. (1972) Variations in Children's Services Amongst British Urban Authorities: A Causal Analysis. Bell, London, 157 pp. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27435) Davies, Bleddyn P., Barton, A., McManus, I.C. (1971) Variations in the provision of local authority health and welfare services: a comparison between counties and county boroughs. Social and Economic Administration, 5 (2). pp. 100-124. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1971.tb00020.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26801) Davies, Bleddyn P., Barton, A., Young, K. (1968) The Child Care Services. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27137) Davies, Bleddyn P., Bebbington, Andrew (1983) Equity and efficiency in the allocation of personal social services. Journal of Social Policy, 12 (3). pp. 309-330. ISSN 0047-2794. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26765) Davies, Bleddyn P., Bebbington, Andrew, Charnley, Helen (1990) Resources, Needs and Outcomes in Community-Based Care. Ashgate, Aldershot, 512 pp. ISBN 1-85628-130-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27399) Davies, Bleddyn P., Challis, David J. (1980) Experimenting with new roles in domiciliary service: the Kent Community Care Project. Gerontologist, 20 (3). pp. 288-299. ISSN 0016-9013. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26761) Davies, Bleddyn P., Challis, David J. (1986) Matching Resources to Needs in Community Care. Ashgate, Aldershot, 690 pp. ISBN 1-85742-113-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27392) Davies, Bleddyn P. and Challis, David J. (1981) A production relations evaluation of the meeting of needs in the community care project. In: Goldberg, E.M. and Connelly, N., eds. Evaluating Social Care. Heinemann, London, pp. 177-198. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26479) Davies, Bleddyn P., Chesterman, John (1995) How case management confers benefits: estimates of the direct and indirect effects from the channeling, the Kent Community Care Project and its replications. In: III European Congress of Gerontology, Amsterdam. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27100) Davies, Bleddyn P., Chesterman, John (1996) How does case management improve efficiency? Effects of case management inputs on the impacts of home care service inputs. Hong Kong Journal of Gerontology, 10 (Suppl). pp. 255-259. ISSN 1608-2346. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26790) Davies, Bleddyn P. and Chesterman, John (1996) Will similar case management arrangements achieve similar costs of outcomes? Within- and between-project learning in the British Kent Community Care Project and its replications. In: Public Health Association, American, ed. Empowering the Disadvantaged: Social Justice in Public Health. American Public Health Association, New York, p. 465. ISBN none. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26514) Davies, Bleddyn P., Chesterman, John, Baines, Barry (1993) How does care management improve efficiency? The effects of case management inputs on the productivities of home care services. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27271) Davies, Bleddyn P. and Chesterman, John and Fernández, José-Luis (1997) How much case management should users have: an economic/econometric analysis based on British experiments. In: Davies, G.R. and Mykita, L.T. and Andrews, M.M. and Pearson, S.A. and Gregory, A.J. and Hagger, J.C., eds. Ageing Beyond 2000: One World, One Future. World Congress of Gerontology, Adelaide, p. 384. ISBN 0-646-32782-8. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26510) Davies, Bleddyn P., Chesterman, John, Fernández, José-Luis (1996) Implications of unmet need (UM), welfare gains [G), and gain/cost (G/C) bases for targeting criteria. In: BSG Conference. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27091) Davies, Bleddyn P., Coles, Oliver (1981) Electoral support, bureaucratic criteria, cost variations and intra-authority allocations: a home help case. Political Studies, 29 (3). pp. 415-425. (doi:10.1111/j.1467-9248.1981.tb00506.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26762) Davies, Bleddyn P., Coles, Oliver (1981) Towards a territorial cost function for the home help service. Social Policy & Administration, 15 (1). pp. 32-42. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1981.tb00664.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26763) Davies, Bleddyn P., Darton, Robin, Goddard, M. (1987) The effects of alternative targeting criteria and demand levels for the opportunity costs to the SSD of care in local authority homes. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27289) Davies, Bleddyn P., Davies, K. (1971) The use of time by academics. Times Higher Education Supplement, . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26808) Davies, Bleddyn P., Ferlie, Ewan B (1982) Efficiency-promoting innovation in social care social services departments and the elderly. Policy and Politics, 10 (2). pp. 181-203. ISSN 0305-5736. (doi:10.1332/030557382783182378) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26764) Davies, Bleddyn P., Ferlie, Ewan B (1984) Patterns of efficiency-improving innovation: social care and the elderly. Policy and Politics, 12 (3). pp. 281-295. ISSN 0305-5736. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26766) Davies, Bleddyn P. and Fernández, José-Luis (2000) Choices! Choices! Techniques for analysing the dilemmas of balancing the interests of stakeholders in community care. In: Dickinson, A. and Bartlett, H. and Wade, S., eds. Old Age in a New Age. Proceedings of the British Society of Gerontology 29th Annual Conference, Oxford, 8-10 September 2000. School of Health Care, Oxford Brookes University, pp. 69-73. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26517) Davies, Bleddyn P., Fernández, José-Luis (1999) Does social work improve the impact during the set-up stage of post-reform case-managed community care of elderly users? In: Asia-Pacific Conference, 1999-04-01T00:00:00, Hong Kong. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27085) Davies, Bleddyn P. and Fernández, José-Luis (2000) Empowerment in post-reform community care in England and Wales. In: Heumann, L.F. and McCall, M.E. and Boldy, D.P., eds. Empowering Frail Elderly People: Opportunities and Impediments in Housing, Health and Support Service Delivery. Greenwood Publishing, Connecticut, pp. 81-100. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26516) Davies, Bleddyn P. and Fernández, José-Luis (2001) How care management arrangements affect outcomes. In: Tester, Susan and Archibald, C. and Rowlings, C. and Turner, S., eds. Quality in Later Life: Rights, Rhetorics and Reality. British Society of Gerontology, London, pp. 274-277. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26518) Davies, Bleddyn P. and Fernández, José-Luis (1997) Impact of British community care reforms on outcomes incidence, outcome costs and service productivities. In: Davies, G.R. and Mykita, L.T. and Andrews, M.M. and Pearson, S.A. and Gregory, A.J. and Hagger, J.C., eds. Ageing Beyond 2000: One World, One Future. World Congress of Gerontology, Adelaide, p. 390. ISBN 0-646-32782-8. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26511) Davies, Bleddyn P. and Fernández, José-Luis (1997) Targeting services in UK community care: are the post-reform patterns more defensible? In: Davies, G.R. and Mykita, L.T. and Andrews, M.M. and Pearson, S.A. and Gregory, A.J. and Hagger, J.C., eds. Ageing Beyond 2000: One World, One Future. World Congress of Gerontology, Adelaide, pp. 547-548. ISBN 646327828. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26512) Davies, Bleddyn P. and Fernández, José-Luis (1997) The impact of UK community care reforms on equity and efficiency in targeting services for the elderly. In: Heumann, L.F., ed. Managing Care, Risk and Responsibility. Proceedings of the Sixth International Conference on Systems Sciences in Health-Social Services for the Elderly and the Disabled (SYSTED). Systems Sciences in Health-Social Services for the Elderly and the Disabled (SYSTED), Chicago, pp. 390-396. ISBN none. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26507) Davies, Bleddyn P., Fernández, José-Luis, Milne, Alisoun (1996) Care management arrangements. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27263) Davies, Bleddyn P., Fernández, José-Luis, Nomer, B. (2000) Equity and Efficiency Policy in Community Care: Service Productivities, Efficiencies and their Implications. Ashgate, Aldershot, 494 pp. ISBN 754612813. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27426) Davies, Bleddyn P., Fernández, José-Luis, Nomer, B. (1998) Productivities. Efficiency, and Three Policy Propositions. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27260) Davies, Bleddyn P., Fernández, José-Luis, Saunders, R. (1998) Community Care in England and France. Ashgate, Aldershot, 228 pp. ISBN 1-84014-584-6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27416) Davies, Bleddyn P., Fernández, José-Luis, Saunders, R. (1995) Effects of benefits in cash and kind and entry to institutions for long-term care in England and France. Personal Social Services Research Unit ISBN 0144686X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27267) Davies, Bleddyn P., Fernández, José-Luis, Warburton, R.W. (1996) Evaluating community care for elderly people: who gets how much of what service. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27261) Davies, Bleddyn P., Fernández, José-Luis, Warburton, R.W. (1996) Post-reform community care for elderly people: who gets how much of what service? Care Plan, 3 (2). pp. 25-30. ISSN 1355-0454. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26788) Davies, Bleddyn P., Goddard, M. (1987) The brokerage-only Britsmo; the Britsmo concept. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27290) Davies, Bleddyn P., Goddard, M. (1987) The insurability of the risk of long-term care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27291) Davies, Bleddyn P., Knapp, Martin R J. (1988) British Journal of Social Work, Supplement. , 187 pp. ISBN 453102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27394) Davies, Bleddyn P. and Knapp, Martin R J. (1988) Costs and residential social care. In: Wagner, G., ed. Published in Residential Care: The Research Reviewed. HMSO, London, pp. 293-378. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26483) Davies, Bleddyn P., Knapp, Martin R J. (1978) Hotel and dependency costs of residents in old people's homes. Journal of Social Policy, 7 (1). pp. 1-22. ISSN 0047-2794. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26758) Davies, Bleddyn P., Knapp, Martin R J. (1988) Introduction: the production of welfare approach: some new PSSRU arguments and results. British Journal of Social Work, 18 (S). pp. 1-11. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26771) Davies, Bleddyn P., Knapp, Martin R J. (1981) Old People's Homes and the Production of Welfare. Routledge & Kegan Paul, London, 265 pp. ISBN 70007000. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27391) Davies, Bleddyn P., Milne, Alisoun, Warburton, R.W. (1995) Do different case management approaches affect who gets what? Preliminary results from a comparative British study. Care Plan, 2 (2). pp. 26-30. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26787) Davies, Bleddyn P., Missiakoulis, Spyros (1988) Heineken and matching processes in the Thanet Community Care Project: an empirical test of their relative importance. British Journal of Social Work, 18 (S). pp. 55-78. ISSN 0045-3102. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26767) Davies, Bleddyn P., Noronha, V. (1989) A base-weighted index number of the value of dwellings owned and occupied by elderly people. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27280) Davies, Bleddyn P., Warburton, R.W., Fernández, José-Luis (1996) At home or into a home. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27262) Davies, Bleddyn P., Warburton, R.W., Fernández, José-Luis (1996) Case management, other community care costs, needs and outcomes. Hong Kong Journal of Gerontology, 10 (Suppl). pp. 250-254. ISSN 1608-2346. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26789) Davies, Bleddyn P., Williamson, Valerie (1968) School meals - short-fall and poverty. Social Policy & Administration, 2 (1). pp. 3-19. ISSN 0144-5596. (doi:10.1111/j.1467-9515.1968.tb00076.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26796) Dawson, Laura, Williams, Jacquetta, Netten, Ann (2006) Extra care housing: is it really an option for older people? Housing, Care and Support, 9 (2). pp. 23-29. ISSN 1460-8790. (doi:10.1108/14608790200600013) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26812) Di Terlizzi, M. (1994) Life-History - the Impact of a Changing Service Provision on an Individual with Learnig-Disabilities. Disability & Society, 9 (4). pp. 501-517. ISSN 0968-7599. (doi:10.1080/09687599466780481) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:20167) Dickinson, Helen, Glasby, Jon, Forder, Julien E., Beesley, Lucinda (2007) Free personal care in Scotland: A narrative review. British Journal of Social Work, 37 (3). pp. 459-474. ISSN 0045-3102. (doi:10.1093/bjsw/bcm018) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2351) Dolan, Paul, Netten, Ann, Shapland, Joanna, Tsuchiya, Aki (2007) Towards a preference-based measure of the impact on well-being due to victimisation and the fear of crime. International Review of Victimology, 14 (2). pp. 253-264. ISSN 0269-7580. (doi:10.1177/026975800701400205) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2810) Donnelly, Maureen A., McGilloway, S., Perry, S., Knapp, Martin R J., Kavanagh, Shane M., Beecham, Jennifer, Fenyo, Andrew J., Astin, Jack (1994) Opening New Doors: An Evaluation of Community Care for People Discharged from Psychiatric and Mental Handicap Hospitals. HMSO, Belfast ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27408) Donnelly, Michael, McGilloway, S., Mays, Nicholas, Knapp, Martin R J., Kavanagh, Shane M., Beecham, Jennifer, Fenyo, Andrew J., Astin, Jack (1996) Leaving hospital: one- and two-year outcomes of long-stay psychiatric patients discharged to the community. Journal of Mental Health, 5 (3). pp. 245-255. ISSN 0963-8237 (Print) 1360-0567 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26813) Drummond, M., Knapp, Martin R J., Burns, Tom, Miller, K., Shadwell, P. (1998) Issues in the design of studies for the economic evaluation of new atypical antipsychotics: the ESTO study. Journal of Mental Health Policy and Economics, 1 (1). pp. 15-22. ISSN 1091-4358 (Print) 1099-176X (Online). (doi:10.1002/(SICI)1099-176X(199803)1:1<15::AID-MHP2>3.0.CO;2-O) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26815) d'Amico, Francesco, Knapp, Martin R J., Beecham, Jennifer, Sandberg, Seija, Taylor, Eric, Sayal, Kapil (2014) Use of services and associated costs for young adults with childhood hyperactivity/conduct problems: 20-year follow-up. British Journal of Psychiatry, 204 (6). pp. 441-447. ISSN 0007-1250. (doi:10.1192/bjp.bp.113.131367) (KAR id:42636) Preview ## E Eklund, Hanna, Cadman, Tim, Findon, James, Hayward, Hannah, Howley, Deirdre, Beecham, Jennifer, Xenitidis, Kiriakos, Murphy, Declan, Asherson, Philip, Glaser, Karen and others. (2016) Clinical service use as people with Attention Deficit Hyperactivity Disorder transition into adolescence and adulthood: a prospective longitudinal study. Bmc Health Services Research, 16 (248). ISSN 1472-6963. (doi:10.1186/s12913-016-1509-0) (KAR id:57226) Emerson, Eric, Robertson, Janet, Gregory, Nicola, Hatton, Chris, Kessissoglou, Sophia, Hallam, Angela, Järbrink, Krister, Knapp, Martin R J., Netten, Ann, Noonan Walsh, Patricia and others. (2001) The quality and costs of supported living schemes and group homes in the UK. American Journal of Mental Retardation, 105 . pp. 401-415. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26819) Emerson, Eric, Robertson, Janet, Gregory, Nicola, Kessissoglou, Sophia, Hatton, Chris, Hallam, Angela, Järbrink, Krister, Knapp, Martin R J., Netten, Ann, Linehan, Christine and others. (2000) The quality and costs of community-based residential supports and residential campuses for people with severe and complex disabilities. Journal of Intellectual and Developmental Disability, 25 . pp. 263-279. ISSN 1366-8250 (Print) 1469-9532 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26817) Emerson, Eric, Robertson, Janet, Gregory, Nicola, Kessissoglou, Sophia, Hatton, Chris, Hallam, Angela, Knapp, Martin R J., Järbrink, Krister, Noonan Walsh, Patricia, Netten, Ann and others. (2000) The quality and costs of village communities, residential campuses and community-based residential supports in the UK. American Journal of Mental Retardation, 105 . pp. 81-102. ISSN 0895-8017. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26818) Evans, S., Vallelly, S., Callaghan, Lisa (2010) A Directory for Promoting Social Well-Being in Extra Care Housing andf Other Settings. Housing LIN Report No. 44, Updated January 2010. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27138) Evans, Simon, Atkinson, Teresa, Darton, Robin, Cameron, Ailsa, Netten, Ann, Smith, Randall, Porteus, Jeremy (2017) A community hub approach to older people’s housing. Quality in Ageing and Older Adults, 18 (1). pp. 20-32. ISSN 1471-7794. E-ISSN 2042-8766. (doi:10.1108/QAOA-02-2015-0008) (KAR id:60643) Preview Evans, Simon, Darton, Robin, Porteus, Jeremy (2014) What is the housing with care ‘offer’ and who is it for? Written for the Housing Learning and Improvement Network (LIN). Housing Learning and Improvement Network (LIN), 4 pp. (KAR id:45765) Preview Evans, Simon, Whitehurst, Teresa, Darton, Robin, Smith, Randall, Cameron, Ailsa, Porteus, Jeremy, Netten, Ann, Bäumker, Theresia, Dutton, Rachael (2012) Adult Social Care in Housing with Care Settings: A review of the literature. Adult Social Services Environments and Settings, 61 pp. (KAR id:37658) Preview ## F Farmer, Elaine, Dance, Cherilyn, Beecham, Jennifer, Bonin, Eva-Maria, Ouwejan, Danielle (2010) An investigation of family finding and matching in adoption – Briefing Paper, DFE-RBX-10-05. Personal Social Services Research Unit ISBN 978-1-84775-768-5. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27294) Fattore, G., Percudani, Mauro, Pugnoli, C, Beecham, Jennifer (2000) Mental health care in Italy: organisational structure, routine clinical activity and costs of a community psychiatric service in the Lombardy Region. International Journal of Social Psychiatry, 46 (4). pp. 250-265. ISSN 0020-7640. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26820) Felce, David, Lowe, Kathy, Beecham, Jennifer, Hallam, Angela (2000) Exploring the relationships between costs and quality of services for adults with severe intellectual disabilities and the most severe challenging behaviours in Wales: A multivariate regression analysis. Journal of Intellectual and Developmental Disability, 25 (4). pp. 307-326. ISSN 1366-8250 (Print) 1469-9532 (Online). (doi:10.1080/13668250020019593) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26821) Felce, David, Lowe, Kathy, Perry, J., Baxter, Helen, Jones, Edwin, Hallam, Angela, Beecham, Jennifer (1998) Service support to people in Wales with severe intellectual disability and the most severe challenging behaviours: processes, outcomes and costs. Journal of Intellectual Disability Research, 42 (5). pp. 390-408. ISSN 0964-2633. (doi:10.1046/j.1365-2788.1998.00153.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:17723) Fenyo, Andrew J. (1989) Canterbury and Herne Bay citizens advice bureaux - ward database 1981 census. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27296) Fenyo, Andrew J. (1990) Using Vector' in SPSS-x. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27295) Fenyo, Andrew J. and Knapp, Martin R J. and Baines, Barry (1989) Foster care breakdown: a study of a special teenager fostering scheme. In: Hudson, Joe and Galaway, BUrt, eds. The State as Parent. Kluwer, Dordrecht, The Netherlands, pp. 315-330. ISBN n.a.. (doi:10.1007/978-94-009-1053-9_25) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26520) Ferlie, Ewan B (1986) Innovatory intent and achievement in the community care of the elderly. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27298) Ferlie, Ewan B, Challis, David J., Davies, Bleddyn P. (1989) Efficiency-Improving Innovations in the Social Care of the Elderly. Ashgate, Aldershot, 224 pp. ISBN 566070499. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27397) Ferlie, Ewan B, Challis, David J., Davies, Bleddyn P. (1988) Efficiency-improving innovations in the community care of frail elderly people - Volume II: Individual schemes and their characteristics. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27297) Ferlie, Ewan B and Challis, David J. and Davies, Bleddyn P. (1985) Innovation in the care of the elderly: the role of joint finance. In: Butler, A., ed. Recent Advances and Creative Responses. Croom Helm, London, pp. 137-160. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26521) Ferlie, Ewan B, Challis, David J., Davies, Bleddyn P. (1984) Models of innovation in the social care of the elderly. Local Government Studies, 10 (6). pp. 67-82. ISSN 0300-3930. (doi:10.1080/03003938408433174) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26823) Ferlie, Ewan B, Judge, Ken F. (1981) Retrenchment and rationality in the personal social services. Policy and Politics, 10 (4). pp. 311-330. ISSN 0305-5736. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26822) Fernández, José-Luis, Davies, Bleddyn P. (1997) Has increasing service inputs per client improved equity and efficiency in UK community care? In: UNSPECIFIED, 1997. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27089) Fernández, José-Luis and Davies, Bleddyn P. (2000) Optimising modal choice: what would be the consequences of cost minimisation? In: Dickinson, A. and Bartlett, H. and Wade, S., eds. Old Age in a New Age: Proceedings of the British Society of Gerontology 29th Annual Conference. Oxford Brookes University, Oxford, UK, pp. 74-80. ISBN 1-902606-08-6. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26522) Fernández, José-Luis, Forder, Julien E. (2008) Consequences of local variations in social care on the performance of the acute health care sector. Applied Economics, 40 (12). pp. 1503-1518. ISSN 0003-6846. (doi:10.1080/00036840600843939) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:34725) Fernández, José-Luis, Forder, Julien E. (2010) Equity, efficiency, and financial risk of alternative arrangements for funding long-term care systems in an ageing society. Oxford Review of Economic Policy, 26 (4). pp. 713-733. ISSN 0266-903X. (doi:10.1093/oxrep/grq036) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32423) Fernández, José-Luis and Forder, Julien E. (2011) Impact of Changes in Length of Stay on the Demand for Residential Care Services in England: Estimates from a Dynamic Microsimulation Model. Project report. Personal Social Services Research Unit (KAR id:34672) Preview Fernández, José-Luis, Forder, Julien E. (2015) Local variability in long-term care services: local autonomy, exogenous influences and policy spillovers. Health Economics, 24 (S1). pp. 146-157. ISSN 1057-9230. E-ISSN 1099-1050. (doi:10.1002/hec.3151) (KAR id:48010) Preview Fernández, José-Luis, Forder, Julien E. (2012) Reforming long-term care funding arrangements in England: international lessons. Applied Economic Perspectives and Policy, 34 (2). pp. 346-362. ISSN 2040-5790. (doi:10.1093/aepp/pps020) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:33550) Fernández, José-Luis and Forder, Julien E. and Knapp, Martin R J. (2011) Long-term care. In: Smith, Peter and Glied, Sherry, eds. The Oxford Handbook of Health Economics. Oxford University Press, Oxford, pp. 578-601. ISBN 978-0-19-923882-8. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32406) Fernández, José-Luis, Hughes, Andrew, Watson, Kevin, Forder, Julien E., Fitzpatrick, Ray (2013) Report on using the GPPS to assess trends in EQ-5D scores for people with long-term conditions. Quality and Outcomes of Person-Centred Care Research Unit (KAR id:41641) Preview Ford, Gary A and Bhakta, Bipin B and Cozens, Alastair and Cundill, Bonnie and Hartley, Suzanne and Holloway, Ivana and Meads, David and Pearn, John and Ruddock, Sharon and Sackley, Catherine M and Saloniki, Eirini-Christina and Santorelli, Gillian and Walker, Marion and Farrin, Amanda (2019) Dopamine Augmented Rehabilitation in Stroke (DARS): a multicentre double-blind, randomised controlled trial of co-careldopa compared with placebo, in addition to routine NHS occupational and physical therapy, delivered early after stroke on functional recovery. Project report. National Institute for Health Research (KAR id:75637) Preview Forder, Julien E. (2008) The Costs of Addressing Age Discrimination in Social Care (PSSRU Discussion Paper 2538). Personal Social Services Research Unit, University of Kent, 36 pp. (KAR id:15678) Preview Forder, Julien E. (2011) Immediate Needs Annuities in England. Discussion paper. PSSRU, Canterbury (KAR id:34667) Preview Forder, Julien E. (2009) Long-term care and hospital utilisation by older people: an analysis of substitution rates. Health Economics, 18 (11). pp. 1322-1338. ISSN 1057-9230. (doi:10.1002/hec.1438) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:24899) Forder, Julien E. (2007) Self-funded social care for older people: an analysis of eligibility, variations and future projections. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27139) Forder, Julien E. (2007) Self-funded social care for older people: an analysis of eligibility, variations and future projections (PSSRU Discussion Paper 2505). Commission for Social Care Inspection, 39 pp. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2909) Forder, Julien E. (2007) Social care for older people: the challenge of future funding. Connect: Long-term Conditions Alliance, 33 (Spring). p. 13. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27036) Forder, Julien E., Allan, Stephen (2011) Competition in the Care Homes Market. OHE Commission (KAR id:34348) Preview Forder, Julien E., Allan, Stephen (2012) Competition in the English nursing homes market. In: European Conference of Health Economists, July 2012, Zurich, Switzerland. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78649) Forder, Julien E., Allan, Stephen (2012) The effects of competition in the English care homes market. In: 3rd joint CES-HESG meeting on health economics, 11-13 January 2012, Inserm-IRD & Aix-Marseille Université, Aix-en Provence, France. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78651) Forder, Julien E., Caiels, James (2011) Measuring the outcomes of long-term care. Social Science and Medicine, 73 (12). pp. 1766-1774. ISSN 0277-9536. (doi:10.1016/j.socscimed.2011.09.023) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32424) Forder, Julien E., Caiels, James, Harlock, Jenny, Wistow, Gerald, Malisauskaite, Gintare, Peters, Michele, Marczak, Joanna, d'Amico, Francesco, Fernández, José-Luis, Fitzpatrick, Ray, and others. (2018) A system-level evaluation of the Better Care Fund: Final Report. Quality and Outcomes of Person-centred Care Policy Research Unit., 140 pp. (KAR id:77827) Preview Forder, Julien E., Fernández, José-Luis (2009) Analysing the costs and benefits of social care funding arrangements in England: technical report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27300) Forder, Julien E., Fernández, José-Luis (2009) Analysing the costs and benefits of social care funding arrangments in England: technical report (PSSRU Discussion Paper 2644). Personal Social Services Research Unit, University of Kent, 51 pp. (KAR id:23320) Preview Forder, Julien E. and Fernández, José-Luis (2011) Funding social care for older people: The implications of extending the current means-test. Project report. Personal Social Services Research Unit, Canterbury and London (KAR id:34679) Preview Forder, Julien E. and Fernández, José-Luis (2011) Geographical differences in the provision of care home services in England. Project report. Personal Social Services Research Unit, Canterbury and London (KAR id:34674) Preview Forder, Julien E. and Fernández, José-Luis (2011) Immediate Needs Annuities and the Dilnot Limited Liability System. Project report. Personal Social Services Research Unit, Canterbury and London (KAR id:34673) Preview Forder, Julien E. and Fernández, José-Luis (2015) Using a ‘wellbeing’ cost-effectiveness approach to improve resource allocation in social care. Project report. Personal Social Services Research Unit, Canterbury and London (KAR id:47609) Preview Forder, Julien E., Fernández, José-Luis (2010) The impact of a tightening fiscal situation on social care for older people. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27299) Forder, Julien E., Fernández, José-Luis, Fitzpatrick, Ray (2013) Some considerations relating to the attribution of NHS activity to outcomes for people with long-term conditions. Quality and Outcomes of Person-centred Care Policy Research Unit (KAR id:41643) Preview Forder, Julien E., Gousia, Katerina, Saloniki, Eirini-Christina (2019) The impact of long-term care on primary care doctor consultations for people over 75 years. European Journal of Health Economics, 20 (3). pp. 375-387. ISSN 1618-7598. (doi:10.1007/s10198-018-0999-6) (KAR id:68984) Preview Preview Forder, Julien E. and Jones, Karen C. (2014) Evaluation of the personal health budget pilot programme. In: Needham, Catherine and Glasby, Jon, eds. Debates in Personalisation. Policy Press, Bristol, UK, 125 -132. ISBN 978-1-4473-1342-7. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:77824) Forder, Julien E., Jones, Karen C., Glendinning, Caroline, Caiels, James, Welch, Elizabeth, Baxter, Kate, Davidson, Jacqueline, Windle, Karen, Irvine, Annie, King, Dominic, and others. (2012) Evaluation of personal health budget pilot programme. Personal Social Services Research Unit, University of Kent, 215 pp. (KAR id:77833) Preview Forder, Julien E., Kavanagh, Shane M., Fenyo, Andrew J. (1995) A comparison of sertraline versus tricyclic antidepressants in primary care. I: Efficacy and effectiveness. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27303) Forder, Julien E., Kendall, Jeremy, Knapp, Martin R J., Matosevic, Tihana, Hardy, Brian, Ware, Patricia, Wistow, Gerald (2001) Mixed Modes of Governance and Mixed Economies of Care. Funded/commissioned by: Nuffield Institute for Health. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27140) Forder, Julien E. and Knapp, Martin R J. (1993) Social care markets: The voluntary sector and residential care for elderly people in England. In: Saxon-Harrold, Susan and Kendall, Jeremy, eds. Researching the Voluntary Sector. Charities Aid Foundation, Tonbridge. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26523) Forder, Julien E., Knapp, Martin R J. (1992) The social care market: researching the voluntary sector. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27304) Forder, Julien E., Knapp, Martin R J., Wistow, Gerald (1996) Competition in the English mixed economy of care. Journal of Social Policy, 25 (2). pp. 201-221. ISSN 0047-2794. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26825) Forder, Julien E., Malley, Juliette, Rand, Stacey, Vadean, Florin, Jones, Karen C., Netten, Ann (2016) Identifying the impact of adult social care: interpreting outcomes data for use in the Adult Social Care Outcomes Framework. Personal Social Services Research Unit, University of Kent, 67 pp. (KAR id:77830) Preview Forder, Julien E., Malley, Juliette, Towers, Ann-Marie, Netten, Ann (2014) Using cost-effectiveness estimates from survey data to guide commissioning: an application to home care. Health Economics, 28 (3). pp. 979-992. ISSN 1057-9230. (doi:10.1002/hec.2973) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:35436) Forder, Julien E., Netten, Ann (1997) The price of placements in residential and nursing home care. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27302) Forder, Julien E., Netten, Ann, Caiels, James, Smith, Jan E., Malley, Juliette (2007) Measuring outcomes in social care: conceptual development and empirical design. Quality Measurement Framework Project. PSSRU interim report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27301) Forder, Julien E., Towers, Ann-Marie, Caiels, James, Beadle-Brown, Julie, Netten, Ann (2008) Measuring Outcomes in Social Care: Second Interim Report (PSSRU Discussion Paper 2542). Personal Social Services Research Unit, University of Kent, 46 pp. (KAR id:15679) Preview Forder, Julien E. and Vadean, Florin (2014) Estimating relative needs formulae for new forms of social care support: using an extrapolation method. Discussion paper. PSSRU, University of Kent and London School of Economics (KAR id:49206) Preview Forder, Julien, Allan, Stephen (2013) The impact of competition on quality and prices in the English care homes market. Journal of Health Economics, 34 (1). pp. 73-83. ISSN 0167-6296. E-ISSN 1879-1646. (doi:10.1016/j.jhealeco.2013.11.010) (KAR id:38515) Preview Forder, Julien, Vadean, Florin, Rand, Stacey, Malley, Juliette (2017) The impact of long-term care on quality of life. Health Economics, 27 (3). e43-e58. ISSN 1057-9230. E-ISSN 1099-1050. (doi:10.1002/hec.3612) (KAR id:63581) Preview Forrester-Jones, Rachel, Beecham, Jennifer K., Barnoux, Magali, Oliver, David, Couch, Elyse, Bates, Claire (2017) People with intellectual disabilities at the end of their lives: The case for specialist care? Journal of Applied Research in Intellectual Disabilities, 30 (6). pp. 1138-1150. ISSN 1360-2322. E-ISSN 1468-3148. (doi:10.1111/jar.12412) (KAR id:63362) Preview Forrester-Jones, Rachel, Carpenter, John, Cambridge, Paul, Tate, Alison, Hallam, Angela, Knapp, Martin R J., Beecham, Jennifer (2002) The quality of life of people twelve years after resettlement from long-stay hospitals: users' views on their living environment, daily activities and future aspirations. Disability & Society, 17 (7). pp. 741-758. ISSN 0968-7599. (doi:10.1080/0968759021000068469) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26829) Forrester-Jones, Rachel, Carpenter, John, Coolen-Schrijner, Pauline, Cambridge, Paul, Tate, Alison, Hallam, Angela, Beecham, Jennifer, Knapp, Martin R J., Wooff, David (2012) Good friends are hard to find? The social networks of people with mental illness 12 years after deinstitutionalisation. Journal of Mental Health, 21 (1). pp. 4-14. ISSN 0963-8237. (doi:10.3109/09638237.2011.608743) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32425) Fox, Diane, Baumker, Theresia, Netten, Ann (2013) Informing service development through analysis of survey data: Improving carer’s quality of life? In: Joint event of the Social Services Research Group and the NIHR School for Social Care Research: Informing and improving policy and practice through research and evaluation, 30th January 2014, London. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78042) Fox, Diane, Bäumker, Theresia, Netten, Ann (2014) Factors associated with unpaid carers' quality of life: the similarities and differences between ethnic groups. In: 3rd International Long-term Care Policy Network Conference, 01-03 Sep 2014, London, UK. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:78040) Fox, Diane, Holder, Jacquetta, Netten, Ann (2010) Personal Social Services Survey of Adult Carers in England - 2009-10: Survey Development Project (PSSRU Discussion Paper 2643/2). Personal Social Services Research Unit, University of Kent (KAR id:23322) Preview Fox, Diane, Holder, Jacquetta, Smith, Nick (2010) Developing a survey of carers’ experience of services and quality of life. In: 5th International Carers Conference: New Frontiers in Caring 2010 and beyond, July 2010, Leeds. (KAR id:45018) Fox, Diane, Rand, Stacey (2014) Developing a social care outcomes measure for unpaid carers. In: International Long-term Care Policy Network Conference 2014, 31st August - 3rd September 2014, LSE, London. (KAR id:44377) Preview Francis, Jennifer, Netten, Ann (2003) Home care workers: careers, commitments and motivations. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27306) Francis, Jennifer, Netten, Ann (2002) Homecare services in one local authority: client and provider views. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27308) Francis, Jennifer, Netten, Ann (2003) Quality in home care: client and provider views. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27307) Francis, Jennifer, Netten, Ann (2004) Raising the quality of home care: a study of service users views. Social Policy & Administration, 38 (3). pp. 290-305. ISSN 0144-5596. (doi:10.1111/j.1467-9515.2004.00391.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:806) Francis, Jennifer, Netten, Ann, Fenyo, Andrew J. (2003) Extension to the Home Care User Experience Survey: Initial Feedback to Councils. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27305) French, Diane (1992) Canterbury and Thanet report. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27310) French, Diane, Ring, Chris (1993) The voluntary social welfare sector in Camden. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27309) ## G Ganna, Vikhitova, Nizalova, Olena, Nizalov, Denis (2013) Social Assistance System Modernization and Participation of the Poor. In: Global Development Network Annual Conference, June 2013, Manila, Philippines. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:35640) Gardiner, Laura and Hussein, Shereen (2015) As if we cared: the costs and benefits of a living wage for social care workers. Project report. The Resolution Foundation (KAR id:68314) Preview Gardner, Frances, Leijten, Patty, Melendez-Torres, G.J., Landau, Sabine, Harris, Victoria, Mann, Joanna, Beecham, Jennifer, Hutchings, Judy, Scott, Stephen (2018) The Earlier the Better? Individual Participant Data and Traditional Meta-analysis of Age Effects of Parenting Interventions. Child Development, 90 (1). pp. 7-19. ISSN 0009-3920. E-ISSN 1467-8624. (doi:10.1111/cdev.13138) (KAR id:69228) Preview Gilchrist, S., Knapp, Martin R J. (1994) Economics and mental health. Current Opinion in Psychiatry, 7 (2). pp. 167-172. ISSN 0951-7367. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26831) Glendinning, Caroline, Arksey, Hilary, Jones, Karen C., Moran, Nicola, Netten, Ann, Rabiee, Parvaneh (2009) The Individual Budgets Pilot Projects: Impact and Outcomes for Carers. Social Policy Research Unit, University of York, 123 pp. ISBN 978-1-903959-07-7. (KAR id:20522) Preview Glendinning, Caroline, Challis, David J., Fernández, José-Luis, Jacobsen, Sally, Jones, Karen C., Knapp, Martin R J., Manthorpe, Jill, Moran, Nicola, Netten, Ann, Wilberforce, Mark and others. (2008) Evaluation of the Individual Budgets Pilot Programme: Final Report. Social Policy Research Unit, University of York, 327 pp. ISBN 978-1-871713-64-0. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:13322) Glendinning, Caroline, Challis, David J., Fernández, José-Luis, Jones, Karen C., Knapp, Martin R J., Manthorpe, Jill, Netten, Ann, Stevens, Martin, Wilberforce, Mark (2006) Evaluating the individual budget pilot projects. Journal of Care Services Management, 1 . pp. 123-128. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26832) Glendinning, Caroline and Jones, Karen C. and Baxter, Kate and Rabiee, Parvaneh and Wilde, Alison and Arksey, Hilary and Curtis, Lesley A. and Forder, Julien E. (2010) Home Care Re-ablement Services: Investigating the Longer-term Impacts (prospective longitudinal study). Project report. Social Policy Research Unit, University of York, York (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32456) Glendinning, Caroline, Moran, Nicola, Challis, David J., Fernández, José-Luis, Jacobs, Sally, Jones, Karen C., Knapp, Martin R J., Manthorpe, Jill, Netten, Ann, Stevens, Martin, and others. (2011) Personalisation and partnership: competing objectives in English adult social care? The Individual Budget Pilot Projects and the NHS. Social Policy and Society, 10 (2). pp. 151-162. ISSN 1474-7464. (doi:10.1017/S1474746410000503) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32426) Goddard, Elizabeth, Raenker, Simone, Macdonald, Pamela, Todd, Gillian, Beecham, Jennifer, Naumann, Ulrike, Bonin, Eva-Maria, Schmidt, Ulrike, Landau, Sabine, Treasure, Janet and others. (2013) Carers’ assessment, skills and information sharing: theoretical framework and trial protocol for a randomised controlled trial evaluating the efficacy of a complex intervention for carers of inpatients with anorexia nervosa. European Eating Disorders Review, 21 (1). pp. 60-71. ISSN 1099-0968. (doi:10.1002/erv.2193) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:32427) Gosling, Amanda, Saloniki, Eirini-Christina (2014) Correction of misclassification error in disability rates. Health Economics, 23 (9). pp. 1084-1097. ISSN 1057-9230. E-ISSN 1099-1050. (doi:10.1002/hec.3080) (KAR id:56819) Preview Gostick, C., Davies, Bleddyn P., Lawson, Robyn, Salter, C. (1997) From Vision to Reality: Changing Direction at the Local Level. Arena, Aldershot, 200 pp. ISBN hbk: 1857424085 / pbk: 1857424093. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27415) Gousia, Katerina (2014) Financial literacy and long-term care insurance coverage. In: International Long-term Care Policy Network Conference 2014, 31 August - 3 September 2014, LSE, London. (KAR id:46760) Preview Gousia, Katerina (2016) Financial literacy and long-term care insurance coverage. Working paper. SHARE Working Paper Series 26-2016 (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64671) Gousia, Katerina (2016) The impact of long-term care on primary care doctor consultations for people over 75. In: 11th EuHEA Conference, 13-16 Jul 2016, Hamburg, Germany. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64676) Gousia, Katerina (2016) The impact of long-term care on primary care doctor consultations for people over 75. In: ILPN International Conference, 4-7 September 2016, London. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:64677) Gousia, Katerina and Nizalova, Olena and Malisauskaite, Gintare and Forder, Julien E. (2019) Heterogeneity in the effect of obesity on future long-term care use in England. Working paper. PSSRU, Kent, UK (KAR id:77853) Preview Gousia, Katerina, Yang, Wei (2015) A literature review of Commissioning on Quality and Innovation payment framework on maternity and children’s services. NHS South East Commissioning Support Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:47748) Graham, Katherine, Norrie, Caroline, Stevens, Martin, Moriarty, Jo, Manthorpe, Jill, Hussein, Shereen (2016) Models of adult safeguarding in England: A review of the literature. Journal of Social Work, 16 (1). pp. 22-46. ISSN 1468-0173. (doi:10.1177/1468017314556205) (KAR id:68310) Preview Graham, Katherine, Stevens, Martin, Norrie, Caroline, Manthorpe, Jill, Moriarty, Jo, Hussein, Shereen (2017) Models of safeguarding in England: Identifying important models and variables influencing the operation of adult safeguarding. Journal of Social Work, 17 (3). pp. 255-276. ISSN 1468-0173. (doi:10.1177/1468017316640071) (KAR id:68300) Preview Greco, Veronica, Sloper, Patricia, Webb, Rosemary, Beecham, Jennifer (2005) An Exploration of Different Models of Multi-Agency Partnerships in Key Worker Services and Disabled Children: Effectiveness and Costs. Funded/commissioned by: Department for Education and Skills. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27141) Greco, Veronica, Sloper, Patricia, Webb, Rosemary, Beecham, Jennifer (2007) Key worker services for disabled children: the views of parents. Children and Society, 21 (3). pp. 162-174. ISSN 0951 0605. (doi:10.1111/j.1099-0860.2006.00039.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2815) Greco, Veronica, Sloper, Patricia, Webb, Rosemary, Beecham, Jennifer (2006) Key worker services for disabled children: the views of staff. Health & Social Care in the Community, 14 (6). pp. 445-452. ISSN 0966-0410. (doi:10.1111/j.1365-2524.2006.00617.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26833) Green, Jonathan, Jacobs, Brian, Beecham, Jennifer, Dunn, Graham, Kroll, Leo, Tobias, Catherine, Briskman, Jackie (2007) Inpatient treatment in child and adolescent psychiatry – a prospective study of health gain and costs. Journal of Child Psychology and Psychiatry, 48 (12). pp. 1259-1267. ISSN 0021-9630. (doi:10.1111/j.1469-7610.2007.01802.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:2816) Guisset, M.J. and Saunders, R. (1992) Un exemple anglais: le Community Care. In: de France, Fondation, ed. La coordination gérontologique: démarche d'hier, enjeu pour demain. Fondation de France, Paris, pp. 119-123. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26524) ## H Haddad, Peter, Knapp, Martin R J. (2000) Health professionals’ views of services for schizophrenia: fragmentation and inequality. Psychiatric Bulletin, 24 (1). pp. 47-50. ISSN 0955-6036. (doi:10.1192/pb.24.2.47) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26834) Hajji, Assma, Trukeschitz, Birgit, Batchelder, Laurie, Saloniki, Eirini-Christina, Kieninger, Judith, Burge, Peter, Hui, Lu, Linnosmaa, Ismo, Malley, Juliette, Forder, Julien E. and others. (2017) Understanding best-worst experiments for calculating quality-of-life preferences in long-term care settings: how they work and what to consider. In: 3rd Austrian Health Economics Association Conference, 25 Nov 2017, Vienna, Austria. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:65319) Hajji, Assma, Trukeschitz, Birgit, Kieninger, Judith, Malley, Juliette, Batchelder, Laurie, Saloniki, Eirini-Christina, Burge, Peter, Linnosmaa, Ismo, Forder, Julien E. (2018) Using best-worst experiments to elicit preferences for long-term care related quality of life states: a closer look at sub-group differences and design effects. In: European Health Economics Association Conference 2018, 11-14 July 2018, Maastricht, Netherlands. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:68547) Hajji, Assma, Trukeschitz, Birgit, Malley, Juliette, Batchelder, Laurie, Saloniki, Eirini-Christina, Linnosmaa, Ismo, Lu, Hui (2020) Population-based preference weights for the Adult Social Care Outcomes Toolkit (ASCOT) for Service Users for Austria: findings from a best-worst experiment. Social Science & Medicine, . ISSN 0277-9536. (doi:10.1016/j.socscimed.2020.112792) (KAR id:79592) Preview Hallam, Angela (1995) Affording community care: lessons from the Friern and Claybury psychiatric reprovision programme. Mental Health Research Review, 2 . pp. 29-32. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27071) Hallam, Angela (1998) Care package costs for people with mental health problems. In: Netten, Ann and Dennett, Jane and Knight, J., eds. Unit Costs of Health and Social Care 1998. PSSRU, University of Kent, Canterbury, pp. 145-150. ISBN n.a.. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26527) Hallam, Angela (1996) Costs and outcomes for people with special psychiatric needs. Mental Health Research Review, 3 . pp. 10-13. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27079) Hallam, Angela (1994) Psychiatric reprovision in North London. Mental Health Research Review, 1 . ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27073) Hallam, Angela (1997) Through a glass darkly: media images of mental illness. Mental Health Research Review, 4 . pp. 10-11. ISSN 1353 2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27060) Hallam, Angela, Beecham, Jennifer (1998) Service packages for people with severe challenging behaviour. Funded/commissioned by: Welsh Centre for Learning Disabilities Applied Research Unit. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27142) Hallam, Angela, Beecham, Jennifer (1994) The costs of residential and training services for people with sensory impairments and learning disabilities. Funded/commissioned by: Hester Adrian Research Centre. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27143) Hallam, Angela, Beecham, Jennifer, Fenyo, Andrew J., Knapp, Martin R J. (1993) Reprovision costs: comparing a detailed approach with interpolations. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27311) Hallam, Angela, Beecham, Jennifer, Knapp, Martin R J., Baines, Barry (1997) Early indications of the long-term costs of psychiatric reprovision. Mental Health Research Review, 4 . pp. 38-39. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27057) Hallam, Angela, Beecham, Jennifer, Knapp, Martin R J., Carpenter, John, Cambridge, Paul, Forrester-Jones, Rachel, Tate, Alison, Coolen-Schrijner, Pauline, Wooff, David (2006) Service use and costs of support twelve years after leaving hospital. Journal of Applied Research in Intellectual Disabilities, 19 (4). pp. 296-308. ISSN 1360-2322. (doi:10.1111/j.1468-3148.2006.00278.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:33555) Hallam, Angela, Darton, Robin, McIntosh, E., Netten, Ann (1994) The Personal Social Services Research Unit. Healthcare Financial Management Association Newsletter, . pp. 8-11. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26836) Hallam, Angela and Knapp, Martin R J. (1997) Costing services in family centres. In: Netten, Ann and Dennett, Jane, eds. Unit Costs of Health and Social Care 1997. PSSRU, Canterbury, pp. 103-106. ISBN 0904938875; 09694226. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26526) Hallam, Angela and Knapp, Martin R J. and Beecham, Jennifer and Fenyo, Andrew J. (1995) Eight years of psychiatric reprovision: an economic evaluation. In: Knapp, Martin R J., ed. The Economic Evaluation of Mental Health Care. Arena, Aldershot, pp. 103-124. ISBN 1-85742-211-2. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26525) Hallam, Angela, Knapp, Martin R J., Järbrink, Krister, Netten, Ann, Emerson, Eric, Robertson, Janet, Gregory, Nicola, Hatton, Chris, Kessissoglou, Sophia, Durkan, J. and others. (2002) Costs of village community, residential campus and dispersed housing provision for people with intellectual disability. Journal of Intellectual Disability Research, 46 (5). pp. 394-404. ISSN 0964-2633. (doi:10.1046/j.1365-2788.2002.00409.x) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26839) Hallam, Angela, Schneider, Justine (1999) Sheltered work schemes for people with severe mental health problems: service use and costs. Journal of Mental Health, 8 (2). pp. 171-186. ISSN 0963-8237 (Print) 1360-0567 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26837) Hamilton-West, Kate E., Quine, Lyn (2003) Coping with Ankylosing Spondyltis. In: Annual Conference of the BPS Division of Health Psychology, September, 3-5 2003, Staffordshire University. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:47873) Hamilton-West, Kate E., Quine, Lyn (2001) Coping with Stress and Illness. In: Annual Conference of the BPS Division of Health Psychology, September 6-8 2000, Canterbury. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:47875) Hamilton-West, Kate E., Quine, Lyn (2001) Coping with Stress and Illness II. In: British Psychological Society Centenary Conference, March 28-31 2001, Glasgow. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:47874) Hampson, R., Judge, Ken F., Renshaw, J. (1985) Caught in the turbulence. Community Care, . pp. 20-22. ISSN ISSN: 03075508. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27038) Hancock, Ruth, Juarez-Garcia, Ariadna, Comas-Herrera, Adelina, King, Derek, Malley, Juliette, Pickard, Linda, Wittenberg, Raphael (2007) Winners and Losers: Assessing the Distributional Effects of Long-Term Care Funding Regimes. Social Policy and Society, 6 (3). pp. 379-396. ISSN 1474-7464. (doi:10.1017/S1474746407003703) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:12619) Hancock, Ruth, Juarez-Garcia, Ariadna, Wittenberg, Raphael, Pickard, Linda, Comas-Herrera, Adelina, King, Derek, Malley, Juliette (2006) Projections of Owner Occupation Rates, House Values, Income and Financial Assets among Older People, UK, 2002-2022. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27312) Harissis, K., Knapp, Martin R J. (1981) Staff vacancies and turnover in British old people's homes. Gerontologist, 21 (1). pp. 76-84. ISSN 0016-9013. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26841) Harlock, Jenny, Caiels, James, Marczak, Joanna, Peters, Michele, Fitzpatrick, Raymound, Wistow, Gerald, Forder, Julien, Jones, Karen C. (2019) Challenges in integrating health and social care: the Better Care Fund in England. Journal of Health Services Research & Policy, . ISSN 1355-8196. (doi:10.1177/1355819619869745) (KAR id:75901) Preview Hazel, H., Fenyo, Andrew J. (1993) Free To Be Myself: The Development of Teenage Fostering. Human Service Associates, Minnesota ISBN 0-9637696-0-X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27418) Healey, Andrew T., Knapp, Martin R J. (1995) Economic appraisal of psychotherapy. Mental Health Research Review, 2 . pp. 13-16. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27068) Healey, Andrew T. and Knapp, Martin R J. (1998) Economic evaluation of mental health service and treatment initiatives. In: Brooker, Charles and Repper, Julie, eds. Serious Mental Health Problems: Policy, Practice and Research. Balliere Tindall, London, pp. 130-156. ISBN 0-7020-2127-X. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26528) Healey, Andrew T., Knapp, Martin R J. (1996) The costs and benefits of treating drug misuse: an overview of the evidence. Mental Health Research Review, 3 . pp. 14-17. ISSN 1353-2650. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27080) Healey, Andrew T., Knapp, Martin R J., Astin, Jack, Beecham, Jennifer, Kemp, R., Kirov, G., David, Alessia (1998) Cost-effectiveness evaluation of compliance therapy for people with psychosis. British Journal of Psychiatry, 172 (5). pp. 420-424. ISSN 0007-1250 (Print) 1472-1465 (Online). (doi:10.1192/bjp.172.5.420) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26842) Healey, Andrew T., Knapp, Martin R J., Astin, Jack, Gossop, Michael, Marsden, John, Stewart, Chris, Lehmann, P., Godfrey, Christine (1998) Economic burden of drug dependency. British Journal of Psychiatry, 173 . pp. 160-165. ISSN 0007-1250 (Print) 1472-1465 (Online). (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26844) Healey, Andrew T., Knapp, Martin R J., Astin, Jack, Gossop, Michael, Marsden, John (1997) Social costs associated with illegal drug use. In: UNSPECIFIED, 1997-03-01T00:00:00, Venice. (Unpublished) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27087) Heath, Clara (2016) Creating reports of analysis findings to engage with decision-makers in your organisation. N/A. (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59011) Heath, Clara (2016) Exploring the differences between respondent groups. In: MAX toolkit webinar series, 2 Dec 2016, Canterbury, UK (hosted via webinar). (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59617) Heath, Clara (2016) Exploring the relationships between variables. In: MAX toolkit webinar series, 7th November 2016, 7th November 2016, Canterbury, UK (hosted via webinar). (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:58986) Heath, Clara (2017) Getting to know your carers. . Blog. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:61382) Heath, Clara (2016) Measuring impact using adult social care survey data. In: MAX toolkit webinar series, 2 Dec 2016, Canterbury, UK (hosted via webinar). (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59621) Heath, Clara (2016) Measuring impact using carers survey data. In: MAX toolkit webinar series, 2 Dec 2016, Canterbury, UK (hosted via webinar). (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59624) Heath, Clara (2017) Measuring the impact of services on carer-reported outcomes. . Blog. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:61381) Heath, Clara (2017) Using Carers' survey (PSS SACE) data to inform local decision-making: introducing the PSS SACE blog series. . Blog. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:61379) Heath, Clara (2016) The importance of planning and stakeholder engagement. In: MAX toolkit webinar series, 7th November 2016, 7th November 2016, Canterbury, UK (hosted via webinar). (Unpublished) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:58974) Heath, Clara (2016) The rocky road to developing a toolkit. . Blog. (KAR id:59071) Preview Heath, Clara, Jones, Karen C. (2016) Introducing the MAX toolkit. . Blog. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59039) Heath, Clara, Jones, Karen C. (2016) Using focused analysis to fulfil local information needs with ASCS and PSS SACE data. . Blog. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59041) Heath, Clara, Jones, Karen C. (2016) Why survey reports matter. . Blog. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59042) Heath, Clara, Jones, Karen C. (2016) The value of stakeholder engagement. . Blog. (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:59040) Heath, Clara, Malley, Juliette, Fox, Diane, Caiels, James (2013) Emerging themes and provisional ideas for how the toolkits developed during the MAX project can help local authorities maximise the use of data in adult social care. . Blog. (KAR id:59038) Preview Heath, Clara, Malley, Juliette, Fox, Diane, Caiels, James (2013) Local authority views and use of the adult social care and carers survey. . Blog. (KAR id:59019) Preview Heath, Clara, Malley, Juliette, Fox, Diane, Caiels, James (2013) The factors that affect local authority use of adult social care and carers survey data. . Blog. (KAR id:59028) Preview Heath, Clara, Malley, Juliette, Razik, Kamilla (2014) Is outcomes-based management and policy?making a reality for local government? In: ILPN Conference, 01-03 September 2014, London, UK. (KAR id:58872) Preview Heath, Clara and Malley, Juliette and Razik, Kamilla and Jones, Karen C. and Forder, Julien E. and Fox, Diane and Caiels, James and Beecham, Jennifer (2015) How can MAX help local authorities to use social care data to inform local policy? Maximising the value of survey data in adult social care [MAX] project [Executive Summary]. Working paper. Personal Social Services Research Unit, Kent, UK (KAR id:58852) Preview Heath, Clara, Malley, Juliette, Razik, Kamilla, Jones, Karen C., Forder, Julien E., Fox, Diane, Caiels, James, Beecham, Jennifer (2015) How can MAX help local authorities to use social care data to inform local policy? Maximising the value of survey data in adult social care [MAX] project. Quality and Outcomes Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:45475) Heath, Clara and Malley, Juliette and Razik, Kamilla and Jones, Karen C. and Forder, Julien E. and Fox, Diane and Caiels, James (2015) How can MAX help local authorities to use social care data to inform local policy? Maximising the value of survey data in adult social care [MAX] project [Full report]. Working paper. Personal Social Services Research Unit, Kent, UK (KAR id:59017) Preview Heath, Clara and Razik, Kamilla and Jones, Karen C. and Beecham, Jennifer and Forder, Julien E. and Malley, Juliette (2015) Further analysis of ASCS and PSS SACE data: Case studies of local authority (LA) practice. Discussion paper. Personal Social Services Research Unit, Kent, UK (KAR id:58870) Preview Heath, Clara, Razik, Kamilla, Jones, Karen C., Beecham, Jennifer, Forder, Julien E., Malley, Juliette (2015) Meeting local information needs with ASCS and PSS SACE data. . Blog. (KAR id:59068) Preview Henderson, Catherine, Knapp, Martin R J., Fernández, José-Luis, Beecham, Jennifer, Hirani, Shashi, Cartwright, Martin, Rixon, Lorna, Beynon, Michelle, Rogers, Anne, Bower, Peter, and others. (2013) Cost effectiveness of telehealth for patients with long term conditions (Whole Systems Demonstrator telehealth questionnaire study): nested economic evaluation in a pragmatic, cluster randomised controlled trial. British Medical Journal, 346 . ISSN 0959-535X. (doi:10.1136/bmj.f1035) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:33545) Henderson, Catherine, Malley, Juliette, Knapp, Martin R J. (2007) Maintaining Good Health for Older People with Dementia Who Experience Fractured Neck of Femur. Funded/commissioned by: National Audit Office. Personal Social Services Research Unit (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:27144) Henderson, Julie, Kavanagh, Shane M., Fraser, W., Mlika-Cabanne, N., Maillard, F., Breart, G. (1998) Economic evaluation of early amniotomy in low risk women. British Journal of Midwifery, 6 (1). pp. 28-32. ISSN 0969-4900. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:26845) Hibbs, Rebecca, Magill, Nicholas, Goddard, Elizabeth, Rhind, Charlotte,	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30458691716194153, "perplexity": 14029.720516317991}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347439019.86/warc/CC-MAIN-20200604032435-20200604062435-00085.warc.gz"}
http://mathhelpforum.com/pre-calculus/127769-reverse-second-order-logrithm.html	# Thread: Reverse Second Order Logrithm 1. ## Reverse Second Order Logarithm Hi all I hopw this is the correct forum for this post, but here goes. I have the following equation :- $\displaystyle y=a+blog(x) + clog(x)^2$ This basically describes the power conversion efficiency from AC Watts input (x) to DC Watts output (y) for a power supply. I need to reverse the equation so that I can find the AC Watts input (x) from the DC Watts output (y). This has got me stumped Cheers Andy 2. Originally Posted by Duckfather Hi all I hopw this is the correct forum for this post, but here goes. I have the following equation :- $\displaystyle y=a+blog(x) + clog(x)^2$ This basically describes the power conversion efficiency from AC Watts input (x) to DC Watts output (y) for a power supply. I need to reverse the equation so that I can find the AC Watts input (x) from the DC Watts output (y). This has got me stumped Cheers Andy Let u= log(x). Then your equation is $\displaystyle y= a+ bu+ cu^2$. Solve that with the quadratic formula and then $\displaystyle x= 10^u$ (I am assuming that your log is the "common log" to base 10).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8477817177772522, "perplexity": 1693.8446558133085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267862929.10/warc/CC-MAIN-20180619115101-20180619135101-00561.warc.gz"}
http://www.mathcounterexamples.net/differentiability-multivariable-real-functions-part2/	# Differentiability of multivariable real functions (part2) Following the article on differentiability of multivariable real functions (part 1), we look here at second derivatives. We consider a function $$f : \mathbb R^n \to \mathbb R$$ with $$n \ge 2$$. Schwarz’s theorem states that if $$f : \mathbb R^n \to \mathbb R$$ has continuous second partial derivatives at any given point in $$\mathbb R^n$$, then for $$(a_1, \dots, a_n) \in \mathbb R^n$$ and $$i,j \in \{1, \dots, n\}$$: $\frac{\partial^2 f}{\partial x_i \partial x_j}(a_1, \dots, a_n)=\frac{\partial^2 f}{\partial x_j \partial x_i}(a_1, \dots, a_n)$ ### A function for which $$\frac{\partial^2 f}{\partial x \partial y}(0,0) \neq \frac{\partial^2 f}{\partial y \partial x}(0,0)$$ We consider: $\begin{array}{l\|rcl} f : & \mathbb R^2 & \longrightarrow & \mathbb R \\ & (0,0) & \longmapsto & 0\\ & (x,y) & \longmapsto & \frac{xy(x^2-y^2)}{x^2+y^2} \text{ for } (x,y) \neq (0,0) \end{array}$ $$f$$ is defined and indefinitely differentiable at all points different from the origin. For $$(x,y) \neq (0,0)$$: \begin{align} \left\vert \frac{f(x,y)}{\sqrt{x^2+y^2}} \right\vert & = \left\vert \frac{xy(x^2-y^2)}{\sqrt{x^2+y^2}(x^2+y^2)} \right\vert\\ & \le \left\vert \frac{xy}{\sqrt{x^2+y^2}} \right\vert\\ & \le \frac{(\vert x \vert + \vert y \vert)^2}{2 \sqrt{x^2+y^2}}\\ & \le \sqrt{x^2+y^2}\end{align} as we have for all $$(x,y) \in \mathbb R^2$$: $$\vert xy \vert \le \frac{(\vert x \vert +\vert y \vert)^2}{2}$$ and $$(\vert x \vert +\vert y \vert)^2 \le 2 (x^2+y^2)$$. This proves that $$f$$ is continuous and differentiable at the origin, with $$f(0,0)=0$$ and a Fréchet derivative equal to zero at the origin. So $\frac{\partial f}{\partial x}(0,0)=\frac{\partial f}{\partial y}(0,0)=0$ We now look at the mixed partial derivatives. For $$(x,y) \neq (0,0)$$:$\begin{cases} \frac{\partial f}{\partial x}(x,y) = \frac{(x^2+y^2)(3x^2y-y^3)+2x^2y(y^2-x^2)}{(x^2+y^2)^2}\\ \frac{\partial f}{\partial y}(x,y) = \frac{(x^2+y^2)(x^3-3x y^2)+2xy^2(y^2-x^2)}{(x^2+y^2)^2} \end{cases}$ In particular: \begin{align} \frac{\partial f}{\partial x}(0,y) &= -y \text{ for } y \neq 0\\ \frac{\partial f}{\partial x}(x,0) &= x \text{ for } x \neq 0 \end{align} which implies $\frac{\partial^2 f}{\partial x \partial y}(0,0) =1 \neq -1 = \frac{\partial^2 f}{\partial y \partial x}(0,0)$ The mixed second derivatives of $$f$$ are not symmetric.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9973458647727966, "perplexity": 428.940902226159}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514708.24/warc/CC-MAIN-20181022050544-20181022072044-00507.warc.gz"}
https://socratic.org/questions/how-do-you-factor-40-2y-w-2	Algebra Topics # How do you factor 40-(2y-w)^2? Aug 4, 2017 $\left(2 \sqrt{10} - 2 y + w\right) \left(2 \sqrt{10} + 2 y - w\right)$ #### Explanation: $\text{factorise using the method of "color(blue)"difference of squares}$ •color(white)(x)a^2-b^2=(a-b)(a+b) $\text{note that } {\left(2 \sqrt{10}\right)}^{2} = 40$ $\Rightarrow 40 - {\left(2 y - w\right)}^{2}$ $= {\left(2 \sqrt{10}\right)}^{2} - {\left(2 y - w\right)}^{2}$ $\text{with "a=2sqrt10" and } b = 2 y - w$ (2sqrt10-(2y-w))(2sqrt10+(2y-w) $= \left(2 \sqrt{10} - 2 y + w\right) \left(2 \sqrt{10} + 2 y - w\right)$ ##### Impact of this question 216 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1510057896375656, "perplexity": 24614.7234429604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347409337.38/warc/CC-MAIN-20200530133926-20200530163926-00081.warc.gz"}
https://2020.euro-par.org/program/conference/	# Program Program is also available in Google Calendar [ical] Conference proceedings are available on Springer website. Show abstracts ## Wednesday 26.08.2020 14:00 - 14:30 14:30 - 15:30 ### Multicore and Manycore Parallelism (A) Chairs: Witold Rudnicki NVPhTM: An Efficient Phase-Based Transactional System for Non-Volatile Memory Alexandro Baldassin, Rafael Murari, João Paulo Carvalho, Guido Araujo, Daniel Castro, João Barreto and Paolo Romano Non-Volatile Memory (NVM) is an emerging memory technology aimed to eliminate the gap between main memory and stable storage. Nevertheless, today’s programs will not readily benefit from NVM because crash failures may render the program in an unrecoverable and inconsistent state. In this context, the use of durable transactions has been proposed so as to ease the adoption of NVM. It leverages on the well-know semantics of database transactions to simplify the task of programming NVM systems. This is achieved by logging NVM writes using software (SW) or hardware (HW) transaction primitives. Although SW transactions are flexible and unbounded, they may significantly hurt the performance of short-lived transactions. On the other hand, HW transactional memories provide low-overhead but are resource-constrained. In this paper we present NVPhTM, a transactional system for NVM that delivers the best out of both HW and SW transactions by dynamically selecting the best execution mode according to the application's characteristics. NVPhTM is comprised of a set of heuristics to guide online phase transition. Furthermore, a careful design of the phase transition step is devised to guarantee persistency when transitioning between HW and SW phases. To the best of our knowledge, NVPhTM is the first phase-based system to provide durable transactions. Experimental results with the STAMP benchmark show that the proposed heuristics are efficient in guiding phase transitions with low overhead. In particular, the NVM-aware heuristics provided an average speedup of up to 10\% when compared to a system using NVM-oblivious heuristics, with only 1\% of transition overhead in the worst case. Enhancing Resource Management through Prediction-based Policies Antoni Navarro, Arthur F. Lorenzon, Eduard Ayguadé and Vicenç Beltran Task-based programming models are emerging as a promising alternative to make the most of multi-/many-core systems. These programming models rely on runtime systems, and their goal is to improve application performance by properly scheduling application tasks to cores. Additionally, these runtime systems offer policies to cope with application phases that lack in parallelism to fill all cores. However, these policies are usually static and favor either performance or energy efficiency. In this paper, we have extended a task-based runtime system with a lightweight monitoring and prediction infrastructure that dynamically predicts the optimal number of cores required for each application phase, thus improving both performance and energy efficiency. Through the execution of several benchmarks in multi-/many-core systems, we show that our prediction-based policies have competitive performance while improving energy efficiency when compared to state of the art policies. Accelerating Overlapping Community Detection: Performance Tuning a Stochastic Gradient Markov Chain Monte Carlo Algorithm Ismail Elhelw, Rutger Hofman and Henri Bal Building efficient algorithms for data-intensive problems requires deep analysis of data access patterns. Random data access patterns exacerbate this process. In this paper, we discuss accelerating a randomized data-intensive machine learning algorithm using multi-core CPUs and several GPUs. A thorough analysis of the algorithm’s data dependencies enabled a 75% reduction in its memory footprint. We created custom compute kernels via code generation to identify the optimal set of data placement and computational optimizations per compute device. An empirical evaluation shows up to 245x speedups compared to an optimized sequential version. Another result from this evaluation is that achieving peak performance does not always match intuition: e.g., depending on the GPU architecture, vectorization may increase or hamper performance. ### Cluster, Cloud and Edge Computing (B) Chairs: Paweł Czarnul TorqueDB: Distributed Querying of Time-series Data from Edge-local Storage Dhruv Garg, Prathik Shirolkar, Anshu Shukla and Yogesh Simmhan The rapid growth in edge computing devices as part of Internet of Things (IoT) allows real-time access to time-series data from 1000's of sensors. Such observations are queried to optimize the health of the infrastructure. Recently, edge-local storage are helping retain data on the edge rather than move them centrally to the cloud. However, such systems do not support flexible querying over the data spread across 10--100's of devices. There is also a lack of distributed time-series databases that can run on the edge devices. Here, we propose TorqueDB, a distributed query engine over time-series data that operates on edge and fog resources. TorqueDB leverages our prior work on ElfStore, a distributed edge-local file store, and InfluxDB, a time-series database, to enable temporal queries to be decomposed and executed across multiple fog and edge devices. Interestingly, we move data into InfluxDB on-demand while retaining the durable data within ElfStore for use by other applications. We also design a cost model that maximizes parallel movement and execution of the queries across resources, and utilizes caching. Our experiments on a real edge, fog and cloud deployment show that TorqueDB performs comparable to InfluxDB on a cloud VM for a smart city query workload, but without the associated costs. Data-Centric Distributed Computing on Networks of Mobile Devices Pedro Sanches, João A. Silva, António Teófilo and Hervé Paulino In the last few years we have seen a significant increase both in the number and capabilities of mobile devices, as well as in the num- ber of applications that need more and more computing and storage re- sources. Currently, in order to deal with this growing need for resources, applications make use of cloud services. This brings some problems: high latencies, considerable use of energy and bandwidth, and the unavail- ability of connectivity infrastructures. Given this context, for some ap- plications it makes sense to do part, or all, of the computing locally on the mobile devices themselves. In this paper we present Oregano, a framework for distributed computing on mobile devices. Oregano is capable of processing sets or streams of data generated on mobile de- vice networks, without requiring centralized services. Contrary to the current state of the art, where computing and data are sent to worker mobile devices, our Oregano performs the computation where the data is located, significantly reducing the amount of data exchanged. WPSP: a multi-correlated weighted policy for VM selection and migration for Cloud computing Sergi Vila Almenara, Josep Lluis Lerida, Fernando Cores, Fernando Guirado and Fabio Verdi Using virtualization, cloud environments satisfy dynamically the computational resource necessities of the user. The dynamic use of the resources determines the demand of working hosts. Through virtual machine (VM) migrations, datacenters perform load balancing to optimise the resource usage and solve saturation. In this work, a policy, named as WPSP (Weighted Pearson Selection Policy), is implemented to choose which virtual machines are more suitable to be migrated. The policy evaluates, for each VM, both the CPU load and the Network traffic influence on the assigned host. The corresponding Pearson correlation coefficients are calculated for each one of the VMs and then weighted in order to provide a relationship between the values and the host behaviour. The main goal is to clearly identify and then migrate the VMs that are responsible of the Host saturation but also considering their communications. Using the CloudSim simulator, the policy is compared with the rest of heuristic techniques in the literature, resulting in a reduction of 85% in the number of migrations, and thus reducing the use of bandwidth (6%), network saturation (24%) and over-saturated hosts (50%). Additionally, it is presented an improved VM allocation technique to reduce the distance the VMs must travel in order to be migrated, obtaining an average reduction of 43% in the quantity of migrated data. 15:30 - 16:00 16:00 - 17:20 ### Support Tools and Environments (A) Chairs: Bartosz Baliś Skipping Non-essential Instructions Makes Data-dependence Profiling Faster Nicolas Morew, Mohammad Norouzi, Ali Jannesari and Felix Wolf Data-dependence profiling is a dynamic program-analysis technique to discover potential parallelism in sequential programs. Unlike purely static analysis, which may overestimate the number of dependences because it does not know many pointers values and array indices at compile time, profiling has the advantage of recording data dependences that actually occur at runtime. But it has the disadvantage of significantly slowing down program execution, often by a factor of 100. In our earlier work, we lowered the overhead of data-dependence profiling by excluding polyhedral loops, which can be handled statically using certain compilers. However, neither does every program contain polyhedral loops, nor are statically identifiable dependences restricted to such loops. In this paper, we introduce an orthogonal approach, focusing on data dependences between accesses to scalar variables - across the entire program, inside and outside loops. We first analyze the program statically and identify memory-access instructions that create data dependences that would appear in any execution of these instructions. Then, we exclude these instructions from instrumentation, allowing the profiler to skip them at runtime and avoid the associated overhead. We evaluate our approach with 49 benchmarks from three benchmark suites. We improved the profiling time of all programs by at least 38%, with a median reduction of 61% across all the benchmarks. A toolchain to verify the parallelization of OmpSs-2 applications Simone Economo, Sara Royuela Alcázar, Eduard Ayguadé Parra and Vicenç Beltran Querol Programming models for task-based parallelization based on compile-time directives are very effective at uncovering the parallelism available in HPC applications. Despite that, the process of correctly annotating complex applications is error-prone and may hinder the general adoption of these models. In this paper, we target the OmpSs-2 programming model and present a novel toolchain able to detect parallelization errors coming from non-compliant OmpSs-2 applications. Our toolchain verifies the compliance with the OmpSs-2 programming model using local task analysis to deal with each task separately, and structural induction to extend the analysis to the whole program. To improve the effectiveness of our tools, we also introduce some ad-hoc verification annotations, which can be used manually or automatically to disable the analysis of specific code regions. Experiments run on a sample of representative kernels and applications show that our toolchain can be successfully used to verify the parallelization of complex real-world applications. ### High Performance Architectures and Compilers (A) Chairs: Bartosz Baliś Modelling Standard and Randomized Slimmed Folded Clos Networks Cristóbal Camarero, Carmen Martinez, Ramon Beivide and Javier Corral Fat-trees (FTs) are widely known topologies that, among other advantages, provide full bisection bandwidth. However, many implementations of FTs are made slimmed to cheapen the infrastructure, since most applications do not make use of this full bisection bandwidth. In this paper Extended Generalized Random Folded Clos (XGRFC) interconnection networks are introduced as cost-efficient alternatives to Extended Generalized Fat Trees (XGFT), which is a widely used topological description for slimmed FTs. This is proved both by obtaining a theoretical model of the performance and evaluating it using simulation. Among the results, it is shown that a XGRFC is able to connect 20k servers with 27% less routers than the corresponding XGFT and still providing the same performance under uniform traffic. OmpMemOpti: Optimized Memory Movement for Heterogeneous Computing Prithayan Barua, Jisheng Zhao and Vivek Sarkar The fast development of acceleration architectures and applications has made heterogeneous computing the norm for high-performance computing. The cost of high volume data movement to the accelerators is an important bottleneck both in terms of application performance and developer productivity. Memory management is still a manual task performed tediously by expert programmers. In this paper, we develop a compiler analysis to automate memory management for heterogeneous computing. We propose an optimization framework that casts the problem of detection and removal of redundant data movements into a partial redundancy elimination (PRE) problem and applies the lazy code motion technique to optimize it. We chose OpenMP as the underlying parallel programming model and implemented our optimization framework in the LLVM toolchain. We evaluated it with ten benchmarks and obtained a geometric speedup of 2.3$\times$, and reduced on average 50\% of the total bytes transferred between the host-GPU. ### Scheduling and Load Balancing (B) Chairs: Joanna Berlińska Xiao Meng and Lukasz Golab Workloads with precedence constraints due to data dependencies are common in various applications. These workloads can be represented as directed acyclic graphs (DAG), and are often data-intensive, meaning that data loading costs are the dominant factor and thus cache misses should be minimized. We address the problem of parallel scheduling of a DAG of data-intensive tasks to minimize makespan. To do so, we propose greedy online scheduling algorithms that take load balancing, data dependencies, and data locality into account. Simulations and an experimental evaluation using an Apache Spark cluster demonstrate the advantages of our solutions. A Makespan Lower Bound for the Scheduling of the Tiled Cholesky Factorization based on ALAP scheduling Olivier Beaumont, Julien Langou, Willy Quach and Alena Shilova Due to the advent of multicore architectures and massive parallelism, the tiled Cholesky factorization algorithm has recently received plenty of attention and is often referenced by practitioners as a case study. It is also implemented in mainstream dense linear algebra libraries and is used as a testbed for runtime systems. However, we note that theoretical study of the parallelism of this algorithm is currently lacking. In this paper, we present new theoretical results about the tiled Cholesky factorization in the context of a parallel homogeneous model without communication costs. Based on the relative costs of involved kernels, we prove that only two different situations must be considered, typically corresponding to CPU and GPU costs. By a careful analysis on the number of tasks of each type that run simultaneously in the ALAP (As Late As Possible) schedule without resource limitation, we are able to determine precisely the number of busy processors at any time (as degree 2 polynomials). We then use this information to find a closed form formula for the minimum time to schedule a tiled Cholesky factorization of size n on p processors. We show that this bound outperforms classical bounds from the literature. We also prove that ALAP(p), an ALAP-based schedule where the number of resources is limited to p, has a makespan extremely close to the lower bound, thus proving both the effectiveness of ALAP(p) schedule and of the lower bound on the makespan. Olivier Beaumont, Lionel Eyraud-Dubois and Alena Shilova Training Deep Neural Networks is known to be an expensive operation, both in terms of computational cost and memory load. Indeed, during training, all intermediate layer outputs (called activations) computed during the forward phase must be stored until the corresponding gradient has been computed in the backward phase. These memory requirements sometimes prevent to consider larger batch sizes and deeper networks, so that they can limit both convergence speed and accuracy. Recent works have proposed to offload some of the computed forward activations from the memory of the GPU to the memory of the CPU. This requires to determine which activations should be offloaded and when these transfers from and to the memory of the GPU should take place. We prove that this problem is NP-hard in the strong sense, and we propose two heuristics based on relaxations of the problem. We perform extensive experimental evaluation on standard Deep Neural Networks. We compare the performance of our heuristics against previous approaches from the literature, showing that they achieve much better performance in a wide variety of situations. Improving mapping for sparse direct solvers: A trade-off between data locality and load balancing Changjiang Gou, Ali Al Zoobi, Anne Benoit, Mathieu Faverge, Loris Marchal, Grégoire Pichon and Pierre Ramet In order to express parallelism, parallel sparse direct solvers take advantage of the elimination tree to exhibit tree-shaped task graphs, where nodes represent computational tasks and edges represent data dependencies. One of the pre-processing stages of sparse direct solvers consists of mapping computational resources (processors) to these tasks. The objective is to minimize the factorization time by exhibiting good data locality and load balancing. The proportional mapping technique is a widely used approach to solve this resource-allocation problem. It achieves good data locality by assigning the same processors to large parts of the elimination tree. However, it may limit load balancing in some cases. In this paper, we propose a dynamic mapping algorithm based on proportional mapping. This new approach relaxes the data locality criterion to improve load balancing. In order to validate the newly introduced method, we perform extensive experiments on the PASTIX sparse direct solver. It demonstrates that our algorithm enables better static scheduling of the numerical factorization while keeping good data locality. 17:20 - 17:30 17:30 - 18:20 ### Keynote Ewa Deelman (A) Automating Science Workflows:Challenges and Opportunities Chair: Rizos Sakellariou Abstract available on keynotes page 18:20 - 19:00 ## Thursday 27.08.2020 13:00 - 14:30 ### industry: Huawei (A) Details available on "industry: Huawei" page 14:30 - 15:00 ### Best paper (A) Chairs: Morris Riedel Maximizing I/O Bandwidth for Reverse Time Migration on Heterogeneous Large-Scale Systems Tariq Alturkestani, Hatem Ltaief and David Keyes Reverse Time Migration (RTM) is an important scientific application for oil and gas exploration. The 3D RTM simulation generates terabytes of intermediate data that does not fit in main memory. In particular, RTM has two successive computational phases, i.e., the forward modeling and the backward propagation, that necessitate to write and then to read the state of the computed solution grid at specific time steps of the time integration. Advances in memory architecture have made it feasible and affordable to integrate hierarchical storage media on large-scale systems, starting from the traditional Parallel File Systems (PFS) to intermediate fast disk technologies (e.g., node-local and remote-shared Burst Buffer) and up to CPU main memory. To address the trend of heterogeneous HPC systems deployment, we introduce an extension to our Multilayer Buffer System (MLBS) framework to further maximize RTM I/O bandwidth in presence of GPU hardware accelerators. The main idea is to leverage the GPU’s High Bandwidth Memory (HBM) as an additional storage media layer. The objective of MLBS is ultimately to hide the application’s I/O overhead by enabling a buffering mechanism operating across all the hierarchical storage media layers. MLBS is therefore able to sustain the I/O bandwidth at each storage media layer. By asynchronously performing expensive I/O operations and creating opportunities for overlapping data motion with computations, MLBS may transform the original I/O bound behavior of the RTM application into a compute-bound regime. In fact, the prefetching strategy of MLBS allows the RTM application to believe that it has access to a larger memory capacity on the GPU, while transparently performing the necessary housekeeping across the storage layers. We demonstrate the effectiveness of MLBS on the Summit supercomputer using 2048 compute nodes equipped with a total of 12288 GPUs by achieving up to 1.4X performance speedup compared to the reference PFS-based RTM implementation for large 3D solution grid. 15:00 - 15:30 ### Best artifact (A) Chairs: Maciej Szpindler A Prediction Framework for Fast Sparse Triangular Solves Najeeb Ahmad, Buse Yilmaz and Didem Unat Sparse triangular solve (SpTRSV) is an important linear algebra kernel, finding extensive uses in numerical and scientific computing. The parallel implementation of SpTRSV is a challenging task due to the sequential nature of the steps involved. This makes it, in many cases, one of the most time-consuming operations in an application. Many approaches for efficient SpTRSV on CPU and GPU systems have been proposed in the literature. However, no single implementation or platform (CPU or GPU) gives the fastest solution for all input sparse matrices. In this work, we propose a machine learning-based framework to predict the SpTRSV implementation giving the fastest execution time for a given sparse matrix based on its structural features. The framework is tested with six SpTRSV implementations on a state-of-the-art CPU-GPU machine (Intel Xeon Gold CPU, NVIDIA V100 GPU). Experimental results, with 998 matrices taken from the SuiteSparse Matrix Collection, show the classifier prediction accuracy of 87% for the fastest SpTRSV algorithm for a given input matrix. Predicted SpTRSV implementations achieve average speedups (harmonic mean) in the range of 1.2-2.4x against the six SpTRSV implementations used in the evaluation. 15:30 - 15:40 15:40 - 16:40 ### Data Management, Analytics and Machine Learning (A) Chairs: Morris Riedel Accelerating Deep Learning Inference with Cross-Layer Data Reuse on GPUs Xueying Wang, Guangli Li, Xiao Dong, Jiansong Li, Lei Liu and Xiaobing Feng Accelerating the deep learning inference is very important for real-time applications. In this paper, we propose a novel method to fuse the layers of convolutional neural networks (CNNs) on Graphics Processing Units (GPUs), which applies data reuse analysis and access optimization in different levels of the memory hierarchy. To achieve the balance between computation and memory access, we explore the fu- sion opportunities in the CNN computation graph and propose three fusion modes of convolutional neural networks: straight, merge and split. Then, an approach for generating efficient fused code is designed, which goes deeper in multi-level memory usage for cross-layer data reuse. The effectiveness of our method is evaluated with the structures from state- of-the-art CNNs on two different GPU platforms, NVIDIA TITAN Xp and Tesla P4. The experiments show that the average speedup is 2.02 × on representative structures of CNNs, and 1.57× on end-to-end inference of SqueezeNet. Optimizing FFT-based convolution on ARMv8 multi-core CPUs Qinglin Wang, Dongsheng Li, Xiandong Huang, Siqi Shen, Songzhu Mei and Jie Liu Convolutional Neural Networks (CNNs) are widely applied in various machine learning applications and very time-consuming. Most of CNNs' execution time is consumed by convolutional layers. A common approach to implementing convolutions is the FFT-based one, which can reduce the arithmetic complexity of convolutions without losing too much precision. As the performance of ARMv8 multi-core CPUs improves, they can also be utilized to perform CNNs like Intel X86 CPUs. In this paper, we present a new parallel FFT-based convolution implementation on ARMv8 multi-core CPUs. The implementation makes efficient use of ARMv8 multi-core CPUs through a series of computation and memory optimizations. The experiment results on two ARMv8 multi-core CPUs demonstrate that our new implementation gives much better performance than two existing approaches in most cases. Distributed Fine-Grained Traffic Speed Prediction for Large-Scale Transportation Networks based on Automatic LSTM Customization and Sharing Ming-Chang Lee, Jia-Chun Lin and Ernst Gunnar Gran Short-term traffic speed prediction has been an important research topic in the past decade, and many approaches have been introduced. However, providing fine-grained, accurate, and efficient traffic-speed prediction for large-scale transportation networks where numerous traffic detectors are deployed has not been well studied. In this paper, we propose DistPre, which is a distributed fine-grained traffic speed prediction scheme for large-scale transportation networks. To achieve fine-grained and accurate traffic-speed prediction, DistPre customizes a Long Short-Term Memory (LSTM) model with an appropriate hyperparameter configuration for a detector. To make such customization process efficient and applicable for large-scale transportation networks, DistPre conducts LSTM customization on a cluster of computation nodes and allows any trained LSTM model to be shared between different detectors. If a detector observes a similar traffic pattern to another one, DistPre directly shares the existing LSTM model between the two detectors rather than customizing an LSTM model per detector. Experiments based on traffic data collected from freeway I5-N in California are conducted to evaluate the performance of DistPre. The results demonstrate that DistPre provides time-efficient LSTM customization and accurate fine-grained traffic-speed prediction for large-scale transportation networks. ### Accelerator Computing (B) Chairs: Łukasz Szustak cuDTW++: Ultra-Fast Dynamic Time Warping on CUDA-enabled GPUs Bertil Schmidt and Christian Hundt Dynamic Time Warping (DTW) is a widely used distance measure in the field of time series data mining. However, calculation of DTW scores is compute-intensive since the complexity is quadratic in terms of time series lengths. This renders important data mining tasks computationally expensive even for moderate query lengths and database sizes. Previous solutions to accelerate DTW on GPUs are not able to fully exploit their compute performance due to inefficient memory access schemes. In this paper, we introduce a novel parallelization strategy to drastically speed-up DTW on CUDA-enabled GPUs based on using low latency warp intrinsics for fast inter-thread communication. We show that our CUDA parallelization (cuDTW++) is able to achieve over 90% of the theoretical peak performance of modern Volta-based GPUs, thereby clearly outperforming the previously fastest CUDA implementation (cudaDTW) by over one order-of-magnitude. Furthermore, cuDTW++ achieves two-to-three orders-of-magnitude speedup over the state-of-the-art CPU program UCR-Suite for subseqeunce search of ECG signals. We plan to make cuDTW++ publicly available upon acceptance of this paper. Heterogeneous CPU+iGPU Processing for Efficient Epistasis Detection Rafael Campos, Diogo Marques, Sergio Santander-Jiménez, Leonel Sousa and Aleksandar Ilic Epistasis detection represents a fundamental problem in bio-medicine to understand the reasons for occurrence of complex phenotypic traits (diseases) across a population of individuals. Exhaustively examining all possible interactions of multiple Single-Nucleotide Polymorphisms provides the most reliable way to identify accurate solutions, but it is both computationally and memory intensive task. To tackle this challenge, this work proposes a modular and self-adaptive framework for high-performance and energy-efficient epistasis analysis on modern tightly-coupled heterogeneous platforms composed of multi-core CPUs and integrated GPUs. To fully exploit the capabilities of these systems, the proposed framework incorporates both task- and data-parallel approaches specifically tailored to enhance single and multi-objective epistasis detection on each device architecture, along with allowing efficient collaborative execution across all devices. The experimental results show the ability of the proposed framework to handle the heterogeneity of an Intel CPU+iGPU system, achieving performance and energy-efficiency gains of up to 5x and 6x in different parallel execution scenarios. SYCL-Bench: A Versatile Cross-Platform Benchmark Suite for Heterogeneous Computing Sohan Lal, Aksel Alpay, Philip Salzmann, Biagio Cosenza, Alexander Hirsch, Nicolai Stawinoga, Peter Thoman, Thomas Fahringer and Vincent Heuveline The SYCL standard promises to enable high productivity in heterogeneous programming of a broad range of parallel devices, including multicore CPUs, GPUs, and FPGAs. Its modern and expressive C++ API design, as well as flexible task graph execution model give rise to ample optimization opportunities at run-time, such as the overlapping of data transfers and kernel execution. However, it is not clear which of the existing SYCL implementations perform such scheduling optimizations, and to what extent. Furthermore, SYCL's high level of abstraction may raise concerns about sacrificing performance for ease of use. Benchmarks are required to accurately assess the performance behavior of high-level programming models such as SYCL. To this end, we present SYCL-Bench, a versatile benchmark suite for device characterization and runtime benchmarking, written in SYCL. We experimentally demonstrate the effectiveness of SYCL-Bench by performing device characterization of the NVIDIA TITAN X GPU, and by evaluating the efficiency of the hipSYCL and ComputeCpp SYCL implementations. 16:40 - 17:00 17:00 - 18:00 ### Keynote Geoffrey Fox (A) Advancing Science with Deep Learning, HPC, Data Benchmarks and Data Engineering Chair: Christan Lengauer Abstract available on keynotes page 18:00 - 19:10 ### Parallel Numerical Methods and Applications (A) Chairs: Hatem Ltaief Efficient Ephemeris Models for Spacecraft Trajectory Simulations on GPUs Fabian Schrammel, Florian Renk, Arya Mazaheri and Felix Wolf When a spacecraft is released into space, its start condition and future trajectory in terms of position and speed cannot be precisely predicted. To ensure that the object does not violate space debris mitigation or planetary protection standards, such that it causes potential damage or contamination of celestial bodies, spacecraft-mission designers conduct a multitude of simulations to verify the validity of the set of all probable trajectories. Such simulations are usually independent, making them a perfect match for parallelization. The European Space Agency (ESA) developed a GPU-based simulator for the exact purpose and achieved reasonable speedups in comparison with the established multi-threaded CPU version. However, we noticed that the performance starts to degrade as the spacecraft trajectories diverge in time. Our empirical analysis using GPU profilers showed that the application suffers from poor data locality and high memory traffic. In this paper, we propose an alternative data layout, which increases data locality within thread blocks. Furthermore, we introduce alternative model configurations that lower both algorithmic effort and the number of memory requests without violating accuracy requirements. Our experiments show that our method is able to achieve speedups of up to 2.6x. Multiprecision block-Jacobi for Iterative Triangular Solves Fritz Goebel, Hartwig Anzt, Terry Cojean, Goran Flegar and Enrique S. Quintana-Orti Recent research efforts have shown that Jacobi and block- Jacobi relaxation methods can be used as an effective and highly parallel approach for the solution of sparse triangular linear systems arising in the application of ILU-type preconditioners. Simultaneously, a few orthogonal (independent) works have focused on designing efficient high performance adaptive-precision block-Jacobi preconditioning (block-diagonal scaling), in the context of the iterative solution of sparse linear systems, on manycore architectures. In this paper, we bridge the gap between relaxation methods based on regular splittings and preconditioners by demonstrating that iterative refinement can be leveraged to construct a relaxation method from the preconditioner. In addition, we exploit this insight to construct a highly-efficient sparse triangular system solver for graphics processors that combines iterative refinement with the block- Jacobi preconditioner available in the Ginkgo library. Parallel Finite Cell Method with Adaptive Geometric Multigrid S. Saberi, A. Vogel and G. Meschke The generation of appropriate computational meshes in the context of numerical methods for partial differential equations is technical and laborious and has motivated a class of advanced discretization methods commonly referred to as unfitted finite elements. To this end, the finite cell method (FCM) combines high-order FEM, adaptive quadrature integration and weak imposition of boundary conditions to embed a physical domain into a structured background mesh. While unfortunate cut configurations in unfitted finite element methods lead to severely ill-conditioned system matrices that pose challenges to iterative solvers, such methods allow for optimized data patterns and for a scalable implementation. In this work, we employ linear octrees for handling the FCM discretization that allow for parallel scalability, adaptive refinement and efficient computation on the commonly regular background grid. We present a parallel adaptive geometric multigrid with Schwarz smoothers for the solution of the resultant system of the Laplace operator. We focus on exploiting the hierarchical nature of space tree data structures for the generation of the required multigrid spaces and discuss scalable and robust extension of the methods across process interfaces. We present both weak and strong scaling of our implementation up to a billion degrees of freedom on distributed-memory clusters. ### Parallel and Distributed Programming, Interfaces, and Languages (B) Chairs: Phil Trinder Managing Failures in task-based parallel workflows in distributed computing environments Jorge Ejarque, Marta Bertran, Javier Álvarez Cid-Fuentes, Javier Conejero and Rosa M. Badia Current scientific workflows are large and complex. They normally perform thousands of simulation whose results combined with searching and data analytics algorithms, in order to infer new knowledge, generate a very large amount of data. To this end, workflows are composed by large amounts of tasks from which is very likely that, at least, one of them fails. Most of the work done about failure management in workflow managers and runtimes focuses on recovering from failures caused by resources (retrying or resubmitting the failed computation in other resources, etc.) However, some of these failures can be caused by the application itself (corrupted data, algorithms which are not converging, etc.), and these fault tolerance mechanisms are not sufficient to perform a successful workflow execution. In these cases, the developer has to add some code in their applications to prevent and manage the possible failures. In this paper, we propose a simple interface and a set of transparent runtime mechanism to simplify how scientists deal with application-based failures in task-based parallel workflows. We have validated our proposal with a set of e-science use cases to show the benefits of the proposed interface and mechanism in terms of programming productivity and performance. Accelerating Nested Data Parallelism: Preserving Regularity Lars B. van den Haak, Trevor L. McDonell, Gabriele K. Keller and Ivo Gabe de Wolff Irregular nested data-parallelism is a powerful programming model which enables the expression of a large class of parallel algorithms. However, it is notoriously difficult to compile such programs to efficient code for modern parallel architectures. Regular data-parallelism, on the other hand, is much easier to compile to efficient code, but too restricted to express some problems conveniently or in a manner to exploit the full parallelism. We extend the regular data-parallel programming model to allow for the parallel execution of array level conditionals and iterations over irregular nested structures. We present two novel static analyses to optimise the code generation to reduce the costs of this more powerful irregular model. We present benchmarks to support our claim that these extensions are effective as well as feasible, as they enable to exploit the full parallelism of an important class of algorithms, and together with our optimisations lead to an improvement in absolute performance over an implementation limited to exploiting only regular parallelism. Alexandre Denis, Emmanuel Jeannot, Philippe Swartvagher and Samuel Thibault A Compression-Based Design for Higher Throughput in a Lock-Free Hash Map Pedro Moreno, Miguel Areias and Ricardo Rocha Lock-free implementation techniques are known to improve the overall throughput of concurrent data structures. A hash map is an important data structure used to organize information that must be accessed frequently. A key role of a hash map is the ability to balance workloads by dynamically adjusting its internal data structures in order to provide the fastest possible access to the information. This work extends a previous lock-free hash-trie map design to also support \emph{lock-free compression}. The main goal is to significantly reduce the depth of the internal hash levels within the hash map, in order to minimize cache misses and increase the overall throughput. To materialize our design, we redesigned the existent search, insert, remove and expand operations in order to maintain the lock-freedom property of the whole design. Experimental results show that lock-free compression effectively improves the search operation and, in doing so, it outperforms the previous design, which was already quite competitive. ## Friday 28.08.2020 13:00 - 14:45 14:45 - 15:00 15:00 - 15:50 ### Keynote Piotr Sankowski (A) Breaking the PRAM O(log n) complexity bounds on MPC Abstract available on keynotes page 15:50 - 16:00 16:00 - 17:00 ### Theory and Algorithms for Parallel and Distributed Processing (A) Chairs: Marek Klonowski On the Power of Randomization in Distributed Algorithms in Dynamic Networks with Adaptive Adversaries Irvan Jahja, Haifeng Yu and Ruomu Hou LCP-Aware Parallel String Sorting Jonas Ellert, Johannes Fischer and Nodari Sitchinava When lexicographically sorting strings, it is not always necessary to inspect all symbols. For example, the lexicographical rank of "europar" amongst the strings "eureka", "eurasia", and "excells" only depends on its so called relevant prefix "euro". The distinguishing prefix size D of a set of strings is the number of symbols that actually need to be inspected to establish the lexicographical ordering of all strings. Efficient string sorters should be D-aware, i.e. their complexity should depend on D rather than on the total number N of all symbols in all strings. While there are many D-aware sorters in the sequential setting, there appear to be no such results in the PRAM model. We propose a framework yielding a D-aware modification of any existing PRAM string sorter. The derived algorithms are work-optimal with respect to their original counterpart: If the original algorithm requires O(w(N)) work, the derived one requires O(w(D)) work. The execution time increases only by a small factor that is logarithmic in the length of the longest relevant prefix. Our framework universally works for deterministic and randomized algorithms in all variations of the PRAM model, such that future improvements in (D-unaware) parallel string sorting will directly result in improvements in D-aware parallel string sorting. Mobile RAM and Shape Formation by Programmable Particles Giuseppe Antonio Di Luna, Paola Flocchini, Nicola Santoro, Giovanni Viglietta and Yukiko Yamauchi In the distributed model Amoebot of programmable matter, the computational entities, called particles, are anonymous finite-state machines that operate and move on an hexagonal tasselation of the plane. In this paper we show how a constant number of such weak particles can simulate a powerful Turing-complete entity that is able to move on the plane while computing. We then show an application of our tool to the classical Shape-Formation problem, providing a new much more general distributed solution. Indeed, while the existing algorithms allow to form only shapes made of arrangements of segments and triangles, our algorithm allows the particles to form also more abstract and general con- nected shapes, including circles and spirals, as well as fractal objects of non-integer dimension. In lieu of the existing impossibility results based on the symmetry of the initial configuration of the particles, our result provides a complete characterization of the connected shapes that can be formed by an initially simply connected set of particles. Furthermore, in the case of non-connected shapes, we give almost-matching necessary and sufficient conditions for their formability. ### Performance and Power Modeling, Prediction and Evaluation (B) Chairs: Arnaud Legrand Operation-Aware Power Capping Bo Wang, Christian Terboven, Matthias Mueller and Julian Miller Once the peak power draw of a large-scale high-performance-computing (HPC) cluster exceeds the capacity of its surrounding infrastructures, the cluster's power consumption needs to be capped to avoid hardware damage. However, power capping often causes a computational performance loss because the underlying processors are clocked down. In this work, we developed an operation-aware management strategy, called OAM, to mitigate the performance loss. OAM manages performance under a power cap dynamically at runtime by modifying the core and uncore clock rate. Using this approach, the limited power budget can be shifted effectively and optimally among components within a processor. The components with high computation activities are powered up while the others are throttled. The overall execution performance is improved. Employing the OAM on diverse HPC benchmarks and real-world applications, we observed that the hardware settings adjusted by OAM are having near to the optimal results of the static-setting approach. The achieved speedup in our work amounts to up to 6.3%. Towards a Model to Estimate the Reliability of Large-scale Hybrid Supercomputers Elvis Rojas, Esteban Meneses Rojas, Terry Jones and Don Maxwell Supercomputers stand as a fundamental tool for developing our understanding of the universe. State-of-the-art scientific simulations, big data analyses, and machine learning executions require high performance computing platforms. Such infrastructures have been growing lately with the addition of thousands of components, making them more prone to fail. It is crucial to solidify our knowledge on the way supercomputers fail. Several recent studies have highlighted the importance of characterizing failures on supercomputers. This paper aims at modelling component failures of a supercomputer based on Mixed Weibull distributions. The model is built using a real-life multi-year failure record from a leadership-class supercomputer. Using several key observations from the data, we designed an analytical model that is robust enough to represent each of the main components of supercomputers, yet it is flexible enough to alter the composition of the machine and be able to predict resilience of future or hypothetical systems. A Learning-Based Approach for Evaluating the Capacity of Data Processing Pipelines Maha Alsayasneh and Noel De Palma Data processing pipelines are made of various software components with complex interactions and a large number of configuration settings. Identifying when a pipeline has reached its maximum performance capacity is generally a non-trivial task. Metrics exported at the software and at the hardware levels can provide insightful information about the current state of the system, but it can be difficult to interpret the value of a metric, or even to know which metrics to focus on. Considering a popular pipeline composed of Kafka, Spark Streaming, and Cassandra, this paper proposes a learning-based approach to automatically infer the state of such a pipeline solely by analyzing metrics. Our results show that we are able to achieve a high prediction accuracy when predicting on new configurations and when the number of data sources changes. Furthermore, our analysis demonstrates that the best prediction results are obtained when metrics of different types are combined. 17:00 - 17:30 17:30 - 18:10 ### Theory and Algorithms for Parallel and Distributed Processing (A) Chairs: Marek Klonowski Approximation Algorithm for Estimating Distances in Distributed Virtual Environments Tobias Castanet, Olivier Beaumont, Nicolas Hanusse and Corentin Travers This article deals with the issue of guaranteeing properties in Distributed Virtual Environments (DVEs) without a server and without global knowledge of the system state and therefore only by exchanging messages. This issue is particularly relevant in the case of online games, that operate in a fully distributed framework and for which network resources such as bandwidth are the critical resources. In the context of games, players typically need to know the distance between their character and other characters, at least approximately. Players all share the same position estimation algorithm but, in general, do not know the current positions of others. We provide a synchronized distributed algorithm Alc to guarantee, at any time, that the estimated distance between any pair of characters A and B is always a 1 + ε approximation of the current distance. Our result is twofold: (1) we prove that if characters move randomly on a d-dimensional grid, or follow a random continuous movement on up to three dimensions, the number of messages of Alc is optimal up to a constant factor; (2) in a more practical setting, we also observe that the number of messages of Alc for actual game traces is much less than the standard algorithm sending actual positions at a given frequency. 3D Coded SUMMA: Communication-Efficient and Robust Parallel Matrix Multiplication Haewon Jeong, Yaoqing Yang, Christian Engelmann, Vipul Gupta, Tze Meng Low, Pulkit Grover, Viveck Cadambe and Kannan Ramchandran In this paper, we propose a novel fault-tolerant parallel matrix multiplication algorithm called 3D Coded SUMMA that is communication efficient and achieves higher failure-tolerance than replication-based schemes for the same amount of redundancy. This work bridges the gap between recent developments in coded computing and fault-tolerance in high-performance computing (HPC). The core idea of coded computing is the same as algorithm-based fault-tolerance (ABFT), which is weaving redundancy in the computation using error-correcting codes. In particular, we show that MatDot codes, an innovative code construction for distributed matrix multiplications, can be integrated into three-dimensional SUMMA (Scalable Universal Matrix Multiplication Algorithm [22]) in a communication-avoiding manner. To tolerate any two node failures, the proposed 3D Coded SUMMA requires ~50 % less redundancy than replication, while the overhead in execution time is only about 5-10 %. ### Performance and Power Modeling, Prediction and Evaluation (B) Chairs: Arnaud Legrand A Comparison of the Scalability of OpenMP Implementations Tim Jammer, Christian Iwainsky and Christian Bischof OpenMP implementations must exploit current and upcoming hardware for performance. Overhead must be controlled and kept to a minimum to avoid low performance at scale. Previous work has shown that overheads do not scale favourably in commonly used OpenMP implementations. Focusing on synchronization overhead, this work analyses the overhead of core OpenMP runtime library components for GNU and LLVM compilers, reflecting on the implementation's source code and algorithms. In addition, this work investigates the implementation's capability to handle current CPU-internal NUMA structure observed in recent Intel CPUs. Using a custom benchmark designed to expose synchronization overhead of OpenMP regardless of user code, substantial differences between both implementations are observed. In summary, the LLVM implementation can be considered more scalable than the GNU implementation, but the GNU implementation yields lower overhead for lower threadcounts in some occasions. Neither implementation reacts to the system architecture, although the effects of the internal NUMA structure on the overhead can be observed. 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http://www.archive.org/stream/actinologiabrita00gossuoft/actinologiabrita00gossuoft_djvu.txt	# Full text of "Actinologia Britannica : a history of the British sea-anemones and corals" ## See other formats Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/actinologiabritaOOgossuoft M -? 9P^ (^ THE BRITISH S E A - A N E M O N E S CORALS. LONDON : a. CLAY, PRINTER, BREAD STREET HILL. VLATE V. ■ : J LOU us sr BOLOCERA TUEDI>€. 3. AIPTASIA COUCHII, ANTHEA CEREUS. 4-. SACARTIA COCCINEA 5 . S. TROCLODYTES. A CTINOL O GIA BRITANNICA. A HISTORY OF THE BRITISH SEA-ANEMONES AND CORALS. WITH COLOURED FIGURES OF THE SPECIES AND PRINCIPAL VARIETIES. PHILIP HENRY GOSSE, F.R.S. LONDON: VAN VOORST, PATERNOSTER ROW. 1860. PREFACE. In TrritiBg the following pages, I have laboured to produce such a " History of the British Sea- Anemones and Corals," as a student can work with. Having often painfully felt in studying works similar to the present, the evil of the vagueness and con- fusion that too frequently mark the descriptive portions, I have endeavoured to draw up the characters of the animals which I describe, with distinctive precision, and with order. It is said of Montagu that, in describing animals, he constantly wrote as if he had expected that the next day would bring to light some new species closely resembling the one before bim ; and therefore his diagnosis can rarely be amended. Some writers mistake for precision an excessive minuteness, which only distracts the student, and is after all but the portrait of an individual Others describe so loosely that half of the characters would serve as well for half-a-dozen other species. I have sought to avoid both errors : to make the diagnoses as brief as possible, and yet clear, by seizing on such characters, in each case, as are txxdj distinc- tive and discriminative. Further to aid the student, I have given the characters in a regular and definite order, so that he may at a glance compare species with species, or genus with genus, La their several parts and organs. In this I have received little aid — I may say almost literally none — from my predecessors. The " History of British Zoophytes " VI PREFACE. by Dr. Johnston has hitherto been the EngUsh naturalist's only guide to the study of these creatures ; and notwithstanding the value of this work in many points, the almost utter worthless- ness of their specific characters has been often confessed. That excellent zoologist lived on a coast where the Anemones are feebly represented ; and hence his personal acquaintance with species was very small, or the result would doubtless have been different. The elaborate " Histoire Naturelle des Coralliaires " of M. Milne-Edwards is liable to the same objection, A work of immense research, labour, and patience, it bears evidence in every page of being the produce of the museum and the closet, not of the aquarium and the shore. With those species which possess no stony skeleton, the learned author evidently had no acquaint- ance, — or next to none ; — and hence he has merely reproduced the words of his authorities in all their vagueness ; while the distribution of the species into genera and families appears so full of manifest error to one personally familiar with the animals in a living state, that I have not attempted to follow his arrangement. I have been compelled, therefore, to draw up the characters of my subjects de novo ; and in doing so I have resorted to nature itself; I have studied the living animals. For the last eight years I have searched the most proUfic parts of the British shores, — the coast of Dorset, South and Korth Devon, and South Wales ; and have moreover, as the following pages show, had poured into my aquaria the productions of almost every other part of our coasts, — from the Channel Isles to the Shetlands, For these last I am indebted to the kindness of many zealous scientific friends, whose names appear in this volume, and to whom I here express my grateful obligation ; especially distin- guishing Mr. F. H. West of Leeds, and the Eev. W. Gregor of Macduff, as pre-eminent in their contributions. The result is that seventy-five species 'find their places in these pages, five of which are merely indicated, leaving seventy good species, exclusive of the LtLcemariadoe. Of these twenty- PREFACE. Vll four only are described in Johnston, — the rest of his species being either synonyms or resting on insufficient evidence. Fifty -four British species have been examined by myself, perhaps a larger number than have come under the notice of any other natiiraHst ; by far the greater part in life and health ; and thirty-four of these have been added to the British Fauna by mysel£ A new featiire in works of this sort, which will strike the student, perhaps needs a word of explanation ; — I mean the dis- tinguishing of the prominent varieties of each species by a diagnosis, and the assigning of a trivial name to each. Consider- ing the variability of many of the forms, I trust the convenience of this procedure will excuse the innovation. The analytical tables of the families, genera, and species, hitherto scarcely known in English zoological works, will, I think, be found useful ; nor wiU the attempt to tabulate the geographical distribution of the species be devoid of interest to the philosophic student. The plates must speak for themselves : they have been printed in colours by Mr. W. Dickes, who has spared no effort to make them, as nearly as possible, fac-similes of my original drawings, which were made from the Ufe. Nearly two years have been occupied in the progressive publi- cation of the work, as it has been issued in bi-monthly parts. Among the former may be reckoned that the information is brought down to the latest period, and that the successive parts stimulate the zeal and co-operation of feUow-labourers ; the book thus embodying the knowledge of many, rather than of one. Among disadvantages must be put down, incongruities between the earlier and the later portions, statements made and opinions hazarded which are subsequently corrected, and omissions which are finally supplied. For these defects the author must cast himself on the kind consideration of his readers, who must be aware that no branch of science is at one stay even for a single month. Till PREFACE. My labour tas been performed con amove; I have looked forward to it for many years past; and it is with no small grati- fication that I see it completed. I send forth the result as one more tribute humbly offered to the glory of the Triune God, " who is wonderful in counsel, and excellent in working." P. H. GossB. ToBQUAT, December, 1859. LIST OF PLATES. I. — 1. Actinoloba dianthus. 2. Sagartia bellis. 3. S troglodytes. 4, 5, 6. S. rosea. 7. S. venusta. 8, 9. S. sphyro- deta To face page 12 II. — 1, 8. Sagartia nivea. 2, 3, 4. S. miuiata. 5. S. troglodytes. 6. S. parasitica. 9, 10. S. omata 42 III. — 1, 2. Sagartia troglodytes. 3. S. yiduata. 4, 5. S. pallida. 6. S. pura. 7, 8 Adamsia palliata 106 IV. — 1. Tealia crassicornis. 2, 3. BuBodes gemmacea. 4. B. BaUiL 5, 6. B. thallia 190 V. — 1. Bolocera Tuedise. 2. Anthea cereus. 3. Aiptasia CouchiL 4. Sagartia coccinea. 5. S. troglodytes Front. VI. — 1 to 6. Actinia mesembryanthemum. 7. A. chiococca. 8. Sa- gartia chrysosplenium. 9. Anthea cereus. 10. Tealia digitata. 11. S. vidoata 206 VII. — 1. Phellia gausapata. 2. P. murocincta. 3. Gregoria fenes- trata. 4. Bunodes coronata. 5, 6. Edwardsia camea. 7. E. callimorpha. 8. Cerianthus LloydiL 9, 10. Hal- campa chrysanthellum. 11. H. microps 228 VIII. — Hormathia Margaritje. 2. Phellia BrodriciL 3. Peachia hastata. 4. P. undata. 6. Stomphia Churchiae. 6. Ily- anthufi Mitchellii 234 IX.— 1 to 5. Corynactis viridis. 6. Bolocera eques. 7. Zoanthus sulcatus, 8. Z. Alderi. 9, 10. Z. Couchii. 11. Aure- liania augusta. 12. A. heterocera. 13. Capnea san- guinea 282 LIST OF PLATES. X. — 1. Lophohelia prolifera. 2. Peachia triphylla. 3. Sphenotro- chus Wrightii. 4. S. Macandrewanus. 5. Zoanthus Couchii. 6. Paracyathus Taxilianus. 7. P. pteropus. 8. P. Thulensis. 9. Hoplangia Durotrix. 10, 11. Bala- nophyllia regia. 12, 13. Caryophyllia Smithii. To face p. SOS XI. — Anatomical details. 1. Ideal deini-section of a Sagartia. a. septum ; 6. septal foramen ; c. stomach ; d. liver ; e. ovarian mesentery ; /. ovary ; g. craspedal mesentery ; h. craspedum ; i. acontia. 2. Fragment of craspedum {S. hellis) with its mesentery {magnified). 3. The same craspedum under pressure {more highly magnified). 4. Fragment of acontium {S. bellis). 5. Portion of column containing cinclides {A. dianthus). k. fully open ; I. slightly open ; m. closed. 6. Chambered cnida {Ca- ryophyllia) before discharge. 7. Chambered cnida (Tealia) discharged, n. ecthoraeum ; o. strebla ; p. pterygia. 8. Chambered cnida discharging, showing the ecthorseum in process of evolving. (N.B. — The strebla and pterygia are here omitted, for the sake of greater clearness.) 9. Tangled cnida {Coi-ynactis). 10. Spiral cnida {Tealia) discharging. 11, 12. Globate cnidse {S. parasitica). q. peribola 348 XII. — Magnified Figures. 1. Phellia picta. 2. Zoanthus sulcatus. 3. Edwardsia carnea. 4. Caryophyllia (tentacle). 5. Zo- anthus Alderi. 6. Halcampa microps. 7. Gregoria fenestrata. 8. Phellia murocincta 358 INTRODUCTION. Though the following " History of the British Sea- anemones and Corals " is intended for general readers, it seems desirable that it should be accompanied by a brief rSsume of what is known concerning the anatomy and physiology of this order of animals. I have commenced the text of the work with a general description of the con- stituent parts of their bodies, in order to establish a determinate orismology for the class, and shall here assume that the reader is sufficiently familiar with the various organs, and the terms by which they are indicated. The Sea-anemones present a low grade of animal existence, and are commonly represented as exceedingly simple in structure. The term " Animal-flowers," by which they were known to the early observers, and which has been perpetuated in the Greek equivalent " Anthozoa," applied to the class by some modem naturalists, has been thought to express the fact, that a vegetable type of organization is scarcely less proper to them than an animal one. It is, however, to the accidental resemblance which these beautiful forms often bear to a highly-coloured and many-petal^d flower, that the name owes its appropriate- ness, rather than to any close assimilation to the vegetable structure. The Sea-anemone is an indubitable animal, and its organization is more complex than is usually supposed. This will be seen as we proceed with the successive ex- amination of the organs.* * In all cases in which I do not adduce any other authority, the following statements may be considered af» given on the authority of my own dissec- tions and observations. XU INTRODUCTION. 1. Tegumentary System. The skin is sufficiently distinct. After a few hours' maceration in fresh water {Sag. helUs), the epithelial and pigmental cells are easily removed with a hair-pencil, leaving the outer layer of muscular fibre bare. If the specimen be immersed in spirit for a day or two {A. dtanthus), the integument may be separated in flakes, which, under the microscope, are seen to be composed of a multitude of short corrugated fibrillee, set in no definite direction, interspersed with clear granules, pigment grains, and cnidae. An examination of the living animal {dtanthus, hellis, crassicornis, Hale. chrysantheUum, Cor. viridis, &c.) shows that the skin is composed of three elements, though these cannot always be separated. A layer of epithelial ciliated cells forms the first tunic : these are constantly in process of being thrown ofi" from the true skin, in the form of mucus ; but in some cases {Phellia, Edwardsia) they entangle foreign matters, and retain their cohesion as an investment more or less dense, and more or less firmly adherent to the skin. Below this is the true skin, of a more granular character, and carrying, imbedded in its thickness, a multitude of cnidae, whose discharging points are directed outwards. Intimately connected with this layer, but still lying sufficiently beneath it to be regarded as a distinct stratum, are the pigment-cells, which impart the colours to the animal. The tentacles of Aiptasia and Anthea (less conspicuously also of S. belh's) are lined with a dense layer of cells, forming to the naked eye a dark brown lining. Some peculiarities of these cells I have detailed (at page 187, infra) : it is probable that this layer may have some special function yet unrecognised. 2. Muscular System. In most species the muscular frame- work of the body is beautifully distinct, and the tissue is readily isolable. The column is a cylinder of muscular tissue, consisting of two layers, the outer composed of transverse, the inner of longitudinal, fibres. The trans- verse fibres are the more strongly marked : they average about "0001 inch in diameter, and are never striate. The cylinder which forms the column, is closed in most species by two extremities, which are flat, like the top and bottom of a tin canister : the former is the disk, the latter the base. Each of these is but a continuation of the same INTRODUCTION. Xin two layers of fibre that compose the columu-wall, — the outer transverse fibres becoming concentrically circular; the inner longitudinal ones converging to, or towards, a centre. In general, the boundaries of these divisions are distinctly marked by an abrupt angular change of the direction of the inner fibres ; but in some species (Ilyan- THiD^, Turhinolia, &c.), the body tapers gradually to a point below, without any angular change of direction. The fibres of the inner layer meet at a central point in the base, except in those species which have a central foramen there ; but in the disk they sustain another change of direc- tion, bending abruptly down at right angles, so as to form an inclosure in the axis of the column, parallel to the outer wall — the fibres of the outer layer still coating them. This downward prolongation forms the stomach, which will be presently described. In T. crassicomis the angle which is formed by the in- bending of the fibres to form the disk, is strengthened by a muscular cord, about half a line in thickness, consisting of annular fibres, and evidently acting as a sphincter : it is this band that forms the parapet. In Sagartia {Jbellis, miniata, nivea, &a) the muscular tunic, in contraction, corrugates into a reticulate or honey- comb-like pattern, inclosing shallow cells of much regu- larity. It is, I think, these inclosed areas, any one of which may be considered as a cell, with perpendicular walls of muscular tissue, that constitute the sucking warts, by means of which minute fiagments of shell or gravel are grasped, and retained with considerable force. If this exposition is correct, all of the corrugated cells are capable of becoming suckers at the will of the animal ; but, in fact, only a few are so used at a time. The cells {nivea, miniata) are about 014 inch in depth and longitudinal diameter, while their transverse diameter may average about '084 inch. It is the outer layer of muscles that constitutes these corrugations. The sucking warts in the Bunodidce, are of similar character ; but here the elevation of the muscular tunic is more permanent, and the walls of the individual cells are thicker, and are incurved towards each other. To the muscular system belong the Septa. These are thin plates of muscular tissue, comprising the two layers of transverse and longitudinal fibres, doubled on each other, XIV INTRODUCTION. and stretching vertically through the cavity inclosed by the column. Each principal septum (Plate XI. fig. 1, a), in any of the normal species, is inserted, by its outer edge, into the column-wall throughout its entire height ; by its lower edge, into the base^ from the wall to the centre ; by its upper edge, into the disk, from the margin to the mouth ; and, by its inner edge, into the stomach, from the lip, almost to the free bottom of that viscus. From thence the inner edge recedes with an arching outline, and is free, until it is gradually merged in the lower edge at the centre of the base. Between these primary septa, others are developed in succession, partitioning off the imperfect chambers thus formed. But the septa of each successive cycle, while still inserted in the column-wall throughout, spring fi-om the stomach at higher and higher points, and terminate at points more and more remote from the centre of the base. The number of septa depends, to a certain limit, on the age of the individual, but in Peachia it never exceeds twelve, and in Halcampa microps, eight. In Peachia, the tissue of the septa is very dense, and still more so in T. crassicornis, where it assumes a firmness almost cartilaginous, and a decided blue colour. The muscular tissue of the disk protrudes in the form of hollow cones, which are the tentacles : each of these springs from an interseptal chamber, and hence their deve- lopment is in cycles corresponding to that of the septa. The fibres which compose their walls are very delicate. 3. Nervous and Sensory System. I have been as unsuc- cessful as my predecessors, in my search for nervous threads or ganglia; still, I have little doubt that such exist. I should expect their presence in the form of a ring, sur- rounding the mouth, perhaps with a pair of ganglia at the foaidial tubercles, distributing threads to the tentacles, have never observed any trace of auditory vesicles or otolithes, nor any organs that I could regard as eyes ; not even in the rudimentary form of those aggregations of pig- ment-cells, that occur on the margin of the Naked-eyed Medusae. A delicate sense of touch certainly exists, dis- tributed over the entire surface, but specially localized in the lips and the tentacles. The occasional elongation of one or more of these latter organs, and their employment (as described at pp. 34 — 36, infra) , indicate the existence of an active tactile faculty, and not merely of passive INTRODUCTION. XV irritability. The tips of the tentacles are bristled with the minute points, called by Dr. T. S. Wright palpocils which he considers as delicate tactile organs. These are specially conspicuous on the globose heads of the tentacles of Corynactis and Caryophyllia. I am not sure whether I ought to regard, as an organ of taste, the surface of the lower part of the stomach, which in T. crassicornis I find covered with innumerable papillae, not quite uniform in size or shape, some being more pointed, others more round, and averaging about 0003 inch in diameter. 4. Digestive System. This is very simple, consisting essentially of a short tube descending from the centre of the disk, with an open extremity hanging loose in the body-cavity (Plate XI. fig. 1, c). I have already observed that the inner edges of the septa are inserted into its outer wall, and these maintain it in place, while by their trans- verse contraction they can draw asunder its surfaces, and by their longitudinal contraction they can either lengthen or shorten it. The stomach-wall itself, however, is muscular ; possessing at least the layer of transverse fibres, though I have not quite satisfied myself of the presence of the longi- tudinal layer. The form of the stomach is not that of a cylinder, but of a flattened sac, or of a pillow-case unsewed at both ends. This form may be well seen in pellucid specimens of A. diunthus, and in the smaller Ilyanthid^, and it may be examined by dissection in others. The excessive contrac- tion of the parts, and the copious excretion of mucus, do, however, present great obstacles to satisfactory demonstra- tions under the scalpel. I have therefore resorted to accessory means. A specimen of T. crassicornis folly expanded I treated with laudanum, drop by drop. It immediately expelled the water contained in the tentacles, causing these organs to shrink and shrivel, but not re- tracting them. The mouth, which had been pursed together, began slowly to open, and dilated greatly, almost to the concealment of the tentacles, the summit of the now flattened animal being almost wholly occupied by the gaping orifice. An excellent opportunity was thus afforded for examining the structure of the stomach, which was revealed without the excretion of mucus. The languor, too, induced by the narcotic, allowed the parts to be freely * See Edin. New Phil. Joum., April, 1857. XVI INTRODUCTION. touched with instruments without much effort at con- traction. The gular tube is remarkably corrugated longitudinally, the folds being so full, that a transverse section would present a series of figures 8. In the present state of con- traction there were horizontal corrugations also. At a short distance below the mouth the stomach ends abruptly, the edge, thin and delicate, hanging freely like a much folded curtain into the cavity. At each angle of this flattened sac the gonidial groove was conspicuous from top to bottom, inclosed by two slender columns of the firm cartilage-like muscle. The diameter of the digestive tube is, when at rest, not greater than that of the mouth ; indeed, the walls are in contact; nor, so far as my observation extends, are they ever separated except for the reception of food. It has been customary to represent the stomach as a sac pierced at the bottom " by one or more valvular openings which communicate with the cavity of the body."* But the case is as I have stated it : the free folded membrane hangs perpendicularly ; nor is there any thickening of the edge, nor any structure which at all resembles a sphincter. In tall specimens, I have observed, through the semi- transparent integuments, food pass into the stomach, and have marked that the morsel is invariably retained, never passing through to the general cavity ; but I am persuaded that this is effected by the common contractility of the walls, and not by a sphincter. When morsels of food, such as fragments of butchers' meat, are swallowed by Anemones, they are retained for some hours, and then vomited ; and because little change has passed upon the solid parts it has been rashly concluded that no process of digestion takes place in these animals. On this foolish hypothesis it is difficult to see why food should be swallowed at all, or what need the animal has of mouth or stomach. Their ordinary food, however, is not mammalian muscle, but the far softer and more fluid flesh of Crustaceay Mollusca, and Annelida. Nothing is more common than to find large specimens of A. mesemhryan- themum or T. crassicornis discharge, soon after their capture, » Siebold's Comp. Anat. § 37. " The stomach with its circular aperture at the oase " (Teale). Johnston, indeed, denies it any aperture at all : — " There is no — other visible exit from the stomach than the mouth." INTRODUCTION. XVU the shell of a crab, or a limpet, from which the entire flesh has been removed and replaced by a tenacious glaire. No doubt the first part of the process consists largely of ma- ceration, and continued pressure, by means of which the juices of the food are extracted. The nutritive matters thus obtained are then subjected to the action of the bile. No anatomist, I believe, has as yet attributed a liver to these animals, but I have little doubt that such is the character of a structure which I am about to describe. In dianthiis, crassicomis, Peachia undata, and others, the stomach- wall is lined on the interior side of its upper portion (the side, I mean, which is within the interseptal chambers) with a thick highly -coloured sub- stance. In the first two named this is yellow or orange, in the last salmon-red. This lining is {dianthus) about half a line in thickness, of a pulpy tissue, arranged in irregular lobules, covered with a ciliated epithelium (^Plate XI. fig. \, d). On being crushed down, the pulp is found to be composed of a nearly uniform mass of yellow fat-cells, the largest of which are about '0003 inch in diameter, and the smallest immeasurable points. Cnidae occur numerously in the true stomach-wall, but none in this lining-coat. I am justified, then, in presuming this organ, from its colour, form, position, and structure, to be a liverJ^ In Aiptasia I find what I think an analogous structure, but with a slightly varied position. The septa, instead of being inserted into the stomach-wall from the point where they spring ofi" to the summit, recede from it at their upper part, where their edges carry rounded pulpy lobes, which under pressure consist of a clear tenacious sarcode, carrying a moderate number of brown pigment-cells. The sarcode is composed of globose cells, averaging "0005 inch in diameter, each containing more or fewer oil-globules, * As an example of the need of caution in such observations as these, I may be pardoned for mentioning the following circumstance : — WhUe viewing the surface of the pulpy tissue above described under a good reflected light with a power of 133 diameters, I saw it forming irregular lobes, with deep narrow sinuous depressions. Over the surface, and chiefly following the lines of the sinuosities, I noticed meandering white lines, like very slender branching threads. The thought that I had dis- covered veritable nerves immediately occurred to me ; but turning the mirror of the microscope to test the observation with a different angle of the light, I found I had been looking at merely the light reflected frvm the edge of the smooth lobules I h XVlll INTRODUCTION. averaging "0005 inch, but some attaining '0003. These are very numerous in the mass. 5. Circulatory and Eespiratory systems. These exist in so simple a condition that we can scarcely separate them in our investigations. Dr. Williams has distinguished by the term Chylaqueous fluid, " that fluid which occupies the gastric and perigastric cavities of all animals below the Annelida."* It is far less vitalized than true blood, but still it is not mere water, being impregnated with organized corpuscles and slightly albuminized. In the animals of the class before us there is no blood, and no vascular system, but the cavity of the body is ample, and is copiously occupied by a transparent fluid, which has by some been mistaken for sea-water. I have, however, proved by ex- periments, recorded elsewhere,! on numerous species, that this fluid is copiously provided with organic corpuscles, circular or ovate disks, granulose in character, of a clear yellow colom, varying from '0001 to -0008 inch in diameter, the larger ones inclosing oil-globules. The fluid coagulates on the addition of ilitric acid, showing that it holds albu- men in solution. It would appear that the action of the stomach is confined to the solution and extraction of albumen and oil, which are carried with sea-water into the general cavity, the com- pound being a chylaqueous fluid ; and that it is in the upper part of the interseptal chambers that it is acted upon by the biliary secretion. For the free circulation of this fluid to every part of the interior, the whole body is lined with a delicate, strongly ciliated epithelium. The ciliary current is upward : when a pellucid dianthus has its fosse much exposed, it is quite easy to see the current driving up from every part of the interior along the whole inner wall, and passing into the tentacles, up which the atoms are then hurled. I believe there is no change in the set of this current : for though atoms are seen, especially at the bottom of the tentacles, occasionally to pass annularly or diagonally; and though of course there must be a return of the fluid driven up- ward — for there does not appear, with the closest watching, a trace of exit at the tip of the tentacles; and though, indeed, atoms are seen, though rarely, to pass downward, — I think these irregular and retrogTade movements are • Phil. Trans. 1862. f Annals of Nat. Hist.; March, 1858. INTRODUCTION. XIX merely the mechanical result of the impact of the ciliary current on the closed tip. If so, the current runs upward on the whole inner surface of the walls, and then returns down the centre. And this, I am persuaded, is the case. That the tentacles are perforated at the tip is, however, certain : but it is closed or opened at the will of the animal, the outer annular layer of fibres acting as a sphincter. Nothing is more common than to see a fully expanded indi- vidual of T. crassicornis, when suddenly alarmed, eject slender streams of water from the tips of its tentacles ; and I have seen an instance in which, the animal being but just covered with water, the jets were projected to a height of three inches above the sm"face. In S. helUs, after macera- tion, the slightest pressure on these organs causes the pigment to ooze out at the tip. In many that I so treated, not one allowed it to escape at the side ; nor in any case was there the least appearance of resistance, suddenly yielding as if by a rupture ; nor did the aperture in any case enlarge, nor was it in any case otherwise than at the precise extremity. From which circumstances I infer a natural foramen there ; and think that it exists in all species, except those (as Corynactis and Caryophyllia) which have a globose appendage at the extremity of the tentacle. The circulation of the nutrient fluid is aided by a curious apparatus of foramina, of which I have met with no description. It is diflficult to find them in dissection, for they appear to close with contraction ; but in hellis, on making a transverse section just below the disk, I have found a small round aperture in each primary and secon- dary septum, through which I could thrust a probe without laceration. It is during life, however, that, under certain favourable circumstances (for they cannot at all times be detected), they must be studied. In dianthiis, when very much distended, I have seen the principal septa perforated with a large circular foramen in the midst of their broadest part, resembling iron girders supporting a floor, excavated for lightness (Plate XI. fig. l,b). In Anthea cereus they are conspicuous; but I have been unable to detect them in T. crassicornis or in CoryncLCtis. * The most satisfactory observations I have made on these perforations were on a specimen of Anthea cereus, var. sulphurea. Being very much expanded, and distended to translucency, the base adherent to the side of a glass tank, the column greatly exceeding the base, the -window opposite, h 2 XX INTRODUCTION. That the function of Respiration should be widely dif- fused and very simple in these animals will follow from what has been said. The ehylaqueous fluid, consisting largely of sea-water admitted freely from without, is itself a reservoir of oxygen, and thus its organized elements are perpetually aerated. We have already seen how the ciliary currents within maintain a constant succession of the bathing fluid upon every part ; and there can be no doubt that some mode of exit is provided for the effete water. What this is, however, I know not. In Cerianthus, which has a posterior foramen to the body-cavity, I have seen the water forcibly ejected from this aperture (see infra, p. 272) ; I have also marked a sudden jet d'eau from the disk (pro- bably from the mouth, but of this I was not sure) ^ of T. crassicorm's, which shot up some mucous shreds with force to the surface, a height of some five inches. Perhaps these expulsions, and those from the tentacle-tips already alluded to, may be set doAvn as so many expirations (per- haps periodical) of deoxygenated water. Ancillary to respiration, as renewing the water in the vicinity of the animal, is the ciliation of the external sur- face. This is strong and uniform on the tentacles, but I have never been able satisfactorily to trace it on the column. It is first visible at the margin, flowing in an even current up the tentacle, on every side, from the foot to the I saw with a lens, for an hour together, with the utmost distinctness, a small circular (oval in perspective) foramen in each septum. That is, I saw them in a dozen or more successive septa, without interruption. The diameter of the foramen was about the same as that of a tentacle near the tip, in its ordinary state of extension. That the foramina were in films whose surfaces were coincident with the line of vision, and not transverse to it, I proved, by moving my eye to the i-ight and left, by which the foramen became more and more round, or more and more linear, the line in the latter case being that of the axis of the column. Hence they must have been in films running from the column-wall towards the axis perpen- dicvilarly, as regards the position of the animal; — conditions which agree with the septa, and with them only. The next day, with a very favourable sight, I traced the foramina conse- cutively for half the circumference of the animal. In this space there were 49 septa (perhaj^s one more than the half, for I bisected only with mj- eye) ; and I found that the foramina are pierced through those which are entire (by far the greater ntimber), but that the series is interrupted irre- gularly by those imperfect septa, which span the cavity like an arch The latter were invariably two together, differing much in the height of the arch, and gra<luated in this respect. The detail of the numbers of the consecutive septa, in the half-animal, stands thus : — Perforate— 13 . 2 , 10 . 4 . 2 . 2 . 2 . Imperforate— . 2.2 . 2.2.2.2.2 INTRODUCTION. XXI tip, where it passes off. BalanophylUa presents an excep- tion to this rule, which I have found to hold good in all other examined cases. In this instance, the tentacles, which are densely clothed with palpocils, seem to me destitute of external cilia, while all the scarlet parts are furnished with these latter. The ciliary currents flow doicn the sides of the column, and up the conical mouth from the whole circumference of the disk. 6. Reproductive System. The Actixaeia increase by spontaneous fission, by gemmation, and by generation. Fission takes place either by a longitudinal di^-ision of the entire animal from above downwards, or by separation of small fragments fi-om the edge of the base, which soon develop themselves into minute and apparently young indi- viduals. The former mode appears to be not uncommon with Anthea cereus (see infra, p. 169) ; and an imperfect form of the same produces double-disked individuals of Actinoloha and Actinia. The latter mode is common with several of the Sagartiadce (see pp. 19, 66, 86, 110). Gemmation, — the production of buds from the parent individual — occurs largely in the order before us, but prin- cipally in those which have a stony skeleton. According to Mr. Dana, whose classification I have followed, the Aste^- ACEA always bud from the disk, the Caeyophtlliacea invariably from the side or base. But a specimen of A. dianthus has come into my possession, — through the kindness of L. Winterbotham, Esq. of Cheltenham. — which has two young individuals projecting one from each side, at about mid-height, — an indubitable example of lateral gemmation. The animal has continued in the same condi- tion for nearly a year, with no tendency to separate its progeny. Generation is of com-se the normal mode of increase of the race. The sexes are sometimes united in one indi- vidual [S. troglodytes, p. 100) ; sometimes separate {Stom- phia CTiurchioe, p. 225). The testes and the ovaries cannot be distinguished from each other by a cursory examination ; each consists of a pulpy mass, usually of an orange or pale salmon-colour, attached to the free edges of the septa. The peritoneal membrane which invests each side of the septum IS produced beyond the muscular layers in the form of a mesentery of two films in contact (Plate XI. fig. 1, e). At some distance from the edge of the septum, the films xxii INTRODUCTION. separate, and inclose the reproductive organ (/), uniting again beyond it into a second mesentery {g), which is bounded by the craspedum (A) presently to be described. Both mesenteries are full and plaited, especially the eras- pedal one. The spermatic fluid is discharged in a turbid cloud through the mouth, and is diffused through the surrounding water (pp. 99, 100). The ova are also discharged through the mouth, or through the gonidial grooves (pp. 97, 98, 99). The development of the egg is into an infusorium-like germ, differing in shape in different species, but always covered with vibratile cilia, and freely locomotive. Exam- ples of the occurrence of these will be found mfra (passim), and many highly interesting details have been recorded in the magnificent works of Sir J. G. Dalyell. The manner in which the development of the Anemone proceeds has been illustrated by Dr. Cobbold;* a depression in the surface of the globose embryo becomes the general cavity ; the edges then become incurved and descend into the cavity, forming the stomach ; septa spring from the inner wall, beginning from the summit and extending downwards, and tentacles bud from around the mouth. Eggs, germs, or fully formed young, are discharged indifferently through the mouth : in the latter two cases the embryos have passed their earlier developments within the general cavity. 7. Teliferous System. In common with some nearly allied forms the Act in aria are furnished with a system of armature of most extraordinary character. It is compara- tively a recent discovery that their tissues contain exces- sively minute bodies, in the form of oblong or oval transpa- rent vesicles, which have the power of shooting out a long thread of extensive tenuity. vVagner first drew the atten- tion of physiologists to these organs, thougli he mistook their functions for that of spermatozoa ; an error which was participated by Dr. Wyman, in his observations recorded in Dana's magnificent work on Zoophytes. Their true cha- racter has, however, been sufficiently established by many observers, including Wagner, Erdl, Quatrefages, Kolliker, Agassiz, and myself. These bodies I have called cmdoe, The cmdce, in the Actinoid Zoophytes, are not confined to one organ or set of organs. They are found in various * Annals Nat. Hist, for Feb. 1853. INTRODUCTION. XXIU tissues, and in different regions of the body. Thej abonnd in the -walls of the tentacles, in the marginal spherules (of Actinia proper), in the conngated integument that sur- rounds the mouth, in the walls of the stomach, and in the epidermic mucus that is thrown off from these last-named parts on the stimulus of irritation. But there are certain special organs in which they are crowded to an extraor- dinary degree, and which, so far as I know, have no other function than that of being magazines of the cnidce. These organs are of two kinds, which I hare designated respec- tively as craspeda, and acontia. The Craspeda. The peritoneal membrane of the septa, having formed, by the contact of its two laminae, a kind of mesentery, separates again to inclose the ovary; again unites into a second mesenteiy, the edge of which is greatly puckered, and thickened in the form of a cylindrical cord, closely resembling the bolt-rope of a ship's sails, or still more the cording in the hem of a flounced garment. This marginal cord, bound throughout its length to the ovary, or to the septum, by a mesentery, I call the Craspedum (Plate XI. fig. 2). So far as my examinations have gone, the craspeda are found in all AcxiNARiA, and for the most part in great profusion. In T. crassicornis, for instance, they constitute an inextricable tangle of white frilled cords, seen every- where below and behind the stomach, and protruding through every wound of the integuments. The thickness of the cord does not, as has been stated, "increase from above downward." Nor does it "terminate in the coats of the stomach :" if we gradually cut away the stomach, piece- meal, until the free edge has disappeared, we still find the craspeda bordering the mesenteries of the septa, until the latter are lost at the point of then- convergence in the centre of the floor of the visceral cavity. The craspedum, under pressure, displays the following elements. (1.) A clear, colourless, highly refractile sar- code, which, under extreme pressure, has a tendency to draw out into strings, and long-tailed drops, like a thick oil on a wetted surface. (2.) Minute scattered granules, very irregular in shape. (3.) Mulberry-like aggregations of granules, of a clear yellow hue, compactly built together, and firm, which have the appearance of being inclosed in a definite cell- wall. These are generally ovate, but are some- XXIV INTEODUCTION, what irregular in form. (4.) Cnidse, in greater or less abundance, according to the species. As the craspedum flattens under pressure, these are crowded at the edges, and are seen to he arranged, more or less distinctly, side by side ; their long axes set at right angles to the axis of the crasj>edum, and their emitting extremities either close to its edge, or projecting from it. The more dense their aggrega- tion, the more definitely is this arrangement maintained; doubtless because displacement of their original position is more readily effected by the flattening action of the com- pressorium, when the cnideB are more loosely scattered in the fluid sarcode. The peritoneal membrane which invests the whole is richly ciliated on its entire surface. (Plate XI. fig. 3.) The Acontia. Certain species of the Zoophytes under consideration have the faculty of shooting forth from the mouth, as well as from minute orifices scattered over the surface of the body, slender flexible filaments, usually of an opaque white hue, but sometimes, as in Adamsia palliata, of a brilliant lilac tint. In some instances, as in Sagartia parasitica, S. mi^iiafa and Adamsia palliata, these threads are protruded in great profusion, coiled up in irregular spirals, and forming tangled masses that resemble bundles of sewing cotton. It appears to be a means of defence ; and any of the species just mentioned may readily be excited to display these weapons by a slight irritation of the surface of the body. The slightest touch is usually a sufficient stimulus to the extension, which will often continue to proceed for some time, the filaments shooting forth from various points with great force and rapidity. They have a strongly adhesive power, which, however, is not dependent on any superficial viscosity, but on the projectile power of the contained oiidce, of which I shall presently speak. If we carefully watch one of these threads, we shall perceive that after a time it is gradually withdrawn again into the body, by the orifice at which it was protruded. In the case of 8. parasitica, a large species, these filaments, which I designate by the term acontia, sometimes extend six inches from the body, in a straight line. Yet in a few minutes the whole has disappeared. It is gradually cor- rugated into small irregular coils, at the end which is attached to the animal ; and these little coils are, one after INTRODUCTION. XXV another, sucked in, as it were, through an imperceptible orifice. Acontia are less universal than craspeda, for whereas the latter are always present, so far as I know, in this order, the former are found onlj in the Sagartiad(£, and perhaps in the BunodidcR. In Sagartia hellis thej spring from the mesenteries that carry the craspeda; generally two acontia from each mesentery, and most frequently in pairs. Their point of insertion may be anywhere in the length of the mesentery, great irregularity prevailing in this respect. Though at first it seems a solid cylinder, the acontium is really a flat narrow ribbon, with involute and approximate edges, which can at pleasure be brought into contact, and thus constitute a tube (Plate XI. fig. 4). Like the craspeduin, of which it seems to be a form modified for a special use, its surface is richly ciliated ; and the ciliary currents not only hurl along whatever floating atoms chance to approach the surface, but cause the detached fragments themselves to wheel round and round, and to swim away through the water. Though there is not the slightest trace of fibrillge in the structure of the acontium, even under a power of 800 diameters, the clear sarcode, of which its basis is composed, is endowed with a very evident contractility. Under pressure, the edges of the flattened acontium appear to be thronged with clear viscous globules, over- lapping one another, and protruding ; indicating one or more layers of superficial cells, doubtless forming the peritoneal epithelium. As the pressm-e is increased, these ooze out as long pear-shaped drops, and immediately assume a perfectly globular form, with a high refractive power. Below these is packed a dense crowd of cnidce, arranged transversely. The Cinclides. The emission of the acontia is provided for by the existence of special orifices, which I term Cinclides. The integument of the body, in the Sagartia, is perforated by minute foramina, having a resemblance in appearance to the spiracida of insects. They occur in the interseptal spaces, opening a communication between these and the external water. The appearance of the cinclides may be compared to that which would be presented by the lids of the human XXVI INTEODUCTION. eye, supposing these to be reversed ; the convexity being inwards. Each is an oval depression, with a transverse slit across the middle. When closed^ this slit may some- times be discerned merely as a dark line (Plate XI. fig. 5, m], the optical expression of the contact of the two edges ; but, when slightly opened {T), a brilliant line of light allows the passage of the rays from the lamp to the beholder. From this condition the lids may separate in various degrees, until they are retracted to the margin of the oval pit, and the whole orifice is open (Jc). The dimensions of the cinclides vary not only with the species, and probably also with the size of the individual, but with the state of the muscular contraction of the integu- ments, and, as I think, with the pleasure of the animal. In a small specimen of S. dianthus, I found the width of a cinch's, measured transversely, ^th of an inch ; but that of another, in the same animal, was more than twice as great, viz. jsnth of an inch. This was on the thickened marginal ring, or parapet, which in this species surrounds the tentacles, where the cinclides are larger than elsewhere. Watching a specimen of S. nivea under the microscope, I saw a cinclis begin to open, and gradually expand till it was almost circular in outline, and a^sth of an inch in diameter. I slightly touched the animal, and it in an instant enlarged the aperture to 2^th of an inch. In a specimen of S. hellis, less than half grown, I found the cinclides numerous, and sufficiently easy of detection, but rather less defined than in dianthus or nivea. They occurred at about every fourth intersept, three intersepts being blind for each perforate one, and about three or four in linear series, but not quite regularly, in either of these respects. In this case they were about eVth of an incli in transverse diameter, a large size, — and I measured one which was even g^th of an inch. By bringing the animal before the window, I could discern the light through the tiny orifices with my naked eye. From several good observations, and especially from one on a cinclis, widely opened, that happened to be close to the edge of the parapet of a dianthus, I perceived that the passage is not absolutely open, at least in ordinary, but that an excessively thin film lies across it. By delicate focusing, I have detected repeatedly, in different degrees of expansion, and even at the widest, the granulations of a INTBODUCTION. XXVll membrane of excessive tenuity, and one or two scattered cnida, across the bright interval. On another occasion, in the case of a cinch's at the edge of the parapet — a position singularly favourable for observation — I sa"w- that this subtle film was gradually pushed out until it assumed the form of a hemispherical bladder, in which state it remained as long as I looked at it. At the same time the outline of the cinch's itself was sharp and clear, when brought into focus farther in. The film, whatever it be, is superficial, and does not appear to be a portion of the integument S roper. I take it to be a film of mucus (composed of eorganized epithelial cells), which is constantly in process of being sloughed from all the superficial tissues in this tribe of animals, and which continues tenaciously to invest their bodies, until, corrugated by the successive contractions of the animals, it is washed away by the motions of the waves. As, however, one film is no sooner removed than another commences to form, one would always expect external pores so minute as these to be veiled by a mucus- film in seasons of rest. That the cinch'des are the special orifices through which those missile weapons, the acontta, are shot and recovered, rests not merely on the probability that arises firom the coexistence of the two series of facts I have above recorded, but upon actual observajtion. In a rather large S. diantkus, somewhat distended, placed in a glass vessel between my eye and the sun, I saw, with great dis- tinctness, by the aid of a pocket-lens, many acoiitia protruded from the cinch'des, and many more of the latter widely open. The acontta, in some cases, did not so accurately fill the orifice but that a line of bright light (or of darkness, according as the sun was exactly opposite or not) was seen, partially bordering the issue of the thread, while the thickened rim of the cinch's surrounded all. The appearance of the orifices whence the acontta issued was that of a tubercle or wart, and the same appear- ance I have repeatedly marked in examples observed on the stage of the microscope ; namely, that of a perforate pimple, or short columnar tube. Tliis was clearly manifest, when the animal, slowly swaying to and fro, brought the sides of the cinch's into partial perspective. On another occasion I witnessed the actual issue of the acontia from the cinch'des. I was watching, under a low XXVin INTRODUCTION. power of the microscope, a specimen of >S'. nivea, while, by touching its body rudely, I provoked it to emit its missile filaments. Presently they burst out with force, not all at once, but some here and there, then more, and yet more, on the repeated contractions of the corrugating walls of the body. Occasionally the free extremity of a filament would appear, but more frequently the hight of a lent one, and very often I saw two, and even three, issue from the same cinclis. The successive contractions of the animal under irritation, caused the acontia already protruded to lengthen with each fresh impetus, the bights still streaming out in long loops, till perhaps the free end would be liberated, and it would be a loop no longer ; and sometimes a new thread would shoot from a cmclis, whence one or two long ones were stretching already ; while, as often, the new- comers would force open new cincUdes for themselves. The suddenness and explosive force with which they burst out, appeared to indicate a resistance which was at length overcome : — perhaps — in part at least — due to the epithelial film above mentioned, or to an actual epiderm, which, though often ruptured, has ever, with the aptitude to heal common to these lowly structures, the power of quickly uniting again. It appeared to me manifest, from this and other similar observations, that no such arrangement exists as that which I had fancied ; — that a definite cinclis is assigned to a definite acontium, or pair of acontia, and that the extremity of the latter is guided to the former, with unerring accu- racy, by some internal mechanism, whenever the exercise of the defensive faculty is desired. What I judge to be the true state of the case is as follows : The acontia, fastened by one end to the septa or their mesenteries, lie, while at rest, irregularly coiled up along the narrow interseptal fossae. The outer walls of these fossae are pierced with the cinclides. When the animal is irritated, it immediately contracts ; the water contained in the visceral cavity finds vent at these natural orifices, and the forcible currents carry with them the acontia, each through that cinclis which happens to lie nearest to it. The frequency with which a loop is forced out shows that the issue is the result of a merely mechanical action; which is, however, not the less worthy of our admiration because of the simplicity of the contrivance, nor the less manifestly the result of Divine INTRODUCTION. XXIX wisdom working to a given end "bj perfectly adequate cnidffi in every part of their length, carry abroad their fatal powers not the less surely, than if each had been provided with a proper tube leading from its free extremity to the nearest cinch's. The Cnidce. — I come now to describe those minute but potent organs which constitute the object of all the mecha- nism above described. Four distinct forms of these cap- sules have occurred to my investigations ; and these I shall treat of in turn. (1.) Chambered Cnid(B {Cnid(B cameratcB). This is perhaps the most generally distributed fonu, as it is manifestly the most elaborately armed. It may be well examined in CaryophylUa SmitTiii. The globular heads of the tentacles seem, under pressure, to be literally com- posed of these capsules, the ends of which project side by side, as close as they can be packed, one against another. The form of these is long and slender, almost linear. The craspeda are also similarly studded with cnidee, which are, however, of longer dimensions, and of fuller form. As I have seen no chambered cnidae, in any species, so large as these, I shall take them as a standard for description, alluding to those of other species only when they differ from these. They are perfectly transparent, colom-less vesicles, of a lengthened ovate figure, considerably larger at one end than at the other (Plate XI. fig. 6). One of average dimensions measures in length "004 inch, and in greatest diameter '0005. In the larger (the anterior) moiety, is seen, passing longitudinally through its centre, a slender chamber, fusiform or lozenge-form, about '00015 inch in its greatest transverse diameter, and tapering to a point at each extremity. The anterior point merges into the walls of the cn{d(B at its extremity, while the posterior end, after having become attenuated like the anterior, dilates with a fonnel-shaped mouth, in which the eye can clearly see a double-infolding of the chamber-wall. After this double fold the structure proceeds as a very slender cord, which, passing back towards the anterior end of the capsule, winds loosely round and round the chamber, with some regularity at first, but becoming involved in contortions more and more intricate as it fills up the posterior moiety of the cavity. The fusiform chamber appears to be marked on XXX INTRODUCTION. Its inner surface with regularly recurring serrations, which are the optical expression of that peculiar armature to be described presently. Under the stimulus of pressure, when subjected to micro- scopical examination, and doubtless under nervous stimulus, subject to the control of the will, during the natural exer- cise of the animal's functions, the cnid(B suddenly emit their contents with great force, in a regular and prescribed manner. It must not be supposed, however, that the pres- sure spoken of is the immediate mechanical cause of the emission : the contact of the glass-plates of the compres- sorium is never so absolute as to exert the least direct force upon the walls of the capsule itself; but the disturbance produced by the compression of the surrounding tissues excites an irritability which evidently resides in a very high degree in the interior of the cnidee ; and the pro- jection of the contents is the result of a vital force. In general the eye can scarcely, or not at all, follow the lightning-like rapidity with which the chamber and its twining thread are shot forth from the larger end of the cnida. But sometimes impediments delay the emission, or allow it to proceed only in a fitful manner, a minute portion at a time ; and sometimes, from the resistance of friction (as against the glass-plate of the compressorium), the elongation of the thread proceeds evenly, but so slowly as to be watched with the utmost ease ; and sometimes the process, which has reached a certain point normally, be- ' comes, from some cause, arrested, and the contents of the cell remain permanently fixed in a transition state. Thus a long continued course of patient observation is pretty sure to present some fortuitous combinations, and abnormal conditions, which greatly elucidate phenomena that nor- mally seemed to defy investigation. In watching any particular cnida, the moment of its emission may be predicted with tolerable accuracy by the protrusion of a nipple -shaped wart from the anterior extremity. This is the base of the thread. The process of its protrusion is often slow and gradual, until it has attained a length about equal to twice its own diameter, when it suddenly yields, and the contents of the cnida dart forth. At this instant I have, in many instances, heard a distinct crack or crepitation, in the examination of cnidce both of this species and of S, parasitica. INTRODUCTION. XXXI When fuUj expelled, the thread or wire, which I distin- guish by the term ecthoroium (Plate XI. tig. 7, n), is often twenty, thirty, or even forty times the length of the cnida ; though, in some species, as in most of the Sa^artice, it frequently will not exceed one-and-a-half, or two times the length of the cnida. The ecthoraa, which are discharged by chambered cnidce, are invariably furnished with a peculiar armature. The basal portion, for a length equal to that of the cnida, or a little more, is distinctly swollen, but at the point indicated it becomes (often abruptly) attenuated, and runs on for the remainder of its length as an excessively slender wire of equal diameter throughout. In the short ecthorcea of Sagartia, the attenuated portion is obsolete. It is chiefly upon this ventricose basal portion that the elaborate armature is seen, which is so characteristic of these remarkable organs. For around its exterior wind one or more spiral thickened bands, varying in different species as to their number, the number of volutions made by each, and the angle which the spiral forms with the axis of the ecthorceum. The whole spiral, formed of these thickened bands, I designate the screw, or strehla (fig. 7, o). In the ecthorcea emitted by chambered cnidce from the craspeda of T. crassicornis, the screw is formed of a single band, having an inclination of 45° to the axis, and be- coming invisible when it has made seven volutions. In those from the same organ in S. parasitica we find a screw of two equidistant bands, each of which makes about six turns, — twelve in all, — having an inclination of 70° from the common axis. In those similarly placed in Caryojjhyllia, the strehla is composed of three equidistant bands, each of which makes about ten volutions — thirty in all — with an inclination of about 40° from the axis. In every case the spiral runs from the east towards the north, supposing the axis to point perpendicularly upwards. Sometimes, especially after having been expelled for some time, the wall of the ecthorceum becomes so attenu- ated as to be evanescent, while the strehla is still distinctly visible. An inexperienced observer would be liable, under such circumstances, to suppose that the screw, when formed of a single band, as in T. crassicornis, is itself the wire ; an error into which I myself had formerly fallen. An XXXll INTRODUCTION. error of another kind I fell into, in supposing that the triple screw of the wire in C. Smitkii was a series of imbricate plates : the structure of the armature is the same in all cases (with the variations in detail that I have just indicated) ; and the structure is, I am now well assured, a spiral thickened band, running round the wall of the ecthoreeum on its exterior surface. I have been able, when examining such large forms as those of Corynactis and Caryojphyllia, with a power of 750 diameters, to follow the course of the screw, as it alternately approached and receded from tlie eye, by altering the focus of the objective, so as to bring each part successively into the sphere of vision. These thickened spiral bands afford an insertion for a series of firm bristles, which appear to have a broad base and to taper to a point. Their length I cannot determin- ately indicate, but I have traced it to an extent which considerably exceeds the diameter of the ecthoraium. These barbed bristles I denominate ^>ferj/^^«. (See fig. l,p.) The number of pterygia appears to vary within slight limits. As well as I have been able to make out, there are but eight in a single volution of the one-banded strebla in T. crass icornis ; while in the more complex screws of S. parasitica, Cor. viridis, and Gary, Smith ii there appear to be twelve in each volution. The barbs, when they first appear, invariably ^ro;ec< in a diagonal direction from the ecthoraum ; and sometimes they maintain this posture ; but more commonly, either in an instant, or slowly and gradually, they assume a reverted direction. From some delicate observations, made with a very good light, I have reason to conclude that the strebla, and even the pterygia, are continued on the attenuated portion of the ecthoreeum, perhaps throughout its length. In Corynactis and Caryophyllia I have succeeded in tracing them up a considerable distance. In the latter I saw the continuation of all these bands, with their bristles; but the angle of inclination had become nearly twice as acute as before, being only 22° from the axis. The appearance of tlie attenuate portion, as also of the base of the ventricose part, is exactly that of a three-sided wire, twisted on itself; the barbs projecting from the angles. (2.) Tangled Cnidce (Cnidce glomifera;). This form is very generally distributed, and is mingled with the former INTRODUCTION. XXXUl in the various tissues. In the genus Sagartia, however, it is by far the rarer form, while in Actinia and Anthea, it seems to be the only one. The pretty little Corynactis viridis is the best species that I am acquainted with for studying this kind of cnida. Their figure is near that of a perfect oval (Plate XI. fig. 9), but a little flattened in one aspect, about 004 inch in the longer, and 'OOlo in the shorter diameter. Their size, therefore, makes them peculiarly suitable for observations on the structure and functions of these curious organs. Within the cavity is a thread (ectJioraum) of great length and tenuity, coiled up in some instances with an approach to regularity, but much more commonly in loose contor- tions, like an end of thread rudely rolled into a bundle with the fingers. The armature of this kind does not differ essentially firom that already described. It is true, I have detected it only in Coryndctis, where the short ecthoraum of the tangled cnida is surrounded throughout its length by a barbed strebia of three bands. The barbs are visible under very favourable conditions for observation, even while the tangled wire remains enclosed in the cnida, but their optical expression is that of serratures of the walls, without the least appear- ance of a screw. This is the only species in which I have actually seen the armature of the ecthoreeum in this kind of cnida, but I infer its existence from analogy, in other species, where the conditions that can be recognised agree with those iu this, though the excessive attenuation of the parts precludes actual observation of the structure in question. (3.) Spiral OntdcB [Cnida cochleatce). In a few species, as S. parasitica, T. crassicornis, &nd Cerianthus Lloydii, I have found very elongated fusiform c«i'c/<e which seem composed of a slender cylindrical thread, coiled into a very close and regular spiral. In some cases the extremities are obtuse, but in others, as in T, crass icoryi is, the posterior extremity runs off to a finely attenuated point, the whole of the spire visible even to the last, the whole bearing no small resem- blance to a multispiral sheU, as one of the Ceritkiadce or Turritelladcs (Plate XI. fig. 10). The ecthorceum is dis- charged reluctantly from this form, and I have never seen an example in which the whole had been run off. So ex- cessively subtle are the walls of the cnida^ that it was not e XXXIY INTRODUCTION. until after many observations that I detected them, in an half of the wire ; I have not seen the slightest sign of arma- ture on the cethoraum. So far as my investigations go, these spiral cmdce are confined to the walls of the tentacles, in which, however, they are the dominant form. (4.) Ohhate Cnid<B {cnid<B globatoi) ? In the acontium of T. parasitica flattened under pressure, and finally ex- pressed from its substance, are numerous more or less globose or ovate vesicles, which' gradually push out a cylindrical protuberance at each end, sometimes to a length equal to that of the original form (figs. 11, 12). These vesicles appear filled with a fluid of dificrent refractive power from that of the clear sarcode in which they are lodged ; but no sign of contained thread have I been able to detect, nor have I seen any discharge beyond the pro- trusion above spoken of. I am not at all sure that these vesicles are consimilar in function with the true cnidce ; and I am still more doubtful about the bacillar bodies ound in the acontioid filaments of T. crassicornis. In the indubitable cnidce, — those which I have distin- guished as (1) Chambered and (2) Tangled, — the emission of the ecthoraum is a process of distinct eversion. This is not a solid but a tubular prolongation of the walls of the cnidce, turned in, during its primal condition, like the finger of a glove drawn into the cavity. Some of the observa- tions on which I ground this conclusion I have already published, but it may not be impertinent to repeat them here, with others which have since occurred to me, all proving the same fact. In the discharge of the ecthorceum of the tangled cnid<s, it frequently runs out, not in a right line, but in a spiral forai ; whenever this is the case, each band of the spire is made, and stereotyped, so to speak, in succession, while the tips go on lengthening : the tip only progresses, the whole of the portion actually discharged remains perfectly fixed ; which could not be on any other supposition than that of evolution. In the discharge of the chambered kind, the ventricose or basal portion first appears ; the lower barbs fly out before the upper ones, and all are fully expanded before the attenuated portion begins to lengthen. This again is consistent only with the fact of the evolution of the whole. On several occasions of observation on the chambered cnidm of CaryophyUia, I INTRODUCTION. XXXY have actually seen the unevolved portion of the ecthoraum running out through the centre of the evolved ventricose portion. But perhaps the most instructive and convincing example of all was the following. One of the large tangled \dth rapidity, when a kind of twist, or " kink," occurred against the nipple of the cm'da, whereby the process was suddenly arrested. The projectile force, however, continuing, caused the impediment to yield, and minute portions of the thread flew out, piecemeal, by fits and starts. By turning the stage-screw I brought the extremity of the discharged portion into view, and saw it slowly evolving, a little at a time. Turning back to the cnida I saw the kink gradually give way, and the whole of the tangled wire quickly flew out through the nipple. I once more moved the stage, fol- lowing up the ecthorcBum, and presently found the true extremity, and a large portion of the wire still inverted ; slowly evolving indeed, but very distinct throughout its whole course, "vvithin the walls of the evolved portion (fi-^)- . , ... From all these observations, there cannot remam a doubt of the successive eversion of the entire ecthoreeum. It may be asked, What is the nature of the force by which the contained thread is expelled'? That it is a potent force, is obvious to any one who marks the sudden explosive violence with which the nipple-like end of the cnida gives way, and the contents burst forth; as also the extreme rapidity with which, ordinarily, the whole length is evolved. A curious example of this force once excited my admiration : the ecthoraum trom a cnida of Corynactis viridis was in course of rapid evolution, when the tip came full against the side of another cnida already emptied. The evolution was momentarily arrested, but the wall of the empty capsule presently was seen to bend inward, and suddenly to give way, the ecthorcemn forcing itself in, and shooting round and round the interior of the cnida. The most careful observations have failed to reveal a lining membrane to the cnida. I have repeatedly dis- cerned a double outline to the walls themselves — the optical expression of their ' diameter ; but have never detected any, even the least, appearance of any tissue starting from the walls, as the ecthormim bursts out. My first supposition, reluctantly resigned, was, that some such I XXXYl INTRODUCTION. lining meml^rane of high contractile power, lessened, on irritation, the volume of the cavity, and forced out the wire. The cnida is filled, however, with a fluid. This is very distinctly seen, occupying the cavity, when from any im- pediment, such as above described, the wire flies out fitfully — waves, and similar motions, passing from wall to wall : sometimes, even before any portion of the wire has escaped, the whole mass of tangled coils is seen to move irregularly from side to side, within the capsule, from the operation of some intestine cause. The emission itself is a jprocess of injection ; for I have many times seen floating atoms driven forcibly along the interior of the ecthorauriiy sometimes swiftly, and sometimes more deliberately. Nothing that I have seen, would lead me to conclude that the wall of the cnida is ciliated. I consider, then, that this fluid, holding organic cor- puscles in suspension, is endowed with a high degree of expansibility ; that, in the state of repose, it is in a con- dition of compression, by the inversion of the ect]ior<Bum ; and that, on the excitement of a suitable stimulus, it forcibly exerts its expansile power, distending, and con- sequently projecting, the tubular ecthorcBum, — the only part of the wall that will yield without actual rupture. The cnida cannot, I think, be regarded in the light of cells, since they are but the contents of other vesicles, which thus present a higher claim to the character of cell- wall. In the craspeda of S. parasitica, may be seen many of the chambered ciiidae, bearing this outer envelope, which, without determining anything concerning its nature, I shall distinguish as the pe7'ihola. Many of the cnida have ruptured their investing membrane, which gives way at no special point, sometimes at the anterior end, sometimes at the posterior, and as frequently, all down the side. The peribola thus ruptured, may be seen in many instances still iianging about the cnida, while others are quite free from any remains of it, and in some cases I have seen the cnida still enveloped in its peribola, unruptured. The peribola I have seen investing, and hanging around the cnida of the spiral and globate kinds, and this circum- stance has afforded me an additional ground for presuming the latter to belong to this category of organs (figs. 11, 12, g). It appears necessary that the cnida should set itself free INTEODUCTION. XXXVU by the rupture of its peribola, before it can effect the emission of its ecthor(Bum. At least I have never met ^vith an example of the contrary. It has long been known, that a very slight contact with the tentacles of a polype is sufficient to produce, in any minute animal so touched, torpor and speedy death. Since the discovery of these cntda, the fatal power has been supposed to be lodged in them. Baker, a century ago, in speaking of the Hydra, suggested that " there must be something eminently poisonous in its grasp;" and this suspicion received confirmation from the chcumstance that the Entomostraca, which are enveloped in a shelly covering, frequently escape unhurt after ha^'ing been seized. The stinging power possessed by many Medusce, which is suf- ficiently intense to be formidable even to man, has been reasonably attributed to the same organs, which the micro- scope shows to be accumulated by millions in their tissues. Though I cannot reduce this presumption to actual certainty, I have made some experiments, which leave no reasonable doubt on the subject. First — I have proved that the ecthormim when shot, has the power of penetrating, and does actually penetrate, the tissues of even the higher animals. Several years ago, I was examining one of the used, but a considerable number of cni'dee had been spon- taneously dislodged. It happened, that I had just before been looking at the sucker-foot of an Asterina, which remained still attached to the glass of the aquatic box, by means of its terminal disk. The cilia of the acontium had, in their rowing action, brought it into contact with the sucker, round which it then continued slowly to revolve. The result I presently discerned to be, that a considerable number of the cnidce had shot their ecthorcea into the flesh of the sucking disk of the Echinoderm, and were seen sticking all round its edge, the wires imbedded in its sub- stance even up to the very capsules, like so many pins stuck around a toilet pin-cushion. To test this power of penetration still farther, as well as to try whether it is brought into exercise on the contact of a foreign body with the living Anemone, I instituted the following experiment. With a razor I took shavings of the cuticle, from the callous part of my own foot, as from the ball of the toe, and from the heel. One of these shavings I XXXVm INTRODUCTION. presented to tlie tentacles of a fally expanded T. crassicornis. After contact, and momentary adhesion, I withdrew the cuticle, and examined it under a power of 600 diameters. I found, as I had expected, cnidce studding the surface, standing up endwise, the wires in every case shot into the substance. They were not numerous — in a space of "01 inch square, I counted about a dozen. I then irritated a S. parasitica till it ejected an acontium, and taking up with pliers another shaving of the cuticle, allowed it to touch the acontium, which instantly adhered across its surface. I now drew away the cuticle gently, so as not to rupture the acontium, and examining it as before, immediately saw dense groups of cnidce, standing endwise on the surface, the ecthorcea all discharged and inserted in the substance almost to the very capsules. The groups were set in a sinuous line, across the cuticle, where the on the same line. In one of these groups I counted thirty- five cnidce in an area about '0025 inch square. These examples prove that the slightest contact with the proper organs of the Anemone is sufficient to provoke the discharge of the cnidce; and that even the densest condition of the human skin offers no impediment to the penetration of the ecthorcea. As to the injection of a poison, it is indubitable that pain, and in some cases death, ensues even to vertebrate animals from momentary contact with the capsuliferous organs of the ZoorHYTA. The very severe pain, followed by torpor, lasting for a whole day, which Mr. George Bennett has described as experienced \)j himself, on taking- hold of PhysaUs pelagica, was produced by the contact of the tentacles. The late Professor Edward Forbes has graphically depicted the "prickly torture " which results to " tender-skinned bathers," from the touch of the long filamentous tentacles — "poisonous threads" — of the Cyanoea capillata of our own seas ; and observes that these ampu- tated weapons severed from the parent-hody, sting as fiercely as if their original proprietor itself gave the word of attack. I have been assured by ladies that they have felt a distinct stinging sensation, like that produced by the leaves of the nettle, on the tender skin of the fingers, from handling our common Anthea cereus ; while, on the otlier hand, I have myself handled the species, scores of times, I . INTEODUCTION. 2XX1X ■with impunity. And I have elsewhere recorded an in- stance, in which a little fish, swimming about in health and vigour, died in a few minutes with great agony, through the momentary contact of its lip with one of the emitted acontia of Sagartia parasitica. It is worthy of observation, that, in this case, the fish caiTied away a portion of the acontium sticking to its lip ; the force w4th which it ad- hered being so great, that the integrity/ of the tissues yielded first. The Acontium severed, rather than let go its hold.^ Now, in the experiments which I have detailed above, we have seen that this adhesion is efiected by the actual impenetration of the foreign body, by a multitude of the ecthor<Ra, whose barbs resist withdrawal. So that we can with certainty associate the sudden and violent death of the little fish with the intromission of barbed ecthorcea. I have instituted some experiments with a view to try whether acid or alkaline properties could be detected in the (presumed) fluid which is discharged. First with a solu- tion of indigo, and afterwards with the expressed juice of violets, I occupied the plate of the compressorium ; and in the flattened drop made the cnid(B in the acontium of S. parasitica to emit. In the case of the indigo, the colouring matter remained in the form of masses, but the juice of violets afibrds an apparently homogeneous fluid, even when reduced by pressure to an excessively thin film. I could not detect, even with the most careful scrutiny, the slightest tinge of discoloration of the blue fluid, — not the most delicate shade of red or green — along the side of the emitted ecthoraa, nor in the vicinity of the cnida. And * " The Aquarium," ed. 1. p. 115. + Dr. Waller has recently recorded an interesting experiment which he made with Act. mesembryanthemum. He allowed its tentacles to touch the tip of his tongue. " The result was such as to satisfy the most scep- tical respecting the offensive weapons with which it is furnished. The animal seized the organ most vigorously,, and was detached from it with some difficulty after the lapse of about a minute. Immediately a pungent acrid pain commenced, which continued to increase for some minutes, until it became extremely distressing. The point attacked felt inflamed and much swollen, although to the eye no change in the part could be detected. These symptoms continued unabated for about an hour, and a slight temporary relief was only obtained by immersing the tongue in cold or warm water. After this period the symptoms gradvially abated, and about four hours later, they had entirely disappeared. A day or two after, a very minute ulceration was perceived over the apex of the tongue, which disappeared after being touched with nitrate of silver."— (Proc. Roy. Soc. April 14, 1859.) xl .INTRODUCTION, though, in order to obtain a greater intensity of colour, I allowed a drop of violet-juice to dry on each plate of the compressorium, so that with a power of 800 diameters, the whole field was of a deep uniform translucent blue — still the ejected wire produced no change of tint. Such a test as this is not sufficient to prove that no acid or alkaline property exists in the discharged fluid, and still less that no poisonous fluid at all is efi'used ; since that most concentrated poison, the venom of the rattlesnake, is said to change vegetable blues to reds, in so slight a degree as to be scarcely perceptible.* Admitting the existence of a venomous fluid, it is diffi- cult to imagine where it is lodged, and how it is injected. The first thought that occurs to one's mind is, that it is the organic fluid which we have seen to fill the interior of the cnida, and to be forced through the everting tubular ectho- rceum. But if so, it cannot be ejected through the ex- tremity of the ecthwoeum, because if this were an open tube, I do not see how the contraction of the fluid in the cnida could force it to evolve; the fluid would escape through the still inverted tube. It is just possible that the barbs may be tubes open at the tips, and that the poison-fluid may be ejected through these. But I rather incline to the hypothesis, that the cavity of the ectliorceum in its primal inverted condition while it yet remains coiled up in the cnida, is occupied with the potent fluid in question, and that it is poured out gradually within the tissues of the victim, as the evolving tip of the wire penetrates farther and farther into the wound. Perhaps it is not too much to say that the whole range of organic existence does not affi)rd a more wonderful example than this, of the minute Avorkmanship and elaboration of the parts, the extraordinary mode in which certain pre- scribed ends are attained, and the perfect adaptation of the contrivance to the work which it has to do. • In a communication made by Dr. M'Donnell to the Royal Society, Bome experiments were detailed, which had led the observer to believe that electricity was the power in question. In a subsequent paper, however, that gentleman gave up his hypothesis. (Proc. Roy. Soc. Jan. 14, and Nor. 18, 1858.) BRITISH SEA-ANEMONES. GENERAL DESCRIPTION AND EXPLANATION OF TERMS. As it is of great importance in scientific description to employ precise terms for the various parts of the ohjects described, and for the conditions of those parts, and to use the same terms always in the same sense, I here define the terms which I propose to use in this work. The principal parts of the body of a Sea- Anemone are the following : — the base ; the column ; the disk ; the tentacles; the mouth; the cavity, 1. The Base {Basts). This is the lowest part of the animal, usually forming a flat area, by means of which it adheres to other bodies. It is often expanded (expansa), its outline being consi- derably broader than a section of the column. In some cases, as in Edwardsia, it becomes very small, loses its function, and finally, as in Cerianthus, disappears. In Adamsia, it is greatly extended laterally into two wings, which, curving round, meet and unite by their edges, forming a complete circle. This form of base may be distinguished as ANNULAR [annularis). B 2 GENERAL DESCRIPTION 2. The Column {Golumna). The body rises in a more or less cylindrical shape, when the base is attached, like the trunk of a tree, often grace- fully and rapidly diminishing from the basal expansion, and sometimes dilating towards the upper extremity : — this I call the COLUMN. At the summit {vertex), the column is, as it were, cut off transversely, forming a distinct margin {margo). In some cases, as in Actinoloha, the margin rises into a thickened PARAPET (ti'cMum) or low wall, separated from the tentacles by a groove or fosse (fossa). In others, there is neither parapet nor fosse. The margin may be NOTCHED {crenata) ; or, instead of notches there may be distinct tentacles, constituting the outer row of these organs; in this case the margin is tentaculate {tentaculata). The suj-face of the column may be quite smooth {Icbvis) ; studded with low warts, — warty [verrucosa) ; or marked with longitudinal sunken lines, — FURROWED [sulcata). When the furrows are deep and the intermediate spaces swell out in a rounded outline, it is invected [invecta) ; when the column is surrounded by transverse wrinkles, it may be called insected [insecta) ; when these insections are so deep as to seem to cut-off or divide the body into parts, it is constricted [constricta) ; when the surface is crossed by numerous longitudinal and transverse wrinkles, it is can- cellated [cancellata) ; when minutely and very irregularly wrinkled, like the bark of a rough tree, it is corrugated [corrugata). Some of these conditions are not permanently characteristic of any species, but are assumed temporarily during the changes of form induced by contraction. As to substance, the column may be tough and resisting, approaching a leathery consistence [coriacea) ; fleshy [carnosa), when soft but moderately firm ; or pulpy [pulposa), when very soft and yielding. AND EXPLANATION OF TERMS. 3 The WARTS {yerrucce)^ in some species, are hollow, and furnished with a muscular arrangement hj which a vacuum is formed, and the edges adhere firmly to foreign bodies ; these may be called suckers {acetahula). Other species have the skin and the muscular beds beneath it pierced with minute orifices, for the emission of armed threads ; these may be called loop-holes (cmclides). This is the flattened upper extremity of the column, as the base is the flattened lower extremity. Its outline is circular ; and this is recognised without diflSculty when, as is usually the case, the edge is plane [plana) ; but some- times the edge is wavy [undulata), as in hellis ; or even deeply frilled [sinuosa), as in dianthus. In Actinia proper, the disk bears, just within its margin, a row of SPHERULES [sphcerulce marginales) ; and, in every species, it carries the tentacles, and is pierced at the centre by the MOUTH. Converging lines [radii) cover the surface of the disk, starting from each tentacle-foot and meeting around to each mouth-angle [gonidium), is often more marked gonidiales). 4. The Tentacles [Tentacida). These are hollow cones springing from the surface of the disk, and arranged in one or more series of circles towards its margin. When there are more circles than one, that circle which is nearest the centre may be called the first ROW [series prima) ; that which stands next to it towards the margin the second [series secunda) ; and so on till we reach the outermost [series extima). With respect to each individual tentacle, its front [antica) is that aspect a 2 '4 GENERAL DESCRIPTION which is next to the centre ; its back {postica), that which is next to the margin ; its RIGHT and left sides (Jatiis dextrum, I. simstrum) , those which depend upon these indications. Each tentacle has a foot [radix] and a tip (apex). 5. The Mouth {Os). The entrance to the stomach is placed, as has been stated above, in the centre of the disk. It is surrounded by a generally thickened lip (labium), which is sometimes elevated on a cone (coUiculits) , and sometimes level. The lip may be smooth (Iceve), or furrowed (sulcatum) ; at each of two opposite points, — the mouth-angles [gonidia), — there are placed two tubercles {lentigines) ,'htt\fee.n which opens an imperfect tube or groove formed by the approxi- mation of two cartilaginous bands : these grooves, one at each mouth-angle, may be termed GONIDIAL grooves [canales gonidiales). Their function appears to be that of oviducts. (In Actinoloha, there is but a single mouth- angle, and a single groove).* From the lip descends into the cavity of the body a membranous veil, much gathered into folds, but free at the lower edge, like a sack without a bottom ; this is the stomach {stomachus), of which the portion immediately below the lip may be conveniently termed the throat [gula). 6. The Cavity [Venter]. The whole of the region included between the walls of the column and the stomach-wall, and between the free edge of the stomach and the base, may be indicated by this term. It is divided into imperfect chambers by • In AcHnopsis, a singular form recently described by Messrs. Danielssen and Koren from the Norwegian coast, the gonidial tubercles are prolonged into a pair of long and rigid semi-cylinders, the sides of which are bent downwards, and the tips of which are cleft. AND EXPLANATION OF TEEMS. 5 perpendicular muscular partitions [septa), all of which are inserted into the column-wall, but advance into the cavity in various degrees. Some are inserted by their inner edge into the stomach-wall, completely dividing-off the cavity : these may be called primary septa [septa primordialia). Others are placed intermediately between these, which do not reach the stomach-wall ; these are secondary septa [s. secundaria). Others, again, are intermediate between these and the former, whose height is still lower (these may be distinguished as tertiary [s. tertiaria) ; and so on, if there be any series beyond this. The spaces thus parted off in the cavity, I would call intersepts [intersepta) . The free edges of the secondary and tertiary septa, and also of the primary ones below the stomach, carry a thin membrane which encloses the ovaries [ovaria), and is terminated by a sort of CORD [craspediim), much twisted and involved. Long missile cords [acontia) are in some species attached by one end to the partitions, and lie coiled- up, or float freely, in the intersepts : these are, by the volun- tary contractions of the animal, forcibly ejected through the loop-holes, into which they are then gradually withdrawn. Both the cra^peda and the acontia are almost wholly com- posed of THREAD-CAPSULES [cnidie), which contain a coiled WIRE [ecthorceum). This wire is shot out under particular stimulus, and is an efficient weapon of offence ; it is usually surrounded with one or more spiral bands composing the SCREW [strebla), each of which cames a series of barbs [pterygia) ; and the whole apparatus is a vehicle for the infusion of some highly venomous fluid. The different conditions assumed by the animal, may be distinguished as the FLOWER [anthus), when the disk with its tentacles is expanded; the button [oncus), when these are retracted and concealed by the closing over them of the summit of the column. CLASS ZOOPHYTA. Animals of radiate structure; of gelatinous or fleshy substance ; more or less column-shaped ; having, in general, one end permanently attached or temporarily adherent to foreign bodies ; the other end forming a flat disk surrounded by one or more circles of tentacles, and pierced in the centre by a mouth opening into the digestive cavity ; furnished with offensive weapons in the form of capsules imbedded in the tissues, each of which encloses a projectile poisoning dart ; possessing no special organs of sense. ORDEK ACTINOIDA. The visceral cavity inclosing the stomach, and divided into compartments by perpendicular partitions of membrane which support the reproductive organs ; germs ejected through the mouth. SUB- ORDER AGTINARIA. Tentacles twelve or upwards, rarely warty ; membranous partitions sometimes simple, sometimes depositing solid calcareous plates, which, with the surrounding walls, con- stitute the corallum. TRIBE I.— ASTR^ACEA. Tentacles many, in imperfect series, or scattered ; coral- lum (when present) calcareous, consisting of cells containing many radiating plates; the plates prolonged outward beyond the cells which enclose them. (N.B. No known British species of this Tribe deposits a corallum.) TRIBE II.— CARYOPHYLLACEA. Tentacles many, in two or more series ; mostly increasing by lateral buds ; generally depositing a coraUum^ which is invariably calcareous, and many-rayed. Tentacles in a single series, twelve (rarely more), some- times obsolete : gemmiparous ; gemmation lateral : coral- ligenous : corallum calcareous ; cells [cali/cesl quite small : rajs (septa) six to twelve, or obsolete : interstitial surface not lamello-striate. {Not British.) TRIBE IV.— ANTIPATHACEA. Animals with six tentacles, forming at the base homy secretions (fleshy, enveloping a homy axis). [Not British.) TRIBE I.— ASTR^.\CEA. ANALYSIS OF THE NON-CORALLIGENOUS FAMILIES. Tentacles simple. Column imperforate. Column smooth. Column waited Biinodida. Lower extremity rounded, simple Ilyanthida. Lower extremity inclosing an air-chamber (A'W^niiaA) Minyadidcc. TRIBE I.— ASTR^ACEA. All tlie members of this Tribe with which we are fami- liar on the European shores are simple, and destitute of a corallum. But when those of all seas are taken into con- sideration, we find that the majority are compound and coralligenous. The increase of these is effected by the budding forth of new polypes from the single primary polype ; and it is in the manner of this gemmation that the tribe Astraeacea differs from the Caryophylliacea. In the former, increase invariably takes place by the extension of the summit, and not of the side or base. The process of widening, in budding polypes, may be confined to the parts exterior to the disk and visceral cavity below, or the disk and cavity may continuously enlarge ; in the latter case, the buds open in the disks, the process of budding being the cause of their enlargement (Dana). The greater part of the Astrceacea increase by disk-buds, and spontaneous subdivision ; the disk of the polype, and the cell of the corallum, gradually widening by growth, and finally separating into two portions, which become in- dependent. A few only widen exteriorly to the disk, or in the interstitial spaces between the cells of aggregate corals (Dana). The polypes in both this and the following tribe are many-tentacled ; but, while this character distinguishes them from the two other tribes, it is of no assistance in discriminating those species with which we have to do. Moreover, as our Astraacea are all simple, it is difficult to apply the rule derived from the manner of gemmation. The spontaneous fission of some species, however, as Actlnoloba diantJnis, partially, and Antliea cereus completely, may help us to assign their affinities ; and their general resemblance, inter se, and that of tlie whole to the polypes of the coralligenous Astrceacea, leave little room for un- certainty. 9 {No European species.) I have thought fit to associate in this group those genera of the Tribe, which have the following characters : — Thej do not deposit a corallum. They have a broad base, capable, at the pleasure of the animal, of firmly adhering to foreign bodies, such as rocks, stones, and shells ; or of being used as a foot, on which to creep, somewhat in the manner of a snail. They have always simple, smooth tentacles, arranged in (generally) uninterrupted circles at the margin of the disk, but often encroaching far upon its surface. Their body is for the most part pulpy or fleshy, generally lubri- cated on the surface with copious mucus ; its exterior is often studded with sucking cavities, hich have the power of adliering to foreign bodies, by the formation of a vacuum within the cavity, its muscular edges being appressed by the weight of the supeirincumbent atmosphere and water. The margins of these cavities do not rise into conspicuous warts when inactive. The integument is pierced with loop-holes (cinclides), — special orifices, through which are emitted and retracted fleshy cords (acontia), which have their origin in the membranous partitions of the body- cavity. These are filled with capsules {cmda;), which are generally chambered, and which shoot a very short, but densely-armed wire {ecthor^um). 10 ANALYSIS OF THE GENERA. Tentacles moderately long, slender. Disk perfectly retractile. Column destitute of suckers Actlnoloha. Column furnished with slickers Sagartia. Column clothed with a rough epidermis . . . Phellia. Disk imperfectly retractile. Base annular; parasitic on shells ..... Adamsia. Base entire ; not parasitic Gregoria. Tentacles mere warts ; set in radiating bands {Not British) Jjiscosoma. 11 GENUS I. ACTINOLOBA (Blainv.). Actinia (Linn.). Cribrina (Ehbenbebo). Sagartia (GtOsse). Base considerably broader than the column; its outline often undulate, but entire. Column pillar-like, in the expanded state ; the margin forming a thickened parapet, or low wall, separated from the tentacular disk by a groove or fosse. Surface perfectly smooth, without suckers, but pierced with loop-holes. Substance approaching to pulpy. Disk deeply frilled at the margin; thinly mem- branous. Tentacles short, slender, not arranged in distin- guishable circles, scattered at their commencement smaller, more numerous and densely crowded as they approach the border. JSlouth surrounded with a thick lip ; furnished with only a single gonidial groove, surmounted by a single pair of tubercles. Acontia emitted somewhat reluctantly, but copi- ously upon occasion. Only one British species. THE PLUMOSE ANEMONE. Actinohha dianthus, Plate I. Fig. 1. Specific Character. Body smooth, columnar when distended ; five inches and upwards in height : mouth strongly furrowed, rufous : tentacles marked with a ring of white. Actinia dianthus. Ellis, Phil. Trans. Ivii. 436 ; tab. xix. fig. 8. Johnston, Br. Zooph. Ed. 2. i. 232; pl.xliii. Dalyell, Anim. of Scotland, 235 ; pi. xlviii. figs, 6. 7; xlix. Gosse, Aquarium, Ed. 2. 182 ; pi. V. TuQWELL, Manual of Sea Ane- mones, 56 ; pi. i. senilis. Linn. Syst. Nat. 1089. judaica. Ibid. Syst. Nat. 1088. pentapetala. Penn. Br. Zool. iv. 104. plwmoaa. Muller, Zool. Dan. iii. 12 ; tab. ixxxviii. ; figs. 1, 2. aurantiaca. Jordan, Annals. N.H.Ser. II. vol. xv. 85. (juv.) Actinoloba dianthus. Blainville, Actinologie, 322. Sagartia dianthus. Gosse, Man. Marine Zool. i. 28. GENERAL DESCRIPTION. Form. Base, Adherent to shells and stones : expanded considerably beyond the diameter of the column. Column. Smooth, lubricated profusely with mucus; destitute of suckers, warts, wrinkles, furrows, and corrugations. Substance fleshy, approaching to pulpy. Form cylindrical, terminating in a siniple thickened parapet, which is separated from the outer tentacles by a fosse. Disk. Widely expanded, thin, greatly overhanging the column, deeply frilled. Tentacles. Exceedingly numerous, moderately large and scattered at about the middle of the semi-diameter of the disk, but becoming smaller and closer outward, until they are excessively crowded, and very minute at the margin. In extreme youth they are comparatively few, and much longer in proportion. Mouth. Not raised on a cone ; lip thick, divided into lobes by strongly marked furrows, A single groove only at one of the mouth-angles, guarded by a pair of tubercles. THE PLUMOSE ANEMONE. 13 Colour. Column. Olive, olive-brown, umber-brown, red-lead, pale-orange, salmon- red, flesh-colour, cream- white, pure white. ["Lemon-yellow," "peach- blossom." — Daltell.] I>isk. Agrees with the column. Tentacles. Generally agree with the column, but in the oUve and brown varieties, they are sometimes almost wholly pellucid-white, and in all cases they are marked with a single transverse bar of white, near their middle ; most conspicuous in youth. Lip. Always rufous, or orange-red ; whatever the hue of the body. Size. Specimens occaaionally attain six inches in height, and three in thickness. LOCALITT. All rovmd the coasts of Europe, in deep water, and on dark rocks between tide-marks. Varieties. These might be made as nximerous as the various shades of colour above- mentioned ; but for practical purposes it may be sufficient to distinguish the following : — a. Brunnea. Including the shades of brown, from dingy blackish olive, to warm umber, or fawn-colour. Sometimes, as in examples that have fallen under my own observation, the tentacles, in these brown specimens, are almost white, marked with the more opaque white bar. There is not the sUghtest reason to assign these, as has been suggested, to another species. i8. Ruhida. The various tints of red, from the full minium-scarlet to the peach-blossom and flesh-colour, may be classed under this variety, which is perhaps the most abundant of all. y. Flava. Sir John Dalyell enumerates " lemon-yellow " among the hues of this species; but it must be a very rare variety. I have never seen it. 5. Sindonea. Perhaps this is the most elegant variety ; the animal being clad in translucent white — " simplex munditiis," as if arrayed in the finest Coan vestments. It is not vmcommon. This noblest of our native Sea-anemones seems to be entitled to generic separation from the Sagartice, with which I have hitherto associated it. Its form and habit, its puckered disk, its crowded and fringe-like tentacles, its thickened parapet and deep fosse, and the presence of only a single mouth-groove, are well-marked characters peculiar to it among our British species. This last peculiarity- isolates the species from every other with which I am acquainted. The generic appellation Actinoloha, I have adopted from De Blainville, who formed the genus in his " Actinologie" (1834). It is sufficiently expressive; but objectionable on account of its construction. It is a good canon that no generic name ought to form a part of a second generic name. In this case the word is constructed out of Actinia, and \oj3o<i, a lobe or flap : it means, therefore, " the lobed Actinia." If it had been formed of the element a/crtV, a ray, the construction would have been unobjectionable, though the word would have been false in signification ; for what the French zoologist wished to express was " a lobed Actinia," not " a creature with lobed rays (= tentacles)." The specific name, dianthus, is due to a pretty fancy of Ellis, the father of English Zoophytology. Observing the resemblance which the Actinice bore to composite or many- petaled flowers, — a resemblance which is perpetuated in the popular appellation, Sea- Anemones, — he named such as were known to him after those lovely objects ; belh's, the daisy ; mesemhryanthemum, the fig-marigold ; dianthus, the pink. I do not know that we are to seek for special resemblances to the particular flowers chosen ; one poly- petalous flower might have served as well as another : still less shall we find any etymological significance in the appropriation. For the latter we must go back to the flower. In the present case, the pink and carnation, genus is named dianthus, some say, for its great beauty (Sto?, divine, dvdo^, Jlower) ; but it may be from its tendency to become double [hi, the sign of duplication, 8Lav0r)<;, having full or double flov^ers) : the lexicons moreover give Biavdio) (from Blo), to bloom. THE PLUMOSE ANEMONE, 15 Muller has called dianthus the most beautiful of all the Anemones, — ^^ Actintarum pulcTierrima f^ and his verdict is surely correct, so far as it refers to European species. When we see a full-grown specimen of some of the more delicately coloured varieties, — the pale orange, the flesh- coloured, or the clear white, — rising erect from its broad base like the stem of a massive tree, crowned with its expansive disk of myriad tentacles, we cannot but consider it a most noble, as well as a most lovely object. It is only in expansion that it is beautiful. The button will some- times shrink down to an abject flatness, scarcely more than an eighth of an inch in height in the centre, the cir- cumference spread out on every side to cover an irregularly outlined area of some five or six inches in diameter, but no thicker than a card. In this condition it is almost a repulsive object, but, perhaps in a quarter of an hour, you look at it again, and the change seems magical. The animal has risen, and swollen, and distended its body with clear water, till the tissues appear plump, and almost transparent ; it now forms a noble massive column, some five inches high, and three thick, irom which the delicate frilled disk expands, and arches over on every side, like the foliated crown of a palm tree. Then again, on some cause of alarm, real or supposed, it will suddenly draw in its beautiful array of frills, contract around them its parapet, and assume a distended bladder-like figure, with the clustering tentacles just protruding from the slightly open aperture. It is imder the veil of night that the Anemones in general expand most readily and fully. While the glare of day is upon them, they are often chary of displaying their blossomed beauties ; but an hour of darkness will often suflSce to overcome the reluctance of the coyest. The species before us is not particularly shy ; it may often be seen opened to the full in broad daylight ; but if you would make sure of seeing it in all the gorgeousness of its magnificent bloom, visit your tank with a candle an hour or two after nightfall. The membranous disk appears to be truly circular in outline, but so fully frilled that it is impossible to expand it on a plane. There are commonly from five to eight broad and deep involutions, which are sometimes simple, sometimes compound ; in the latter case forming a semi- globular head of close slender tentacles, almost furry in character. Mr. W. A. Lloyd has favoured me with the following- note, on a tentacular peculiarity in this species : — " In a marine tank belonging to a customer of mine, there is an Act. dianthus having one single long slender tentacle, high overarching the great fleecy mass of ordinary tentacles, and acting independently of them, very different from anything I have ever before seen in this species, and similar to the one solitary tentacle sometimes present in A. hellish When very young, neither the frilled involution of the disk, nor the smallness of the tentacles, nor their crowded condition, is characteristic of the species. It is then very likely to be mistaken by an inexperienced observer for another form, or to be described as new. Professor Jordan has, I feel sure, fallen into this very excusable error ; for the specimens which he has described* under the name of Actinia aurantiaca were certainly none other than infant diantJiuses. Their size, — about half-an-inch high ; their hue, — orange or almost salmon-colour; their tentacles, — of a greyer tint, with a whitish bar ; their locality, — the under surface of an inclined mass of rock ; their numbers, — • In the Annals of Nat. Hist, for Feb. 1855, THE PLUMOSE ANEMONE. 17 many of the same size associated together ; their habit, — hanging pendent from the midst of the acorn-shells and sponges, " like a rain-drop ready to fall ;" — all agree exactly with the young of dianihus. My friend, in a private letter, tells me, moreover, that he is certain they were immature, from the length of the tentacles ; and that his brother suspected them to be the young of di'anthus, because he found old di'anthus at the same spot. There can be no doubt that Mr. Charles Jordan is right. A very heterodox notion seems to have obtained cur- rency, that this species differs from other Actmicein that it is incapable of altering its place, when once it has selected it. Dr. Johnston says, — and his statement is the more surprising since he had seen " several hundreds of indivi- duals," — " As A. dianthus is a. pei'manentli/ attached species, and cannot be removed without organic injury to the base, it has some claim to be made the type of a genus." (Brit. Zooph. p. 234). If this were correct, the claim (which I have allowed on other grounds) would indeed be well founded ; but the statement is erroneous. Sir John Dalyell, again, while allowing that dianthus shifts its position spon- taneously, affirms that it cannot be compelled to do so with impunity. In illustration of this assertion he mentions the case of a very large one, which was attached to a stone too wide to be put into any of his vessels. In this emergency he reversed the stone, laying it across the top of a jar, so that the Anemone should hang suspended in the sea-water. He had hoped that the animal would voluntarily quit its hold, and descend into the jar, but it did not ; and, after stretching itself for some days, it ruptured its body across the centre, apparently by its own weight, and died. Notwithstanding these excellent authorities, however, I * Rare and Rem. Anim. of Scotl., 235. C can unhesitatingly affirm, both that the species travels as freely as any in captivity, and that it may be removed from its attachment with the utmost ease and impunity. In " The Aquarium" (p. 192) I had given evidence of both these facts, and experience has since confirmed them in number- less instances. Instead of repeating my own observations, however, I will fortify them with the authority of my friend Mr. Merriman, of Bridgnorth, who has favoured me with the following remarks on this subject : — " Dr. Johnston's statement is not confirmed by my experience any more than yours. I have a very fine speci- men of dianthus, which persisted in crawling up the side of ray glass, — a circular one, — until part of its disk was actu- ally above ' high-water level.' A few days ago it became necessary to empty my glass. Accordingly I drew off the water, and the dianthus hung in the most disconsolate way, looking very like an old wet kid-glove. Finding I could not finish my operation without entirely removing him, I worked him off" with the back of my nail. Of course, at the first rude touch on his base, he shrank up into a ball, in which shape he continued, when I dropped him into some water to remain until I could restore him to his own home. While here he became quite like a ball of cotton, so many were the nettling-threads that he threw out on all sides. In two hours' time I put him back into the glass, having taken the precaution to place a bit of slate upright behind liim, that I might not have the same difficulty again. In less than six hours he had stuck as firmly to the slate as he had previously done to the glass, and he has continued most magnificent ever since." In spite of Sir John Dalyell's assertion, that this species is " less hardy than most," the fuller aquarian experience of the present day enables us to affirm that no British species is more readily preserved in confinement than dianthus. I THE PLUMOSE ANEMONE. 19 There are probably thousands of specimens of this fine Anemone now living in the aquariums of Great Britain and Ireland ; and a large number of these have been several years in captivity. They continue to live and flourish, expanding and erecting themselves with the greatest free- dom ; nor do they seem at all afiected by the turbidity of the water, provided it be free from impurity. I have had some specimens of rather large size continue for many months in water so loaded with green Alga spores as to be almost opaque, yet during the whole period they appeared perfectly at ease, and even increased their number by fissiparous division. It is the frequent habit of the species to crawl up the perpendicular side of the tank which it inhabits, till it reaches the water's edge, a situation which seems particularly grateful to it ; for there it remains from week to week, daily (or rather nightly) projecting its columnar form in a horizontal direction, at the very surface, and then expanding its beautiful frills, so that the air bathes a part both of its body and its tentacles. I have never seen this Anemone increase its kind by proper generation, that is, by the discharge of ova, or of young. But no species more freely increases by sponta- neous division. When a large individual has been a good while adherent to one spot, and at length chooses to change its quarters, it does so by causing its base to glide slowly along the surface on which it rests ; — the glass side of the tank, for instance. But it frequently happens that small irregular fragments of the edge of the base are left behind, found it easier to tear its own tissues apart than to over- come it. The fragments so left soon contract, become smooth, and spherical or oval in outline, and in the course of a week or fortnight may be seen each furnished with a margin of tentacles and a disk — transformed, in fact, into c 2 perfect though minute Anemones. Occasionally a separated piece, more irregularly jagged than usual, will, in contract- ing, constringe itself, and form two smaller fragments, united by an isthmus, which goes on attenuating until a fine thread-like line only is stretched from one to the other ; this at length yields, the substance of the broken thread is rapidly absorbed into the respective pieces, which 'soon become two young dianthuses. It is to this tendency to spontaneous division that I would attribute the frequent occurrence in this species of monstrosity, such as two disks uniting into a single column. This is very common. Dr. Johnston supposes that such cases are produced by the coalescence of two individuals which happened to be in contact, and he accounts for its frequency by the gregarious habit of the species.* The possibility of two individuals thus uniting, remains, how- ever, to be proved ; while the fissiparous habit, which is patent, is quite sufficient to produce the phenomenon. I have been informed of a case, in which a young one was produced by gemmation from the base of the adult, without previous separation of the fragment. When erect, and fully distended with water, the integu- ments and tissues become translucent, and, in parts, even transparent. In this condition, when favourably placed, — as when in front of a window, or with a candle just behind it, — an excellent opportunity is afforded of examining the internal arrangement of the organs, free from the confusion wliich the excessive contraction consequent upon dissection induces. The septa are seen stretching away into the general cavity, and the acontia lying in many coils along the inter- septs ; while ever and anon a minute coiled fragment, torn from some acontium, is seen driven to and fro along the * Br. Zooph. 2nd Ed., 233. THE PLUMOSE ANEMONE. 21 intersepts, hj the action of the cilia with which the inte- rior membranes are covered. Occasiouallj, such a spiral frasnnent is driven into the interior of a tentacle, which is indeed but a continuation of the interseptal chambers — and here it is hurled to and fro in the ciliary currents, now shooting forward to the tip, then slowly retrograding, then again whirled towards the tip, which it appears to make the most strenuous efforts to reach; the combination of the twofold ciliary action, — that which is dependent on the cilia that line the interior of the tentacle, and that which results from its own richly ciliated surface, — imparting a vacilla- tion and ever-varying impetus to its movements that may easily be mistaken for independent life. I have myself fallen into this error.* The proper habitat of dtantkus is the coralline zone. The trawlers in West Bay and Torbay bring up populous colonies from a depth of twenty fathoms. In Weymouth Bay it is specially abundant ; and yet this apparent pre- eminence may be rather due to the fact that this celebrated locality has been so perseveringly dredged. Be it so or not, I can testify to the profusion with which the bottom of this bay, from the deep sea of the offing to three fatlioms or less, is stocked with this fine Anemone. The oyster and scallop- banks of Portland and Brixham are favourite haunts. It is the habit of the species to live in society ; and both the dredge and the trawl are constantly bringing to light clustered groups, as well as single individuals. Family groups are sometimes very numerous, as many as twenty being not uncommonly crowded on a single oyster-shell, f • Devonsh. Coast, 116. t Dr. Battersby informs me thai, in the eummer of 1 856, one of the trawlers brought into Torquay a water-lojrged board, about two feet long by one broad, on which were crowded between four and five hundred specimens of A. dianthiu, of all sizes, but a considerable proportion of them large. What was curious was, that all on one side the board were white, all on the other orange. Of course, in so limited a space, a large proportion of this number must consist of small individuals ; and specimens in several gradations of development may often be observed, suggestive of as many generations, from the gigantic fore- father of the family to the tiny great-grandchildren that crowd around his foot, no larger than split peas. From the fissiparous tendency above noticed, it is probable that these multiplications are but essential parts of one individual, not his descendants ; analogous to the multiplication of a plant hj cuttings as distinguished from that by seeds. There is no real process of generation in either case. What confirms my suspicion, that such is the true explanation of these congregated groups of dianthus, is the fact that, in general, all the members of each colony are of the same variety of colour. Now and then, however, we do see in the cluster a specimen of quite a different hue, as, for example, a dark olive one in the midst of a flesh-coloured group. In this case we must presume that there has been the deposition of a real germ, — the product of a really generative function — either from one of the individuals already settled there, or from some stranger. Flat stones, but more commonly large bivalve shells, such as oysters, pectens, and pinnae, are the sites usually selected for the colonies of dtanthus. But though the floor of the sea is the proper home of the species, it is found, in certain favourable localities, to con- gregate in great numbers within tide marks. Where a breadth of semi-cavernous rock, honeycombed by mollusks, and studded with Alcyonia, Tunicata and Sponges, darkly overhangs a tide-pool, as around Petit Tor, and in the caves of Tenby and Lidstep ; or where an immense boulder has so fallen upon others as to present a broad under-sur- face to the flowing tide ; I have seen scores on scores of dianthuses hanging, dank and flaccid, from the rock, each with a globule of crystal water, suspended like a dew-drop THE PLUMOSE ANEMONE. 23 from its drooping head. In general these are young indi- viduals : I have never met with one between tide-marks, that exceeded an inch in diameter when contracted. What becomes of them as they attain riper years I do not know ; I can only conjecture that they may retire, during the flow of the tide, to a more genial seclusion at a tideless depth. Mr. Peach tells me that he finds the species in pools be- tween tide-marks at Peterhead ; — " hundreds I have seen, some white and others brilliant red, side by side in the same pool." The same excellent observer assures me that he has obtained it four inches in height between tide-marks in that vicinity. The following list of British localities will show the general distribution of this species. Peterhead, Keith Inch (plentiful), C. W.Peach: Frith of Forth, Bir J. G. Daly ell: Berwick Bay, Dr. G. John- ston : Northumberland and Durham, J. Alder : Scar- borough, Filey, F. H. West: Sandgate (rare), E. L. Wil- liams: Guernsey, E. W. H. Holdsicorth: Plymouth, G. Spence Bate : Selsey, Bognor, G. Gatehouse : Weymouth Bay, P. £r. Gosse : Teignmouth (young), R. C. Jordan: Torquay (young) ; Torbay, P. H. G. : Dartmouth, and up the Dart as far as Dittisham, E. W. H. H. : Falmouth, W. P. Cocks: Lundy, Morte, Bev. G. Tugwell: Tenby, P. H. G. : Liverpool (under the pontoons of the landing- stage), F. H. W. : Mersey Estuary, Hilbre Island, E. L. W.: Morecambe Bay, F. H. W. : Clyde, near Glasgow (at low ebbs). Miss Anne Church : Curabrae, Bev. D. Landsho- rough: Belfast and Strangford Loughs, Dublin Bay, Dr. E.P. Wright.'^ * Most of the above references rest on the authority of private commu- nications made to me by friends ; whose names, having been once given at length, I shall thenceforward cite by their initials. Perhaps the most magnificent Actinia known is A. Pau- motensis, described and figured in Dana's "Zoophytes." It was found at the Isle of Raraka, in the Paumotu group, bj the naturalists attached to the American Exploring Expe- dition. It is twelve inches in diameter of disk, which is deeply frilled. A. reticulata, from Terra del Fuego, is another fine and richly coloured species ; with a frilled disk, and tentacles very numerous and fringe-like. Both these must doubtless be assigned to the genus Actinoloha. A. Achates, a species dredged by the same Expedition, in thii-ty fathoms, on the east coast of Patagonia, has the frilled character of dianthus, with but three rows of ten- tacles, which are not specially crowded. It is evidently intermediate between dianthus and hellis ; but further examination is necessary to determine to which genus it rightly belongs. I may, however, venture to exhibit the affinities of our Anemone in the following gradation ; distinguishing exotic species by [ ] : — [Paumotensis.] [reticulata.] DIANTHUS. [Achates.] bellis. 25 GENUS II. SAGARTIA (Gosse). Actinia (LiNy.)- Cribrina (Ehrenb.). Actinocereus (Blainv.). Base broader than the column ; its outline often undulate, but entire. Column in the expanded state pillar-like, sometimes low and thick, sometimes tall and slender ; the margin notched or tentaculate, without parapet or fosse. Surface studded with suckers, which do not form permanent warts ; pierced with loopholes. Sub- stance fleshy, or pulpy. Disk sometimes wavy ; more commonly plane, some- times slightly turned-over at the edge. Tentacles varying in number, form, and arrangement in the different species. Mouth generally elevated on a more or less con- spicuous cone; furnished with two gonidial grooves, each with its pair of turbercles. Acontia emitted freely and copiously. NATURAL ORDER OF TEE BRITISH SPECIES. 1. bellis. 2. miniaia. 3. rosea. 4. ornala, 5. ichthygtmaa. 6. renusta. 7. nivea. 8. sphyrodeta. 9. pallida. 10. troglodytes. 11. liduata. 12. ^jara«/fica. 26 ARTIFICIAL ANALYSIS OF THE SPECIES. Body salver-shaped ; disk strongly waved bellis. Body of the usual form; disk nearly plane : — Tentacles without markings : — Disk and tentacles white nivea. Disk orange ; tentacles white venusta. Tentacles with characteristic marks : — With a B-like mark at the foot troglodytes. A broad black bar above a narrow one at the foot : — ■ Outer tentacles scarlet miniata. All the tentacles rose-purple rosea. Two broad black bars at the foot omcUa. Two narrow black bars at the foot ichthy stoma. A dark line down each side : — The lines unbroken viduata. The lines broken into several fragments .... parasitica. Tentacle foot enclosed — Within a purple circle sphyrodeta. Within two unconnected purple curved lines . . . pallida. ASTR^ACEA. THE DAISY ANEMONE. Sagartia bellis. Plate I. Fig. 2. Specific Character. — Body salver-shaped, the disk forming a shallow cir- cular cup, often wavy at the margin, of which the column is the foot. Tentacles small, numerous, in six rows, the outer ones mere crenations of the margin. Actinia bellis. pedunciUata. Templetonii. AetiTWceretu peduncvlata. Crihrina bellis. Sagartia bellis. Ellis and Solandeb, Zooph. 2. Johnstox, Br. Zooph. Ed. 2. i. 228; pi. xlii. figs. 1, 3 — 6. GossE, Devonsh. Coast, 25; pi. i. figs. 1, 2. Pennant, Br. Zool. iv. 102. Cocks, Rep. Comw. Polyt. Soc 1851. 8, pi. ii. figs. 10, 14. Blainv., Diet. Sci. Nat. 1830; Ix- 194. Ehbenb., CoraU. 41. GossE, Linn. Trans. xxL 274 : Man. Mar. Zool. L 28 ; fig. 41. GENERAL DESCRIPTION. FOBM. Base. Adherent to rocks; expanded considerably beyond the diameter of the column ; the outline often undulate. Column. Smooth on the lower half, on the upper studded with suckers, to which in freedom arp often firmly attached minute fragments of shell, gravel, &c. ; generally without wrinkles, furrows, or corrugations ; but occasionally invected. Substance firmly fleshy. Form exceedingly variable, sometimes being thick and low, nearly equalling the disk in diameter; bat, when expanded to the utmost, the column generally takes the form of a comparatively slender, lengthened, and perfectly cylindrical footstalk, abruptly expanding to a great circle, the margin of which is cut into minute notches which form the outermost row of tentacles. Disk. In the condition just mentioned, this ia a broad horizontal plate, or a slightly concave saucer, of which the rim is perfectly circular, though this form is often disguised by its being thrown into undulations, some- times approaching to frillings. Tentacles. Small, but numerous, arranged in about six rows ; the first as many ; the fourth again doubled ; the fifth increasing in about the same proportion; and the sixth insluding about thrice as many as the fifth. Thus the total number may be about five hundred. Those of the first row usually stand erect, the others decline more and more as they recede, until the last two or three rows lie quite horizontally on the disk, to which the sixth row forms an exquisite fringe. Those of the first row rarely exceed one-fourth of an inch in height, and the others diminish regularly ; those of the sixth are very minute ; the longest (for they are not equal) scarcely exceeding the sixteenth of an inch in length, and some being mere tubercles ; these are slender, and set so close together, that sixty are contained within an inch. Those of the inner rows are usually marked with a depressed line or groove, down the middle of the front. Mouth. Not raised on a cone. Lip moderately thin, finely furrowed. Colour. Colvmm,. Lower part flesh-colour, often flushing into pink; gradually paling upward to white, drab, or buff in the middle part : this as gradually becoming dull violet on the upper third, where the suckers usually are conspicuous as pale spots. Bisk. Dark brown or black, the radii separated by fine lines of rich vermilion, commencing at the mouth, and diverging till they meet the tentacles, passing a little way up the sides of each. TENTACLE OF S. BELLIS {front). Tentacles. Yellowish-brown, studded with whitish specks, and varied with white or grey patches. There is commonly a dark-brown space near the base, bounded, above and below, by a band of pure white. Frequently groups of tentacles thus mottled alternate with equal groups of uniformly dull-brown ones; the regions of the discal border from which they re- spectively spring, corresponding in some measure, being either brown or lavender grey. In many specimens a single tentacle, or sometimes two opposite ones, of the first series, are rather larger than the rest, and of an unspotted cream-white ; when these occur, it is generally in connexion with one or two white gonidial radii. In other specimens there is no trace of such a distinction. Mouth. Lip and throat white. * The student will please to observe that the specific description is the description of but one condition, or variety. It is convenient to have a starting-point or standard of comparison, but it must not be supposed that this particular condition is the one proper to the species, aad that the other THE DAISY ANEMONE. 29 SiZK. The average diameter of the disk is about one inch and a half; but large specimens attain a breadth of two inches. The height is dependent on the depth of the hole which they inhabit ; in general it is about an inch, but sometimes it is as much as three inches, the column in this case being about three-eighths of an inch in thickness. Locality. The south and west coasts of England and Ireland, abundant ; almost unknown in Scotland. Crevices, and holes in rock, chiefly in tide-pools. Varieties. a, Tyrientis. The condition described above, which is perhaps the most common ; at least on our south-western coasts. black ; tentacular border alternately pale blue and dull black. One large tentacle of first row pellucid horn-brown ; the rest dark grey, or white, in alternate groups. Column rose-pink on lower half, purple-grey on upper. Thus there are seven distinct colours in this variety, which yet is not at all showy. y. Eburnea. Disk ivory-white (Tugwell). 8. Modesta. Disk deep umber-brown, mottled with grey at the first row of tentacles, and merging into gi"ey, lavender, or white, towards the third or fourth row. Tentacles mottled with brown and grey. e. Sordida. Column dull wainscot-yellow, paler at the basal region. Disk blackish-brown, freckled with grey and white spots. Tentacles similarly coloured. General form thick and clumsy, without the usual tendency to assume a salver-shape. " varieties " are deviations from it. Those which I name versicolor or modts'.a, for example, might as well have been selected for the standard as Tyriensis. Indeed the only true idea of the species must include all its variations. " We may attempt," observes a master in science, " to reach what is called the typical form of a species, in order to make this the subject of a conception. But even within the closest range of what may be taken as typical charactt>rs, there are still variables ; and, moreover, no one form, typical though we consider it, can be a full expression of the species, so long as variables are as much an essential part of its idea as constants. The advantage of fixing upon some oiie variety as the typical form of a species is this, — that the mind may have an initial term for the laws embraced under the idea of the species, or an assumed centre of radiation for its variant series, so as more easily to comprehend those laws." — (Dana's " Thoughts on Species.^) (. Stellata. Disk pale buflF; a broad darker circle at the commencemeut of the tentacular border. Tentacles long and pointed ; very pale stone- drab, each varied with pellucid patches, which give a pretty and deUcate effect. But what is most peculiar is the alternate depression and elevation of the margin, a kind of frilling, which imparts to the disk a star-like form, usually of seven rays. This is a large and well-marked variety. The genus Sagartia was established hy me in a Memoir* read before the Linnean Society, March 20th, 1855. I then included in it dianthus, as well as the species to which I now confine it. The character on which I mainly relied in constituting it, appears to me, on maturer consideration, to mark a group of higher value than that of a genus ; and I have accordingly used it to characterise a family. Hence it became necessary to make a fresh diagnosis of the genus, which, though large, appears a very natural one. The name I have chosen alludes to the peculiar mode of dis- abling their prey, by means of missile cords, which is possessed pre-eminently by the species of this group, re- calling to my mind a graphic passage in the writings of the Father of History. In the army of Xerxes, he says, — " there was a certain race called Sagartians. The mode of engaged an enemy, they threw out a rope with a noose at the end ; whatever any one caught, whether horse or man, he dragged towards himself, and those that were entangled in the coils were speedily put to death," f The specific appellation of the present subject is the botanic name of a favourite flower, — the modest Daisy ; — helUs, from bellus, 'pretty. Though the Daisy Anemone is, as I have shown, subject to considerable variety, and has no one very strongly * ■" Description of Peachia hastafa, &c." Linn. Trans, xxi. 267. t Herodotus, vii. 86. THE DAISY ANEMONE. 31 marked, and at the same time constant, specific character, there is scarcely any of our species more readily or more certainly recognisable. Its variations are circumscribed' within appreciable limits, both of colour and form, and it has little tendency to merge into the characteristic con- dition of any other (British) species. Indeed, but ^for the needless multiplication of genera, I should be tempted to separate it from the other Sagartice, constituting for it, in association with two or three closely allied forms from the southern hemisphere, a distinct genus. From the elegance of its form, and its ready power of accommodating itself to captivity, few of our native species are more favourite tenants of an aquarium than this. Its habits, too, render it easily accessible. Within the limited range of its habitat it is for the most part "abundant. The rugged, indented, rocky shores of Devon and Cornwall seem to be the metropolis of the species : and here the tide-pooLs, fissures, and honeycomb-like burrows of the SaxicavcB, are densely crowded with the pretty Daisy. The broad front of Capstone Hill, at Ilfracombe, is broken, within the range of the tides, into a succession of narrow horizontal shelves, the angles of which run down into long fissures. The limestone promontory, known as Petit Tor, on the south-east coast of Devon, presents many ledges very similar in character, but more eroded into irre- gular holes and cavities. In both of these localities, hellis abounds, generally of the beautiful scarlet-lined variety, Tyriensis. Each usually occupies a little hollow, being attached by its base to the bottom, and expanding its beautiful disk over the edge. In the broader basins, moreover, which the waves have worn, " hollows of the tide-worn ree^" overshadowed by ribbon-shaped sea-weeds, — which are the very counterparts, in the sea, of the hart's-tongue fern fronds which overarch the green hedge-banks just above, — larger and finer specimens occur, apparently each broad coin-like disk stuck on to the smooth wall of the cavity, but really, as you find when you attempt to capture it, imbedded in its own proper cranny, into which it can retire out of danger. But it is as common to find colonies of the species, inhabiting the long narrow fissures, covered with but an inch or two of water when the tide is out ; five, ten, or even twenty individuals crowded together in a line as close as their bases, firmly planted side by side, will admit. Here, of course, when expanded, the puckered edges of each disk press upon and fit into the mutual irregularities of the others ; and the effect is very attractive, when the variety is that patched one, pale blue and black, which I have named versicolor. I have much admired them in this condition along the foot of the lofty overhanging cliffs at Watcombe, between Teignmouth and Torquay. Huge masses of the red con- glomerate have fallen from above, and are piled in con- fusion along the whole sea-line. And these seem to have formed a natural breakwater, protecting the base of the cliff from the action of the waves. Hence the lower part of the rock remains in situ, while all the upper and middle portions have been detached by the influence of rains and frosts, and have fallen ; and this lower part forms a suc- cession of sloping terraces, averaging perhaps some twenty feet above low-water mark. Each successive terrace dips to the northward at a very gentle angle with the horizon, so that the explorer has to mount from one to another in turn, while he pursues the line of coast, as each slope successively brings him to the water^s edge. These ter- races are very rough, but not unpleasant to walk iipon; and their angles are occupied with water, forming long THE DAISY ANEMONE. 33 narrow shallow pools, the bottoms of which run down into thin crevices. In these crevices reside the Daisies in question, in great numbers, and some of them of very large dimensions, as three inches in diameter, when follj ex- panded. They are, however, as I have said above, mostly so crowded together, that they are not able to spread their blossom-disks fully, but are fain to accommodate each other, by allowing the protrusions of one sinuous and frilled margin to fit into the recesses of another. They thus con- stitute lines of variegated frills, in which the individuals cannot be separated by the eye of the beholder ; and though no brilliant hues appear, there is sufficient contrast between the black and the white, the blue and the grey, all puckered and convoluted as the fringed outlines are, to gratify the eye. Nor are these very difficult of possession. For the con- glomerate, though hard, yields readily to the chisel, and the edges of the crevices present in many cases fair angles for the blows of the experienced collector. The Daisy is not unfrequently brought up in the dredge from a few fathoms' depth. In Weymouth Bay I have repeatedly obtained it thus, but still maintaining its wonted troglodyte habit ; for its favourite domicile is one of the deep angular chambers formed by the leafy expansions of that fine coral-like Polyzoan, Eschara foliacea. But Weymouth possesses a breed of the species which deviates much more widely from the normal habit. It is the variety which I have called sordida, having an eye not less to its filthy dwelling-place than to its dirty colour. The broad expanse of fetid mud, either wholly bare at low tide, or covered only with a foot or two of water, that floors the two inlets called the Fleet and the Backwater, is studded with multitudes of these dingy Anemones. The soft slimy mud affi)rds no proper surface for adhesion ; I and hence the animals can scarcely be said to adhere in the manner of the family, but simply to rest on the broad base. This is not, however, indicative of any defect in the power of adhesion ; for on being removed to a basin of sea-water, they are soon found firmly attached to the bottom and sides. mouth ; which is the more remarkable since the long ledges of low rock, broken into fissures, and excavated into num- berless hollows, would seem to present a favourable site for it. But since my residence there, it has yielded, in con- siderable abundance, the beautiful variety stellata ; which, as I understand, occurs to the north-east of the town. In Dr. Johnston's Brit. Zooph. (p. 231) may be found some curious figures by Mr. Cocks, illustrative of the pro- tean mutability of shape manifested by this species. This depends on the power of distending the body generally with water, together with that of strongly constringing some part, the constriction ever moving its place. Several of the Sagartice (as S. helUs, miniata, and troglodytes) have a singular habit of elongating to an im- mense extent one of the tentacles, while all the rest remain in the ordinary condition. The phenomenon has once or twice fallen under my own observation, but I will describe it in the words of some of my kind correspondents, who have from time to time directed my attention to it. It seems to have been first noticed in S. troglodytes by Mr. Hugh Owen of Bristol, who, in May, 1856, mentioned the fact in a letter to me. Soon afterwards he observed the same phenomenon in " a loosely-formed hellis, with longer tentacula than usual, found in a cave at Tenby." " I was, a few days since," he writes, " watching it closely, when one tentacle began to extend itself; and for an hour I watched its motions. The animal is about an inch and a half in I THE DAISY ANEMONE. S5 extreme diameter, and it threw out its tentacle to a dis- tance of three inches from the margin. Of course all colour disappears, and it requires one to he looking for the fact to ohserve the transparent memhranous nature of the ex- tended limb. I tried if its object was seeking for food, by dropping a scrap of meat in the way of the tentacle : it was seized and carried to the oral disk instantly." The same gentleman in a subsequent letter (dated 7th July, 1856) thus continues his observations : — " Another specimen of bellis, from Ilfracombe, of a dark self-colour (chocolate or umber-brown), is constantly extending the tentacles to full four times their length imder ordinary cir- cumstances ; and on one occasion I have seen a tentacle on each side thrown out so long as to command fully a circle of six inches in diameter. After the extension, I observe that the tentacle assumes for several hours a white appearance, increasing in intensity towards the extreme tip. This ex- treme extensility is interesting, as showing the resources of the animal in commanding a larger range for feeding : and the modus operandi is no less curious; for, after having reached the utmost length, any nearer spot is examined by curling the tentacle into a variety of elegant curves and rings." Mr. E. W. H. Holdsworth has also favoured me with some interesting observations on the same curious habit. me in the case of S. miniata, and which will be detailed in its place,* this excellent observer says : — " Since my last letter I have seen the elongation of one of the tentacles of the first row in hellis. The ordinary shape and proportions were retained, but the arm was stretched to more than twice its natural length, yet without any appearance of imnatural tension or straining : it was constantly in motion, * See infra, p. 44. d2 apparently feeling about for something, but assumed its usual size after a few hours. It was altogether very dif- ferent from what I have observed in the case of miniatay The Daisy is prolific in captivity. Mr. Holdsworth tells me that he has known 146, 160, and nearly 300 thrown out from single individuals in one day. They appear be- tween the tubercles at the summit of the gonidial grooves ; these grooves evidently acting as ducts for the transmission of the fully-formed young from the intersepts to the exter- nal world, and doubtless for that of the ova, when these are discharged. The characteristic form and markings are dis- tinctly recognisable in the newly-bom young ; their prin- cipal distinction, besides size, consisting in the fewness of their tentacles, which are commonly twelve in number, and in the comparative length of these organs, which is much greater than in the adult. Mr. Holdsworth says : " I have observed in this species, as well as in dianthus, and \Bunodes] gemmacea, that the size of the young varies with that of the parent, — large parents producing large young ones, and vice versa. I have noticed it repeatedly ; and the fact may perhaps be accounted for by the greater capacity of the larger parent affording room for a further development of the young before they are expelled than could be admitted of in the case of a smaller individual ; for the mature ova, I imagine, are always of the same size in the same species." I have already remarked that this species is easily kept in the Aquarium. It requires, however, some caution and skill in the manner of its capture ; for, as it resides in holes and crevices of the solid rock, it cannot be worked off with the nail, like some others, but must be cut out with a steel chisel. And, unless this operation be carefully performed, there is danger of tearing away the animal from its base, the central portion of which may be left behind. In this THE DAISY ANESfONE. 37 case it will expand in captivity, and look healthy to the eye of the tyro ; but, when examined, it will be seen to be perforate, a stick thrust in at the mouth coming out at the base. Specimens so mutilated never recover. Little more than ordinary treatment is required for S. belUs. It is desirable that it should be gently pushed, base downward, into a hole of a piece of rock ; — flints are often found suitable for it ; — or, if such cannot be readily obtained, two pieces of stone may be set side by side, and the Daisy dropped between them. Then it will soon attach itself to the bottom or sides of the crevice, and expand its beautiful disk, like a broad coin, at the top. S. bellis appears to be essentially a southern form. Sir John Dalyell, in his twenty years' experience, seems never to have met with it on the Scottish Coast ; nor has it, so far as I know, occurred on the Scandinavian or Danish Coasts, nor on either shore of the German Ocean. On the south- western shores of Scotland, however, it has recently been found in some numbers. On the other hand, it has recently been obtained near Boulogne; Mr. Holdsworth finds it "by myriads" near Oporto ; Rapp and Lamarck give the Mediterranean gene- rally as its habitat ; and De Blainville, more specially, la Mer de Naples. The following list of British localities is as complete as I have been able to make it. Guernsey (abundant), E. W. H. H.: Selsey, G. G.: Weymouth, P. H. G. : Torquay, P. ff. G. : Dartmouth, E. W. H. H.: Falmouth, W. P. C: Mount's Bay, Gaertn^: Lundy, G. T. : Ilfracombe, P. E. G.: Tenby (rare), P. H. G. : Holyhead, E. L. W. : Man, F. H. W.: Puffin Island, E. L. W. : South CorrigiUs, Arran, T. S. ^ Wright: Cumbrae, D. Robertson: Eathlin, /. Tetnpleton: K Balyholme Bay, W. Thompson : Dublin Bay, E. P. W. L Of foreign species the beautiful >S^. decorata (Dana), found in tlie Lagoon of Honden Island, is closely allied to our "bellis, S. Fuegensis (Dana), from Terra del Fuego, a very fine species with rich yellow column and disk, and grass-green tentacles, has much in common with the subject of this article, but it has far more prominently the characters, that the tentacles are short, and spring isolatedly from the disk. 8. invpatiens. (Dana) has the habit of elongating the column pillar-wise, and of variously constringing and writh- ing the body ; thus appearing to be intermediate between hellis and viduata. It seems to be through hellis and Fuegensis, that the genus Sagartia leads off to the curious Discosoma nummifonne of the Ked Sea, in which the column has no appreciable height, the animal being a very thin, flat, circular plate, with the tentacles reduced to minute warts, arranged in groups which form radiating bands. Of native species >S^. parasitica and B. clavata present, in the expanded character of their disks, marked relations with hellis. But a still closer affinity exists between hellis and Aiptasia amacha, in the characters both of the disk and of the column, as I shall notice more particularly when I come to describe the latter. It ought never to be forgotten that the order of sequence which we are compelled to adopt in treating of creatures in a book — that of placing each species between two others — can by no means express all their relations. Every species stands in the midst of many others, some closer to it, some more remote, to which it is linked more or less obviously. " Ten or twenty links would often be insuffi- cient to express these numerous relations."* To obviate * Cuvier. THE DAISY ANEMONE. 9B in some measure the false impressions liable to be pro- duced by this unavoidable order of linear succession, I endearour to represent some of the radiations of relation, in the following manner, observing that more direct affinity iS expressed by the perpendicular order. dianthus [Achates] A. amacha parasitica bellis B. clavata [Fuegensis] ? [impatiens] pDiscosoraa] miniata yidnata. rosea The late Edward Forbes described* what he considered to be " the Actinia hellis of British authors, not of Bapp," but which certainly cannot be referred to the species as now recognised. He obtained several specimens by dredg- ing on the Manx coast in September ; and it would be worth while to examine that prolific locality afresh for the animal, which will probably prove an unnamed species. " The body is cylindrical, of a reddish, or reddish white colour, regularly and finely striated longitudinally and transversely ^ and having glands of a Iright yellow colour ^ small and not very numerous, scattered over the surface. At the oral end the body bulges, forming a calyx [cup], on which the furrows are fewer but more granulose. When the disk is expanded, this calyx laps back, and is then almost even with the expanded tentacula. Disk angidar, in my speci- mens square, surrounded by three or four rows of short tentacula, thickly set, of a white or brownish colour, varie- gated ; having generally a tcMte line down the centre of each. The disk is broad, brownish, or orange, with white • In the Annals N. H. for May, 1840. lines. The margin of the mouth is bright orange. The animal can project its disk forward in a pouting manner. Tentacula and disk retractile. The specimens described were about one inch long when expanded, but I have seen larger." I have marked with italics the principal points in the above description, which seem inconsistent with the suppo- sition that bellis can be the species intended. The figures (which are engraved from the late Professor's drawings, in Johnston's Brit. Zooph., 2d Ed. pi. xlii. figs. 3 to 6) can no more be reconciled with our bellis than the description. i THE SCAKLET-FRINGED ANEMONE. Sagartia miniata. Plate II. Figs. 2, 3, 4. Specific Character. Tentacles with two sub-parallel dark lines along the front : a white space at foot, crossed by a broad black bar, and a narrow one below it. Outer row of tentacles with a scarlet core. Actinia miniata. GossE, Annals N. H. Ser. 2, vol. xiL 127. omata. T. S. Wright, Proc. Roy. Phys. See. Edinb. 1855. Bunodes (?) miniata. Gosse, Man. Mar. Zool. i. 29. GENERAL DESCRIPTION. Form. Base. Adherent to rocks and shells : slightly exceeding the column. Column. Minutely corrugated, studded on the upper half with large suckers. Substance fleshy. Form thick, the height rarely exceeding the Disk. Undulate, scarcely exceeding the diameter of the colunon ; radii strongly marked, and covered with transverse striae. Tentacles. Moderately numerous, arranged in about four rows. Those of the first row average in length about half the diameter of the disk ; the others diminish outwards, the last row being not more than one-fourth as long as the first. They are lax, and are usually arched over the margin, or thrown into sigmoid curves. Mouth. Not raised on a cone. Lip strongly crenate. Acontia. Emitted freely and copiously. Colour. Column. Deep rich brown, of a tint intermediate between burnt sienna and scarlet, sometimes merging into deep orange, paling into buff or light red towards the base, and often deepening into purplish-brown towards the summit. Suckers pale buff, which in the button-state become con- fluent, and form pale radiating bands, around the pursed aperture. Disk. Yellowish or greenish-grey, the radii distinctly mottled with darker grey or brown ; very variable. Sometimes one, or a pair, of broad 42 Tentacles. Pellucid pale-brown, or yellowish, indistinctly annulated with dusky. The front face of each (except the outer row) is marked with two longitudinal dusky lines, parallel with the sides, and meeting at the summit : these are some- times interrupted by a pale band ci'oseing the middle of the tentacle. Below them, at the tentacle-foot, is a large space of white, which is crossed by two bars of black; the upper one thick and very constant, the lower slender, and sometimes thinned away to a mere shade in the middle. Groups of tentacles often occur of a more or less opaque white, but barred like the others, with which they form alternate clusters. Those of the outer row consist each of a pellucid sheath investing a core of scarlet or brilliant orange, resembling in appearance the central gland in the papilla of an Eolis. This effect seems to depend on the pig- ment being spread over the interior surface of the wall of the tentacle, which is unusually thick and colourless. TENTACLE ^^'^^'''' Orange-red. o^ Size. S. MINLVTA {front). Specimens attain a height of two inches, with an equal width of disk. Locality. The south and west coasts of England, from Deal to Arran. Rock-pools and deep water. Varieties. a. Ornata. To the state above described, which may be considered as the normal colouring, I appropriate this name, which was applied by my friend Dr. T. Strethill Wright, to the species, which he described, believing it to be new. (Plate ii. fig. 4.)* /3. Yenustoides. Disk rich orange. Tentacles opaque yellowish- white or pure white, marked, however, with the two characteristic black bars ; the outer row showing traces, more or less conspicuous, of the orange lining. This variety, from Ilfracombe and Torquay, has much pnma-facie re- semblance to S. venusta ; but the specific marks of the tentacles, the strong crenation of the mouth, and the well-defined and concentrically striate radii are good signs of distinction. (Plate ii. fig. 3.) * My friend Mr. F. H. "West has received a specimen from the vicinity of Boulogne, with the disk more variegated than is usual with our specimens, and which had this peculiarity, that one-half of the disk was flushed with a dslicate rose-pink, and the opposite half with sax equally lovely shade of green. FT.ATE It I. 8. SAGARTIA NIVEA . 2.3.4. S. MINIATA 5. S. TROGLODYTES 6. S PARASITICA , IC 7. S. IGTHYSTOMA. 9. 10. S. ORNATA. THE SCARLET-FKINGED ANEMONE. 43 7. Roxoides. Column orange-brown ; disk pale yellowish-grev ; ten- tacks rose-coloured, with the proper markings ; and the outer row either wholly or partially scarlet -cored. Dartmouth, Plymouth. This is exceed- ingly like S. rosea. (See the article on that species.) 5. yireoides* Column drab-oliYe. All the tentacles opaque white, except five groups sub-symmetrically arranged, each group comprising a few tentacles of a pale orpnge-buff hue. A single specimen in the possession of" Mr. G. H. King, of Torquay, obtained by him in the vicinity. €. Coccinea, Column deep pellucid crimson : tentacles crimson. TMb " proaches a common state of A. mesembryanthemum in its appearance and . louring : its suckers, however, will in a moment distinguish it on exa- mination, and the usual row of orange-cored t«ntacle3 determines its true character. (Plate iL fig. 2.) f. Brunnea. Column umber- or even bistre-brown, with pale suckers : tentacles with the characteristic bars much disguised, and almost lost in a general cloud of dusky black occupying the lower half of the tentacle : this is divided by a naiTow whitish band from the terminal half, which is pellucid umber. The tentacles are unusually long. Those of the outer row are not all scarlet, some being white; aU, however, have the cored appearance. Torquay. It maj suffice to particularise tliese varieties, but spe- cimens are frequently found combining the characters of several, and running into one another bj imperceptible gradations. I obtained a very young individual at Wey- mouth, which I assign to this species, in -which the ten- tacles of all the four rows were cored with the richest orange. I first became acquainted with this very fine species in the summer of 1853, at Weymouth, where I found several specimens adhering to the shells of oysters and pectens, brought to market by the trawlers. Since that time I have met with it in some abundance in the neigh- bourhood of Tenby, especially on the eroded surface of some dangerous rocks, known as the Woolhouse Rocks, lying about a mile off shore, and exposed only at low water. In the pools and hollows of this reef, open to • In these compounds I take the liberty of using the elements " ventutet," '•■ rosea," and " nivea," not as Latin adjectives, but as words now having the force of proper names. investigation only under favourable circumstances of wind and weather at the equinoctial spring-tides, this, with other lovely kindred species, as rosea, nivea, &c,, expands its beautiful blossom, in charming abundance. But still more profusely does it occur in certain situations in the vicinity of Torquay. The line of shore between the Baths and Meadfoot is very bold, and a great number of precipitous insular and peninsular rocks fringe the sea- margin. When the tide is very low, and when the sea is very smooth, a small boat can penetrate into the narrow straits and caverns formed by these fragments : and there, on their landward sides, where the rays of the sun never reach, may be seen myriads of Anemones, chiefly of this species, but mingled with dianthus, rosea, and nivea, and varied by a vast number of Alcyonium digitatum, which beneath the surface of the clear water are seen blossoming with their lovely polypes. The finest specimens I have seen are those whicli Mr. W. A. Lloyd obtains from the Menai Straits. The species seems to be specially abundant in that locality, and specimens two inches in diameter are not at all rare. The varieties ornata and hrunnea are the prominent forms. The habit referred to, under S. helUs, of greatly lengthen- ing one of the tentacles, is possessed by this species also. Mr. E. W. H. Holdsworth has favoured me with tlie fol- lowing note. " In two specimens of the Rosy-armed miniata [var, roseoides] I have observed a remarkable elongation of one of the tentacula, apparently of the second row. Under the microscope the surface appeared corru- gated [or transversely annulated], but mostly so when tlie arm was fully distended, and the corrugations were most decided at the free end, which was enlarged, truncate, and slightly dimpled at the centre. No use was made of this long arm when the animal was feeding : it hung down as THE SCARLET-FRINGED ANEMONE. 45 if it did not possess any paxticnlar function. It had the same colour as the others ; but was not, like them, wholly withdrawn when the animal was closed. In fact, it appeared as if rather in the way, and not easily disposed of by its possessor. After about a week [the phenomenon] disappeared, and I have seen nothing of the lengthened them." Those curious missile filaments which I have named €Uiontta* are discharged by this species in great profusion. They are, as usual, white, but appear to possess the power of discharging a pigment. A large specimen, which I had irritated by forcibly detaching it (in the usual way) from a stone, diffused a copious mucus. Acontia were also abundantly protruded, and spread to double the diameter of the body on all sides, on the bottom of a saucer in -which I had placed it. After a while the whole of this mucus over the same area icas of a delicate but decided roseate hue, as seen on the white china. The acontia are very densely filled with cnidce, of two kinds, chambered and unchambered. The former are Tonsil ^f ^^ inch in length, linear-ovate, of a clear pale yellow hue, highly refi-actile, with a long parallel-sided chamber, extending through three- fourths of the cnida. It discharges a wire (ecthoroeum) about one and a half times its own length, furnished for the distal two-thirds with a screw of two (or three) spiral bands, closely set, and forming an angle with the axis of 30" : the bands are clothed with reverted barbs. The imchambered cnidse are ^^th of an inch long, of a similar shape, shooting a wire to eight times its own length, which is attenuated to a fine point, and is furnished with a single screw-band, unbarbed. When out of water, miniata has the habit of protruding * See the Qeneral IntroductioD, for a full description of these organs. the wall of the stomach, almost to as great an extent as B, crassicomia. This is specially seen when the specimens hang from the perpendicular face of a rock. According to Mr. Holdsworth, S. miniata increases by spontaneously separated fragments of the base, like A, dianthus. He says, — " I have had two young ones of miniata produced from hits of the base detached from a large specimen, which had been fixed for a long time. It was anchored too firmly ; so it cut its cable, and started for fresh quarters." According to the same careful observer, double individuals are not uncommon — a fact which points to a more decidedly fissiparous habit. The following note contains all the original information that I possess of the generative process. Examining a small specimen, about the middle of August, I found that it had given birth to several ova or gemmules. I had just removed it from a stone in one of my tanks, to which it filaments copiously, and these were now partially retracted and coiled up, forming a white coat almost entirely in- vesting it. Under a one-inch objective, as these were twining and twisting, I saw among them several olive- yellow bodies, which seemed to have a motion independent of the filamental currents ; and I isolated one. It was of a sub-nautiloid form, irregularly convolute, much like a Bursaria, about xTnnjths of an incli in long diameter, Y?jWths in lateral, and about -ij^ths in transverse; of a dull clear olive, but granular, riclily clothed everywhere with small cilia, by means of which it revolved freely in all directions. Others which I saw were much less than this one. Dr. T. S. Wright, however, seems to have witnessed the birth of perfectly-formed young. " Four young ones," he observes,* " produced by as many specimens of Actinia * Proc. Roy. Phys. Soc. i THE SCARLET-FEINGED ANEMONE. 47 omata [= Sag. miniatd\ in the last six months, were horn with a double row of tentacles, the inner long, the outer short, and tinged "with orange-red as in the adult." This beautiful species is easily reconciled to captivity, and is hardy. I have kept individuals for long periods. It expands freely. It ought to be placed on a worm-eaten piece of rock, but it does not require so deep a hole as hellis. The rich hue of the column, in some varieties, makes it desirable that this should be visible. The following list of localities marks the range of the species as at present known. I am not aware that it has been found out of Great Britain. Deal, Rev. H. H. Dombrain : Weymouth, P. H. G. : Torquay, P. H. G. : Dartmouth, E. W. H. H. : Plymouth, Dr. G. Dansey: Ilfracombe, W. A. Lloyd: Tenby, P.H. G.: Menai Stiait, W. A. L. : Hilbre Island, E. L. W. : Arran, T. S. W,: Cumbrae, D. R. bellis. MINIATA. rosea. ornata. ichthystoma. THE ROSY ANEMONE. Sagartia rosea. Plate I. Figs. 4, 5, 6. Specific Character. Tentacles all rose-coloured ; the first row sometimes with a broad dusky bar above a narrow one at the foot. Actinia rosea. Gosse, Devonshire Coast, p. 90, pi. i. figs. 5, 6 (var. vinosd). ptdcherrima. Jordan, Ann. N. H. Ser. 2, vol. xv. p. 86 (var. pidcherrima). vinosa. Holdsworth, Proc. Zool. Soc. 1856 (var. vinosa). Sagartia rosea. Gosse, Tenby, p. 365. Frontisp. (var. De- metana). GENERAL DESCRIPTION. Form. Base, Adherent to rocks : scarcely exceeding the column. Column. Minutely corrugated, studded on the upper half with suckers, to which fragments of gravel or shell occasionally adhere. Substance fleshy. Form in expansion elongate, cylindrical. Disk. A shallow cup, the margins occasionally undulate. Radii strongly marked, and covered with transverse striae. Tentacles. Moderately numerous, in four or five rows, nearly equal in length (but this varies according to the variety) ; often arching regularly over the margin, but sometimes very small and forming a fine fringe. Mouth. Not raised on an obvious cone, often apparently four-lobed. Lip crenate. Acontia. Emitted copiously. Colour. Column. Deep brown, inclining more or less to dark red, paling to buflf at the base. Suckers pale bufi" or whitish. Dink. Pale silvery olive, without markings, except an ill-defined dusky margin, produced by the blending of the bands that cross the foot of each tentacle. Tentacles. Clear rose-red or rose-purple, very brilliant; those of the outer row showing a slight tendency to lilac. Those of the first and THE ROSY ANEMONE. 49 second rows are crossed at the foot by two undefined dusky ban, some- times obsolescent, of which the upper is the thicker. Mouth. Lip white ; or light pink. It occasionally rises to a height of an inch and a half; and the diameter of the tentacular flower is about an inch. LocAiiirr. The south-west comer of Great Britain : in holes and rock-pools at low water-mark. Vabibties. a. Vinota. The condition described above, which is that to which the specific name rosea was first applied, and which appears to be the most widely-spread variety. (Plate L fig. 4.) 0. PultJurrivka. Column cream-white, merging towards the sununit into pale olive. Disk cresun-white, with dark lines between the radii Tentacles crimson-lake, with several (more or less distinct) darker bars ; those of the first row thicker, usually carried erect, or arching inwards. (Plate L fig. 6, which is copied from a beautiful drawing with which Professor Jordan has favoured me.) y. Erythropi. Column dark brown, inclining to olive, with conspicuous pale suckers. Disk brilliant orange-scarlet. Tentacles rather short, stout, bright rose-lilac, the bands across the foot well defined. A very lovely Tariety, which I have found near Torquay. 8. DetMtana. Small and low, rarely exceeding half an indi in height or diameter. Coliunn rich red-brown, with inconspicuous suckers. Disk crimson, oft^ with a tinge of orange, usually more or less puckered at the margin. Tentacles crimson, short, crowded, resembling a compact fringe. (Hate L fig. 5.) For the first and second of these varieties, I have retained the names proposed respectively by Mr. Holdsworth and Professor Jordan, who described them as species under Ihese appellations. I am quite sure that both must be referred to this species. The fourth is the form so abun- dant on the Pembroke coast ; a very marked variety, to which I have assigned a name alluding to the Aij/tiTrai, the ancient inhabitants of that part of Wales. All are beautifal ; but perhaps pulcherrima, as its name imports, is the loveliest of all. I There Is no doubt tliat S. miniata and >S'. rosea approxi- mate in some of their varieties very closely ; and I have separate. I have seen, in the vicinity of Tenby, specimens, in which some of the small tentacles of the outer row had a scarlet or orange core, and yet in no other respect could I distinguish them from the true rosea. Normal rosece and normal mimatoi were abundant on the same rock (the Woolhouse-rock) within a few feet ; which fact suggests the possibility of hybridization. Besides the scarlet-cored tentacles, miniata may be described, in those varieties which come nearest to rosea, as darker externally; as growing to a far larger size ; as being lower and less pillar- like; and as having a much more lax, flaccid habit of body. The qucestio vexata, — What constitutes a species ? what a variety? is one which it is much easier to answer theo- retically than practically. Some have proposed certain arbitrary canons, such as that assumed by Mr. Tugwell, \hsdform distinguishes the species, colour only the variety. But this is quite untenable. In many instances colour is not only specific, but even generic ; — as black, "(^hite, and red, in well-recognised patterns and in certain fixed regions of the body, in the Woodpeckers ; black, yellow and red, again in certain patterns, in Paj>ilio ; yellow, red and white in the Pteridoi. Indeed, our entomological friends would be sorely puzzled to define their species, if colour were denied them as a distinction. In the Butterflies alone, hundreds of indubitable species rest exclusively on colouring. The fact is, anything may be a specific character, provided it be constant. Constancy, permanency, is what we require ; let us only indicate any mark that is invariably found,— no matter whether it be colom', form, pattern, surface, sculpture, or any thing else ; or any combination of THE EOSY ANEMONE. &1 these, and we have a good specific character. I believe, with Mr. Wallace, that " the two doctrines of * permanent varieties' and of 'specially created unvarying species' are inconsistent with each other."* In other words, I would say a species is permanent, a variety transitory. There is no doubt, however, that the latter may be maintained within certain limits by breeding in and in ; though there will always be a tendency to revert to the original and normal character, which marks the permanent species. Though I believe this distinction to be a good one, it does not therefore follow that we can put it in practice without any difficulty. We find a specimen; — we know nothing of its antecedents ; — at most we can trace it only through a few generations ; and thus we are precluded from applying our test of permanency to it. The only resom'ce is the practical skill and judgment which expe- rience and observation gradually give ; and these, as they cannot be communicated to another, nor be reduced to formula, differ indefinitely in individual cases. In the present work I must beg my readers to believe that I use the best light I have, to arrive at right conclusions. Under all its variations, which are not very numerous, JS. rosea is a lovely little species. When left by the receding tide, it protrudes from its tiny cavity in the over- hanging rock, and droops, a pear-shaped button of orange- brown, with a cluster of brilliant purple tentacles just showing their tips from the half-opened centre, and a drop of water sparkling like a dew-drop, hanging from them. Then it is beautiful. But a more charming sight is seen when, as at the rock near Lidstep, or on the Woolhouse reef, you gaze down into a narrow basin worn by the waves of ages in the solid limestone, and, having first care- fully lifted the broad fronds of Laminaria and Bhodymema * Zoologist, p. 5S88. E 2 pahiata that spring from the edges, you see the dark brown walls and bottom of the pool, — which is filled to the brim with quiet crystal water, — all studded over with the expanded disks of roseoe, nivece, and venustce. Then indeed the sloping sides and bottom resemble a parterre, of which these are the lovely flowers; while the tufts of green, brown and purple Algse that spring up everywhere around, some like moss, some like fantastically cut leaves, may well serve for the foliage of the " fairy paradise." " In hollows of the tide-worn reef. Left at low water, glistening in the sun, Pellucid pools, and rocks in miniature, With their small fry of fishes, crusted shells, Rich mosses, tree-like sea-weeds, sparkling pebbles, Enchant the eye, and tempt the eager hand It is equally attractive in those imitations of such rock- pools, which we make in glass tanks and china pans for our drawing-rooms. But, like the other species of the group to which it belongs, it is a somewhat precarious tenant of the Aquarium. I have kept at different times a large number of specimens ; but none of them, so far as I can remember, survived a twelvemonth's captivity. A dark-coloured mass of rock suits it best, serving as a back- ground for its rich crimson blossom. It loves the shadow, too; and should therefore be placed on the side farthest from the light. A rough perpendicular surface is very appropriate for it. The Rosy Anemone occasionally protrudes the walls of the stomach, like B. crassicorms, which then overlap the disk in large furrowed pellucid lobes. It sometimes -distends the tentacles till they are translucent, and then it is not uncommon to see the free ends of the acontia, lying within these organs in coils, having penetrated through the open base of the tentacle from the intersepts of the body- THE ROSY ANEMONE. 53 cavity. One may sometimes also discern fragments of the same filaments, which have become accidentally detached, driven to and fro at the tip of the interior of the tentacle. The proper ciliary motion of these twisted atoms combining with the motion produced by the lining cilia of the tentacle- wall, gives them the fitful vacillating action of spontaneous volition ; so that they may readily be mistaken for living worms accidentally imprisoned. The acontia are emitted from the pores of the body in great profusion upon irri- tation. The form and armature of their cnidce do not differ from those in the species last described. The following are the localities of the Rosy Anemone known to me : — Guernsey, E. W. H. E. : Teignmouth, R. C. R. J. : Torquay, P. H. G. : near Paignton, Rev. W. F. Short : Dartmouth, E. W. H. H. : Tenby, Lidstep, St. Gowan's Head, P. H. G: Bantry Bay, E, P. W. miniata. EOSEA. venusta. nivea. A SIR jE ACE A. THE ORNATE ANEMONE. Sagartia ornata, Plate II. Figs. 9, 10. Specific Character. Basal region of the tentacles, and the outer region of the radii blackish : a white bar across the former, and a white cordate spot on the latter. Actinia ornata. Holdsworth, Proc. Zool. Soc. 1856. PI. v. figs. 5, 6, 7, 8. GENERAL DESCRIPTION. Form, .Base. Adherent to the roots of Zaminaria : slightly exceeding the column. Column. Minutely corrugated ; studded on the upper half with suelsers, more numerous as they approach the summit. Form in expansion elon- gate, cylindrical. Tentacles. Moderately numerous, in five rows ; those of the first row rather stoutly conical, comparatively short ; the rest diminishing rapidly as they approach the margin. Mouth. Not raised on an obvious cone. Lip tumid. Aconlia. Emitted freely. Colour. Column. Dark orange-brown, paler at the base. Suckers pale. Dish. Central moiety pale orange, changing to a rich purplish brown on the outer moiety. The radii of the first and second rows of tentacles separated by narrow yellow bands slightly diverging " as they proceed outwards, and at their extremities partially surroundmg the bases of the tentacles, according to the following arrangement. The first tentacle may be said to arise from the space between two pairs of bands, tbe second being situated within the pair ;* the band bifurcates near its extremity, and incloses the third tentacle : these branches again divide and form a similar inclosure for the tentacles of the fourth row : + beyond these is a set of * The apparent distribution of the bands in x>airs is merely a necessary result of the fact that the secondary radii ai'e narrower than the primary. + Hence the yellow bands are doubtless the united radii of the tertian and quartan series. THE OBNATE ANE3I0XE. Tery short tentacles ; these, as far as I have been able to examine them, are not connected with the yellow bands." On each primary radius is a large heart-shaped spot of cream-white, well defined, in the midst of the daxk-brown ; and on each secondary radius a similar spot, but more elon- g;ated, and situate a little more remote from the common centre. TetUacles. Dark brown at the base, becoming paler toward the tip, en- circled by three white rings, of which the basal one is very distinctly defined. Mouth. Lip pink ; frequently conspicuous. Size. About three-fourths of an inch in height when extended ; flower half an inch in diameter. Locality. The entrance of Dartmouth harbour, in the laminarian zone. Yarietles. a. Fusca. The condition above described. TENTACLE ^3 Ruiida. The brown on the tentacles and cei-tain parts (front view). ^^ ^^^ ^^^ replaced by various shades of red. This attractive little Anemone appears to have been seen only hy Mr. HoldsTvorth, who described it in detail, "with accompanying di'awings, in a Memoir read before the Zoological Society of London, Dec. 11th, 1855. From those details, as published in the Society's proceedings, I have compiled the above description, merely throwing them into that order of arrangement, which, for convenience of reference, I have adopted in this work. I have been aided, however, by the original beautiful drawings, which my friend has liberally placed in my hands. From these, the figures in Plate II. have been likewise copied ; fig. 9 re- presenting the flower, fig. 10 the button. " This species," as its discoverer observes, " is chiefly remarkable for the beauty of its oral disk, which, for colom'ing and elegance of marking, will bear comparison with that of any of the larger kinds. . . . Several ex- amples were obtained at extreme low-water mark, from a large mass of detached rocks known as the Mewstone, near the entrance to Dartmouth Harbour. Thej were met with on two or three occasions, but were always found nestling among the roots oi Laminaria digitatar The variety riihida was described in the same paper. Six specimens were found among the roots of a Laminaria sent to Mr. Holdsworth from the same locality. He could find no other difference of importance, than the substitu- tion of red for brown above-mentioned. From a private communication with which he has recently favoured me, I learn that he failed to discover any more specimens of either variety, though he subsequently searched the same locality. rosea. ORNATA. ichthystoma. THE FISH-MOUTH ANEMONE. Sagcwtta ichthy stoma. {Sp. nov.) Plate II. Fig. 7. Specific Character. Tentacles minute, mar-ginal; each having two narrow black bars across the foot. GENERAL DESCRIPTION. Form. Bate, Adherent to rocks or shells : not exceeding the column. Column. Coarsely corrugated, with no (observed) suckers. Form (in button) low, nipple-like, with a coarsely-puckered involution ; (in flower) cylindrical, in height about equal to its diameter. Diik. A shallow saucer; with radii strongly marked; the margin slightly exceeding the diameter of column. Tentacles. Moderately numerous, arranged in three rows, set very close to the margin of disk ; nearly equal in size, very small, short, and conicaL MoiUh. Set on a large cone. Lip very tumid, coarsely furrowed. Colour. Column. Brownish-scarlet, becoming pale towards the top, and tinged with purple at the very summit. Disk. Pale fawn or bay, with numerous radiating lines of black, so thick at the outer half of the area as to give the effect of a broad, black, dightly-interrupted ring. A pair of gonidial radii, opposite, white. • Tentacles. Pellucid white, marked at the foot with two dose-set, narrow bars of black, and a broad ill-defined ring of dusky near the middle. The radial lines of black wind sinu- ously among the tentacles, on the pale ground of the disk, with a distinct and pretty effect. Mouth. Lip deep rich scarlet. Size. Button half an inch in height Flower three-fourths of an inch in diameter. tektaclk (fhtU). I Locality. The soutli coast of England : deep water ; low rocks. Varieties. o. Stihista. The condition above described. ;8. Astimma. Disk dull olive-grey. Lips dull brick-red. I know this little Anemone only by two specimens. The first (of the variety stihista) I found on an oyster in the fish-market at Weymouth, in the summer of 1853. As the oysters with Avhich the market was supplied were brought in by a trawler, whose fishing grounds were West Bay, and the offing of Weymouth Bay, we may safely set down one of these as the native locality of my little prize. The second specimen, which exhibited that measure of diversity in colour, that I have set down as distinctive of the variety astimma, but exactly agreed with the former in all its other characters, and was manifestly, at the first glance, of the same species, was sent me from Torquay, in April, 1856, by the Kev. W. F. Short. I understand it was taken at the insular rock knoAvn as the Ore Stone. Though less showy than the former specimen, whose black-lined face and pouting scarlet lips made it very attrac- tive, this latter was still very pretty ; and it proved to be easily reconciled to captivity, for it remained in one of my tanks, — sometimes under rather unfavourable conditions of the water, — from the 10th of April, 1856, to the middle of August, 1857, a period of sixteen months. Nor have I any reason to believe that it would have died then, but for my own carelessness ; for having taken it out of the tank to examine it, I incautiously left it, after my observations, exposed in a saucer to the midday beams of a hot August sun, and found it, of course, killed, when I looked at it again. THE FISH-MOUTH ANEMONE. 59 The (icontia contained, as usual, both unchambered and chambered cnidce. The former were linear-oblong, gioth of an inch in length, discharging an ecthorceum, four times as long as themselves, surrounded with a single spiral band. The latter were of the same form, but twice as long and wide, discharging an ecthorceum very little longer than themselves, in which I could not discern the least trace either of barbs or screw. The acontium was taken, certainly, from the specimen last mentioned, when it was either dying or dead, decomposition having commenced ; but the invest- ing cilia were in parts still active, and the cnidce dis- charged vigorously, just as when alive. In both varieties the small, conical, pointed tentacles projecting very regularly from the margin, impart a pecu- liar and well-recognised character to the species. These organs so strongly resembled the little sharp teeth crowded round the jaws of some fishes, that I was induced to borrow a nomen trivicde from that resemblance. The appellations of the varieties allude, as my classical readers will have perceived, to the long-standing custom among the Oriental ladies (nor altogether unknown to the dandies of ancient Kome) of staining the eyelids with stibium, a preparation of antimony, for the purpose of imparting a soft voluptuous languor to the eyes. Jezebel "put her eyes in painting " (2 Kings ix. 30 ; marg.). ornata. ? ICHTHTSTOMA. B. crassicomis. ? miniata. • See Pliny, Nat. Hist xi. 37 ; Juv. Sat. il 93. A S TR^A CEA . . SAO A RTIA D^. THE ORANGE-DISKED ANEMONE. Sagartia venusta. Plate I. Fig. 7. Specific Character. Disk orange ; tentacles white. Actinia venusta. Gosse, Ann. N. H. Ser. 2, xiv. 281. Sagartia venusta. Ibid., Linn. Trans, xxi. 274. Tenby, 358 ; pi. xxiiL figs, a, b. GENERAL DESCRIPTION. Form. Base. Adherent to roots ; little exceeding the column. Column. Smooth, or very minutely corrugated ; studded on the upper half with suckers, which are not raised on conspicuous warts. Substance fleshy. Form cylindrical, the height rarely exceeding the diameter. Dish. Flat or slightly concave ; the margin somewhat undulate. rows; the inner ones about as long as the diameter of the disk, the outer- most small and close-set ; slender, acute, somewhat flaccid. Mouth. A simple orifice without cone, or distinct lip ; frequently thrown into lobes. Throat ribbed. Acontia. Emitted copiously and freely. Colour. Column. Warm brown, varying from deep buff, to full rich brown- orange, often paler towards the lower half, where traces of alternate lon- gitudinal bands of pale and dark tint are sometimes visible. Suckers whitish. Disk. Wholly of a most brilliant orange, without markings. Tentacles. Pure white, without markings, except that the colour is generally pellucid at the foot and at the tip, and more or less opaque in the middle. Mouth. Paler than the disk. Ribs of throat white. SiZB. A full-si^ed specimen well expanded is about three-fourths of an inch in diameter of disk , but the extended tentacles may increase this to an THE ORANGE-DISKED ANEMONE. 61 in jh and a half, or rather more. The height rarely exceeds three-fourths of an inch. LOCALTTT. Various points in the south and west of Great Britain and Ireland. In Scotland it has not been recognised. Hollows in perpendicular and over- hanging rocks, exposed at low water : dark tide-pools. Vakieties. The variation seems to be limited to the greater or less depth of tint in the column. This most elegant species was first met with by myself in the neighbourhood of Tenby, where it is so abundant as to be quite characteristic. It has since been found in several other somewhat remote habitats, but nowhere in anything like the profusion in which it occurs in that its first recognised home. I am justified therefore in consider- ing South Wales the metropolis of the species. It occurs all along the south coast of Pembrokeshire, at least from Monkstone Point to St. Go wan' s Head ; but is more than usually numerous in the fine perforate caverns of St. Catherine's Island, that form such an attraction to Tenby visitors, and in the hollows and erosions of that rich pre- serve of zoophytic game, — the Woolhouse Eocks. The Orange-disk is essentially a cave-dweller ; almost invariably choosing for its residence some crevice or cranny, or one of those little cavities made by boring moUusks, with which the limestone on those coasts is generally honeycombed. Occasionally, indeed, we find it in shallow pools, with a bottom of impalpable mud, the detritus pro- duced by the action of the waves on the surrounding rocks ; but in such cases it will be invariably found that the Actinia is attached to a hollow in the solid floor of the pool, protruding its body through the deposit by elongation, and expanding its beautiful disk on the surface. Owing to this troglodyte habit, it is, like many of its congeners, rather '■ difficult to procure, notwithstanding its abundance, as it must be chiselled out, — an operation, which, from the great hardness of the compact limestone, is both tedious and precarious. Hundreds might be seen* in the largest of the caverns just alluded to, hanging down from the walls during the recess of the tide ; the button elongated to an inch or more. And almost every dark overarched basin hollowed in the sides of the caves, or in similar situations, at Lidstep, at St. Margaret's Island, and under Tenby Head, each filled to the brim with still crystalline water, had its rugged walls and floor studded with the full-blown blossoms of this 9,nd cognate species. As a specimen of the exceeding richness of these " gar- dens of the Nereids," wherewith our iron-bound coasts arc adorned, I shall take the liberty of citing the description of one, as it appeared to myself in the vicinity of which I am speaking. It w^as on the face of the blufi" castle- crowned promontory known as Tenby Head. " After scrambling over many rough ridges, we come to a perpendicular wall of rock some twenty-five feet high, jutting out from the cliff right across our way ; its foot Avashed by the sea, which is evidently of considerable depth, its summit tapered to a sharp edge, and the whole side holed, and furrowed, and honeycombed, and covered with barnacles to the very top. * I use the past tense ; for alas ! it is so no more. WTieu I revisited Tenby in 1856, I found that these caves, and almost every accessible part of the neighbouring coast, were pretty well denuded of the lovely animal- flowers, which, in 1854, had blossomed there, as in a parterre. I fear that the hammers and chisels of amateur naturalists have been the desolating agents ; and my friends tell me, not without a semi-earnest reproachful- ness, that I am myself not guiltless of bringing about the consummation. If the visitors were gainers to the same amount as the rocks are losers, there would be less cause for regret ; but owing to difficulty and unskilful- ness combined, probably half a dozen Anemones are destroyed for one that goes into the aq[uarium. THE ORANGE-DISKED ANEMONE. 63 " On the south side of this wall, almost at its tase, on a rough mass of rock so covered with luxuriant tufts of Dulse {Rhodymenia palmata) as to be richly empm-pled with it, I found a little basin, somewhat irregnlar in outline, but rudely oval, about a foot long, eight inches wide, and sis inches deep ; in other words, about the size of a soup- tureen. It was much obscured by overhanging drapery of Fucus ; but, on lifting this, I was astonished and delighted with the profusion of animal life, whose gay and varied hues gave to the tiny area the appearance of an artist's newly-rubbed palette. "Lest I should seem to exaggerate if I reported the contents of this basin from memory, I took the trouble to count the specimens, noting each sort in my pocket-book on the spot. Their numbers were, — nineteen of the bril- liant Orange-disk [Sagartia veimsta), and twelve of the Snowy [S. ntvea), all fully blown ; besides two large Shore- Crabs [Carciaus moenas), a Shanny {Blennius pholis), a Cynthia J several Sabellce, a gi'oup of Sabellaria alveola fa, some very fine masses of Botrylloides, and many specimens of the Crown Sponge {Grantia ciliata). than in its zoology. Chondrus crispus, finely tipped with Bteel-blue, as usual ; the Common Coralline ( Corallina officinalis), purpling the sides and bottom ; some small fironds oi Rhodymenia palmata, and one or two tiny ones of Laminaria saccharina, — which is particularly pretty while it is young, — were there ; as also two other kinds of superior elegance, namely, Delesseria rascifolia, with its oak-like leaves of fine dark crimson, and the pretty rich-green feathers of Bryopsis plumosa. Besides all these, there were other plants and animals of less note, which I did not enumerate." * * Tenby ; a Sea-side Holiday ; 96, et teq. I think it more than probable that the long deep Atlantic fiords of the sister island, will, on examination, prove at least to equal, if they do not greatlj surpass, in the luxuriance of their marine zoology and botany, any- thing that we can boast in England. As a companion to the above, I gladly give an Irish picture of 8. veniista, in situ, sketched by the graphic pen of my friend Dr. E. Per- cival Wright, the able and energetic Director of the Dubliu University Museum. " Last August, while entomologizing with Messrs. Haliday and Furlong in Killarney and GlengarifF, we made one day's excursion down Bantry Bay — a famed spot, but, with all its fame, it has never been worked. Well ; the weather was bad, — very bad ; a thick mizzling rain soon bespangled us with heavy dew-drops : however, pulled by four good oars, we did get on. The tide being right against us, it was hours ere we reached some remarkable caves, — the chief object of our trip. " Thousands of the dark olive-green Actinia mesembry- antJiemum lined these caves. It was not safe to try to land ; but in places where the sea, owing to shelter, was quiet, I could see the sea-floor covered with an extra- ordinary luxuriance of Actinise, Sponges, &c. ; — their colours, and forms, of course, distorted by every ripple of the waves. " We did land for a few minutes on one spot ; and, even at Tenby, and under St. Catherine's Rock, I never saw so much in the time; and this, though I did not wander from a single rock-pool. In it I saw about four and twenty specimens of Echinus lividus, all comfortably sitting in arm-chairs nicely cut out of stone, and most of them of a lovely purple tint. Down the centre of the pool ran a narrow fissure quite clicked with Bunodes crassicornis, which, as is their wont, had managed to gather all the THE ORANGE-DISKED ANEMONE. 65 little broken debris of shells, and to stick them over their bodies, in the way children stick broken china on heaps of mud, in our Irish villages. " But new to me as was E. lividus, and splendid as the really fine crassicornes were — they were of that pretty healthy white and pink variety — yet they were surpassed by your Sag. venusta, which with S. rosea sprouted out of every fissure. The former is, I think, the most exquisite of our Irish Anemones. In your figure in ' Tenby,' the tentacles are hardly white enough, and no painting can do justice to the clear orange. Book it and S. rosea, both very distinct from any other of our species. I saw other Anemones that I suspect will turn out new species ; but what could twenty minutes and an insect-net effect in 'catching' such things as Sagarts? Why, touch them roughly and — they're gone! If spared, I will visit them again ; and you shall see them, I hope, too : for if I spend a month in Bantry Bay, say next June or July, I can easily send you my Actinia captures ; — that is, if you won't visit Ireland. It is as pleasant as Jamaica." To turn firom these tempting scenes of wild nature ; — our beautiful Orange-disk is easily made happy in captivity : where, indeed, fed daily by fair fingers, and admired by bright eyes, it would argue badly for its temper if it were not. It is soon at home, and becomes one of the most brilliant ornaments of the Aquarium, expanding its lovely disk freely, fringed with its elegant border of snow-white tentacles, and thus making up in beauty what it lacks in size. It will survive an indefinite period, if it receive a moderate degree of attention. The observations which I have made on the treatment of S. rosea will apply with equal force to this species and to the following. Mr. Holdsworth informs me that he has witnessed the production of new individuals from fragments spontaneously detached from the "base, in S. venusta, as before described in the case of A. dianthus. .Miss Loddiges has favoured me with information of the same phenomenon in this species. The following are the localities known to me as inhabited by the Orange-disk : — Guernsey, Dr. J. D. Hilton : (on Laminariee washed up) Miss Guille : Torquay, P. H. G. ; Clovelly (on oysters from deep water), Bev. C. Kingsley : Morte Stone, O. T. : Lundy, G. T. : Tenby, P. H. G. : St. Gowan's Head, P. H. G. : Puffin Island, E. L. W.: Bantry Bay, E. P. W. : Belfast (abundant), C. Bosanquet. This species has close relations with S. nivea. Its colouring, however, so far as I have seen^ is constant, without any approach to albinism ; and its tendency to an ovate outline also distinguishes it, though less satisfactorily. It may possibly be found hereafter that the two constitute but a single species; but in the absence of any intermediate condition, I think it best to consider them distinct. miniata. VENUSTA. nivea. THE SNOWY ANE^^IONE. Sagartia nivea. Plate II. Figs. 1, 8. Specific Character. Disk and tentacles opaque white, without markings. Actinia nivea. GossE, Devonsh. Coast, 93 ; pL i. fig. 8. Sagartia nivea. Ibid. Ti-ans. Linn. Soc. xxi. 274. Tenby, 368, Frontisp. Annals N. H. Ser. 3, toI. i. p. 415. GENERAL DESCRIPTION. Form. Base. Adherent to rocks ; little exceeding the column. Column. Smooth, or slightly coiTugated : studded on the upper half xrith suckers, which form somewhat conspicuous warts. Substance fleshy. Form cylindrical ; the height often exceeding the diameter. Disk. Flat or slightly concave ; the margin scarcely undulate. Outline Tentacles. About two hundred, arranged in four distinct rows ; of which the first and second contain each twenty-four ; the third forty-eight; and the fourth, which ia marginal, about one hundred. Those of the first row, when extended, are about as long as the diameter of the disk ; the others diminish gradually, the outer row being small, and often papillary. Mouth. Sometimes raised on a cone, which at other times disappears ; frequently thrown into lobes. Lip slightly tumid. Throat ribbed. Acontia. Emitted freely and copiously. CoLOrR. Column. A light olive drab, slightly varying in intensity; becoming paler towards the lower half, which is often marked with alternate longi- tudinal bands of white and drab tint. Suckers whitish. Disk. Opaque white without markings, except that, when fully ex- panded, a grey tinge spreads in a circle, near the bases of the tentacles. Occasionally a very faint tinge of yellow surrounds the mouth. Tentacles. Pure snow-white, opaque, except when much distended with water; without any markings either on the body or around the fcot. Mouth. Lip and throat pure white. F 2 Size. Large specimens attain tlie thickness of an inch, the height of an inch and a quarter, and the diameter of an inch and a half, when fully expanded. Locality. The south-west coast of England. Crevices and rock-pools. Varieties. o. Immaculata. The condition above described. j3. Obscurata. Disk tinged with faint greyish-olive ; the tentacular region smoke-grey, undefined. This variety sometimes has the column of that rich orange-brown hue which is characteristic of this group. It was on the north side of the limestone promontory- known as Petit Tor, on the south coast of Devon, that I first met with the Snowy Anemone, in the spring of 1852. The rock here is hollowed into large cavernous pools, isolated only at very low tides, and dark with the shadow of the slimy sponge-covered precipices that arch over them ; where Laminarice grow abundantly, affording many a nidus for profuse forests of parasitic Hydroids of the genera Sertularia, Pliumularia, and Laomedea. The little red siphons of thousands of Saxicavce hang down from the holes which they have excavated in the solid limestone, each terminated by a diamond drop of water, awaiting the moment when the returning tide shall cover their abodes, and restore to them activity ajid enjoyment. It is their season of periodical idleness and repose. Among the roughnesses of the rock, and the conical papillary pores of the sponges, which, olive, yellow, and scarlet, stud the sur- face, — green Nereidous worms glide along, in and out, by means of the curious packets of slender bristles, alternately projected from every segment and withdrawn, that serve them instead of feet. Below the water-line, that is to say, THE SNOWY ANEMONE. 69 the level of the lowest part of the margin of the pool, which of course never varies, such animals and plants as require to be perpetually covered with water enjoy circumstances suited to their wants. In the deepest shadow, fine speci- mens of the fleshy Dulse {Iridaea eduUs), and the lovely leaf-like Delessena sanguinea, display their crimson fronds in copious tufts ; plants that cannot hear the absence of water, their delicate leaves becoming orange-coloured in large patches, which soon die and slough away, — if left nnbathed even for a single tide. The curious white Cows' paps [Alcyoniuvi digitatum), all studded with their clear glassy polypes, project from the rock ; and here I saw several white Acttmce, which at once attracted my notice, though beyond my reach, on the opposite side of the pool. At length, however, by searching in another smaller pool, to which I could gain access, I found, beneath the drooping Oarweeds, one of the white Actinice within reach. It was three or four inches beneath the surface ; so that to procure it, it was needful to bale out the water to that depth, which I effected by the aid of one of my collecting jars, and then to cut out the animal's cell with the steel chisel. I was, however, sufficiently repaid for the labour by the beauty of this snow-white Anemone. After an absence of nearly six years, I visited this inter- esting spot again. It had often been a subject of specula- tion with me whether the minute featm-es of a rocky coast change rapidly under the action of weather and sea ; and I had looked forward to this visit with interest, as likely to afford me data for determining the question. The shore was as if I had left it but yesterday. Everything appeared as if it had been untouched : every tide-pool, every projec- tion, I recognised : the broad cleft that I have described (Devonsh. Coast, p. 34) ; the little basins within it ; the slight projections on the face of the cliff by means of which I scrambled across, just as of old ; tlae fartlier cliasm (p. 39) ; and tlie large dark tide-pool Inwhicli I had seen the Prawn; — all were exactly as when I first made acquaintance with them six years ago. This last pool is still fringed with Oarweeds crowded with Laomedea forests, and the farther walls are still spotted over with daisy-like Snowy Ane- mones, just where I saw them first, and in all probability the very same identical individuals. But in the interim I had become familiar with the fair nivea, in what I may call its metropolitan home. It is in the numerous caverns and dark rock-pools into which the limestone formation on the Pembroke coast is hollowed, that this lovely species is seen to advantage ; especially in the dark holes of Monkstone, the Caves of St. Catherine's and St. Gowan's, and the oversliadowed pools of Tenby Head and Lidstep. Here, as we peer into the clear water of these obscure wells, we see the Snowy Anemone studding the rugged sides by hundreds, like bright stars on the mid- night sky, singly and in constellations. Here, too, swarm its congeners and companions, the equally lovely rosea and venusta / and this trio of graces are the very gems of the Demetian rocks. When covered by water, m'vea expands freely, and con- tinues long unfolded ; but^ in situations where it is left by the tide, it either withdraws into its hole, or, if this be placed on the side of a perpendicular or overhanging rock, it hangs out in the form of a lengthened wart, with a drop of water depending from its drooping head, like a dewdrop, in the centre of which a speck of white reveals the peeping tips of the contracted tentacles. Mr. Holdsworth has observed in this species that curious form of elongation of the tentacles described under S. miniata. Here, however, no fewer than ten or twelve of the tentacles of the first and second rows hung down, THE SNOWY ANEMONE. 71 straight and motionless, to a distance of two inches from the disk, Thej were attenuated towards the middle, enlarging again on nearing the tip, which was truncate in some^ rounded or obtusely pointed in others. Corrugation was present in some, but was rather difficult of detection, owing to the absence of colour. It is probable that this peculiar condition of the tentacles maj be accompanied with func- tions distinct from those of the mere elongation, such as has been described under S. hellis. (See ante, p. 35.) This species bears a far closer resemblance to a daisr, both in size and colour, than that which has obtained pos- session of the name. Indeed, one can scarcely see a group of mvecB and venustce under water, especially among the small mossy growth of grass-green Algfe, — Bryopsis, Con- ferva, Calothrix, Enter omorpha, &c., — without being forcibly reminded of a crop of daisies on a lawn. Mr. Holdsworth finds it "not uncommon at Dartmouth, but usually small ; inhabiting crevices in steep rocks under sea-weeds; at Guernsey, in sheltered nooks, very fine." The young do not difier from the parent, except in size and in the number of the tentacles. An infant specimen that was bom in one of my aquaria, adhered by the base immediately, and presently expanded. It displayed twelve tentacles, set in six pairs ; each pair being nearly parallel, and separated by a marked inteiwal from the pair on either side. Nivea rivals miniata in the profusion with which it shoots forth its poison-bearing acontia, on the slightest irri- tation. They are moderately crowded with cnidce, mostly of the chambered kind, discharging an ecthorceum little longer than themselves, densely armed with reverted barbs, which impart the brush-like form so characteristic of this genus. Most of the recognised habitats of the species have been already mentioned incidentally : they may, however, con- veniently be tabulated. Guernsey, E. W. H. K: Torquay, P. H. G.: Dart- mouth, E. W. H. H. : Clovelly (on oysters trawled), C. K. , Morte, G. T.: Ilfracombe, P. H. G.: Lundy, G. T. : Tenby, P. H. G. : St. Gowan's Head, P. H. G. : venusta. NIVEA. sphyrodeta. THE SANDALLED ANEMONE. Sagartia sphyrodeta. Plate L Figs. 8, 9. Specific Character. Tentacles few, thick, pure white ; the foot of each inclosed within a slender ring of purple, vrhich passes off in a line towards the margin. Actinia Candida^ GossE, Devonsh. Coast, 430; pi. viii. figs. 11, 12, 13 (" The Purple Spotted Anemone"). Sagartia Candida. Ibid. Linn. Trans, xxi. 274 : Man. Mar. ZooL i. 28. tphyrodeta. Ibid. Annals X. H. Ser. 3, vol. i. p. 415. GENERAL DESCRIPTION. Form. Base. Adherent to rocks ; expanded beyond the column. Column. Smooth, without conspicuous suckers. Substance pulpy. Form cylindrical; the height in general slightly exceeding the diameter. Disk. Flat or slightly concave ; the margin entire. Outline circular. Tentacles. About forty-eight, arranged in four rows ; of which the first and second contain each eight, the third and foiu^h each sixteen. Those of the first row are by far the largest, the size diminishing regularly to the external row : their form is stout and conical. They are usually spread horizontally, and have their tips frequently bent downwards. Afoutk. Raised on a conspicuous cone, which, however, is not per- manent. Lip capable of great protrusion and distension. Acontia. Emitted freely and copiously. COLOITB, Column. Marked longitudinally with many bands and narrow lines of opaque white, separated by interspaces, always narrow, of pale semi-pellucid brown, or drab. The summit is occasionally tinged with reddish-brown. Disk. Opaque white, marked with five radiating lines of pellucid white. The tentacular region is marked with the ring-lines to be presently described. Tentacles. Ivory white, without the least appearance of spots or bars : but at the very foot, where each tentacle springs from the horizontal disk, it is svuTouuded by a narrow ring of purplish, reddish, or dusky brown. which is occasionally broken in front, but always passes off behind in a f^lender wavy line to the margin, where it slightly bifurcates. Frequently the ring dilates into an undefined spot at each side of the tentacle-foot. Sometimes the line passing off to the margin can be scarcely discerned beyond the second TENTACLES OF RPHT- ^'°^^' ^^^ Sometimes the whole marking seems BODETA obliterated. {viewed vertically). Mouth. Pure white. Size. Half au-inch in height, and about the same {or occasionally a little more) in expanse. Locality. The south and west coasts of England. Low-water mai-k. Fissures in rocks ; the under surface of stones. Varieties. a. Candida. The condition above detailed, which I originally described in my " Devonshire Coast" under this specific name. fi. Xanthopis. Disk assuming various shades of yellow, from a pale chrome or lemon-colour to a deep orange, or even dull vermilion. This pretty little species was discovered by myself at Ilfracombe, It was during an unusually low spring- tide, in October, 1852. Specimens occurred at that time in two localities, having this in common, that in each case they were adherent to the perpendicular or overhanging surface of the cliif, at the very verge of lowest water. The animals were social : in the one case I found three indi- viduals associated ; in the other many dozens, a numerous colony thronging the approximating sides of a narrow fissure that runs far up into the solid rock at the seaward base of Capstone Promenade. A frequent tendency to a pendent posture was noticed ; for even where the general surface of the rock was perpendicular, many of the Ane- THE SANDALLED AXEMONE. 75 mones were hanging from Taeneatli the little points and projecting ledges. In describing these specimens, I suggested the possibility I tiiat they might be referred to the Actinia alba of 3Ir. W. P. Cocks.* The absence of the bright yellow dots that were found on the mouth of the latter, and the entire want j of visible suckers, induced me to consider mine as unde- scribed. It is true, the repeated occurrence since of specimens with a disk more or less yellow nullifies the force of the former objection, but the latter remains ; and until I see specimens of A. alia from Mr. Cocks's locality, I dare not assume the identity. From original drawings with which that gentleman has kindly favoured me, I per- ceive, moreover, that the tentacles in alba are numerous and slender, whereas in spliyrodeta they are few, tliick, and conical. Besides this, the marking of the ten- tacles in alba, which are described as " barred, having opaque white patches anteriorly," removes the animal from any species with which I am acquainted. I am not, however, without hope, that before this work is closed, the kindness of my Cornish friends may bring me into personal acquaintance with this, and other desiderata of that prolific coast. The substitution of another appellation for that which I had at first assigned to this species was called for on two accounts. First, there was already a species named Candida by Miiller; of which fact I was not aware. Secondly, this name proved objectionable. While no specific name may be rejected on account of its having no significance, every one ought to be rejected which has a false sig- nificance. Mr. Holdsworth's discoveries of the species at Dartmouth and in the Channel Islands have proved, or at • Johnst. Br. Zooph. ; Ed. 2; 217. Eep. Comw. Polyt. Soc. ISal; 6. least rendered it highly probable, that the normal con- dition is to have the disk of a yellow hue, more or less deep, the white variety being nothing more than the albinism to which organic colours so often tend. The term " candidal'' therefore, became inappropriate as a nomen triviale ; and I have sought one which should express a more unvarying character. The word " sj)hy- rodeta''^ signifies sandalled, from a^vpa, the ankles, and Beco, to bind ; and alludes, as I need scarcely say, to the line which, like a narrow ribbon, encircles the tentacle-foot. That the white disk marks a degenerated condition is rendered more probable by some facts that have come under Mr. Holdsworth's observation, and, in part, also under my own. A specimen obtained by that gentleman at Dartmouth was at first of a rich chrome-yellow over the whole disk ; but after having been some time in captivity, it gradually faded to a sort of dull cream- white ; in this condition, my friend submitted it to my care for a few days, during which time it quickly resumed its brilliant face. Another individual, which I think Mr. Holdsworth brought from Guernsey^ fell into a like condition. Writing of this, he observes, ' The animal has been out of sorts, and I have been obliged to administer to it several draughts (of pure sea-water), which have nearly set it to rights again. The beautiful colour of the disk, however, has nearly vanished, but some traces of it are still to be seen around the mouth. When I first had it, the colour was very conspicuous." The Sandalled Anemone is an interesting little captive. It expands its flower-face with great readiness ; rarely remaining long closed, provided the surrounding water be pure. The large conical tentacles stretch out hori- zontally to their utmost, like a star ; and though, on being touched, it will partially contract, it unfolds the instant I THE SANDALLED ANEMONE. 77 the annoyance ceases, and is presently fiill-blown again. It is fond of floating at the surface of its prison, the base dilated at the top of the water, like a swimming Nudi- branch, the body hanging downwards, with the tentacles widely expanded. It cannot be considered a common species ; but where it does occur, it is usually in some numbers. It is easily obtained when discovered, as it does not inhabit holes or crevices, but adheres to the smooth rock; it does not appear to indue its body with gravel, or any extraneous substances. Mr. Holdsworth found it not uncommon at Guernsey, with the unexpected habit of lodging under stones on the beach, at low water. At Dartmouth the same observer records its occurrence on the roots of Laminaria, as well as on the rocks. In my original notice of the species, I have mentioned the readiness and profusion with which the acontia or armed filaments are shot forth from the body on the slightest provocation. Subsequent observation has abun- dantly confirmed this irritable habit. The character and armature of the cnidce, are also there noted. The localities of the species are as yet but few, though they are widely scattered. Jersey, Guernsey, E. W. H. H. : Dartmouth, E. W.H.E.: Hfracombe, P. H. G. : Hilbre Island, E. L. W. nivea. SPHTRODETA. pallida. THE PALLID ANEMONE. Sagartia pallida. Plate III. Fifjs. 4, 6. Specific Character. Tentacles numerous, slender, white, each rising between two bowed blue lines. Actinia pallida. Holdsworth, Proc. Zool. Soc. I80G, pi. v. fig. 4. Sagartia pallida. Gosse, Annals N. H. Ser. 3. vol. i. p. 415. GENERAL DESCRIPTION. Form. Base. Adherent to rocks ; considerably wider than column ; outline undulate. Column. Smooth, without conspicuous suckers. Substance pulpy. Foiin cylindrical, pillar-like, about twice as high as wide, when extended, I but very flat when contracted. Margin a low parapet. Dish, Flat or slightly concave ; the margin entire. Tentacles. Numerous, arranged in four rows ; moderately long,' slender, and slightly tapering to the tips, their length regularly diminishing from -J the first row outwards. They are commonly carried sub-erect, the external rows arching outwards. Mouth. ? Acontia. Emitted from the mouth in some abundance, but not very Colour. Column. Pellucid whitish. White longitudinal lines are sometimes visible, but they are merely the edges of the septa, seen through the ] translucent skin, and not bands of surface-colour. Disk. Pellucid whitish. Tentacles. Pellucid whitish. The foot of each ten- tacle is embraced by two curved lines of dai'k blue, which ai^proach each other without meeting ; and pass off in front towards the centre of the disk, and behind towards the margin, in the form represented in the accompanying figure. The general effect is to tentacle of produce a bluish shade on that region of the disk pallida from which the tentacles spring. (^vieuxd vertically). \ THE PALLID ANEMONE. 79 Size. Diameter of column about one-tlxird of an inch ; height of column two- tiurda J expanse of flower nearly an inch. Locality. South-west coast of England ; rocks between tide-marks. Varieties. a. Cana. The colourless state above described. Plate iiL fig. 5. /5. Riifa. Column of a dull brownish-orange, paler or deeper in tint. Plate iii. fig. 4. I am indebted for mj knowledge of this little form to Mr. Holdsworth, who discoyered about a dozen specimens scattered about the rocks near the entrance to Dartmouth Harbom', " a part of our western coast, which, from its steep rugged character, and its liLxuiiant growth of sea- weeds, presents a fruitful hunting-ground for those in search of marine productions." They were obtained in Julj, 1855, and were described bj their discoverer, in a Memoir read before the Zoological Society of London in the following December, and subsequently published in their Proceedings. All of the individuals were of the variety cana, differing in no respect among themselves except in size. " They were found on the exposed surface of perpendicular rocks at about half-tide mark ; and when out of the water and contracted, were very difficult to dis- tinguish, owing to their great transparency." Some time afterwards the same gentleman obtained several specimens of a little Anemone which agreed with his former captives in every respect, save that their column was of a rufous hue ; the tentacles, however, having the same characteristic foot-marks as before. He concluded • Pi-oc. ZooL Soc. 1S53. that they were but varying phases of the same species ; and, as he kindly gave me an opportunity of forming a judgment by presenting me with a specimen of each colour, I concur with him in this opinion, and have accord- ingly so represented them. Some of my friend's observations on this minute species, — made in the course of a correspondence concerning its claim to be so considered, — will be read with interest. " Pallida is certainly not Candida [= sphyrodetd\. I have now seen, and know both well, and can readily point out the distinctions. Pallida may be easily taken for a young dianthus at first sight, having a smooth skin, with a rather erect body, and long pellucid filiform tentacles The basal rings on [? around] the arms of pallida are even narrower than in Candida, and have no direct communi- cation with the edge of the disk ; nor is there any appear- ance of a spot; their colour is almost black, but with a purplish tinge. The disk is quite transparent. The original specimens were almost colourless, but later captures were of a reddish buff, like some of dianthus ; and one of these, not more than half an inch in expanse, produced about a dozen young ones, about an eighth of an inch in height, — slender little things, with tentacles almost erect. They resembled their parent in form and colour, as far as could be seen in such minute creatures. There was no other Actinia besides the red pallida in the glass at the time, and the young ones adhered to the side of the glass vase, immediately surrounding the larger specimen, so that I had no doubt of their origin I have more than once suspected that pallida was merely the young of dianthus : but surely the latter would not breed when only half an inch high." I may add that the characteristic lines of blue, though minute, are a suflScient distinction of the species. THE PALLID ANEMONE. 81 In my limited opportunities of investigating this Ane- mone, I found it impatient of liglit, and sufficiently loco- motive. A specimen, adhering to the upper surface of a flat stone, I put into a tea-saucer ; it immediately crawled to the edge of its stone, glided round, and passed under, till it was quite out of sight : it thus traversed about thrice its own length in a quarter of an hour. I then turned up the stone, and the animal presently crawled off to the bottom of the saucer : closed all the time, except that the tips of its tentacles were protruding. Its manner of crawling was somewhat curious. It gradu- ally distended a portion of its body, which then was swollen, and quite pellucid, having a strange appearance, owing to the white china shining through the tissues of the distended portion. Then this part, being raised from the bottom so as to be loose, was pushed out and took a fresh hold, and the other half was rapidly pulled up to it, when the ante- rior half began again to distend instantly, and proceeded as before. The progress could be easily watched with a lens^ over the minute specks of the bottom. It was impos- sible to witness the methodical regularity of the process, and the fitness of the mode for attaining the end, without being assured of the existence of both consciousness and will in this low animal form. At night I found it had marched about three inches, or twenty-four times its own diameter, in six hours : but its progress, while I watched it, was much more rapid than this. The only recognised habitat for Sagartia pallida is — Dartmouth, E. W. H. H. sphyrodeta. PALLIDA. dianthus. a THE TRANSLUCENT ANEMONE. Sagartia pura. Plate III. Fkj. 6. Specific Character. Wholly pellucid-white, without markings. Actinia pellucida. Alder, Catalogue of Zooph. of Northumb. and Durh., 43. Sagartia pellucida, Gosse, Annals N. H. Ser. 3, vol. i. p. 415, pura. Aldkb, in litt. GENERAL DESCRIPTION. Form. Base, Adhei'ent to shells from deep water : somewhat exceeding the column. Column. Perfectly smooth, without visible suckers. Substance pulpy. Form cylindrical, a little higher than wide, when extended, but nearly flat when contracted. Dish, Slightly concave ; the margin entire. Tentacles, Thirty or upwards, arranged in about three rows ; the inner ones longest (about twice the diameter of the disk in length) ; diminish- ing regularly outwards, the outermost row being rather short. The inner ones are usually carried more or less erect, the outer arching downwards. Mouth, Set on a small cone. Colour. The animal is wholly without positive colour, except that the tentacles have sometimes a slight tendency to become sub-opaque at each extremity, wlien they assume a white appearance in these parts. Occasionally a few white lines occur on the column ; but these appear to be merely the edges ©f the septa, seen through the transparent integuments. Size. About a quarter of an inch in height, and one-sixth in diameter of column ; expanse nearly half an inch. Locality. The coast of Northumberland. On old shells from deep-water. THE TRAXSLUCEST ANEMONE. 83 This species I know only by the descriptions and figures of Mr. Joshua Alder, who has kindly put into my hands, not only the published '• Catalogue of the Zoophytes of yorthumberland and Durham," in which it first received . name and place among our Anemones, but additional notes in MS., and sereral original drawings. AU these I have used in my diagnosis and figure. The name ** pellucida,'' originally applied to this little animal, having been preoccupied, Mr. Alder proposes that it should be called "^jira." Little is known of its history. Its discoverer observes of it, — " It has occurred to me two or three times at Cullercoats, on old shells, — crusted shells of Fusus anti- quus from deep water, — nestling among the Serpulae and Barnacles with which they were covered- It is so incon- spicuous, when contracted, as to elude observation ; and it was not till the shells had been some time in sea-water, and the Actinia became expanded, that its presence was detected. A specimen kept in a vase was very restless, shifting its place continually, and often changing form." It seems to be somewhat rare. Mr. Alder has seen but three specimens. ^Ir. E. Howse has obtained it once or twice from the five-men boats, on the same coast. His specimens were slightly larger than iMr. Alder's. sphyrodeta. PUEA. pellucida. I g2 THE EYED ANEMONE. Sagartia coccmea. Plate V, fig. 4 : XII. Jig. 4 {magnified). Specific Character. Body rufous, with white lines ; tentacles pellucid, ringed with white, marked at the foot with a black bar, and two triangularj black spots below it. ^ Actinia coccinea. Mulleb, Zool. Dan. Prod.; 231, No. 2792. Zool, Dan. ii. 30 ; pi. Ixiii. figs. 1 — 3. Johnston, Brit, Zooph, 2d Ed. p. 215. Sagartia'coecinea. Gosse, Annals N. H. Ser. 3. vol. i. p. 416. GENERAL DESCRIPTION. Form. Base. Adherent to shells, in deep water : little exceeding the column. Outline irregularly cut and lobed. Column. Smooth, without visible suckers. Substance pulpy. Fomi cylindrical ; the height, when extended, twice the diameter ; the margin tentaculate. Disk. Flat; the margin entire. Outline circular, scarcely exceeding the diameter of column. Radii distinct, smooth. Tentacles. About sixty-four (in my largest specimen), arranged in three indistinct rows, of which the first and second contain each sixteen — the third, which is marginal, thirty -two. The inner rows are the largest, some of the outermost being minute points. Compared with the average of Anemones, they are short and thick, obtusely conical, and stand nearly erect. Mouth. Not raised on a cone. No distinct lip. Acontia, Protruded freely, both from column and mouth. COLOUB. Column. Light brownish orange, marked with many white or whitis longitudinal streaks from margin to base, more numerous below. Thea Btreaks are of varying width, but are in general equal or superior to thfl intermediate red spaces ; their edges are irregvilarly jagged. They < THE EYED ANEMONE. 85 not formed by the edges of the septa, nor always correspondent with them. Disk. Light red. Each radius bears two white lines, — one parallel and close to each edge, but separated from its neighbour by a fine line of the ground colour : this gives an appearance as if every radius were divided from its fellow by a pair of white lines. Among the tentacles the colour of the disk becomes a rich and brilliant orange, which colour extends in short lines between the tentacles over the edge of the margin. Tentacles. Pellucid, colourless, with four broad rings of opaque white, and a white tip : the rings are obsolete on the hinder face. At the foot of the front, a band of dark brown divides the two lower white rings, the lowest of which is succeeded by two triangular clouds of dark brown. Mouth. The radial lines end suddenly at the edge of the mouth, which is sharp and abrupt. The upper part of the throat is orange, but pre- sently becomes a deep red-brown. Size. The largest I have seen is half an inch in height, by abo\it one-third of an inch in diameter when expanded. TENTACLE (viewed endwise and fronivrise). LOCALTTT. The north-west coasts of Europe. Laminarian and coralline zones. I owe my acquaintance with this attractive little species to the kindness of !Mr. Charles W. Peach, who forwarded to me, in April of the present year, four or five living specimens attached to an old pecten-valve from deep water ofi" the Caithness coast. The same gentleman has since favoured me with sketches of manifestly the same species, which he made from the life, during his residence in Cornwall. It was first described hy Mliller, in 1777, and figured in his magnificent work on the animals of Den- mark. Dr. Johnston included it in his second edition of " British Zoophytes," on the authority of Edward Forbes, who found it on the coast of Ireland, " on rocks and sea-weeds;" but added no other information to the description of Miiller, which he quoted in the original Latin. An expression in this, which had puzzled me not a little, became graphically descriptive when I saw the living animal. Miiller says that the tentacles '' seem com- posed of an eye furnished with exceedingly slender rings crowded together," — a comparison which at first seems little applicable to such organs. But, in fact, they are frequently contracted into very low cones or warts ; when, viewed from above, they present the appearance of a number of fine rings surrounding the central point, very much like the eye-spots in a butterfly's wing. (See left- hand figure above.) The colony in my possession consists of one of the size and character that I have described above, and several minute ones around it, none of them so large as a small pea. Since I have had them, two or three more have been produced from the largest, from the size of a grain of sand to that of a poppy-seed. I believe all of these are the result of a spontaneous separation of fragments from the base, and not of a generative process. The most minute displays its circle of tiny tentacles. The outline of the base is exceedingly variable: it projects in ragged promontories and rounded points, which continually, though slowly, change their form and relative proportions. From some of these, minute fragments sepa- i'ate, which soon become independent animals. It is possible that the Actinia lacerata of Sir J. Dalyell may be this species; but I rather incline to identify it with our viduata. The sinuous outline on which he relied rather indicates a condition than a species. Though the short conical form of the tentacles is charac- teristic, yet occasionally they assume a lengthened slender shape, their markings becoming evanescent. Miiller THE EYED ANEMONE. 87 describes the animal as " changing place by the aid of its tentacles ;" I find it rather given to wandering, but not in this manner, which I have never seen an xA.ctinia use (his phrase "utt congeneres" notwithstanding), but by the extension and contraction of the base. Ireland, KF.: Caithness, C. TV.F.: Cornwall, C.W.P. miniata. venusta. COCCINEA. viduata. THE CAVE-DWELLING ANEMONE. Sagartia troglodytes. Plate I. fig. 3 : II. fig. 5 : III. figs. 1, 2 : Y.fig. 5. Specific Character. Tentacles barred transversely ; marked at their foot with a black character resembling the Roman letter B. Actinia viduata. Johnston, Mag. Nat. Hist. viii. 82. fig. 13. E. Forbes, Ann. Nat. Hist. iii. 48. CoucH, Com. Fauna; iii. 75 (nee Miiller). mesembryanthemum, var. fi. Johnston, Brit. Zooph. Ed. i. 211. troglodytes. Johnston (after Price), Brit. Zooph. Ed. 2. 216. fig. 47. Cocks, Rep. Cornw. Polyt. See. 1851. 6. pi. i. fig. 16. ? elegans. Daltell, Anim. of Scotl. 226 ; pi. xlvii. fig. 9. lexplorator. Ibid. Ibid. 227; pi. xlvi. fig. 11. Sagartia troglodytes. Gosse, Linn. Trans, xxi. 274 : Tenby, 365 ; Manual Mar. Zool, L 28 : Annals, N. H. Ser. 3. i. 416. attrora. Ibid. Ann. N. H. Ser. 2. xiv. 280 : Tenby, 356 (Frontispiece). Scolanthm sphceroides. Holdsworth, Proc. Zool, Soc. 1855. pi. v. figs. 1—3. GENERAL DESCRIPTION. Form. Base. Adherent to holes in rocks, frequently detached : somewhat exceeding the column. Column. Smooth towards the base, but beset on the upper two-thirds with suckers, which have a strong power of adhesion. Substance firmly fleshy. Form cylindrical and much lengthened, in full extension, the height many times exceeding the diameter. Margin tentaculate. Dish. Flat or slightly concave : the margin rarely undulate. Outline circular. Radii strongly marked, and crossed by close-set transverse striae. Tentacles. Numerous (amounting to two hundred or upwards in some specimens), arranged in four or five rows ; the first row largest, and decreasing gradually to the outermost ; in extension about as long as the width of the disk, conical, bluntly pointed. The manner in which they are carried varies in the dififerent varieties. Mouth. Generally elevated on a cone. THE CAVE-DWELLING ANEMONE. 89 Aeoniia. Long and very slender. Emitted reluctantly, and only on great irritation. Colour. Column. OUve, of a greener or browner tint in different specimens, marked with pale longitudinal stripes, widest and most conspicuous at the base, where the longer alternate with shorter ones, all generally vanishing towards the summit. The suckers for the most part pale. Disk. Varied with black, white, and grey, in a delicately pencilled pattern, that has justly been compared to the mottling of a snipe's feather. The pattern, which is pretty constant, is produced by the following elements :— each primary radius is greyish-white from the B-mark of the tentacle-foot, about half-way to the mouth ; then there is a patch of black inclosing a spot of white (often very bright), and then a narrow line of pale yellow or drab, edged with black, brings the radius to the lip. The secondary radii have the same pattern, but more attenuated. Tentacles. Pellucid grey, crossed by three (or four) broad rings of pellucid white, of which the lowest is undefined, and is frequently tinged with buff or orange. At the foot of each tentacle is a black mark con- TENTACLE OF S. TROGLODYTES (front). sisting of a thick transverse bar, succeeded by two curves, the whole bearing the form of the Roman capital letter B- This mark is very con- stant and characteristic ; sometimes, though the form is preserved, the outline is wholly filled up with black ; and sometimes, but very rarely, the whole is nearly or even quite obliterated. Mouth. Generally whitish. Size. Large specimens attain a diameter of an inch in the coltunn, and two inches in expanse of flower : the height is sometimes two inches and a half, but more commonly it does not exceed an inch.* * Mr. Holdsworth, in one of his letters, has drawn a pen-and-ink sketch of one which was protruding to a height of two inches from the sand at the bottom of his tank ; and states that, as the sand was full two inches thick, and that, to his belief, the troglodytes was attached,— it must have been four inches long. Locality. The coasts of England and Scotland. Hollows in rocks between tide- marks. Varieties. • With characteristic marks on disk and tentacles. a. Scolopacina. The condition above described. (Tenby : Torquay.) Plate II. fig. 5. j8. Hypoxantha. Disk and tentacles pinkish drab: the latter strongly ^«/«/^/i^.»7''barred, with the JJ indistinct ; each tentacle full orange. (F. H. West in ^ ^ litt.) '^ y. Badifrons. Disk ground-colour pale umber-bro%vn : tentacles wholly pellucid grey. (F. H. West in litt.) 5. Alhicornis Disk, ground-colour French-grey ; tentacles wholly opaque white. (F. H. W. in litt.) ** With characteristic marks on tentacles only. 6. Nigrifrons. Column greenish drab, duskier towards the summit. Disk uniform blackish-grey ; summits of mouth-angles orange-cream- colour. Tentacles pellucid, for the most part marked with an undefined long patch of opaque orange-cream-colour on the lowest third of the front; above this three i-emote spots of opaque white on the front face. The JB distinct when searched for, but nearly merged in the dark hue of the disk. (Morecambe Bay.) ^. Fulvicoitiis. Column drab, blackish at the summit. Disk dull umber; each radius with an undefined centre of black in the exterior half; the interior third wholly drab, separated by black lines. Lip narrow, orange. Tentacles short, remarkably blunt; numerous, in five rows ; uniform opaque pale orange ; the U strong, and distinct. Between the bases of the tentacles black radial lines are continued on a fawn ground, which becomes orange marginally, with a pretty effect. (Morecambe Bay.) ij. Pallidicornis. Column dull grey, blackish above, becoming dull rusty immediately at the summit. Disk dull sepia-brown ; the radii sepa- rated by slender black lines : primaiy radii with a central white spot broadly margined with black. Tentacles short, very blunt, set in five full rows ; opaque dull cream-white, the front with a line of faint orange, and a broad ill-defined stripe of blackish down each side; each tipped with a round dark spot. The 13 separated into its constituent halves, by a dividing line of whitish. (Morecambe Bay.) Plate I. fig. 3. 0. Aurora. Agrees with a in column and disk, and in the form and comparative fewness of the tentacles ; but the colour of these organs is brilliant orange, with the B rather ill-defined. (Tenby : Torquay ) Plate IIL/^s. 1,2. THE CAYE-DWELLING ANEMONE. 91 I. Subicunda. Agrees with o in disk and tentacles (nearly) ; but ground- colotir of tentacles rose-red : column dull buff. (Torqtiay.) K. LUacina. Column greyish-drab with faint longitudinal bands of darker. Disk buff, the radii separated by delicate black lines. Tentacles an exquisite light lilac,* with a white cloud at the lower part, succeeded by a strongly -defined black B- (Boulogne.) X. Melanoleuca. Column greenish drab. Disk whitish, becoming orange on the central region. Tentacles divided into well-defined alter- nate groups of semi-pellucid white and bluish black ; about five groups of each colour, but not quite regular in extent : those of each hue are con- spicuously ringed with a darker tint, and have the B thick and strongly marked. (Morecambe Bay ; Boulogne.) Plate V. Jig. 5. /t. Prasina. Disk and tentacles transparent crown-glass-green ; primary radii with a white spot, secondary with a white line. Lip white. (Tirth of Forth ? Dr. T. S. Wright in litt.) ♦ Without characteristic marks on disk or tentacles. (Column drab.) V. Flaricoma. Disk grey-buff, more positive on the lip ; tentacles warm orange-buff; remarkably short, blunt, and stifiBy set. (Boulogne.) {. Auricoma. Disk pale orange, with an undefined dash of white on 8ome of the radiL Tentacles long, slender, pellucid rich orange. (More- cambe Bay.) o. iMna. Disk warm orange, with the central fourth white. Tentacles elongated, opaque white, with an \inbroken line of pellucid white running down each side. (Boulogne. F. H. W. in litt.) X. yox. Disk and tentacles black : the latter much attenuated, with an unbroken line of grey running down each side. (Boulogne. F. H. W. in litt.) p. Eclipsis. Disk black. Tentacles opaque brilliant orange. (Morecambe Bay. F. H. W. in litt.) ff. Nycthamera, As p in every respect, except that the black of the disk ends abruptly at half-radius, the central portion being light grey. (More- cambe Bay. F. H. W. in litt.) T. Henpei'us. Wholly pure white ; gradually acquiring colour in a con- finement of some months. (Lundy. W. Brodrick in litt.) V. Nobilis. Disk deep violet-blue. Tentacles rich orange. (Cheshire From tlie above list it will be readily perceived that there is no species of our native Anemones that approaches * I describe it as I see it ; but Mr. West, to whose liberality I am indebted for this, as for so many specimens of this species, informs me that it Ls now in a deteriorated condition. Originally it was a very rich full lake or dark lilac. this in Protean variability. And yet there is, in general, no difficulty in determining the species ; the characteristic B is an excellent note of distinction wherever it is present ; and in those varieties in which it is obliterated in the evanescence of the markings, as in vars. fx, v, ^, o, or merged in the abnormal spread of the dark hue of the disk, as in vars. tt, p, cr, v, the true character of the specimen will be betrayed by the form and substance of the body, the drab colouring of the column, or the tendency of the tentacles to assume the orange hue.* It is one of our most generally distributed species, rang- ing apparently all round our coasts, from east to west, and from north to south. It is also tolerably abundant, at least in many of its localities, though less liable than some to be seen by casual observers, from its habits of retirement. Mr. Price well characterised it, when he proposed for it the name of troglodytes (" cave-dweller," from rpcoyKr), a cavern, and 8vv(o, to enter) ; for its favourite habit is to ensconce itself in holes and crevices of the solid rock, into which it retreats on alarm. In the shallow pools that floor the largest of the caves at St. Catherine's, Tenby, the vars. scolojaacina and aurora are abundant, especially the former, spreading their pretty blossom-faces at the bottom of the clear water. And yet it is not easy to discover them even when scores are thus exposed ; for the mottled colouring of the disk and tentacles is so like that of the sand and mud of the pools, that even a practised eye may overlook them without the closest searching. They often protrude the tentacles only, clustered perpendicularly, through the mud, and sometimes only the tips of these organs. Their concealment is aided by the fragments of sand, gravel, and broken shells, that * " In addition to these characteristics, I think the stout firm texture of the base a fair mark, as it is not so readily injured as in most species. Also the comparatively slight adhesion, at least when you can get fairly down to it : I think it generally yields to careful fingering." (F. H. W. in litt.) THE CAVE-DWELLING ANEMONE. 93 adhere to the suckers of the column ; these foreign bodies are often present in considerable quantity, and are pertinaciously retained for a long time, even in captivity. Its general resort is not very low; from ebb neap-tide downward may be considered its range: but the var. aurora affects a much higher level, habitually dwelling near high-water mark, but then it is invariably in some little hollow of the rock in -which the water stands. Several of the varieties have been found at Morecambe Bay, by my friend Mr. F. H. West. He describes the locality as "a low, flat, sandy shore, remarkably dreary and uninviting for the sea-coast, and without so much as a rock in sight. The tide goes out a considerable distance ; perhaps three-quarters of a mile, or even more, laying bare an almost unbroken expanse of what is rather mud than sand, very soft and tenacious. Towards the south side of the Bay is a spit of firmer ground where a few stones are nncovered, which can hardly be dignified with the name of boulders, since any of them may be turned over without assistance. Attached to these we find A. dianthus, both the pure white and orange varieties, mostly young. In the course of an hour we found numerous specimens of these, several varieties of troglodytes, some rather pretty pied sorts oi crassicornis, ?Lndi of course the covarxionmesemhryanthemum. Several kinds of EoUs, as coronata, papillosa, Drummondi, and pelluctda, are found here : — Sabella in abundance ; and Sertularice, various. There are no rock-pools ; but in the sandy hollows are Gobies, Blennies, Fifteen-spined Sticklebacks, and Pipefishes ; not to mention young Con- gers, that flop and flounder about when disturbed with most unpleasant energy. . . . All the troglodytes, including the orange-disked, present themselves through the sand, much elongated, — the point of attachment being sometimes three or four inches below the surface. They are all equally sensitive, shrinking on the slightest alarm." Mr. Holdsworth found the species under circumstances which deceived him into the belief that it was a per- manently free form, and he accordingly named it Scolanthus sphcero'ides* " The specimens were found near low- water mark, imbedded in the fine chalky mud which fills the crevices of the rocks at Seaford, their expanded disks being just level with the surface, but so nearly covered that only a faint star-like outline was visible ; on being touched they instantly disappeared ; and so great was tlieir power of inversion and contraction, that on digging carefully, they were generally found about one-and-a-half inch deep, and having that peculiar bead-like form Avhich has suggested the specific name of sjphceroides. There was usually a depth of six or seven inches of mud below them ; so that they could not have been fastened to the rock ; and since I have had them at home, now nearly five weeks, they have not shown the least inclination to attach themselves to the gravel, or glass sides of the tank in which they are living ; three of them have burrowed into some sand on which they were placed, but the others remain on the sur- face and are but rarely contracted. Soft mud is probably their natural habitat, being the most easily penetrated; and I could find no traces of any of these animals in a con- siderable tract of sand only a few yards from the locality whence these were obtained." My fi-iend was subsequently convinced that he had been misled by the appearance of the specimens : he examined them with me, and kindly gave me one of his original specimens, and we were both convinced that they were of this species. The apparent perforation at the rounded pos- terior extremity could have been nothing more than the * Proc. Zool. Soc. ; May, ] 855. i THE CAVE-DWELLING ANEMONE. 95 contraction and approximation of the column around the retracted base ; and we proved its power of basal adhesion in the specimen which came into my possession ; for it not only attached itself by the entire broad base to the saucer — and that repeatedly after having been removed — ^but during the niarht marched several inches to seek shelter under a O shell. AVhat had appeared to be an epidermis was nothing but a ring of exuviated mucus, which was readily removed, bringing away all the dirt, and leaving a clean smooth Sagartia. The tentacle-feet displayed the B-mark, and there seemed little to distinguish it from the normal colouring, except the dingy drab hue of the column. A specimen of the var. fulvicornis, in my possession, when disturbed, assumed a globular form, with the base contracted to one-sixth of an inch in diameter, and became very buoyant. It thus strongly reminded me of Mr. Holdsworth's sphcero'ides. It seems the habit of the species to be very free ; and this tendency more especially marks the mud-loving kinds with a pale drab exterior. It is a common thing for one of these to lie for weeks in a tank rolling loosely about the bottom, alternately contracting and stretching its column, and folding or expanding its tentacles at pleasure, apparently quite healthy, and yet showing no inclination to choose a settled residence. I have had many examples with this habit, which, by and by, having sown their wild oats, suddenly fix themselves, give up their vagrant ways, and become sober housekeepers. Mr. Holdsworth writes me of one which, after six months' captivity, " has not yet attached itself, but wanders about, like a restless spirit without a home." The suckers are in this species very adhesive ; and in this vagabond condition it is not rare for the Anemone to moor itself temporarily, not by the base, but by these organs ; sometimes by a few of the most anterior ones, when the "base is thrown up at an angle, in a somewhat undignified fashion. Occasionally I have seen a specimen which had attached itself thus to a stone, or the side of a vessel, and had, by its own weight or other cause, removed a little from its attachment, — still fastened by two or three suckers, which were unnaturally stretched out to a length of the sixth of an inch, and a proportionate tenuity, resem- bling the suckers of a Holothuria, Some observed facts indicate a considerable tenacity of life in this species. On the 5th of October last Iilr. West inclosed in a small tin canister three specimens with a little damp weed, but without water. The box was then addressed to me, and committed on the same day to the post-ofiice at Leeds ,• where, however, owing to the oozing forth of a slight wetness, it was detained. In the course of a few days I informed him that it had not arrived ; but my friend residing out of the town, and my letter arriving on Saturday evening, he was not able to obtain from the over-scrupulous postmaster the suspicious missive, until Monday morning, the 12th — a week (within five hours) of the animals' imprisonment. Of course he expected to find them in a pretty advanced state of decomposition ; but, on removing the lid, saw at once that the case was not hopeless. They were immediately treated to the long- foregone luxury of a bath of sea-water ; and though one of them was hors de combat, the other two recovered, and lived to bear the journey to Devonshire under better auspices. To the same kind friend I owe the possession of the lovely var. lilacina, and the following playful note of its endurings : — " It is one of the French consignment, and has led almost a charmed life. Soon after my letter to you [dated Jan. 27], written after their arrival, I fancied the water in one of the vases was becoming foul, and therefore THE CAVE-DWELLINQ ANEMONE. OT removed all the animals save one — the most valuable, — which could not be found, and which I concluded was the source of the mischief. The vase stood, however, in an empty room tiU, last Tuesday [April 20], — so you may guess the strength of the pickle, — when I emptied out the whole kettle of fish, and found Monsieur at the bottom. He is only the shadow of himself, and looks uncommonly seedy ; but is a character, nevertheless." the first time, of seeing the discharge of true ova from an Anemone. In a saucer, containing a Corynactis and some varieties of troglodytes, that was standing on my library table, I found, on the morning of the 28th of April, that there had been deposited during the night an even layer of pale brown substance on the bottom, so placed as to make it uncertain whether it had proceeded from the Coi'ynactis or fi-om one of the troglodytes. The mass was about as large as a fourpenny-piece. A little taken up with a pipette, and examined under a power of 500 diam., proved to be composed of ova, opaque, perfectly globular, varying from .0043 to .0051 inch (but the former was an unusually small one) : they were mostly very uniform in size, viz. .0050 inch. They had a clear weU-defined edge, and not the slightest appearance of cilia. I removed the troglodytes to a clean part of the saucer (it was the beautiftd orange var. auricoma), and after a few hours perceived that it was discharging more ova, which were streaming over its lower tentacles, as it lay on its side, but fully expanded. I therefore immediately transferred it to a straight-sided glass box for closer examination. As soon as it had expanded again after the shock of removal, which it did in a few minutes, I began to watch it. It was lying on its side, with its disk and expanded tentacles near the glass side, and facing my eye. Many of H the tentacles, especially those which were on the in- ferior side, were occupied with more or fewer ova, some having fifty or more, others half-a-dozen, others one or two. In each case they were rolling up the interior of the ten- tacle from the general cavity, and coursing to and fro under the influence of the lining cilia, sometimes accumulating temporarily at the tip, but never, so far as I saw, discharged there. On looking at the mouth, I perceived that the gonidial tubercles of one angle were brought into contact with those of the opposite angle, dividing the mouth into three tem- porary orifices, two lateral and one central. The lateral orifices, however, were at right angles to the ordinary line of extension. Through each of these lateral orifices ova were issuing, somewhat slowly, with an even motion evi- dently ciliary, for the most part not in contact with the sides of the tube, but coming up through its dark centre. As each came into view, and deliberately rolled over the edge of the orifice, it streamed across the disk, and over the face of the expanded tentacles, carried clear of all by means of the ciliary currents of these parts. The ova closely fol- lowed each other, generally in single file ; but occasionally two, or even three, were slightly agglutinated together. Perhaps on an average about three or four in a minute issued, but with many lengthened interruptions of the continuity. The process of egg-discharge did not continue long after I began to watch it ; though the accumulations remained in the tentacles. The next morning, those that had been deposited were for the most part disintegrated, resolving into an undefined mass of minute cells. A few only here and there retained their outline. During the next day or two, especially in the night, a few more were discharged, which were a little larger than the former, averaging .0060 THE CAVE-DWELLING ANEMONE. 99 inch. No result, however, followed the discharge, and they soon decomposed. Dr. Byerly, however, has succeeded in rearing the young of this species ; but from ciliated germs, not from ova. Some specimens which he found numerous on the Leasowe shore of the Mersey, threw off many germs, which could be plainly seen through the skin at the base. These made their exit through " breaches of continuity in the outer envelope near its junction with the basal disk, and some- times through ragged apertures in the base itself." The globular, and had a very sluggish motion. Three or four were put into a wide-mouthed bottle and stopped : after two months, one had developed a perfect Actinia, the ten- tacles being fully expanded. At the time of the record it had lived six months ; but having never been fed, it had not visibly grown.* Since the former observations were made, I have proved this species (contrary to what has been asserted of the Actinoids) to be hermaphrodite. The variety in this case was the exquisite one I have named melanoleuca (see PI. from Morecambe. On the 26th of May, this individual, on being put into fresh sea-water, instantly made it turbid. I took it out in the course of the day, and isolated it in a small glass tank of clear water. Presently this also became quite turbid, as if milk had been mixed with it, while clouds of the white fluid were seen floating about the animal. On the vessel being shaken, and again on my touching the Anemone, it contracted ; and, on each occasion, a stream of white fluid, almost as opaque as milk, shot up from the mouth, and slowly diffused itself in the surrounding water. * Edin. New Phil. Joum.; Jan. 1855. H 2 With a pipette I took up a drop from one of the diffusing clouds, and submitted it to the microscope. It was filled with millions of excessively minute, but vigorously motile atoms, clear and colourless, having an ovate body, and a slender tail, which wriggled their little tails, and rapidly oscillated from side to side, from the tail-tip as a 'point d'appui. This was the first time I had ever seen the sper- matozoa (for such they assuredly were) of the Anemones. The next morning, the water still continuing turbid, I was about to pour it away, when I saw beneath the spot where the Anemone had lain, a thick layer of cream- coloured soft substance, well-defined in its outline. I took up a little of this and examined it. It proved to be a mass of ova. They agreed with those above described, being mostly quite globular (though a few were distorted) ; the majority closely alike in size, viz. .0058 inch ; but a few were manifestly smaller, and measured from .0046 to .0048 inch. They were perfectly defined, with a distinct clear wall, and olive granular contents. When crushed with a graduated pressure to rupture, the whole contents of each ovum were seen to consist of a vitelline mass of minute oil-particles in an albuminous fluid, inclosed in a very thin vitelline membrane. In a few instances I detected the germinal vesicle with its germinal spot, some- times by its clearness when the ovum was flattened, some- times by its escape as a clear bladder from the ruptured membrane : but in many examples I could not find it at all. I removed the Anemone from the vase, leaving the ova alone, in hope that they would develop, but they all decomposed. I may add, that since then I have seen the like discharge of spermatozoa from a specimen of vidnata. I refer with hesitation the Actinia elegans and A. ex- plorator of Sir John Dalyell to this species. The former THE CAVE-DWELLING ANEMONE. 101 he describes as of a reddish-brown or orange hue, with white (snctorial) spots, and well-barred tentacles ; the disk generally crossed with a white line. The latter has more of the ordinary aspect of a troglodytes. Sir John Dalyell observed in the latter (which he named explorator from the circumstance) the occasional elonga- tion of one or two tentacles, which we have seen to be a not uncommon phenomenon in this family. A specimen, not half an inch in diameter, exhibited two tentacles together, each of the length of an inch and three quarters. In general, the elongation took place at night. From its ordinary length of half an inch, each tentacle gradually became two inches long, thickened and distended to transparency. "It is then seen rising from among the rest, curving over to the opposite side of the disk, and as if searching around." After a while, it shrank back to its former state. Both of these (supposed) species were prolific. The latter produced sixty young in one night ; which were pure white, and large in proportion. Of the former, three indi- viduals, in October, produced infusorium-like germs, which were ovoid, and yellow-green in hue : some showed a long transparent horn in front, visible as the animalcule pur- sued a steady course ; behind it was open like a cap. They presented much disparity both in form and size. They swam actively by means of cilia. These germs continued visible throughout October, but, though carefully preserved, they led to no ultimate results.* been favoured with an interesting letter from Miss Gloag, of Queensferry, Fifeshire, who has long been a successful cultivator of Anemones. I regret that limited space forbids my giving her communication in extenso : I am compelled * Rem. Anim. of ScotL ; 226, 227. to select and abridge. This lady finds troglodytes abund- ant on the Fife coast, in several varieties. Of these she specially enumerates lilacina, of which eight specimens have from time to time occurred ; Hesperus, two specimens, and a third well-marked variety. One of the var. Hesperus has been in Miss Gloag's possession fifteen months : " the disk and tentacles are, if possible, whiter than snow ; only at the extreme tip of each tentacle is it quite black. It is a little gem of beauty." This variety frequently elongates two of its tentacles to the length of an inch ; when they lose their opaque white colour, and become transparent, the tip, however, retaining its black hue. The new variety is very showy : it has a bright orange disk, and perfectly black tentacles : thus reversing the colours of EcUpsis. It may be added to the catalogue, as var. <^. Pyromela. of Miss Gloag's experience in collecting. " I find no diffi- culty in digging the troglodytes out of the rocks or mud. The instruments I use are long, thick hair-pins [of iron- wire, 1^6 th of an inch thick]. I am obliged to have them made for the purpose ; but they are splendid, and seldom fail to bring out the treasure unhurt. After getting my fingers nearly skinned, I bethought me of hair-pins. When I see a troglodytes that I wish to possess, I take one of these strong pins in each hand, and as quickly as I can I put the bent ends down the fissure as -close as I dare to the creature : when I think I have reached its base, I work them gently but firmly towards each other, till I feel I have detached the Anemone, when it is easily lifted out either with the fingers or with the pins." More recently still, Mr. D. Robertson has sent me from Cumbrae an exquisite variety, of which I was at first inclined to make a distinct species. It has the charac- THE CAVE-DWELLING ANEMONE. 103 teristic marks of troglodytes, however, on disk and tentacles. Column marked with longitudinal green bands on a pellucid olive ground. Tentacles very short and conical, pellucid, with three transverse white bars, and three longitudinal streaks of fine grass-green, reaching from the middle to the tip ; one frontal, broad, the others lateral, narrower. Disk pellucid olive, with a white lip. This variety I enumerate as y^. Pra^inopicta. All the varieties of this species are hardy in confinement, and accommodate themselves readily to almost any kind of bottom. Many observations (some of which have been already mentioned) concur in showing its tenacity of life under circumstances, ^uch as long imprisonment in a box, foul water, &c., that would prove fatal to other species. It requires attention, however, in the aquarium, to preserve it in condition. The more beautiftd varieties, at least, speedily degenerate both in size and colour, if they be not frequently and regularly fed. They possess a healthy appetite, and will greedily devour fragments of raw fish or flesh, or of univalve or bivalve mollusca. Perhaps the best food for all Anemones, and one that can generally be com- manded, is the uncooked flesh of the oyster or the mussel. It should be cut into small pieces, and guided gently to the disk or tentacles of the Anemone, when fnlly expanded. If the animal shrink from the food, and contract ; or if it be allowed to lie on the disk ungrasped, it will be of little use to allow it to remain: remove the fragment, and wait a hungrier moment. If the food be gradually sucked in, its remains will be disgorged in the course of a period varying from a few hours to several days. Often it will appear little changed ; but it has performed its part, and must be carefully removed, or its decomposition will be likely to spoil the water, and kill, or at least render sickly, the living tenants. The frag- merits may be removed by means of a bent spoon at tlie end of a stick, by boxwood pliers sold for the purpose, or by a glass tube closed at one end by the finger. The following somewhat extensive list includes all the British localities of this species that have come to my knowledge : — Wick, C. W. P. : Moray Frith, A. Robertson: Coast of Fife, ifws (/. C.) Gloag: Frith of Forth, T. 8. W. : Berwick Bay, G. J. : Cullercoats, R. Howse : Guern- sey, E. W. H. H. : Dover, J. B. Mummery : Hastings, a K.; E. a Holwell: Seaford, E. W. H. H. : Selsey, G. G. : Weymouth, W. Thompson : Teignmouth, B. a J.: Torquay, P. H. G.: FalmoUth, W. P. C: Ilfra- combe, G. T. : Tenby, P. H. G. : St. Bride's Bay, H. Owen : Menai Strait, W. A. L. : Mersey Estuary, Hilbre Island, E. L. W. : Birkenhead, /. Price : Morecambe Bay, F. H. W.: Man, E. Forbes; F. H. W.: Frith of Clyde, A. B. a : Cumbrae, B. B. : Belfast, E. P. W. coccinea. TROGLODYTES. yiduata; THE SNAKE-LOCKED ANEMONE. Sagartia viduata. Plate III. Jig. 3 ; VI. fig. 11. Specific Character. Tentacles -very extensile, Tery fleruous, indistinctly barred ; marked with an uninterrupted dark line down each side. Actinia viduata. Muller, Zool. Dan. Prod. 231. No. 2799. Zool. Dan. ii. 31 ; pi. Ixiii. figs. 6 — 8. ? undaiii. Ibid. Zool. Dan. ii. 30 ; pi. LxiiL figs. 4, 5. anguicoma. Price in Johnst. Brit. Zooph. 2nd Ed. p. 218 ; fig. 48. GossE, Devon. Coast, 96 ; pi. i. figs. 9, 10. ? laeerata. Daltell, Rem. Anim. Scotl. 228 ; pL ilviL figa. 12—17. Isacmaea viduata. Ehrexberg, Corall. 34. Sagartia viduata. Gosse, Linn. Trans. xxL 274 ; Tenby 363 ; Man. Mar. Zool. i. 28 ; Ann. X. H. Ser. 3. L 416. GENERAL DESCRIPTION. Form. the column. Column. Smooth, slightly corrugated in contraction ; with distinct suckers on the upper half. Substance fleshy. Form cylindrical; capable of great elongation, in the shape of a tall and slender pillar. Margin tentaculate. Disk. Flat; the margin plane. Outline circular. Radii distinct; crossed by fine striae. Tentacles. About two htindred, arranged in five rows ; of which the first and second contain each twelve, the third twenty-four, the fourth forty- eight, the fifth ninety-six. Those of the first row are longest ; but there is not so much difference between the rows in this respect as is the case with the preceding species : those of the first row, when fully extended, are longer than the width of the disk ; all are slender, tapered to a fine point, and very flexuous. They are usually carried either arching downwards on every side or sub-erect, and thrown into many irregular snaky curves. Mouth. Set on a low cone. Lip thin ; slightly furrowed. Acontia. Emitted from variovis parts of the body, from the base to the summit, occasionally ; but very reluctantly, and in small quantity : short and slender. 106 Colour. Colwnm. Ground tint a ligtt buff, sometimes merging into a warm fawn, or wood-brown, at others into a flesh-bue, or even pale scarlet. This is marked with longitudinal bands of paler hue, sometimes almost white ; the bands being equal to the interspaces. As these bands approach the base they become more defined, and the contrast between the alternate dark and light hues is beautifully distinct, especially as they are separated by slender jagged lines of very dark brown. The whole upper parts are freckled with numerous brown dots; and the suckers are generally inclosed each in a little olive blotch. Bisk. Ground tint a dull whitish-grey, covered with a regular speckled pattern, formed of the following elements. At the point where each tentacle springs from the disk, the radius is marked by a long dash of deep brown, or blackish, at each edge; the intervening space between the dashes is occupied by a transverse band of pellucid greyish-brown ; two other similar bands cross the radius at equal distances, but without the bounding dashes. As the markings of the secondary radii do not coincide in posi- tion with those of the primary, the result is the minutely chequered or dotted pattern above spoken of. Go- Tentacles. Translucent grey, marked on each side with a line of dark brown running through the whole length. Occasionally a very faint ring of pel- lucid white surrounds the tentacle near its middle, and a second just above its foot : the lateral lines are lightened at these places, but their continuity is not interrupted. They end abruptly just above the junction with the disk. n-T.-xrrr»r.r,;. Moutk. Grcylsh white ; with darker (right side). furrows. Size. Average specimens in the button state are about five-eighths of an inch in height, and the same in width of column ; the base covering an area of nearly an inch in diameter. Such a specimen in ordinary expansion would spread an inch and a half from tip to tip of the tentacles. But specimens an inch and a quarter in height and width in the button are not rarely met with. Locality. It is widely scattered over the European coasts. Where found it is generally common, adhering to rocks and loose stones, between tide-marks; PLATJE in PH .C DCL C0LOU1S Br W DCmS- 1.2. SAGARTIA TROGLODYTES. 4- . 5 . S PALLIDA 3 . S . VIDUATA . 6 . S PURA . THE SNAKE -LOCKED ANEMONE. 107 and is especially abundant on a sandy bottom in the laminarian zone, where it appears to be nearly or quite free, since it is washed ashore by hundreds after a gale. Vambtt. The only distinctly marked variety that I have noticed besides those diversities of the genei-al tint that I include in a. Aleurops,* — the mealy-faced condition above described, — is 0. MeIanops;\ which has a broad well-defined band of deep black, crossing the disk and tentacles ; just as if a dash of ink had been struck across the whole flower ; including in its breadth three or four tentacles of each row on each side. The band crosses at right-angles to the line of the mouth ; the gonidial radii of which are white. Sagartia viduata is somewhat liable to be confounded with troglodytes ; and some varieties of the latter approach it very nearly, especially when closed. But an experienced eye will seldom be deceived ; the tint of viduata is a warmer brown, generally mealy, or speckled ; that of troglodytes tends to drab, smoky brown, or olive, and is not speckled : the stripes of troglodytes, when present, are closer, generally narrower, and rarely extend far from the base ; the suckers, too, which are so obvious and so constantly used in troglo- dytes, are inconspicuous in viduata, and rarely used for attachment. Then, when expanded, the peculiar pattern of each disk respectively does not merge into the other, though in troglodytes it is apt to become evanescent : the tentacles in this latter very rarely show obscure lateral lines; in viduxita these marks are constant and conspicuous: the more slender form of these organs, and their tendency to assume irregular curves, in viduata, are also a very good distinction. I have no hesitation in identifying the species which we get so abundantly in Torbay, and which I have described above, with Mr. Price's anguicoma ; though that gentleman has not noticed the characteristic tentacle-lines. Its re- * "AXtvpov, meal; cK^r, the fece. f MeAos, black; <Si\|», the face. raarkable power of elongation in the dark, alluded to by liim, I have often noticed. The finest specimen I have ever seen used to stretch up at night in the form of a perpendicular column, five inches in height, with a thickness of about two-thirds of an inch ; from the summit of which the numerous slender tentacles, arching outward on all sides, and extended to extreme tenuity and translucency, gave to the whole animal somewhat of the appearance of an elegant palm-tree. This form I have endeavoured to imitate in Plate III. fig. 3 ; though the engraver has not succeeded in tentacles, which look like a thin light blue cloud when seen against a dark background. The more ordinary appearance I have given in Plate VI. fig. 11. But as little doubt exists in my mind tliat the species is the viduata of the " Zoologia Danica." I have before me at this moment specimens, which answer almost precisely to Miiller's description, even in such minute characters as the number of the white bands (twenty-six in mine^ " viginti-quatuor" in his) ; the dark brown speck, with a white dot in its centre — " puncto pertuso " — at the summit of each main band; the slender evanescent line between the bands — " inter has strigas alia tenuior et pallidior ;" the longitudinal dark lines of the tentacles — " lineola, duplici longitudinali obscur^ ;" and even the minute depression in the middle of each tentacle at its foot — " foveola versus basin:" all these points I trace readily; and while they do honour to the precision of the great Danish zoologist, they abundantly prove the identity of our species with his. Whether his undata is not a variety of the same, I am not sure. The Actinia lacerata of Dalyell I also incline to identify with the present, — from what he says of the colour, the length, form, and contour of the tentacles, the card-like. THE SNAKE-LOCKED ANEMONE. 109 a"bject flatness of the body in contraction, and the elonga- tion at night.* The name viduata (" widowed ") probably alluded to the white and black lines, which seem to have been remarkably contrasted in Miiller's specimen. Mr. Price's name — anguicoma (" snake-locked ") — is far more suggestive and significant ; and I regret that the law of priority forbids ]^Ir. Holdsworth has found some curious anomalies in the tentacles of a specimen in his possession. He first observed that all these organs assumed a nodulous appear- ance, being abruptly thickened into knobs at regular inter- vals in their length. The phenomenon disappeared and recurred several times, sometimes lasting two or three days. record of this fact, he wrote me again as follows : — " The viduata that had the knobbed arms has taken a new freak, and not being content with a normal number of tentacles, must needs throw out branches from some of them. I inclose a sketch of the most conspicuous." From the drawing it appeared, that while some of these organs were but slightly notched at the tip, others were divided nearly half-way down, the branches diverging in various degrees ; while one bifurcate tentacle had one of its branches cleft. A similar phenomenon has occurred to my own observation in Aiptasia, and in Anthea. It is by no means common for either viduata or troglodytes to emit the filaments, which I call acontia, from the loop- holes of the column ; but I have witnessed the fact on several occasions. From the mouth they are protruded much more readily. In both species they are crowded with long oval cnidce about .002 inch in length, and * Rem. Aiiim. of Scotland, p. 228. under ; which discharge an ecthorceum about one and a half times the length of the cnida, and densely hearded. Of the increase of this species I have no information, unless the lacerata of Sir J. Dalyell be truly identical with it. He observed that this increases by spontaneous sepa- rations of portions of its base. The outline becomes irregu- larly sinuous, and the prominences gradually (in the course of a week or two) become pinched off, maintaining their connexion only by a very slender lengthened filament, not in contact with the glass, hut free above it. Rupture of the connecting thread at length takes place, and the independent fragment develops itself into a young Anemone. The laceration of the outline of the parent was always very irregular and ragged. Above seventy were thus produced in a year from a single adult.* Sir John Daly ell could never detect any embryo or germ inclosed in the portion of margin about to be separated : and the careful experiments of Dr. T. S. Wright appear conclusively to negative that hypothesis which would thus explain the mode of increase by fission of the base. From an attached individual of Actinoloha diantJius, Dr. Wright cut a minute piece of the base, having first ascertained, by careful examination of the part, which was perfectly trans- parent, that no ovum or germ existed there. The part immediately receded from the parent, and in three weeks had become a perfect Anemone, with long tentacles. From this small one he cut two other minute slips, which also assumed the perfect condition ; and from the base of the original adult fourteen other slips yielded the same results. From these experiments it appears that all that is essential to the process is the existence of a portion of each of the three elementary tissues of the animal — the tegumentary, * Op. cit, p. 228. THE SNAKE-LOCKED ANEMONE. Ill the muscular, and the epithelial or ciliated lining-membrane of the cavity.* S. viduata is hardy in an aquarium, and needs no special care or peculiar treatment. It expands principally during the hours of darkness ; a shaded angle suits it best. The following are the British localities in which it has been recognised : — Felixstowe, Miss M. E. GhiiHe: Dover (rare), E. L. W.: Guernsey, E. W. H. H. : Bournemouth, Rev. J. GuUle- inard: Torquay (abundant), P. H. G.: Dartmouth, E. W. H.H.: Falmouth, W. P. C: Ilftacombe, P. If. G.: Tenby, P. H. G.: Menai Strait, J. P.; (abundant) W. A. L.: Puffin Island, E. L. W. : Mouth of the Dee, F. H. W. : Dublin Bay, J. R. Greene ; E. P. W. : Belfast Lough, W. T.: Lahinch (Co. Clare), E. F. troglodytes, bellis. YIDUATA. [impatiens.] coccinea. parasitica. A. amacha. A. cereus. • Edin. PhiL Journal, for 1856. THE PARASITIC ANEMONE. Sagartia parasitica. Plate II. fig. 6. Specific Character. Large, pillar-like ; skin coriaceous ; tentacles in seven rows, marked with a many-broken line down each side. Actinia effceta. Rapp, Polyp. 64; pi. ii. fig. 2 (An Linnaei?). parasitica. Couch, Zooph. Cornw. 34 ; Corn. Fauna, iii. 80 ; pi. XV. figs. 1, 2. JoHNST. Brit. Zooph. Ed. 2. 228, pi. xli. Cocks, Rep. Cornw. Pol. See. 1851, 8, pi. ii. fig. 11. GossE, Aquarium, 144, pi. iv. TUQWELL, Manual, pi. vi. Sagartia parasitica. Gosse, Tr. Linn. Soc. xxi. 274 ;' Ann. N. H. Ser. 3. i. 416. GENERAL DESCRIPTION. Form, Base. Adherent, generally to shells. Little exceeding the column. Column. Minutely corrugated on the upper parts, but studded on the lower half with numerous warts, mostly small, but a few among the rest large and prominent. No apparent suckers. Substance firm, somewhat coriaceous. Form, that of a thick pillar; the height twice or thrice as great as the diameter ; plump and rounded. Margin forming a slightly thickened rim, minutely notched, scarcely rising above the level of the disk, and obliterated when the disk is fully expanded. Disk. Nearly flat, or slightly concave ; the margin somewhat mem- branous, wider than the column, which it overarches; occasionally it is thrown into puckered undulations, but only to a small extent. Radii not prominent. Tentacles. Five hundred or upwards ; arranged in about seven rows, of which the first contains about twenty, the second twenty-four, the third forty-eight, the fourth ninety-six ; those of the other rows are too numerous and too closely set to be enumerated. The first row springs from the disk at about half-radius, — that is, midway between the lips and the margin they occasionally stand erect, but more frequently arch outwards in elegant overhanging curves. When distended, those of the first row are often an inch in length, and one-eighth of an inch in thickness : the others diminish in regiilar gradation, until those of the margin do not exceed a line in length. Their form varies in different individuals, and perhaps at THE PAEASITIC ANEMONE. 113 different timea, sometimes being blunt and nearly cylindrical, at others tapering to a fine point. Mouth. The centre of the disk gradually swells into a stout low cone, in the centre of which is the mouth, edged with a thick furrowed lip. Acontia. White, long, and as thick as sewing-cotton ; projected on the slightest irritation, and in the most copious profusion, both from the mouth and from the loop-holes of the column. Colour. Column. Ground-colour, a dirty white or drab ; often slightly tinged with pale yellow : longitudinal bands of dark wood-brown, reddish- or purplish-brown, run down the body, sometimes very regularly, and set so closely as to leave the intermediate bands of ground-colour much narrower than themselves : at other times these bands are narrower, more separated, or broken into chains of dark spots. Immediately around the base the bands usually sub-divide, and are varied by a single series of upright oblong spots of rich yellow, which are commonly margined with a deeper brown than that of the bands. The whole column is surrounded by close-set faint transverse lines of pale hue, sometimes scarcely distinguishable, except near the summit, where they cut the bands in such a manner as to form, with other similar lines which there run lengthwise, a reticulated pattern. Disk. Pellucid yellowish-white, often tinged with faint of six squarish patches of opaque white. Tentacles. Pellucid, faintly tinged with flesh-colour, cream- yellow, or purplish; each marked with a dark piu-plish or brown line down each side, which is broken into about five — — \| dashes. The sub-marginal rows, which from their minuteness may be compared to a fringe, are frequently divided into alter- nate patches of colour ; — a patch of pale tentacles, then one of purplish, — ^six groups of each colour completing the circle. These alternations do not conceal the lateral lines of the ten- tacles ; and though sometimes beautifully distinct, they are at tentacle others scarcely perceptible. The pale patches correspond to {front]. the square spots of white on the disk. Mouth. Opaque white, or cream -white. Size. It frequently attains a height of four inches, with a diameter of two and a half in column, and three and a half in flower. LOCALITT. The shores of the British Channel, the Mediterranean and Red Seas ; in the coralline zone. For the most part adhering to such shells as are inhabited by the Soldier-crab. I Varieties. Though subject to considerable diversity in colouring, I am scarcely able to select any pattern sufRciently distinct, or sufficiently stable to warrant its registration as a named variety. I have above defined the limits within which, so far as my experience goes, the divergence extends ; it seems mainly to consist in the relative proportions and arrangements of the dark and light bands of the columns. One mentioned to me by Dr. Hilton, of Guernsey, as having been found by him at Herm, seems more worthy than any other of being considered as a distinct variety. It "had a very light coloured body, and was beautifully marked with lilac spots." Perhaps I may venture to call it Amethystina. I have seen a specimen at Torquay, in which the stripes of the column were dark crimson. The keen eye and scientific zeal of old Ellis failed to discover this species, notwithstanding its large size and commanding appearance. Common as it is in some locali- ties, it seems, however, to be quite unknown along the eastern coasts of great Britain and Ireland, whence Ellis's zoophytic treasures were principally gathered. It was left for Mr. K. Q. Couch, of Penzance, to indicate it as a British and Mediterranean seas.* I have found it exceedingly abundant in Weymouth Bay, — extending from the deep water of the offing even into the narrow harbour, — but have never heard of its being found within tide-marks, except in the instance of the var. amethystina, above mentioned, which was found attached to a stone at low-water mark. It is, as its name imports, normally parasitic in its habits ; though not so strictly but that we frequently dredge specimens adhering to stones ; and in captivity it is by no means uncommon for an individual to detach itself from its native site and adhere to the bottom of the vessel, or even to crawl up the perpendicular side. Generally, however, it is found seated » With Dr. Johnston I utterly and indignantly reject Linnasus's specific names in the Actmoida, and with reluctance even cite them. THE PARASITIC AXE3I0NE. 115 on some univalve shell, which is tenanted hy a Soldier-crab : young specimens on Turritella terebra^ Trochus magus, T. ztztphi'nus, &c. ; but adults, which are much more frequently met with than the young, almost invariably on the great Whelk {Buccinum undatum). The dredge, indeed, often brings up shells invested by this Anemone, which are empty ; but I believe that in every such case the shell has recently been vacated by the Soldier, and that the Sagartia never voluntarily selects either an empty shell, or one tenanted by the living Mollusk, for his residence. My friend. Dr. E. Percival Wright of Dublin, has favoured me with a humorous sketch of the ways of this* loving pair, — Crab and Zoophyte, Arcades ambo, — which bears on the matter before us. " The following scene," he observes, " was witnessed by my much lamented friend Dr. R. Ball. One of the specimens referred to, attached to the shell of a Buccinum undatum, which had from its appearance been, in all probability, just deserted by a Pac/urus, was placed in a glass aquarium : in a short time the Anemone left the Buccinum, and attached itself to the side of the tank; it next deserted this position and fixed itself on the side of a large stone that filled the centre part of the aquarium. After the lapse of some weeks, a Hermit Crab was dropped into the tank (I think Pag. hemhardus), WeU, if these Hermits can't live without hiding themselves in the deserted shell of some poor Mol- lusk, I think it is equally true that they can't live happy until they hide both themselves and their shells in some quiet little hole in the rock -work of our aquaria, from whence they can look out ; and, thinking that the supers imposed stone-work adds vastly to the strength of their fortifications, experience sundry intense feelings of safety. Be this as it may, the Hermit in question was not long ere he walked up to a little grotto that was in the rock-work l2 of the aquarium (quite close to the Sag. parasitica) ; and after a slight survey to see that all was right, he turned his left shoulder forward and ' backed in : ' then he began to whisk his antennge and foot-jaws in a dreadful manner, and looked evidently quite content. I suppose this was a state of things the parasite perched on the rock above had long been waiting for ; for it was not long in moving its disk over the top of the small whelk ; and before the Crab knew where he was, the big Sagartia had pitched his tent on the roof of the Hermit's house. Where the Hermit Crab goes, there goes the Sagartia: a quiet life it led before; a restless one it has to lead now. But doubtless it knows what's best for it." The crab who sustains the honourable office of porter to this species is invariably the brawny-limbed Fagurus hern- hardiis, as P. Prideauxii is favoured with the support of Adamsia palliata. In the rude and blundering manner in which the bearer performs his office, it cannot be but that the poor Anemone sustains many a hard knock and many a rough squeeze among the rocks and stones over which his servant travels; but he appears to bear these mis- chances with great philosophy : I know of no species which lives so constantly exposed. A rude shock will, indeed, cause it to withdraw its tentacles, and contract its disk into that button-like shape which is common to the tribe ; but this is only for a moment ; it instantly expands again, and remains full blown in spite of all its draggings hither and thither. Its skin is peculiarly tough and leathery ; a provision, doubtless, against the accidents to which its vagrant life exposes it. Mr. R. Q. Couch says that the favourite site for this Anemone (in the neighbourhood of Penzance ?) is on the claw of the Corwich Crab {Maia squinado). Mr. Cocks, however, says that in the neighbourhood of Falmouth it is THE PARASITIC ANEMONE. 117 never found on this Crab, nor on Pinna mgens, but fre- quently on Pecten maxtmus, as well as on Buccinum un- datum, and on stones.* I do not remember myself to have ever seen it on a bivalve. We bave no species of Sea- Anemone which, to such an extent as this, shoots forth those filaments which I have called acontia, and which are undoubted weapons of offence. On being rudely handled, or otherwise alarmed, from vari- ous points of the body, particularly from the larger warts, the loop-holes [cinclides] give issue to these threads, which exactly resemble in appearance white sewing-cotton. They are often shot forth with force to the length of four or even six inches ; and under circumstances of great irritation an immense bundle of such threads is projected from the mouth. Their interior end remains, however, attached to the cavity whence they issued, and they are soon withdrawn again. Most species of Anemones give out a rank penetrating odour, but it is more than usually offensive in S. parasitica. It is communicated to the fingers on handling thq animal ; and repeated washings with soap, and even scrubbings with a brush, scarcely avail to remove it. It is insufferably nauseous. S. parasitica, like its congeners, is by turns oviparous and viviparous. To the former mode of increase Mr. G. H. Lewes bears witness. " In the water of a pan con- taining, among other animals, specimens oi Actinia para- sitica, I twice noticed abundance of light-purple ova floating at the surface. Some of these were placed in a vase by themselves, and others left in the pan; but no further development took place. One day, dissecting a. parasitica, I found in its ovaries these very purple ova which had attracted my attention. "f ]Mr. Lewes doubts, however, that it is viviparous. This point has been settled by my friend, Mr. F. H. West. " A ' Johnston, Br. Zooph. 228. \ Sea-side Studies, 141. specimen," he writes, " which I received in December from Weymouth, produced a young one on the 1st of March following ; it was most beautifully and distinctly marked, and as dark-coloured as the parent, which was of the dark reddish-brown variety. It was a pretty little creature, and lived for five or six weeks, when I lost sight of it." Mr. Holdsworth also has met with the young of this species, not more than a line in height, yet distinctly marked like As a proof of the tenacity of life of Anemones under the privation of sea-water, provided the skin be preserved from becoming dry by evaporation, I may mention the following fact, which is valuable as bearing on the trans- mission of these animals from distant localities. I inclosed two large specimens of 8. parasitica, two of T. crassicornisy and one of A. dianthus, in a large jar, containing one or two tufts of Chondrus crispus, but no water. The jar was closed with a bung, but was not air-tight. The Anemones remained thus imprisoned for ten days, wallowing in their mucus and discharged water, which from time to time I poured off. At the end of that time they were quite well, and I restored them to the aquarium. Might not the species from North America, or those from the Mediter- ranean, be transmitted to us thus inclosed? I should add that the experiment was performed in December. The following are the known British habitats of this species. Guernsey, Herm, J. D. H.: Jersey, G. H. Lewes: Weymoutli, P. H. G.: Teignmouth, R, C. J.: Torquay, P. H. G.: Falmouth, W. P. G.: Penzance, R. Q. Couch: Bantry Bay, E. P. W. viduata. V bellis. PARASITICA. A. palliata. B. coronata. THE GOLD-SPANGLED ANEMONE. Sagartia (?) chrysospleniwni. Plate VI. Juj. 8. Specific Character. Colmnn green, with lines of golden-yellow dots : tentacles pellucid, with green bars. Actinia ckrysosplenium. Cocks, Rep. Cornw. Polyt. See. 1851, 5; pi. i. fig. 17. Johnston, Brit. Zooph. Ed. 2, 214 ; pi. xxxviL figs. 1—3. Sagartia (?) ckrysosplenium. Gosse, Annals N. H. Ser. 3. L 416, GENERAL DESCRIPTION. Form. Base. Adherent to stones : slightly exceeding the column. Column. Smooth, studded with numerous scattered suckers (or loop- holes), resembling punctures. Form shortly cylindrical, becoming conoid in contraction. IHsh. Smooth. Tentacles. Few, nearly equal in size, rather short, siotit, and obtusely pointed. Mouth. Set on a roundish cone. Lips slightly puckered or imperfectly furrowed. Acontia. None have been obserred. Colour. Column. Green, varying in tint from a bright pea-green, to that of a dark holly-leaf; marked with longitudinal bands of spots of a rich golden yellow ; a line of the same golden hue margins the base. Disk. Yellowish-brown ; gonidial tubercles bright golden yellow. Tentaelea. Pellucid, sometimes nearly white, crossed by transpcui-ent green bars. Size. About an inch in height ; the diameter of the base and of the flower three-quarters of an inch ; that of the column five-eighths of an inch. Locality. The coast of Cornwall. Under-snrfaces of stones at extreme low water, and rock-pools. To Mr. W. P. Cocks, of Falmouth, to whose scientific research our zoology is largely indebted, Dr. Johnston owed the admission of this species into his " History of British Zoophytes." I am under obligations to the kind- ness of the same gentleman, who has favoured me with some additional notes on the species, and a beautiful coloured sketch, which I have copied in Plate VI. The generic position of this beautiful form I indicate not without doubt. The short conical tentacles, crossed with bars, suggest a relationship with Tealia ; and this afiinity had occurred to its discoverer, who in one of his MS. notes has added the words, — " allied to crassicornisy On the other hand, the marginal line around the base, and the gonidial tubercles being distinguished by a different colour from the rest of the animal, while agreeing inter se, suggest Actinia, of which these peculiarities are characteristic. There is, too, a well-known variety of A. tnesembryan- themum, which is green, marked with lines of yellow dots, and of this circumstance I ventured to remind Mr. Cocks. His reply was as follows : "In the A. meserribr. var. the stripes and spots are as in chrysosplenium, but several shades lighter, and the labial tubercles, as well as the edging of the base, are bright blue ; the tentacles are uniformly of one colour, and are much more numerous, slender, and tapering." The character of the surface, however, decidedly separates first written, — " Suctoreals numerous, scattered, embedded ;" but he afterwards added the following particulars : — " When I examined the body of the chrysosplenium with a lens of two inches' focus, the surface appeared to be pierced or punctured, and in appearance resembled a piece of smooth India-rubber when pierced with a pin ; not the slightest trace of tubercles apparent. The body when contracted THE GOLD-SPANGLED ANEMONE. 121 was as smooth as before ; not papillated ; and the apertures were nearly obliterated." Until I have an opportunity of personal examination, I therefore assign to the species a place in the genus Sagartia ; but I consider that it is one of the links which connect this with the neighbouring families. On the history of this lovely little Anemone I can only quote what has already been published. " The old ones are solitary, not more than one on a stone : but there are two or sometimes four growing on the same stone. . . I have had some in my possession for weeks, well supplied with water and air daily ; yet the tubercles and edging were obdurate, determined to keep to their original colour." I must hope that the zeal of our Cornish zoophytologists will before long make me personally acquainted with the pretty Gold-spangle. The following localities are enumerated for it by Air. W. P. Cocks: — Gwyllyn-Yase, Pennance, Helford, St. Ives. mesembryanthemum. chrtsosplenium. crassicomis. 9 ON THE SUBDIVISION OF THE GENUS SAGARTIA. Fifteen species of the genus Sagartia have been described in the preceding pages ; and I possess information more or less definite concerning some five or six others, which I have not seen ; whose history therefore, in hope of a fuller acquaintance with them, I defer writing for the present, but expect to be able to give some account of them in an Appendix to this Volume. The species already described appear to me to be divi- sible into four or five groups, which cannot, however, be properly considered as higher than sub-genera, the charac- ters by which they are distinguished being too vague to aflford a basis for generic rank. The most typical group, and that for which, should the genus be broken up, I would retain the name Sagartia, includes the following species : — mim'ata, rosea, ornata, ichihy stoma, coccinea, venusta, nivea. These have conspi- cuous suckers, discharge acontia freely, attain only a mode- rate elevation, expand the disk only a little beyond the column, are for the most part painted with gay colours, often in striking patterns, and in particular have the column usually of a rich warm brown hue. A group rather less typical than this, I consider to be formed by the following species: — sphyrodeta, pallida, pur a ; to which will probably be added most of the species which I defer to the Appendix. These have no con- spicuous suckers; discharge acontia less and less abun- dantly ; are in general destitute of positive colour, and have a tendency to a colourless transparency. Nivea and sphyrodeta are the links which unite these two groups. Should a generic name ever be required for this group, I propose for it that of Thoe, one of the sea-nymphs. {Hes. Th. 245.) Troglodytes, viduata, and parasitica may be associated as a group departing still more widely from the typical form. Their suckers are distinct, but minute ; their power of emitting acontia varying (feeble in trog. and vid., strong in paras.) ; their tentacles are generally streaked (only occasionally in trog.) with lateral longitudinal lines ; their column is marked with longitudinal bands of lighter and darker colour ; they have the power and habit of greatly elongating the column ; and manifest a proneness to become and to continue detached. In these last two particulars they approach the Ilyanthidce. Coccinea and parasitica are the links of connexion between the first group and this, though not inter se. In the event of re-distribution, this group might receive the name of Ch/ltsta, from KvXio), to Bellis will probably be considered by many as worthy of generic separation. The slenderness and elongation of its column when fully expanded, the salver-like expanse of its disk, the small size, great number, and crowded arrange- ment of its tentacles, the undulation of its margin, as well as the peculiarities of its colouring, isolate it strongly from its fellows. Iltniata, from the undulation of its margin, and. parasitica, from the craterine form of its disk, and the multitude of its tentacles, are connecting links with it in their respective groups ; while bellis looks, as has been already intimated, towards other genera, as Actinoloha, Aiptasia, &c. It might be called Scyphia^ from a-KV(fio<;, a drinking-bowl. Finally, chrysosplenium is the most aberrant form that I have included in the genus, so far as I am able to judge of its peculiarities without personal inspection. Its affinities I have just enumerated. If I had isolated it generically, I would have named it Chrysoela, " that which is studded with golden nails," from •xpva6<i, gold, and 77X09, a nail. 124 Cribrina (Ehrbkb.). Base adhering to the inner hp of univalve shells ; greatly expanded laterally in two wing-like lobes, surround the mouth of the shell, and meet on the outer side of the body-whorl. Column greatly depressed; margin forming a low sharp parapet, with a distinct fosse. Surface smooth towards the summit, striated or irregularly furrowed on the outer (= lower) part ; pierced with loopholes, which, on the outer (= lower) part, form permanent warts. Substance fleshy. Bisk, long-oval, almost hnear, smooth. Tentacles numerous, sub-marginal, short, crowded, imperfectly retractile. Mouth protrusile, large, thrown into loose folds, but not furrowed. Acontia emitted freely and copiously. The genus contains but one known species, A. paU liata. ASTRjEACEA. THE CLOAK ANEMONE. Plate III. figs. 7, 8. Specific Character. Medina palliata. Actinia maculata. picta. Cribrina paUiata. palliata. Body studded with purple spots. BoHADSCH, Anim. Marin. 135 ; pL xi. fig. 1. Adams, Liiin. Trans, t. 8. Coldstream, Edin. New Phil. Joum. ix. 236 ; pi. iv. figs. 6, 7. Otto, Nov. Act. Acad. Nat. Cur. xL 288 ; pi. 40. Risso, L'Europ. mend. v. 286. Ehrenberg, CoralL 41. E. Forbes, Ann. Nat. Hist. v. 183. Johnston, Brit. Zooph. Ed. 2. i. 207 ; fig. 44 ; pi. xlii. figs. 1, 2. GossE, Aquarium, Ed. i. 139 : Man. Mar. Zool. i. 27; fig. 38; Ann, N. H. Ser. 3. L 416. GENERAL DESCRIPTION. FOBJf. Base. Circular in youth ; dilating laterally with age, until the two sides, curving round, meet and unite with a suture, forming a ring; adherent to the mouth of turbinate shells of Gastropoda, which it sometimes invests with a homy membrane. Column. Exceedingly thin, low, and flat : the margin forming a low sharp-edged parapet, with a distinct, but narrow fosse. Substance fleshy, soft. Surface quite smooth for about one-third of the distance from the margin to the edge of the base ; then it begins to be marked with fine radiating depressed lines. These lines meet those from the opposite side, where the two divisions of the body unite on the upper lip of the shell, and alternating with them make a zigzag suture. The outer half of the column is moreover generally thrown into irregular folds and puckers. Loopholes numerous, large, pierced in the centre of slight elevations of the skin, which are most conspicuous on the outer portions. J>isk. Very long and narrow, smooth. ■ Tentacles. Numerous, arranged in four sub-marginal rows ; nearly equal, short, cylindrical, obtusely pointed, crowded, not completely retractile. Mouth. Protrusile, long, oval : the lips thrown into coarse folds, but not furrowed ; throat and stomach marked with close-set white furrows. In the only specimen in which I have had an opportunity of examining the mouth with exactitude, there was only one gonidial groove, with its pair of tubercles. And this was so placed, as to make the bisecting line of which it formed the termination, one at right angles to the lateral develop- ment of the animal into lobes. A contia. Long and thick ; emitted in great profusion, on the slightest irritation. Colour. Column. Sienna-brown, or reddish-brown on the outer portions, marked with bluish longitudinal lines, and gradually melting into the purest white on the upper third ; the whole studded with large round spots of the most brilliant purplish-rose, which are most distinct in the middle third. Margin surmounted by a line of delicate pale scarlet, crowning the parapet. Dish. Pure white. Tentacles. White, with a faintly-dark core. Mouth. White. A contia. Rose-lilac, with the suture, formed by the edges of the infolded ribbon, white. Size. Large specimens attain two inches and a half in diameter, measured from edge to edge along the curve, as they adhere to the shell ; but the long diameter of such individuals, if measured from the suture round the ring, • along the line of the disk, to the suture again, would be not less than five or six inches. The height from the parapet to the surface of the shell is about one-third of an inch. Tentacles one-third of an inch in length. Locality. The coasts of Europe generally. Deep water. " They seem to love a muddy bottom, mixed with gravel and dead shells." (D. R. in litt.) Vabieties. a. Rhodopis. The condition described above. (Plate III. fig. 7.) fi. Crinopis. Whole body pure white ; unspotted (Forbes); or marked with a few scattered, mostly minute, pink dots. "^ (P. H. G.) (Plate III. fig. 8.) The name of this genus was assigned to it by the late Edward Forbes in honour of John Adams, who first described the animal as British. It had, however, been described and figured by Bohadsch before him, and by many since, " both at home and abroad," and by no one more accurately than Dr. Coldstream, the principal parts of whose account are cited in Johnston's " British Zoophytes," (Ed. 2, p. 207.) The true character of the animal has been THE CLOAK ANEMONE. 127 pointed out by myself in ' The Aquarium," in which I have thus explained its manner of growth. — '' The Adamsia is evidently an Actinia of a long-oval form, capable of development in its long diameter into two lengthened wings. Its instinct invariably leads it to select as its support the inner lip of some univalve shell ; having ad- hered to which, the lateral expansions creep along the shell, following its surface until they have surrounded the aper- ture, and meet each other on the outer lip. Here the meeting edges unite by mutual adhesion, and seem to grow together ; yet the suture is always distinctly visible, both by a slight depression, and by a pale line which assumes a zigzag form, owing to the terminations of the body-striae fitting into the interspaces' of the opposite ones." In Plate III. fig. 8, 1 have depicted an individual, adherent to the shell of Buccinum unddtum, in which the lateral lobes, though projected around the edges of the mouth of the shell, have not yet met each other on the outer lip, but are separated by a space of a quarter of an inch. And I have seen a very young specimen, less than half an inch in diameter, the outline of which was exactly like that of a normal Anemone ; the lateral lobes not having yet com- menced their extension. This little individual was adherent to the inner lip of the shell of a Garden Snail {Helix aspersa) , which had been accidentally washed into the sea. A Pagurus Prideauxii had selected the same shell as his abode, and to his wanderings it was probably owing that the shell had found its way into eight fathoms' water, a mile or two from land. This manner of growth is further illustrated by what takes place at the disease and death of the animal. The adhering base begins to peel off, and shrink away from the * Aquarium; Ed. i. p. 139 ; et aeq. shell. This process invariably hegins at the suture, and as it goes on the suture divides, the lateral portions separating more and more from each other by shrinking ; thus reversing the steps by which the annular habit was assumed. So far as my own experience goes, the Adamsia always selects for its support the inner lip of a turbinate shell. Buccinum undatum I have generally seen chosen at Wey- mouth, but not rarely the various species of Trochus ; and Milford Haven, on Murex despectus (= Fusus anttquus) : Thompson, at Belfast, on Bulla lignaria, as well as on the larger Trochi : E. Forbes, at the Isle of Man, on old Fusi and Trochi: Landsborough, at Arran, on Turritella and Buccinum. Mr. D. Robertson sends me specimens from Cumbrae, on Trochus umbilicatus. I believe that the shell chosen is always tenanted by a Hermit Crab, and that the species is invariably Pagurus Prideauxii. In this my observation coincides with those of Dr. Coldstream, Thompson of Belfast, and Mr. D. Eo- bertson. Forbes seems to throw doubt on the constancy of this association ; having taken many specimens on the Manx coast, the shells of which were not tenanted by any crab. Similar examples have occurred to myself at Weymouth ; but when we remember how readily the Pagurus leaves its shell on alarm, and how terrifying the rough action of the dredge-iron must be, it seems the most obvious mode of accounting for the occasional vacancy of the shell, that it has been just deserted by its frightened tenant. The Adamsia itself in early life has the power of shifting its quarters. Forbes observes that it " seems to change its habitation according to its size : " and I have had two young specimens in my aquarium, which crawled sponta- neously from their shells, and attached themselves the one to a stone, the other to the frond of a sea-weed. While THE CLOAK ANEMONE. 129 two in his possession, which manifested the same propensity. Each first detached the two lobes from the shell, which then were thickened, and apparently hollow, being much dis- tended with water. The same evening, both began to adhere to the side of the jar in which they were kept, by their lateral lobes. Three days afterwards, the lobes were of the jar." Mr. Thompson, of Weymouth, has dredged a specimen, which was adherent to a frond of Fucus serratus. It was round, about as large as a shilling, and flat, but " with the appearance of a suture down one side, Yery frequently, there is found intervening between the Adanisia and the shell to which it is affixed, a film of membrane, of a homy texture, somewhat brittle, of a translucent dark greenish-brown colour. After death this film is found adherent to the surface of the shell, from which, however, it easily peels when dry. It invariably extends beyond the margin of the lip, making, as it were, an adventitious continuation of the shell, and following the same general spiral direction. From several specimens from the Frith of Clyde, for which I am indebted to the kindness of Mr. D. Robertson, I have been able to learn the nature and object of this membrane. In one of these the shell of Trochus umbilicatiis, full-grown and perfect, had a great continuation of the membrane into a fictitious body-whorl, as voluminous as the whole shell. In another, the shell was that of Buccinum undatum, an inch and a half in height. Here the membrane was confined to a small film, sub-triangular in outline, continuing the front margin of the outer lip, and a similar one continuing the hind margin of the same ; each the production of a lateral lobe of the animal, the two not having as yet attained th« K point of union. In a third beautiful specimen, sent mc alive, I found, after death, the membrane showing dis- tinct concentric lines of growth. And these took exactly the form of the outer edges of the two lobes, meeting in the centre, where there was a representative suture. The growth-line being curved, there was a delta at the end of the suture ; and this was filled with a much thinner film of membrane, showing that it was the last Mr. Walter Gregor, of Macduff, has sent me a large specimen which had in youth chosen a shell of Natica swdida for its support. The shell is in no direction more than one-third of an inch in diameter, but the adventitious body-whorl of membrane measures (along its curve) two inches and three quarters ! From these and other observations of my own, as well as from information supplied by Mr. Robertson, it appears to me manifest that the membrane is a provision for the support of the growing Adamsia, when it has selected small or broken shells. Experiments, which I have detailed at length else- where,* have satisfied me that the membrane is produced by the Adamsia ; that it is an epidermic slough ; and that it is composed mainly of chitine, having no calcareous element. It cannot, therefore, in any respect, be regarded as a corallum. The membrane is not invariably present. In specimens dredged in the Frith of Clyde, small or broken shells appear to be usually chosen ; and these are enlarged, as I have stated above. In Weymouth Bay, however, where the species was common when I was there in 1853, the shell most commonly selected being the Great Whelk, the ' * Annals and Mag. of Nat. Hist, for Aug. 1858. THE CLOAK ANEMONE. 131 membrane is so miusual that I do not remember to have ever seen it. Pagurus Prideauxii s6ems to be as dependent on the Adanma, as the latter is on it. The only instance in which I have heard of its having been ever found disso- ciated from its friend, is the following, communicated to me by Mr. Robertson : — " Lately I dredged a small Pagurus Prid. unassociated ^vith Adamsia palliata. After a few days I put it into a I saw them six hom'S after ; Pagurus had left his shell, and was perched on the top of Adamsia^ with his fore claws among the tentacles. Next morning Pagurus sheU." This association, however, Uke so many other things that the naturalist is constantly meeting with, is unac- countable. Why one species of Soldier-crab must needs seek the companionship of this Anemone, while other Soldier-crabs are able to live alone ; and why this species of Anemone must needs associate with the Soldier-crab- while other kinds of Anemone are solitary, I can by no means answer. Nor is the difficulty in any wise solved by supposing — what we may easily grant — that each may find advantage from the other's presence. Dr. Lands- borough pleasantly says, — " In all likelihood, they in various ways aid each other. The Hermit has strong claws ; and while he is feasting on the prey he has caught, many spare crumbs may fall to the share of his gentle- looking companion. But, soft and gentle-looking though the Anemone be, she has a hundred hands ; and woe to the wandering wight who comes within the reach of one of them, for all the other hands are instantly brought to its aid, and the Hermit may soon find that he is more K 2 than compensated for tlie crumbs that fall from his own booty." It is probable that Adamsia would be a dainty morsel for the table. I have not essayed it, but the smell of the fresh animal is very agreeable, resembling that of the cooked flesh of the crab. Beautiful as it is, it appears unlikely ever to become an habitual tenant of our aquariums, as it cannot long endure captivity. Its crab, too, seems peculiarly unable to survive confinement ; and I do not think the Cloaklet will ever live long dissociated from its companion. Yet Sir John Dalyell seems to have been more suc- cessful than I have been, if I may judge from the expres- sion " a long time " in the following statement. One which had detached itself from its shell " diffused the base on the bottom of a glass vessel, not unlike the wings of a butterfly. But until it adheres, the base remains a long time with its whole under surface merely folded together." He describes it as feeding readily, and as greedy of worms. According to the same observer, thousands of minute, opaque, bright yellow globular germs are produced by the species in July, August, September, and October ; several these developments in his experience.* Rapp assigns Adamsia palliata to the Mediterranean and North Seas if MM. Koren and Danielssen mention it as common in fifteen to twenty fathoms off the coast of Norway.! The following list includes its known British habitats : — Wick, Peterhead, C. W. P. : Moray Frith, W. Greg&r : Guernsey, J. D. H. : Weymouth Bay, P. H. G. : Torbay, * Rare and Rem. Anim. of Scotl. ; 233. t Polyp. 68. X Faun. Litt, Norv. ii. 87. THE CLOAK ANEMONE. 133 P. H. G. : Falmonth, W. P. G. : Milford Haven, Adams : Isle of Man, E. F. ; F. H. W.: Arran, D.L.: Cumbrae, i>, R. : Bute, Dr. J. Coldstream : Oban, Mrs. A. Murray Memtes : Strangford Lough, Belfast Bay,^ W. T, : Bantry Bay, E. P. W. PALLIATA. S. parasitica, 9 134 QENUS IV. PHELLIA (Gosse). Base adhering to rocks ; little exceeding the column. Coltimn pillar-like in expansion ; the margin ten- taculate, without parapet or fosse. Surface smooth, pierced with loop-holes ; partly clothed with a tough epidermis, which is rough externally, firmly adherent to the skin. Disk concave ; the edge not undulate. Tentacles few, in more than one row ; baiTed. Mouth not raised on a cone ; lip thickened. Acontia discharged, but reluctantly. ANALYSIS OF THE SPECIES. Epidermis dense ; free and tube-like at the upper part ; its surface not warted murocincta. Epidermis dense ; firmly adherent throughout ; warted . gausapata. Epidermis thin ; firmly adherent throughout ; not warted picta. THE WALLED CORKLET. PJiellia murocincta. Plate VII. fig. 2 ; XII. fig. 8 {magn.). Specific Charactev. Epidermis dense; free and tube-like at the summits- its surface not warted. PhtUia murocincta. GossE, Annals N. H. Ser. 3. ii. 193. GENERAL DESCRIPTION. FOBH. Base. Adherent to rocks ; slightly exceeding column. C'Aumn. Cylindrical, pillar-like when expanded, slightly grooved longi- tudinally, smooth, but partly clothed with a dense, rough, membranous skin, which is firmly adherent from the base about half-way up, but there becomes free, forming a loose firm sheath or tube, from which the animal protrudes its fore parts in extension, and into which it retires at will, more or less completely. Surface of epidermis rough, but not warted. Height, in full extension, double the diameter. DUJc. A deep cup, bounded by the thick feet of the inner tentacles. Tentacles. Twenty -foxir, in two rows, twelve in each ; those of the first row twice as large as the others, with which they alternate : variable in form, sometimes strongly conical, stout at the foot, and pointed ; at other times nearly cylindrical and obtuse : they have a tendency to assume a knotted appearance : they are generally carried hanging over the margm with a double curve, like the bi-anches of a chandelier ; but sometime* those of the inner row stand erect Mouth. Not raised on a cone, so far as could be ascertained. Aooniia. Emitted sparingly and reluctantly. Colour. Column. Exposed portion having a mealy appearance, produced by a number of whitish longitudinal lines and dashes, more or less speckled and interrupted by the ground-colour, which is pellucid yellowish grey. Of these lines, twelve are] broader, and between these are about four slender lines in each interspace. The margin becomes deep buff, pro- ducing a depression of that hue when in the button-state. Epidermis. Pale buff, studded with dirty foreign matters. Disk. Dull buff, marked with a white star, which is formed by a foi-ked line proceeding from the front of each primary tentacle towards the Tentacles. Dark brown, pellucid, crossed by three narrow remote rings of white. Where the foot of the tentacle unites with the disk, its radius has a white patch, succeeded by two parallel, longitudinal, black dashes. Mouth. Rich buff. Size. Diameter of column one-eighth of an inch ; height one-sixth ; expanse of flower one-sixth. Locality. Overhanging rocks and sides of caverns near low-water mark, around Torquay. The large dark overhung pool at Petit Tor, which I have more than once described, is a fertile nursery of marine life. Though situated not much lower than half-tide level, yet, from the volume of water which it contains, the constancy of its fulness, the aspect, excluding the sun's rays, and the inclination of the rocks preventing evaporation, the rough worm-eaten surface, both below and above the brim, is always wet, always dark, and always crowded with Algge, Sponges, Zoophytes, Worms, and Mollusks. This pro- fusion of riches is not always, however, easily available ; for though it stands in tantalising proximity to the eye of the naturalist, it is quite beyond the reach of his hands, unless he choose to wade into the pool and work in the water breast-high. On the 29th of June of the present year I essayed in this manner to rifle the promising treasury ; and the result by no means disappointed my expectations, though, from several circumstances, it was diflScult to work with hammer and chisel. Among other things I obtained there this new form. THE WALLED COEKLET. 137 It has been my custom, — and I recommend the plan to brother and sister naturalists, — not to satisfy myself with such creatures as I see on the spot, but to take specimens of the rock at random for examination at home. I look out for the dirtiest, roughest, most corroded parts of the rock, at the lowest level that I can reach, and with the chisel cut off small fragments. These I bring home, and spread out, face upward, in shallow pans of clean sea- water. After a few hours, say perhaps the following morning, I carefully search with my eye, aided at intervals by a lens, but without disturbing the water, the surfaces of the bits of rock, as well as the sides of the vessel ; and thus I have obtained more than one new species, which I might never have known otherwise. For the actual discovery of the present species, I am indebted to my little son, whose keen and well-practised eye detected the tiny atom, as a form with which he was unacquainted, on one of the fragments I had brought home. Presently afterwards I discerned another specimen; and these two are the only examples that have as yet come under my notice. The rough corky appearance of the epidermis in this and the following species, suggested the generic name, which is formed from ^eXXo?, the cork-tree, and also its bark. The specific appellation indicates the chief distinc- tion between this and the following species, the edge of the epidermis encircling the summit of the animal when con- tracted, as if with a wall. The force of the English appellation is obvious.* * I feel that I am arrived at a point where I need the kind consideration of my readers. Popular as the ciitivation of Zoophytes has become, there are still many who prefer to caU them by English names, the ladies in particular. It is a natural and proper desire, and I wish to respond to it. But no vernacular terms exist, by which the hitherto recondite subjects of thia work are known. What shall I do in this case ? Shall I use the term In both this and the preceding genus we find a remark- able development of the epidermic layer ; in Adamsia from the base, to enlarge its support, — in FhelUa from the column, to thicken its investing coat. The investment is, as I have intimated, a tightly-adhering epidermic layer, but free at the upper part, which stands up as a thin, clear, firm tube, when the animal retreats. Its substance is strong and tenacious, yet portions of it can be torn away in shreds with a needle. These, under a power of 600 diameters, show, in the clear parts, a structureless membrane, which has a slightly fibrous appearance, apparently only because of its foldings and wrinkles. The greater part is rendered opaque by the foreign matters entangled in it, consisting largely of irre- " Anemone" tlii'oughout, employing an epithet to discriminate the families from each other, a second epithet to discriminate the genera of each family, and a third epithet for each species ? " The Anemone : " " the Warty Anemone :" "the Lined Wai-ty Anemone : " "the Glaucous Lined Warty Anemone." This would be an available mode, but would it not be repulsive and lumbering? Again, I might make new words — arbitrary aggregations of vowels and consonants, — " Fai"Son," " Toler," — words, if words they might be called, without an etymology, and without a meaning. I do not think this would be generally acceptable, though I might plead precedent in scientific technology, — '•' Rocinela," " Conilera," &c. for example. A celebrated Greek orator is said to have coined only three words in the whole course of his professional eloquence ; and, for the comfort of those who should attempt the same again, it is added that the Athenian public refused to swallow these. Yet it is much easier to make a Greek word than an English one, I manufacture " Aiptasia '" and " Bolocera" boldly ; yet it is not without mistrust that I see " Trumplet" and " Opelet" on my pages. In this dilemma, since the words must be made, I have thought that they ought to be formed according to certain conditions. First, they , should be Saxon : " Ilyanth," " Lucemary," " Cyathine," are no more English than if they retained theu' classical terminations. Secondly, they should be significant : the new word should aid the memory, not tax it. Thirdly, they should be consimilar in structure, since they are intended to designate consimilar objects. Fourthly, they should not, if possible, exceed a dissyllabic length. According to these rules, I have ventured to construct a series of verna- cular names for the genei"a. Allowing " Anemone " to stand for Sagartia, I have formed for each of the others a dissyllable, Saxon in origin, sug- gestive of some prominent character, and having a common termination, — viz, the English diminutive " -let," from lie, little. In accoi-dance with this plan. Plumelet may stand as the English representative of ActinoUhci, and Cloaklet of A damsia. THE WALLED CORKLET. 139 gular, clear granules, with some Alga-spores, Diatoms, and here and there a cuida. I removed, with a fine needle's point and pliers, the epidermis piecemeal. It was tongh, allowing the Anemone and its bit of rock (as large as a filbert) to be lifted out of the water by it, without giving way. Its adliesion to the lower part of the column was very firm. As I removed the loose free tubular portion, (the animal having retreated far in at the exirliest assaults,) I discovered free within its cavity about half-a-dozen egg-like germs, of a rich deep orange colour; these, under the microscope, proved to be covered with vibratile cilia, by means of which the germ slowly swam. They were soft, ovate, '04 inch long, by •025 wide. One, on being crushed, was resolved into a mass of minute round clear gi-anules, — fat-corpuscles ? When the whole epiderm was removed, I detached the animal from its adhesion in a small hollow of the lime- stone ; not without the discharge of a thick mucus from the base, and the emission of a single acontium from the lower part of the column. The animal was now reduced to an abject flatness, and looked like a miniature S. vtduata m its greatest contraction. In a day or two it attached itself to the rock again, and even crawled a little way. It now expanded freely, and looked just like an ordinary Sagartia : but did not renew the epidermis. The only locality as yet known for the species has been already indicated : — Torquay, P. H. G. A. palliata. mukocincta. E. camea. gausapatci. picta. A STRjEA CEA . SA GARTIA DJi. THE WAETED CORKLET. Phellia gauswpata. Plate \ll. fig. 1. Specific Character. Epidermis dense ; firmly adherent throughout ; warted, Phellia gausapata. GossE, Annals N. H. Ser. 3. ii 194, GENERAL DESCRIPTION. FOBM. Base. Adherent to rocks : scarcely exceeding column. Column. Cylindrical, pillar-like when expanded ; smooth in extension, but in contraction becoming coarsely corrugated, so as to present large irregularly rounded knobs or warts. To this a dense epidermis is firmly adherent throughout, having no free margin ; and, being modelled on it, it is covered with coarse warts or knobs ; " resembling, when contracted, a straw bee-hive." (<7. W. P.) Disk. A deep cup or funnel. Tentacles. Sixteen, arranged in two rows, eight in each : those of the first row twice as thick and long as those of the second, with which they alternate ; variable in form, sometimes being conical and pointed, at others short, rounded, and even slightly inflated at the tips. Mouth. Not raised on a cone : lip thickened " as in dianthus." Acontia. Freely discharged from the base ; long and very slender. Colour, Colvmn. Exposed portion pellucid white, with sub-opaque whitish longi- tudinal streaks. Epidermis. Pale yellowish, with darker warts ; the separation of which in extension causes the general tint to appear lighter, and vice versd. Disk. (No note has been taken of its colours.) Tentacles. Pellucid drab, with the lower part and a broad ring near the tip dark brown, undefined : probably there is also an intermediate ring of paler brown. Size. Diameter of column half an inch ; height three-fourths of an inch. Locality. Bocks at low-water : extreme north-east of Scotland. THE WARTED CORKLET. 141 By a curious coincidence, on the very day that I disco- vered the preceding species, the post brought me a living specimen of the present, from Mr. C. W. Peach, of Wick ; and so the extreme north-east of Scotland and the south- west of England conspired, at the same moment, to augment our native Actinologia, each with a species of a genus entirely new to science. The kindness of Mr. Peach had, it is true, sent me a specimen of the same animal before this, viz. in the even to form a conjecture of its characters. Observation of the species is even now very defective ; for though the last specimen sent arrived in health, and continued for upwards of a month to live in my possession, yet, during the whole of that period, I never saw it expand sufficiently to enable me to describe either its tentacles or disk. For the above description I am largely indebted to the notes and sketches of Mr. Peach. The distinction between PhelUa gausapata and P. muro- cincta is slight ; and future observation may resolve the two species into one. The distance of their respective localities, however, renders their identity less probable. The specimens were obtained from very narrow fissures in a rock called Proudfoot, at the entrance of Wick Bay, in Caithness. This rock is accessible only at the low water of spring-tides. The first specimen obtained, which was much larger than the second, remained unattached for several days, while in Mr. Peach's possession, but appeared healthy. The smaller one sent to me remained adherent to its original fragment of rock for more than a month ; at the end of which time I lifted the base from its attachment. It was in doing this that I saw the acontia copiously discharged from the offended base. When received, several young algae, — one apparent] j a minute Laminaria, another a Rliodymenia lyalmata, — were growing from the upper margin of the epidermis; a fact which is of value as showing the persistency of this investment, which, moreover, was not separated during the subsequent period of the animal's captivity. The trivial name of this species I have formed from the gaiisajpe, or rough frieze coat which the Roman soldiers wore in cold weather. The only known locality for this PhelUa is, as above stated,~Wick, C. W. P. ? aAUSAPATA. murocincta. picta. ASTR^ACEA. THE PAINTED CORKLET. Fhellia picia. (Sp. noT.) Plate XII. fig. 1 (piagn.). .'S^cific Character. Epidermis thin ; firmly adherent throughout ; not irarted, GENERAL DESCRIPTION. Foasi. Base. Adherent ; scarcely exceeding column. Colamn. Cylindrical, pillar-like when expanded, capable of great elonga- tion : permanently smooth ; clothed with a very thin memLranons epidermis, which is not warted, but carries minute extraneous matters entangled in it. It is wholly adherent, and extends about half-way up the column. Disk. Nearly flat or slightly concave. Tent<icia. Thirty-two, arranged in three rows, 8, 8, 16, = 32 ; thick, long, and bluntly pointed ; one or two of the first row often much enlarged temporarily, and_ standing erect, the rest sub-horizontaL MoHih. Not raised on a cone ; but the lip very protruaUe ; thin. Acontia. Not observed- COLOUB. Column. Pellucid white, with opaque white streaks. Epidermis. Transparent and colourless. Disk. Delicate yellow ; bounded by an irregular circle of dark brown, formed by a broad band crossing the foot of each tentacle; the whole oossed by radial lines of pure orange which spread between the tentacles : — a beautiful pattern. TcHUtdes. Pellucid white, the front £ace crossed by three bands of opaque white. Sns. Diameter and height, in ordinary expansion, one-eighth of an inch : expanse of flower one-sixth. LOCAUTT. North-east coast of Scotland : old shells in deep wato". Since the preceding article was in type, I have received the little Corklet above described ; which differs so greatly from the others, that I must either regard it as specifically distinct, or else consider all three as constituting a single species, subject to an unusual amount of variation. I have no right to assume the latter conclusion, and therefore prefer the former. The only specimen that I have seen was sent me by the kindness of Mr. Walter Gregor, of Macduff, near Banff, who obtained it, in October, from deep water, adhering to an old shell of Cyprina Islandica. " When put into a basin of water," observes its discoverer, " it lengthened itself to a great extent without throwing out its tentacles. Before doing so, it assumed a globose form, and expanded very slowly, withdrawing its tentacles on the least agitation of the water." When it came into my own possession, it feeding eagerly on raw meat. The epidermis, which is very delicate, can be detached in shreads without difficulty ; it holds minute atoms of sand in its substance. It is a brilliant little species, and I have named it from its beauty of coloration. The only recognised locality is — Banff, W. O, PICTA. gausapata. murocincta. 145 GENUS V. GREGORIA (Gosse). Base not broader than the column. Column a low pillar, strongly invected, very in- flatable in irregular lobes; the margin forming an irregularly undulate parapet, separated from the ten- tacles by a deep but narrow fosse, which is never obhterated. Surface smooth, but becoming trans- versely wrinkled in contraction ; without suckers ; perforated with few, but very conspicuous loop-holes ; these are arranged in longitudinal lines, on the swell- ings, which correspond to the intersepts. Substance pulpy. Disk plane; not exceeding the column; smooth, Tentacles moderately short, blunt, unicolorous ; not perfectly retractile. Mouth set on a cone ; lip thin ; two gonidial grooves, each with a pair of small tubercles. Aconlia emitted sparsely. THE EYELET. Oregoria fenestrata. (Sp. nov.) Plate VII. fig. 3 ; XII. fig. 1 (magn.). Specific Character. Column green, with purple lines ; tentacles red. GENERAL DESCRIPTION. FOEM, In addition to the characters given on the preceding page, I may add, that in my specimen, (which may be immature,) the perforations are very visible, with a lens : there are about six in an intersept, of which five are placed in quick succession near the summit, and one remote near the base : they are not found in all the intersepts, from two to four im- perforate ones intervening between those which are pierced. Under mag- nification the perforations are rounder, and less eyelid-shaped than in the Sagartice; they have a distinct granular layer exterior to them, though their outline is in some cases very clearly defined, and even thickened. The tentacles are about forty-eight, arranged in three rows; all sub- marginal : their form is nearly cylindrical, with very obtuse tips. Colour. Column. Translucent glaucous green, very pale; each longitudinal furrow marked by a line of deep reddish-purple, decided but not well- defined ; the loop-holes are each surrounded by a ring of the same colour. Dish and tentacles. Dull red, pellucid; exactly as in the common varieties of A. mesemlryantheimi/m. Mouth. More decidedly lake-red. Throat glaucous. Size, Colvmn about one-sixth of an inch in height, and one-fifth in diameter : expanse of tentacles one-third. Locality. The Scottish coast near Banff; half-tide level. Mr. Walter Gregor, of BaniF, (after whom I have named the genus,) has just favoured me with this little Anemone, THE EYELET. 147 which is highly interesting, as presenting a link which connects Sagartia with Actinia. The disk and tentacles are exactly those of mesemhryanthemum ; and the texture of the column, and its style of colouring, are such as to give the impression that the most familiar of our Actinioids is before us. Yet, on examination, the perforation of the integument, the presence of acontia, and the absence of spherules, indicate its place among the Sagartiadoe. At the same time, its indiiference to contact, and its permanent expansion, — for it seems not to have the power of retracting the tentacles, — are peculiarities which ally it to the mem- bers of the following family. I have seen but a single specimen, which may be im- mature. The specific and English appellations allude to the perforations of the column-wall, which are veiy striking. It attaches itself readily by the base; is constantly swell- ing out part of its body in lobes ; and generally remains widely expanded, with the tentacles arching outwards and downwards. It feeds eagerly, and appears quite hardy in captivity. Only locality known — Banflf, W. G. A. dianthus. FENESTRATA. A. cereus. A. mesemhryanthemum. 1.2 148 In my " Synopsis of the British Actinije," (see Annals of Nat. Hist, for June 1858,) I had associated Anthea with Actinia, in one family, distinguished by the negative character of lacking suckers, warts, and loop-holes. But groups founded on negative characters are always unsatis- factory ; and maturer consideration has convinced me that the positive diversities of these genera are of sufficient im- portance to warrant their separation into distinct families. The members of the family Antheadce are marked by a great development of the tentacular system. The tenta- cles extend to a remarkable length, — in the typical genus often reaching to twice or thrice the diameter of the disk, — and are very flexuous. These organs have thinner walls than usual, but are lined with a thick coat of comparatively large pigment-grains of a deep brown hue. They show a greater tendency to discharge the water which ordinarily distends them, by contracting in diameter than in length, the effect of which is, that these organs under irritation collapse into a shrivelled or withered condition. Another remarkable peculiarity is the almost total in- ability to retract the disk and tentacles, and to close over them the margin of the column, — the common mode in which Actinioids seek protection from annoyance. It is true that, on rare occasions, and when perfectly undisturbed, I have seen both Anthea and Aiptasia in this retracted condition ; but still, even then, there is a tenseness and globularity in the covering column which is at once seen to be peculiar, and which suggests the notion that the effect is produced more by the distension of the column than by the contraction of the disk. In these cases, too, the slightest touch (which, in a Sagartia or Actinia, under similar circumstances, would only cause a closer contrac- tion) is followed by the instant recession of the column, and the protrusion of the tentacles. The whole body manifests comparatively little contrac- tility. The shrinking of the parts from every touch, which in the Bunodidw and the Sagartiadce is so excessive, going on even after decomposition has set in, and which is so annoying and so baflling to the anatomist, prevails to a far less degree in the Antheadce ; and hence the family presents favourable conditions for dissection. The power of discharging mucus is also comparatively small. Though ordinarily adherent by the base, the power of adhesion is unwontedly feeble in the family ; the animals can be detached with the slightest force, and often spon- taneously free themselves. Both of our British genera have the habit of frequently crawling to the brim of the water, and then expanding their base upon the surface and allowing it to dry, floating by means of it with the body inverted, and the tentacles expanded in mid-water. An attentive observer sees in the habits of the Antheadce, and particularly in the lively and flexuous movements of the tentacles, an indication of superior muscular power in these organs, and also a higher degree of intelligence, or at least of perceptive faculty, than the Sagartiadm possess. Besides our own two genera, Aiptasia and Anthea, one or two exotic genera must belong to the family. If Mr. Dana has correctly described the Act. jkigellifera of Madeira, it must be generically distinct, notwithstanding its very close resemblance in figure and colour to the green variety of Anihea cereus. He speaks of the inner row of tentacles as furnislied with a retractile pencil of hair at the tip.''^ Nothing corresponding to this peculiarity belongs to either of our species ; but it is so utterly abnormal, that I cannot help suspecting some source of illusion : and the more, because I learn from Mr. Holdsworth, that Anthea However, the A. pustulata of the same author f must certainly constitute a distinct genus of this family. It appears to be essentially an Anthea, but with the column covered with warts. It would form, therefore, an osculant form, connecting Antheadce with Bunodidce; as our Aiptasia, in its acontia and cinclides, links the family with the Sagar- The New Norwegian genus, Actznopsis,\ must, I suppose, be referred hither. * Dana's Zoophytes, 126 ; pi. i. fig. 1. f Ibid. 128 ; pi. i. fig. 2. ± Faun. Litt. Norv. ii. 89. ANALYSIS OF THE GENEKA, Mo uth normal. Skin smooth. Column long, trumpet-shaped : furnished with acontia and cinclides Ai^tasia. Column short, broad : destitute of acontia and cinclides Anthea, Skin Vf&vted {]Vot British) " A. piisiulata" Gonidial tubercles elongated Actinopm. 151 GENUS I. AIPTASIA (Gosse). Antkea (Cocks). equaUing the medium diameter of column. Column trumpet-shaped, many times higher than wide, very changeable in shape from irregular disten- sion ; margin tentaculate ; surface minutely corrugated, adhesive, but without distinct suckers ; pierced with loop-holes. Substance pulpy. Disk greatly expanded, membranous, concave. Tentacles in several rows, long, lax, irregularly flexuous, perforate at the tip, the first row longest, scarcely retractile. Mouth not set on a cone ; hp thin ; stomach pro- trusile. Acontia abundant, but not often spontaneously extruded. The genus contains but one known species, A, Couchii. ASTRjEACEA. THE TRUMPLET. Avptasia Couchii. Plate V. fig. 3. Specific Character. Body smoke-brown ; disk marked with pale blue lines. ? A ctinia biserialis. Anihea Couchii. Aiptasia amacha. Forbes, Ann. N. H. Ser. 1, v. 182 ; pi. iii. Johnston, Brit. Zooph. Ed. 2. i. 221 ; pi, xxxviii. fig. 1. Cocks, Eep. Cornw. Pol. Soc. 1851, 6 ; pi. i. fig. 18. Ibid. Rep. Comw. Pol. Soc. 1851, 11 ; pL ii. fig. 30. GossB, Annals Nat. Hist. Ser. 3. L 416. GENERAL DESCRIPTION. FOBH. the middle of the column. Column. Slender just above the base, enlarging upwards, dilating at the summit into a wide hemispheric cup or trumpet-shaped disk ; four or five times higher than wide ; the form susceptible of great and rapid changes from irregular distension. Margin formed by the outer row of tentacles. Substance pulpy. Surface minutely corrugated in the ordinary condition, but smooth when fully distended, pierced with loop-holes ; without visible Dish Thin and membranous, greatly expanded as a broad concave cup. Outline circular, but lax, and often imdulate, or even revolute. Radii strongly marked. Tentacles. Arranged in four rows : the first row containing six, set at half radius, remote from each other, and from the second row ; when fully extended, an inch and a-half long ; the other rows diminish gradually, the outermost being about half an inch in length. All, especially those of the first row, very lax, flexuous, frequently thrown into sinuous curves, perforate with a large terminal aperture. Mouth. Lip thin. Throat irregularly furrowed. Stomach-wall occa- sionally protruded. Two gonidia, scarcely rising into tubercles. THE TRUMPLET. 153 Acontia. Abundant; copiously protruded from the mouth or from wounds ; occasionally also, but sparingly and reluctantly, from loop-holes. Colour. Colum/n. Warm orange-buff, richer at base, blending into a bluish-black hue where it expands into the cup-like disk : the entire length marked with longitudinal faint lines, indicating the insertions of the septa. Disk. Dark iron-grey, becoming ashy towards the centre : each radius bo\mded by lines of pale greyish blue. Tentacles. Sepia brown ; but seen under a low magnifying power to be of a warm umber, more or less decided, minutely mottled with darker : the colour xisually softens into white at the extreme tip of the tentacle. Mouth. Lip and throat ash-grey. Size. When fully extended the coliamn is sometimes four inches in height, and from an eighth to three-fourths of an inch in diameter. Expanse of flower LOOAUTT. The Channel Islands and ComwalL Under surface of stones at low- water mark ; deep water. In the latter part of March of the present year (1858), Dr. Hilton of Guernsey found on the shores of that island, and kindly sent to me, several specimens of an Anemone new to him, and equally so to me. The locality, the colour of the disk, and much in the form and contour of the animal, at once suggested the Actinia hiserialis of Edward Forbes, for which species I was on the look-out. Not long after this, I was indebted to the courtesy of Mr. Sydney Hodges, the Secretary of the Koyal Cornwall Polytechnic Society, for other specimens of the same species from Falmouth, which were sent under the persuasion that they were A. hiserialis. Still so much diversity existed between the specimens (those from Guernsey and Falmouth perfectly agreeing inter se) and Forbes's description, that I could not but consider the point very doubtful. At the same time, if I were quite sure that the specimens in my possession were identical with that described by Forbes, I should be compelled to reject his specific name as involving important error. The tentacles can in no sense be called biserial : there are four distinct rows, which are regularly graduated in length, and which show no other distinction ; the appearance indicated by his figure (supposing it to represent the present species) being quite illusory. But on examination I found peculiarities in the animal, which required its generic separation. The most promi- nent of these were its form, the length and flexuosity of its tentacles, and its permanent expansion. In two of these characters, as well as in several other points which I shall presently notice, it manifested so close an affinity with Anthea cereus, that I should not have hesitated to include it as a second species in that genus, had not the presence of acontm, and their extrusion through cinclides, indicated a nearer approach than is made by that species to the family Sagartiadce. I therefore ventured to describe it under the name oiAiptasia amacTia ; the generic appellation referring to its permanent expansion, from aet, always, and Trerdco, to expand ; and the specific to the patience with which it bore pushings and pokings without unsheathing its weapons, from a, priv., and /Mu-x^ofiac, to fight. The English name refers to its trumpet-like form. Subsequently, however, I have found that the species has been well described and figured by Mr. "W. P. Cocks, in his valuable List of the Actiniae of Falmouth, published in the Report of the Cornwall Society for 1851, under the title of Anthea Couchu, which specific name takes prece- dence of mine. It is true, in his description, mention is made of three white lines extending longitudinally up the column, of which no trace exists in my specimens ; but by a coloured drawing with which Mr. Cocks has THE TBIIMPLET. 165 favoured me, I perceive that these lines were not equidistant and symmetrical, but all close together on one side ; a cir- cumstance which at once shows their presence to have been accidental, and of no value as a character, whUe in every other respect, even in the most minute points, his drawing and description agree with my specimens. At the same time it is interesting to observe that Mr. Cocks did not consider his specimens as the A. hise- rialis ; for he describes this separately in the same list, as " not uncommon." i\Ir. S. Whitchurch, of Guernsey, informs me also that there exist at Herm Actiniae, which are commonly spoken of as " the yellow and blue varieties of A. bisen'ah's" so that a species may yet turn up which will justify the description of that form ; and at all events it would be rash at present to accuse so excellent a zoologist as E. Forbes of incorrectness, on the known premises. The present species seems to be found in considerable abundance iu its recognised localities, especially Guernsey and the contiguous little isle of Herm ; appearing chiefly to affect the under sides of loose stones at the level of lowest tide, to which it adheres with a very slight attachment. When the animal has been some time deprived of water, — as in transmission by post, — it has a very abject appear- ance, shrivelled almost to shreds of blackish membrane, which, when immersed in sea-water, lie helplessly on the bottom, ragged and hideous, discharging brown pigment. Presently the tentacles begin to fill, and one by one to assume plumpness, and to move slowly; and gradually, after some hours, the animal presents a more life-like appearance. The extremities of the tentacles remain collapsed, and apparently withered, long after the greater part of their length has become plump, the division between the one and the other condition being abrupt. The distension begins k from the bottom of the tentacle, and passes up very slowly, occupying many hours. When once it has adhered, and recovered its health, its elegant postures and forms, and its remarkable versatility, make the Aiptasia an interesting occupant of the aquarium. It marches from stone to stone, and around the walls of its tank, frequently creeping to the top of the water, and ex- panding its base upon the surface, almost or even quite floating, while the disk and tentacles, widely expanded, are suspended below in mid-water. In these habits we see a close resemblance to Anihea cereus, as also in the texture of the body, and in the tentacles, which in both genera are lined with a profusion of dark-brown pigment-granules, Occasionally I have noticed that it has the power of adhesion to foreign bodies by the general surface of the column ; a habit common to several of the IlyantMdce, (as the Halcampce, for example,) but which, I think, is not possessed by Anihea. When in full vigour it towers up to the height repre- sented in the figure, when, with its ever-twisting tentacles and semi-pellucid tapering column, it is a very elegant object. When thus greatly elongated, the loop-holes are plainly seen with a lens. I have been able to thrust the point of a fine needle into one and another of these orifices, without meeting any resistance ; and, by using great care, without the animal's being conscious of it ; when it did feel the touch, however, it suddenly contracted. Under these and similar irritations, it contracts in length by successive spasmodic jerks, but makes no attempt to roll in the margin of the disk, or to hide the tentacles in any way. Yet it has the power of involving the disk. It feeds greedily, throwing the margin in folds over the mouth. After a full meal, I have seen it take the shape of a ripe THE TKUMPLET. 157 fig, the lower half of the column greatly attenuated, while the upper half was as greatly distended, but with a con- striction between the swollen part and the trumpet-like expanded disk. The appearance of the animal varies exceedingly. Some- times it lies utterly flaccid and withered, appearing as if quite dead ; not contracted, but emptied of its water, and the lax membranes collapsed. Then, especially at night, it swells up, erects its broad disk, and stands up like a flower after a shower, ^-ith a noble appearance. At such times the tentacles are sometimes much distended, pre- serving their regular conical form, and are of a much lighter hue. They are then occasionally constricted with numerous close rings, and take snaky curves. At times the long inner tentacles are curled in ram's-liom coils over the mouth. One of the individuals in my possession has forked tentacles : one of these organs bifurcated at about half its length; another divided near the tip into three, of which one ramification extended on each side horizontally, and the third, which was much smaller, followed the original direction of the tentacle. This tendency is common to Anthea cereus, and to Sagartia viduata. That our Aiptasia is tenacious of life will appear from the following curious rencontre, to which a specimen in the possession of Mr. Holdsworth was subjected. " Two days ago," writes my friend, " on making my customary morn- ing's inspection of my family, I missed the Aiptasia, A diligent search in all the crevices of the rock- work having failed to discover it, I began to suspect foul play ; and after administering the stomach-pump, in the shape of a stick, down the throats of some fine specimens of hellis, I suc- ceeded in dislodging the poor lost sheep, in a shapeless mass of membrane and acontia, which were largely ex- posed ; but the animal was too much injured to enable me to gay whether these were emitted in the usual manner, or exposed by a rupture of the integuments. The invalid was removed to a separate jar of sea-water, (the best hos- pital for sick Actmice,) and it is now attached to the glass, which is, as you know, a good symptom, but I can hardly pronounce it to be as yet quite convalescent." Some weeks recovered from its involuntary visit to the interior of hellis." As in Anthea, the non-retractile character is not absolute in our Aiptasia ; Mr. Holdsworth has repeatedly see» it with the tentacles quite concealed, the body globose and very pellucid, and the orifice long and linear. The only localities I am as yet acquainted with for Aiptasta Gouchn are the following : — Bordeaux Harbour, Guernsey, J. D. H.: Herm, S. Whitchurch: Gwyllyn Vase, Helford Eiver, W. P. C: Falmouth, S. Hodges. The species before us forms a beautiful link of con- senting very marked resemblances to S. lelUs and 8. viduata in the former, while the preponderance of its characters allies it with Anthea cereus. S. bellis. S. viduata. H. chrysanthellum. A. Couchii. [A. rhodora.] A. cereus. 159 GENUS II. ANTHEA (Johnston). Actinia (Ellis). Enfaemcea (Ehbenb.). its outline irregularly undulate. Column forming a low thick pillar; the summit expanding ; the margin notched, and bearing budding tentacles, with no distinct parapet, or fosse. Surface cancellated by the intersection of longitudinal furrows, and transverse wrinkles. No suckers, warts, nor loop-holes. Substance pulpy. Bisk membranous, very expansile, undulate at the margin. Tentacles numerous in several rows, sub-marginal, very long, lax, irregularly flexuous ; scarcely retractile. Mouth elevated on a low cylindrical wart. Acontia wanting. The genus contains but one British species, A. ceretis. THE OPELET. Anthea ceveus. Plate Y. fig. 2; YI. fig. 9. Specific Cfiaracter. Tentacles smooth, consimilar. Actinia cereus. Ellis and Solandeb, Zooph. 2. Rapp, Polyp. 56 ; pi. ii. fig. 3. Geube, Actin. 11. sulcata. Pennant, Brit. Zool. iv. 102. Anemonia edulis. Risso, L'Eur. M^rid. v. 289. Anthea cereus. Johnston, Brit. Zooph. Ed. 1, 221, Ibid, Ed. 2, 240; pi. xliv. Cocks, Rep. Cornw. Pol. Soc. 1851, 10; pi. ii. figs. 23, 27. Gosse, Man. Mar. Zool. L fig. 37. TuG^yELL, Man. Sea-Anem. pi. vii. Anemonia sulcata. Milne-Edwards, Hist. Nat. Corall, i. 233 ; pi. C. L fig. 1. GENERAL DESCRIPTION. Form. Base. Adherent to rocks, but with a very slight tenacity; dilated considerably beyond the medium diameter of the column; the outline generally undulate, often forming irregular lobes. Column. Shaped like a dice-box, or a pillar, which is much dilated above and below; when expanded, the diameter usually exceeding the height ; the margin greatly overlapping, crenate, with numerous rounded teeth, some of which are usually seen to be rising into incipient tentacles. Surface marked with numerous longitudinal furrows, which are correspond- ent with the insertions of the septa, and whose upper extremities alter- nate with the marginal crenations. In the ordinary state of extension, there are also very numerous and minute transverse wrinkles, which cross the furrows at right angles. Skin imperforate, and destitute of any Disk. Thin and membranous, greatly expanded in the form of a broad, shallow saucer, with the margin lax and undulate, often revolute. Radii strongly marked; two gonidial radii often more conspicuous tLaa the others. J THE OPELET. 161 which the first, second, and third contain thirty-six each, the fourth seventy-two. These numbers are, however, only approximative; for the crowded condition of the tentacles, the irregularity of their serial arrange- ment, and the ever-varying distension of the disk, make it almost impos- sible to count, much less accurately to distribute them into rows. They are sub-equal in length ; but what difiFerence there is, is a diminution out- wards. All are very long ; those of the first row sometimes upwards of four inches in length, and more than doubling the diameter of the disk : they are slender, and taper uniformly to the tip, which is obtuse and as if truncate, or sometimes slightly enlarged ; very lax and flexuous, they are almost always thrown about in irregular, snaky curves, intertwisting in all directions. Their entire surface is very adhesive. Mouth. Seated on an elevation, which more commonly takes a cylin.- drical than a conical form ; sometimes lai^e and tumid, at others small : lip rounded. COLOUB. CMttmn. Pale wood-brown, mnber-brown, pmrplieh-brown, or fiesh- colour, marked with numerous narrow bands alternately paler and deei)er, which correspond to the furrows ; sometimes the lighter bands are dull light lilac, with darker edges. Ditk. Dark bistre-brown, or ambra>brown; the gonidial radii often a lighter shade of the same colour. TcTitadei. Light pea-green or emerald-green, opaque, with a rich, satiny lustre; the extreme tips, for about one-fourth of an inch, rich lilac-crimson ; tiie green gradually blending into the lilac, and the latter hue increasing va brilliancy to the extremity. A faint whitish line usually runs along the back of each tentacle throughout its length. Mouth. Lip agreeing with the disk ; throat ash-brown. Size. Large specimens are sometimes seen covering an area of six inches in diameter, with their tentacles four inches long ; the disk two inches, and the column the same, in diameter. LOCAUTT. The western and southern coasts of Europe generally. Shallow pools between tide-marks, and littoral rocks. Yabietixs. a. Smaragdina. The state described above, with rosy-tipped green tentacles. )6. Sulphiirea. As the preceding, except that the tentacles are pale delicate lemon-yellow, with the slightest shade of green ; lilac-tipped. •^Herm : S. W. Ventnor.) In the Herm specimen, the tentacles were scarlet at the foot. y. Alabastrina. Column and disk light translucent olive; tentacles wholly clear waxy white. (Ventnor. Torquay.) S. Rustica. Column and disk dull brown ; tentacles ash-grey, generally with a paler line down the back. c. Ptinicea. Tentacles mahogany-red. {Gaertner.) Anthea cereus is one of our most abundant species, at least on the south and west coasts of England and Scot- land, and probably all round Ireland. Rapp and Grube indicate it as common in the Mediterranean and Adriatic seas ; but the omission of any allusion to it by Miiller or by Sars implies that it is unknown in the North Sea. Its abundance where it occurs, its habit of congregating in numbers, and its favourite resort, — shallow pools within tide-marks, protected only by a few inches of water from the full glare of the sun, as well as its size and conspicuous colours, — all conspire to make it familiar to the most cur- sory observer. It would, probably, be one of the first species of the whole race to become popularly known ; and hence it is not surprising that old Rondeletius should take notice of it in the middle of the sixteenth century, includ- ing it in his "Libri de Piscibus Marinis," by the descrip- tive epithet of Urtica cinerea. The late Dr. Johnston separated the genus from Actinia in his " Brit. Zooph." Ed. 1 ; giving it the noinQoi Anthea, from avdo<i, a flower. The specific name of cereits seems to have been appropriated to it in accordance with a fancy which Ellis had of naming the Actinioids after many- petaled flowers, — cereus being the name of one of the Cacti, now a genus. The waxy appearance of the tentacles in some of the varieties may have influenced him in the selection. The English name I have formed for it alludes to the habitually open condition of the disk. THE OPELET. 163 This is the species, doubtless, which attracted the notice of the poet Southey, when, in the retirement of our wild western shores, he was meditating his oriental poems, and which he has interwoven into their beautiful imagery. " Meantime, with fuller reach and stronger swell. Each following billow lifted the last foam That trembled on the sand with rainbow-hues : The living flower, that, rooted to the rock. Late from the thinner element Shrank down within its purple stem to sleep, Now feels the water, and, again Awakening, blossoms out All its green anther-necks." * Whether in its native freedom, fringing the edges of some shallow basin in the red sandstone of the Devon coast, or waving its silky tentacles " like streamers wide outflowing," now exposed, now concealed among the black fronds of some undulating Fucus to which it is clinging ; or throwing them into fitful snake-like contortions as it hangs from the rock-work of a well-kept aquarium, — the Anthea, especially the emerald variety, is an exquisitely beautiful object. Its imwonted liveliness also makes it more than usually interesting in the last-named condition ; for not only are its tentacles continually in motion, but the animal itself is very restless, frequently changing its place, and that with so much activity that the process can be For the following graphic note I am indebted to Mr. Eobert Patterson, of Belfast. I also have often marked the beauty of Anthea under similar conditions, but never in such numbers as he describes : — " I had on one occasion the pleasure of seeing the Anthea under circumstances that I shall not readily forget. Out- ♦ Thalaba, xil 3. M 2 side the belt of sand and rocks that is left uncovered at every tide [on the south side of Belfast Lough], is another, where the large sea-weeds, such as the tangle and sea- furbelows {Laminaria sp.), flourish. ... As our boat drew nigh to the shore, the large spreading fronds of the sea- weed became more and more distinct, until each was per- fectly revealed to us, below the unruffled surface of the sea. We had come at the time of low water ; and, as we floated onward, could mark the glorious submarine forest which was beneath our boat. It rose and fell, it heaved and sank, as gracefully as the meadow yields to the breeze, or as the willows bow to the breath of April. As we came weed seemed studded with blossoms. What could they be? A few moments more disclosed the mystery: each blossom was endued with life and motion — it was a living Anthea!'' The power exercised by this species, pre-eminently, of inflating portions of its body, swelling them out in large tumid lobes separated by deep sulci from the rest of the circumference, assists it in crawling. We will suppose the Anthea resting on the bottom of the vessel, when it feels a desire to mount the sides of the glass. Pushing out a great inflated lobe towards that side, the sole of which is free from the surface, it takes hold of the glass with the edge of the lobe ; and when the contact is firm, relaxing its former hold, it slowly drags forward the body, until the lobe is again lost in the general circumference, or even till the body projects in two smaller lobes, one on each side of the principal one. The base being now made firmly to adhere, again the lobe is freed, and again protruded, and the same process is repeated until the animal is satisfied with the position it has gained. Sometimes this is at mid- height, the intertwined tentacles streaming loosely down THE OPELET. 165 by their own weight. At other times it rises to the very- water's edge, and even thrusts out its base in an inverted position upon the surface of the water, as if it would float by the mere contact of the dry base with the air, just as the LimnecB and many other Mollusca do. And not seldom does it boldly break the tie that connects it with the side of the vessel, and actually swim, or at least passively float, with its base in contact with the inferior surface of the superincumbent stratum of air. A little shaking of the vessel, however, causes the water to overflow the frail boat, which had been hitherto dry, when the animal instantly falls prone to the bottom. No very special care is required to maintain the health and vigour of the Opelet in captivity : as to situation, it will select for itself the position its wayward will may most fancy ; and if the water be kept in purity, the lovely crea- ture wiU survive an indefinite period. It needs to be fed at frequent intervals, or it will droop and die ; for it is one of the most voracious of its class. Nothing in the way of flesh or fish comes amiss to it : a day or two ago I had an instructive example of its gluttony. I had just dropped two large ones of the variety Smaragdina into my col- lecting-jar, when I succeeded in capturing a young Conger Eel, about six inches in length and half an inch in thick- ness. I wish that the sciolists who deny a poisoning power to the organs of the Actinioids had seen the result of the introduction of the lithe and vigorous fish to the expectant Anthece. Before it could reach the bottom of the jar^ the green tentacles of one of the Opelets had entwined themselves around its head, and, wrapping the wretch around as if with a cloth, almost in an instant had dragged it to the cavernous mouth, in which it was partially engulphed. My little son, who was with me, begged for the life of the fish ; and I drew it by force from the greeiji embrace, in less than five minutes after its capture. But — de eo actum est! it was all up with the poor Eel; its eyes were already dimmed in death, and it lay in my hand flaccid and helpless, with only a momentary convulsion or two ; — the fatal cnidce of the tentacles had done their work : and when I restored it to the offended gourmand, it was speedily lost to view, coiled up in the capacious maw. Numerous witnesses vouch for the fact, — though others, myself included, are insensible to it, — that the contact of Anthea's tentacles has a perceptible morbific power on the human skin. One of the most distinct statements of the fact that I have met with is contained in the following communication, for which I am indebted to Miss Pinchard, an accomplished naturalist of Torquay : — '* I have myself been repeatedly so affected by their clinging to the back of my hand as to have the skin mottled, and so tender as to induce me to refrain from willingly coming in contact with them. On one occasion the whole of the back of the hand and fingers was covered with white blisters, as if I had thrust it into a bed of nettles, and nearly as painful. The affection did not last above an hour or two, and only occurred when the AntJieas had become flaccid and feeble, as they often do after a short captivity. I have never found any effect arise fiom handling them when they were in an active and healthy state.' Mr. Dana attributes to the kindred species, A. flagelli- fera, a power of making its terrors known even at a distance. " Having a number of Monodontas [a genus of univalve Mollusca allied to our Trochi] too much crowded in a large jar of water, I took out half-a-dozen and placed them in a jar with the Actinia. On looking at them about three hours after, I found that, instead of climbing like the others to the top of the water, they remained just where THE OPELET. 167 they had fallen, closely withdrawn into their shells. Sup- posing them dead, they Avere taken out, when they directly began to emerge ; and Avhen returned to the jar with the other Monodontas, they were all in less than five minutes clustered round its mouth. On placing them again in the jar with the Actinia, though kept there for two hours, they did not once show themselves out of the shell. Once more placing them along with the other shells, they exhibited their former signs of life and activity. The experiment was repeated several times with a large Littorina, with the same result, evincing fear of the Actinia on the part of the Mollusks."* I can only say that Trochus umhilicatus, Littorina littorea, and Chiton fascicidatus have no such iesoc oi Anthea cereus^ for I have just seen these crawl without hesitation by the side of a full-grown and vigorous specimen. Though sensible pain or irritation does not invariably follow the contact of the human skin with the tentacles of Anthea, yet their strong power of adhesion is never lacking. Dissection reveals the cause of both, in the unwonted pro- fusion with which these organs are famished with cnid^R. In the outer of the two layers of which the tentacle- wall is composed reside the en idee, excessively numerous and thickly crowded ; of two kinds, chambered and sjnraL But it is in the crimson tips that the cnid(e exist in the most prodigious profusion. They completely fill the field of the microscope, when a portion of the wall, flattened by the compressorium, is under view, without the least ^ace free of them, not even a line or a point ; but overlying each other like herrings in a barrel, yet maintaining a general uniformity of direction. Within the cnidiferous layer, there is another of pig- ment cells, visible to the naked eye as a dark brown or * Zoophytes, p. 12P. nearly black lining, which can be readily pressed out from a wound in the tentacle. These granules are very regu- larly globular, of a translucent golden-brown hue by trans- mitted light, varying in diameter from '0003 to '0004 inch, and are arranged in bead-like rows running transversely. This pigment-layer does not give the green hue to the tentacle ; for it may be entirely scraped away, leaving the interior surface of the tentacle-wall of the same opaque emerald-green hue as the exterior. This green tint does not appear to be dependent on pigment, but on the arrangement of the primary molecules of the sarcode ; for when pressed to flatness, it yields no transmitted colour, except a very slight yellowish tinge which has no distinct location. It presses to a viscid glaire, full of amorphous refracting granules, and cnidce. The tip exhibits similar phenomena, but the diffused tinge is faintly purple. The larger Eolides tear away and devour the tentacles of Antliea: but I know not of any other animal that can venture on attacking it with impunity. I one day saw an amusing example of its power of passive resistance. A beautiful little specimen of the variety Alabastrtna, which remove from one tank to another. There was a half-grown in captivity rather more than a fortnight. As he had not been fed during that time, I presume he was somewhat sharp-set. He marked the Anthea falling, and before it could reach the bottom, opened his cavern of a mouth and sucked in the bonne louche. It was not to his taste, how- ever; for as instantly he shot it out again. Not discou- raged, he returned to the attack, and once more sucked it in, but with no better success ; for, after a moment's rolling of the morsel around his mouth, out it shot once more; THE OPELET. 169 and now the Bullliead, acknowledging his master, turned tail, and darted into a hole on the opposite side of the tank in manifest discomfiture. But if you, my gentle reader, be disposed for exploits in gastronomy, do not be alarmed at the Bullhead's failure : only take the precaution to " cook your hare," Eisso calls this species " eduUs," and says of it, — " On le mange en Jri'fure" and I can say ^'probatum est." No squeamishness of stomach prevents our volatile friends, the French, from appreciating its excellence; for the dish called Rastegna, which is a great favourite in Provence, is mainly prepared from Anthea cereus. I would not dare to say that an Opelet is as good as an Omelet ; but chacun h son ga&t ; try for yourselves. The The species not unfrequently increases by spontaneous division. I have elsewhere* given the details of a case of this sort; since the publication of which I have received from various correspondents accounts of the same pheno- menon. The fission begins at the margin of the disk, and gradually extends across and downward, until the separa- tion is complete, when each moiety soon closes and forms a perfect animal. It is, perhaps, only another phase of the same tendency, that the tentacles are frequently forked. Anthea cereus has been observed in the following British localities : — Jersey, G. G. : Guernsey, E. W. H. H. : Herm, S. W. : Ventnor, G. G. : Weymouth, P.H. G. : Lyme Regis, J. G. : Dawlish, R. C. J. : Teignmouth, R. C. J. : Torquay, P. K G. : Falmouth, W. P. C. : Fowey, C. W. P. : Pen- zance, R. Q. C. : Scilly, G. H. L. : Ilfracombe, P. K G. Tenby, P.H. G.: Holyhead, E. L. W. : Man, F. H. W. Cumbrae, D. R. : Oban, J. C. G. : Ballyholme Bay, W. T. Newcastle (Co. Down), W. T. : Portrush, E. P. W. : Dublin * Tenby, a Seaside Holiday, p. 373. Bay, E. P. W. : Carnsore Point, E. P. W. : Clew Bay, W. T. A. Coucliii. A. dianthus. cereus. A. mesembryanthemum. [flagellifera.] [A. flava.] • B. Tuediffi. [ — ? pustulata.] The curious little Actinoj)sis flava of MM. Danielssen and Koren,* which appears to be a near ally of this genus, is remarkable for having the two gonidial tubercles greatly prolonged into semicylinders, and terminating in two points. It closely resembles in other respects a small Anthea, and is of a yellow hue. As it has occurred in deep water (250 fathoms) off the southern end of Norway, it may reason- ably be looked for on the opposite coast of Scotland, and in the fShetland Seas. * Fauna Litt. Norveg. ii. 89. 171 The species of this family, though very few in number, are well marked by the single character of being famished with those peculiar organs which il. ]iIilne-Edwards calls (not very felicitously) bourses chromatophores, or tuhercuUs caltcinaux, and which I have named marginal spherules. These are hollow spherical vesicles, with thin walls, situ- ated near the edge of the disk, on the inner side of a sharj) margin, and outside the exterior row of tentacles. For the most part, if not always, these organs are of bright or vivid colours, generally differing from those of the other parts ; and hence they are conspicuous, and impart a peculiar aspect to the physiognomy. What fimction in the economy of the animal is per- formed by these bead-like spherules is as yet unknown, though that they play some important part can scarcely be doubted. In our Actinia mesemhryanthemum, I have ascer- tained that the walls are almost wholly composed of cntdce, of nearly linear form, and about •0025 inch in length. The inclosed thread is with difficulty seen, both before and after extrusion ; it is, however, of considerable length. From this structure I have conjectured that the marginal spherules in this family may represent — functionally, not homo- logically — the acontia of the SagartiadcB, which are here wanting. Sir John Dalyell has an extraordinary observation to the effect that each of these spherules " is pierced by an orifice, which opens and dilates occasionally, some time after the animal has fed." * This fact, however, if fact it be, is confirmed by no other observer that I am aware of. The integuments of the column seem to be imperforate : this is certainly the case in the genus Actinia; and in Phymactis, though the evidence is of a negative character, there is no reason to believe that it is otherwise. The character of the surface varies according to two very dis- tinct types. In Actinia it is remarkably smooth, soft, and fine ; in Phymactis it is roughened with strong and coarse warts. These diversities manifest the osculant position of the group ; for while the former genus , shows a close affinity with the Antheadce, the latter takes no less firm a hold upon the Bunodidce. It is interesting to find an exotic species (the A. primula of Drayton f) with marginal spherules and a smooth skin, which emits long filaments from the mouth. Here, then, we have the ■ representative As regards Geographical Distribution, the Family is extensively spread ; the two principal genera representing it respectively in the northern and southern hemispheres. Actinia ranges from the Red Sea, through the Mediter- ranean, over the western coasts of Em-ope, and the isles of the North Atlantic. Phymactis is widely distributed over the shores of both sides of the South Pacific, and of the South Atlantic, reaching a little way north of the Equator, being represented by no less than three species at the Cape de Verd Isles, where, it is curious to observe, it meets • Rare Anim. of Scotl. ; 203. + Dana, Zoopli. 134 : pi. ii, figs. 12 — 15. At least it is thus represented in one of Mr. Dana's beautiful figures, though no allusion is made to the peculiarity in the text. M. Milne- Edwards has made of it his genus Nemactis, but with a wholly gratuitous assumption of characters. with the beautiful representatives of the northern form, — Actinia tabella, and A. graminea of Dana. ANALYSIS OF THE GENERA. Skin smooth. Possessing acontia {Not BritvJt) Nenutetu. Destitute of acontia Actinia. Skin waited {Xot British) Phymaetia, 174 GENUS L ACTINIA (Linn.). Eniaemcea (Eheenbkeg). JBase adhering to rocks ; considerably exceeding diameter of column. Column pillar-shaped, usuall}'- much wider than high ; margin greatly developed, smooth, separated by a broad, but shallow fosse from the outer tentacles ; a circle of vividly coloured spherules projecting from the inner surface of the wall of the fosse; surface delicately smooth, imperforate, non-adhesive ; sub- stance fleshy. Dish greatly expanded and overarching ; concave. Tentacles in several rows ; moderately long ; nearly equal ; unicolorous ; wholly retractile. Mouth set on a protrusile cone ; two pairs of goni- dial tubercles, brightly coloured. We possess but a single British species, A. meseni- iinvtiltpvnit.wi hryantliemum ASTR^ACEA. Actinia mesemhryanthemum, Plate YL figs. 1 — 7. Specific Character. Colours of column not arranged in transverse zones. A etin ia equina . metembryanthemum. Iiemisphterica. rvfa. purpurea, corallina. margaritifera. Porskdlli. cei'a^um. ehioeocca. 1 tahdla. fragacea. ? graminea. Entacmcea mesemhrrinn- themum at E. ru/a. Linn., Syst. Nat. 1088. Mulleb, Zool. Dan. Prod. 231. Milxe-Edw., Corall. L 238. Ellis and Sol., Zooph. 4. Rapp, Polyp. 52 ; pi. ii. fig. 1. Grube, Actin. 10. CoccH, Com, Faun. iiL 74 ; pi. xiv. fig. 1. Johnston. Brit. Zooph. Ed. 2. i. 210 ; pi. xxxvi. figs. 1 — 3. Dalyell, Rare Anim. of Scotl. ii. 203 ; pi. xliii. and xlvii. fig. 1. Cocks, Rep. Comw. Pol. Soc. 1851. 5. pi. L figs. 7—11, 15. GossE, Aquarium, pi. iL : Tenby, 3/0 : Linn. Trans, xxi. 274 : Manual Mar. Zool. L 30 ; fig. 43. TuGWELL, Man. Sea-Anem. 52, Pennant, Brit. Zool. iv. 104. Ibid. Ibid. iv. 105. Mulleb, Zool. Dan. i. 23 ; pL xxiiL figs. 1 — 3. Lasiakck, Anim. s. vert. iii. 67. RoGET, Bridgew. Tr, i, 198; figs. 86, 87. CuviER, TabL fldm. 653. Risso, L'Eur. M^r. v. 285. Templeton, Mag. Xat. Hist. ix. 304 ; fig. 50. JOHNST. Br. Zooph, L 213 ; fig. 46. Cocks, Rep, Comw. Soc. 1851, 5, M-Edwakds, CoralL 241, Dalyell, Rare Anim. Scotl. iL 219 ; pi. xlvL fig. 1. Cocks, Rep. Comw. Soc. 1851. 5. pi. L fig. 14. JoHNST. Br. Zooph. i. 214; pL xxxvi. figs, 4—6. Dana, Zoophytes, 132 ; pL ii fig. 9. TuGWELL, Man. Sea-Anem. 53 ; pi. 5. Dana, Zooph. 132 ; pi. iL fig. 10. Ehbenb, Corall, Roth. Meeres, 36. GENERAL DESCRIPTION. Form. Base. Adherent to rocks ; considerably exceeding the column, outline often long-oval. Column. Delicately smooth, without much excretion of mucus, wholly imperforate, and non-adhesive. Substance fleshy, approaching to pulpy. Form hemispheric in button, a low column in flower, much expanded at the summit. Margin strongly developed, with a smooth, sharp edge, bounding a wide but shallow fosse, within which are seated a single series of numerous spherules. Disk. Slightly concave, smooth ; the radii faintly marked. Tentacles. About two hundred in full-grown individuals, arranged in six rows thus : — 6, 6, 12, 24, 48, 96:= 192 ; moderately slender, shorter than the diameter of the disk, sub-equal ; flexuous, usually carried arching over the margin. Mouth. Elevated on a blunt cone. COLOUK. Base. Edged wit h a narrow line of bright blue. Column. Liver-brown. Marginal Spherules. Brilliant azure. Bisk and Tentacles. Dull pellucid crimson. Mouth. Rich crimson. Gonidial Tubercles. Blue. Size. Large specimens sometimes cover with their ,base an area four inches long by two wide, attain a height of about an inch, and expand to a flower of three inches in diameter. Locality. The Mediterranean and Atlantic shores of Europe, universally distri- buted, on exposed rocks, from half-tide, or even a higher level, to low- water mark. Varieties. The characteristic colours of the species are crimson and green. The extreme of variation on either hand is produced by either of these two colours prevailing so as to exclude the other. But many intermediate grades are found, either by the blending of the two hues into some inter- mediate tint of olive, brown, or liver-colour, or else by the separation of the two into a pattern of spots on a different ground, or, where the green hue exists alone, by a separation of its constituent elements, blue and yellow. We may distinguish the following varieties : — a. Hepatica. The liver-brown condition above described, which is the most common (fig. 2). * Approaching the red. j3. Rubra. Column dark crimson ; disk and tentacles as before. In youth this and the following variety are of a pellucid light crimson (fig. 5). 7. Chiococca. Column rich scarlet; basal line flesh-colour or non- apparent ; disk and tentacles full crimson ; spherules pure white (fig. 7, labelled A. chiococca). The A. ForskdUi of the Red Sea, the A. ceramm, of the Scottish Coast, the A. chiococca of St. Ives and other parts of Corn- wall, must be conaidered as belonging to this variety ; nor can I separate from it the A. tabeUa of the Cape Yerd Isles, except that this approaches the var. p. Approaching the green. 8. Umbrina. Column, disk, and tentacles, a yellowish umber-brown ; spherules (as in all the following) azure; basal line (as in all of this section) blue (fig. 3). «. Ochracea. Column, disk, and tentacles orange-bufiF. f. Olivacea. Dark olive. ij. Glauca. Pellucid bluish green ; tentacles pale greenish blue (fig. 1). 0. Prasina. Fine leek-green ; tentacles the same, pellucid. * Colours interrupted. u Opora. Leek-green, with longitudinal broken lines of light green or pure yellow ; spherules and basal line blue (fig. 4). K. Tignna. Red, streaked with yellow (Tcgwell). X. Fragacea. Liver-coloured, or dark red, studded with nmnerous spots of light green ; no basal line. Attains a very large size (fig. 6). The most marked of the above varieties is undoubtedly the last, — the Strawberry, as it is familiarly named. Its constancy of colour and pattern, its tendency to an ovate form, and its great size, distinguish it from its fellows ; and yet I cannot, after much consideration of the subject, in the presence of the animals themselves, convince myself that it is entitled to specific distinction. I have found specimens in which the spots were small and crowded, others in which they were large and scattered, others in which they were small and scattered ; sometimes the spots are portions of lines irregularly interrupted, and not seldom considerable regions of the surface are quite des- 17^ titute of spots. The marginal splierules are sometimes large, sometimes minute; now azure, then pearly white. A more marked character is the absence of the coloured line bounding the base; but I am not sure that this is constant. I am glad to fortify my own opinion by that of so acute an observer as Mr. Holds worth. He writes me as follows : — " I have now seen so many connecting links between the typical mesemhryanthemum and the ft'agacea, so called, that I am convinced they are one and the same species; although I have not arrived at this conclusion without devoting considerable time and attention to the subject." Of the supposed species, chiococca, cerasum, and Fors- kdlli, for these are assuredly all the same thing, I would speak with some deference, owing to my having never seen the form in its perfect type, though I have no doubt of its identity with the present subject. Sir John Dalyell, though he gave it a specific name, summed up his obser- vations with the following words : — " On the whole, I am disposed to view it as a variety of mesembryanthemum." Nor do I see how he could do otherwise ; for he tells us that, of his cerasum, which was very prolific, all the young were red hut one, which, red at first, became at five months old ^aZe green. This bred, and all its progeny were green ; though it had upwards of a hundred descendants before it was two years old, and continued to breed for five years more. It is but fair, however, to add, that Mr. W. P. Cocks, "who constituted chiococca a species, and to whom I am indebted for the beautiful drawing which I have copied in my Plate VI. fig. Z, retains his opinion. From one of his letters to me, I cite the following interesting notes : — ' The A. chiococca is certainly a good species. I have never found it associated with the A. mesemhryanthemum, and rarely more than one or two in the same locality (though explored .by me in Cornwall), with one exception. On the under surface of some very large stones used for making a pier near the north-western extremity of the town of St. Ives, I found several colonies of the in- teresting creatures in fall health, enjoying the blessings of freedom in a nook not often disturbed by anything but the rough and boisterous waves from the North Channel. About twenty feet from this spot, and nearer high-water mark, the under surfaces of the stones forming a portion of this abortive construction were covered with old and young members of the beautiftd varieties of the A. mesem.' hryanthemum, dark bottle-green with yellow dots, dark green with yellow stripes^ claret with yellow spots, yel- lowish green, light ochre, amber, scarlet, &c. The blue beaded rim and blue fillet at base were displayed by each member of this group. A specimen of the A. chiococca, which I had in confinement for more than twelve months in my experimental jar, furnished me with a batch of young ones, — all were true to colour and markings." This, how- ever, can by no means outweigh the positive evidence on the other side furnished by Sir J. Dalyell. Nor can the A. inargaritifera of Templeton be allowed any higher rank. The flattened, rigid, corrugated con- dition on which he relied for a specific character, I have not unfrequently seen in individuals, which, in the course of an hour or two, were swollen out to the softness and plumpness normal to the species. Mr. Cocks comes to my aid here with an interesting narrative of two specimeiis which he found in a condition exactly cosi^sponding to Mr. Templeton's description of margaritifera. He was at once convinced that sickness was the cause of their pecu- liar flatness and attenuation, and the shrivelled tesselated n2 180 ACTINIA D^. character of their skin. He treated them accordingly, and in a few days they assumed the usual plump condition. Facts seem to show that even the same individual is liable to considerable change of colour. Mr. Cocks tells me that from some hundreds of experiments he has ascer- tained that " the colour is materially changed by diet, good or bad ; by water, pure or impure ; by attention or neglect; by over- feeding or starvation." And Mr. E. L. Williams, jun. has favoured me with still more precise statements on this very species. He observes : — " A. me- sembryantkemum does change. Bright green in two months has got to dark olive in my tank ; bright amber to dark brown ; brown with vertical yellow spots or dots has lost these markings." Characteristic as are the marginal spherules, they are subject to some irregularities. I found a large specimen of the deep olive variety, which had on the exterior of the margin two azure tubercles; — one of them round, well defined, and in no respect distinguishable from the intra- marginal spherules, — the other somewhat less so. Below these, scattered down the side of the column, were four or five more blue warts ; more irregular in form and shape, but still well defined, and perfectly similar in their azure hue to the spherules. I subsequently obtained a second specimen with exactly the same peculiarities . On the other hand, a specimen of the same variety — which was sent me from Cumbrae by Mr. D. Robertson about six months ago, and is still in my possession — has never 'showed the slightest trace of spherules, though in every other respect perfectly normal; the basal line and the gonidial tubercles being of the usual azure hue. It is * " M. Haime has remarked that these bourses chromatophores, or calycine tubercles, are to the number of 18 in those individuals which have not yet developed the tentacles of the 5th cycle ; of 24 in those which have 5 or 5 J cycles, and of 48 in those which have 6 cycles com- hj no means unusual to see examples of the red varieties, in which the spherules are pale red, — the blue pigment being defective. The name Actinia, originally applied to the whole race of Sea- Anemones, is derived from dicTlv, a ray ; the specific appellation, mesembryanthemum, is the name of the fig- marigold, so called from its opening at noon, [fieaijfi^pca, = fi€(ro^, rjfiipa, mid-day) : the term headlet alludes to the As no species is more abundant, nor more easily pro- cured than this, since it afiects the most exposed rocks, and does not seek the protection of hollows, so none is more easily reconciled to captivity, and few are more beautiful. It requires no special treatment ; a surface for the support of its base, and water sufficient to cover it, are enough ; nor is it essential to its existence that the latter should be very pme, for it will continue to drag on life when its fellows have died out. Yet few species more immediately resent negligence of this kind, or more grate- fully express their appreciation of a pure and limpid element. Widely as the species is distributed in a state of freedom, we scarcely ever see it except where the water is habitually clear. It is a curious fact, for which I am indebted to Mr. E. L. Williams, jun., that " the Mersey estuary is the only place on our coasts in which he has ness of the water. This absence would be less remarkable, were it not that Tealia crassicornis is abundant there ; but Actinia is clean and Tealia is dirty in it? habits. In the plete; that is to say, in the large individuals where 192 tentacles or there- abouts may be counted. He has recognised also that these pouches communicate directly with the sub-tentacular chambers of the first cycles ; and that they contain little muscular fibre, but carry navicular thread-cells of various forms, and of which the interior thread is indistinct, together with transparent vesicles, and pigment-globules." —Milne-Edwards, Hitt. Corallaires, i. 240. neighbouring estuary of tlie Dee, the former is common, as usual. With ordinary attention the pretty Beadlet will attain a good old age in captivity. A veteran, whose portrait is given by Sir John Dalyell, had lived in his possession twenty years (in 1848), and was judged to be not less than seven years old when he obtained it. At Sir John's death the specimen passed into the hands of Professor Fleming, and it was not many months ago that I heard of it as still surviving. If it is alive now, it must be approaching forty years old. This individual was the prolific parent of 334 children. A second specimen had lived about fourteen years under the worthy baronet's care. The species is generally viviparous, producing abun- dantly ; but sometimes it gives birth to ciliated, shapeless embryos, on which tentacles appear in about ten days. Copious details of high interest on the embryology and general economy of this Anemone are furnished in the magnificent volumes of the eminent Scottish naturalist. It is superfluous to give a list of habitats for this species: since it occurs all round the coasts of England, Wales, Scotland, and Ireland, wherever there ,4s rock enough to afford it standing ground. The Actinia Cari of Delia Chiaje (the A. concentrica of Kisso) appears to be a second species of the genus ; at least in none of the recognised varieties of ours do we perceive an approach to the pattern of colouring, — a series of concentric zones or bands, — by wliich that is marked. A. cereus. Mesembryanthemum. [Nemactis.] [Phymactis.] Actinoloba. Bunodes, 183 FAMILY v.— BUNODIDJS. 1 propose to include in this family all those species, the surface of whose column is studded with persistent tuber- cles, and which are not provided with marginal spherules, nor with perforations of the integument. In some instances, certainly, — perhaps in all, — these excrescences have the faculty of adhering with force to foreign bodies ; and thus they agree in function with the suckers of many of the Sagartiadoz ; there is this difference, however, that whereas in those, the margins of the suckers do not rise above the general level when inactive, in these the tubercles are always well developed, and are particularly prominent in those species in which the adhesive function, if it exists at all, is feeble and rarely exercised. The integuments and muscular coats appear to have a much greater density than in any of the previous families, and the movements of the animals manifest a higher degree of vigour, and even of intelligence. The tentacles are generally short, thick, and conical. The typical and sub-typical genera — Bunodes and Tealta — appear to be represented by species which are scattered over the seas of the world, and are for the most part littoral: the genera Cystactis and Echinactis are confined to the southern hemisphere : and the aberrant genera, Bolo- cera, Hormathia, and Stomph'a, inhabit the deep water of the British and Norwegian seas. 184 ANALYSIS OF THE GENERA. Tubercles conspicuous. Disk and tentacles retractile. Tubercles of one kind only. In the form of rounded warts. Irregularly scattered Tealia. Arranged in vertical lines .... Bunodes. Arranged in many horizontal lines (Not British) " A.fusco-rubra.' Arranged in a single horizontal line . Hormathia. In the form of pointed blisters {Not British) Cystactis. Tubercles of two kinds, viz. rounded warts and erectile pointed papillae {Not British) . . Echinactis. Disk and tentacles not retractile Bolocera. Tubercles obsolete Stom/phia. 185 GENUS I. BOLOCERA (Gosse). Anthea (Johnston). Base adherent : not much exceeding the column. Column pillar-like, the diameter and height sub- equal. Surface generally very smooth, studded with small warts, remotely scattered. Substance "fibro- cartilaginous" {JF.P.C). Disk smooth, circular in outline, not overlapping the column. Tentacles short, thick, constricted at foot, obtusely pointed, longitudinally furrowed ; flexuous and motile ; easily separated ; not retractile. Mouth not raised on a cone ; stomach capable of being greatly protruded. There is but a single known species, B. Tuedia. ASTRjEACEA. BUNODIDJi. THE DEEPLET. Bohcera Tuedice. Plate V. fig. 1 . Specific Character. Body dull red ; tentacles chestnut. Actmia Tuedice. Johnston, Mag. Nat. Hist. v. 163 ; fig. 58. Anthea Tuedice. Ibid., Brit. Zooph. Ed. 2. ; i. 242, fig. 53. Landsborouoh, Scott. Chr. Her. 1840, 243. Cocks, Rep. Cornw. Pol. Soc. 1851, 11 ; pi, ii. fig. 33. GossE, Ann. N, H. ; Ser. 3. i. 416. GENERAL DESCRIPTION. Form. Base. Adherent, scarcely exceeding the column. Column. Cylindrical, smooth and as if polished on the general surface, but studded, somewhat sparsely, with minute rounded warts, which are scarcely apparent when the animal is extended, but, on contraction, " re- semble the heads of small pins in a pincushion" ( W. P. C.) ; in this condi- tion the smooth surface is thrown into transverse wrinkles. Substance firm and sub-cartilaginous. Disk. Flat, smooth, without conspicuous radii; outline circulai", not exceeding the column. Tentacles. Numerous, in three rows, close-set ; the innennost remote from the mouth, somebimes two inches in length, and half an inch in diameter ; the other rows diminishing in gradation ; stout, constricted at the foot, then swollen, and tapering to an obtuse point, which is perforate ; marked with longitudinal sulci, which are obliterated when the tentacle is completely distended ; very flexuous and motile ; readily detached, and retaining their irritability and worm-like motions long after the separation. They cannot be retracted within the column, nor are they capable of any considerable elongation or contraction. Mouth. Not raised on a cone. Lip apparently not thickened. Stomaph- wall capable of being protruded in the form of great bladder-like lobes. Colour. Column. An xmiform deep flesh-colour, reddish, or brownish-orange. DisJc. A lighter tinge of the same. Tentacles. Chestnut or reddish flesh-colour. Stomach. When protruded, reddish with paler lines. THE DEEPLET. 187 Size. Three or four inches in height, and from five to eight inches in diameter, when expanded. LOCALITT. Deep water off rocky coasts, from &ftj to two hundred fathoms. It will be evident from the above-mentioned characters that this form must be considered as genericallj distinct from Anthea. It is, in fact, intermediate between that genus and Tealia ; with a preponderance, however, of the features proper to the latter, which has induced me to assign it to this family. In this judgment I cany the con- currence of Mr. W. P. Cocks, who has enjoyed more opportunities of studying it in life than any other naturalist ; and to whom I am indebted for the carefdlly coloured drawing which embellishes my Plate V. as well ag for some interesting notes. Notwithstanding its great size, and somewhat inelegant form, Mr. Cocks calls it " a charming creature ;" and says on another occasion, "this is certainly a beautiful animal when healthy and half-grown ; though the queer move- ments of the peristome and lobed mouth, pouting like an old man with negro lips and toothless jaws, at once pronounce its relationship with crassicornis.^^ It is essentially a deep water species : Messrs. Danielssen and Koren ascribe it to the coralline zone off the coast of Norway, from thirty to fifty fathoms, and, on the authority of Mr. Sars, mention it as ranging to the amazing depth of two hundred fathoms. On the Cornish coast, it is not seldom found among trawl-refuse ;t and Dr. Johnston tells us that in Berwick Bay it occasionally occurs at- tached to the deep-sea lines of the fishermen. " I have often found," he remarks, " the tentacula in a separated * Faona lati. Norv. u. 87. t Cocks in litt. 188 BUN0DIDJ5. state adhering to their lines ; and, as these retain their irritability and motion for a long time, they are apt to be mistaken for independent and perfect worms, which they much resemble." * I have seized so unusual a peculiarity as the ready parting with the tentacles, to create a generic appellation, — Bolocera, from ^dk\(o, to cast, and Kepa<;, the horn. The word Tuedice was applied to the species by Dr. Johnston, because Tuedia was the ancient name of the maritime parts of Berwickshire. The English term I have formed in allusion to its habits. With the exception of some extraordinarily gigantic specimens of A. dianthus, this is the largest of British Anemones. The following are its recorded localities. Peterhead, C. W. P: Berwick Bay, G. J. : Cullercoats, J. A. : Falmouth, W. P. C. : Cumbrae, D. L. A. cereus. TuEDIiE. T. crassicornis. » Br. Zooph. i. 243, 189 GENUS II. BUNODES (Gosse). Actinia (Ellis). Oribrina (Ehbsxbebo). Base exceeding the column ; its outline generally undulate. Column pillar-like; the height in extension consi- derably exceeding the diameter. Surface studded with permanent rounded warts, set in vertical lines, which are separated by bands of plane skin. Margin denticulate. Substance firmly fleshy. Disk flat, circular in outline ; scarcely overlapping Tentacles not very numerous, arranged in several rows, submarginal; moderately long and slender, obtusely pointed, smooth, not very flexuous ; marked (in the more typical species) with irregulai* white spots on the front face ; perfectly retractile. Mouth not raised on a cone; stomach not habi- tually protruded : gonidial tubercles generally conspi- cuous. ANALYSIS OF BRITISH SPECIES. Warts generally distributed. Warta large and small in alternate lines gemmacea. Warts subequal. Warts vertically remote, unicolorous thaUia. Warts vertically contiguous, red-spotted BalUi. Warta only on upper half of column coronata. ASTB^ACEA. BVNODIDM. THE GEM PIMPLET. Bunodes gemmacea. Plate IV. figs. 2, 3, Specific Character. Alternate series of large and small warts. Column grey or flesh-coloured, with six equidistant bands of white. Tentacles thick, marked with white oval spots. Actinia gemmacea. Ellis and Solander, Zooph. 3. Johnst. Brit. Zooph. Ed. 2, i. 223 ; pi. xxxviii. figs. 6—9. Cocks, Rep, Cornw, Soc. 1851. 7 ; pi. i. figs. 24, 25, 28, GossE, Dev, Coast, 168 ; pi. viii. figs. 1 — 4. verrucosa. Pennant, Brit. Zool, iv. 103, Lamakck, Anim. s. vert. iii. 70. Rapp, Polyp. 50, iglandulosa. Rapp, Polyp. 52. Oribrina verrucosa. Ehbenb. Corall. 40. Cereus gemmaceus. M.-Edwards, Corall. i. 265, pi. C 1, fig. 3. Bunodes gemmacea. Gosse, Tr. Linn. Soc. xxi, 274; Ann. Nat. Hist. Ser. 3. i. 417 ; Manual Mar. Zool. i. 29. GENERAL DESCRIPTION. Form. Base. Adherent to rocka ; in general but slightly exceeding the column. Column. Pillar-like, rising to a height twice the diameter. Surface covered with round warts, arranged in forty-eight vertical rows, according to the following arrangement : — six primary rows equidistant, distinguished by theii- white colour, and by their superior size; six secondary rows, intermediate ; twelve tertiary, intercalated between the primary and secon- dary; — the difference in size between these is slight, but is often dis- cernible ; finally a row of quaternary warts (twenty-four in all) is placed between all the above, and these are much smaller and less distinct. All these become indistinct towards the base, being traceable downwards in the ratio of their order; while towards the summit they become larger and bladder-like, the uppermost individuals of all the geries crowning the THE GEM PIMPLET. 191 margin like serried teeth. In contraction the surface is thrown into trans- verse wrinkles, which of course pass between, and not aa'oss, the warts, and thus a latticed or decussate appearance is communicated ; — as if each wart were the centre of a little square. Disk. Flat or slightly conpave ; the outline circular and plane, a little Tentacles. In four rows, containing 6, 6, 12, 24 = 48 ; corresponding to the lines of warts. They are sub-marginal, thick, moderately long, conical, obtuse ; decreasing in size from the first row outwards ; and are generally carried arching over the margin, or bent into a double curve, hke the branches of a candlestick : often, however, they assume a clumsy, thickset form, swollen in the middle (see fig. 3). Mouth. Raised on a blunt cone. Lip furrowed. Gonidial tubercles prominent. Colour. Column, Rose-pink, varying in brilliance, and often becoming brownish towards the summit. Primary warts white, making conspicuous longitu- dinal bands, which in the button state form a beautiful radiating pattern. Secondaiy and tertiary warts bluish- or reddish-grey, the former generally paler. Quaternary warts generally indistinguishable from the ground colour. Sometimes, however, the quaternary row which bounds each primary on each side is also white (see fig. 3). Z>j5i. Ground colour bluish-grey on the outer region, blending into a fine yellow-green around the mouth : each radius is bounded by a scarlet line, lost at about half-disk; the primary radii are often marked with darker and paler portions, sometimes even black and white ; and the result is a briUiant kaleidoscopic star, of varied hues, the blue and scarlet lines in particular miming out among the tentacles. Tentacles. PeUucid grey or whitish, the front face olive, undefined, and deepening into black in the median line, often with a purple reflection : this face is crossed by about half-a-dozen large transversely-oval spots of opaque white, occasionally interchanged with more nar- row and even linear ones. These spots are well-defined, and, though they vary in the tentacles of the same in- dividuals, are never wanting. Mouih. Lip whitish : gonidial tubercles grey, each marked with a central dot of bright rose-colour. „ tehtaclb {lateral new). SiZB. Sorely exceeding an inch in diameter, and an inch and a half or two inches in height. 192 BUNODID^. Locality. The south-western and southern shores of England and Ireland ; the coasts of Portugal, and of the Mediterranean : on exposed rocks and shallow pools between tide-marks. Vabiett. The species is but little subject to variation of form, or of hue, except within the limits mentioned above. Specimens differ a good deal, how- ever, in the intensity and brilliance of the tints. The Gem was first discovered, or at least distinctly described, just a century ago, by Gaertner, who found it on the shores of Cornwall ; but it was not till fifteen years afterwards that it received a name. Pennant then called it Actinia verrucosa ; but this appellation has yielded to that of A. gemmacea, which was conferred upon it by Ellis and Solander, and which has been so generally adopted by British zoologists, that it would be pedantic to attempt to restore the original name. Both epithets are appropriate. Pennant's (signifying warty) is, however, rather generic than specific; while Ellis's, if somewhat more vague, is well fitted to suggest the delicate beauty of this pretty little species, — perhaps unrivalled, among British species, for its painting. The English term by which I designate the genus, alludes to the pimples^ or warts, with which the animals are studded. It is essentially a littoral species. I am not aware that it has ever been brought up from deep water, nor does it much afiect the concealment of holes or crevices. The surfaces of stones, and shallow pools within tide-marks, are the stations it habitually prefers, and it is often found in the latter even when they are but little below the level of high water. It appears to be gregarious ; for, though we do not find individuals crowded together, as is the habit of hellis, a dozen or twenty are often seen occu- pying the shallow basins of an area of rock a yard or two THE GEM PlilPLET. 192^ in extent, though none are to be seen beyond this. In the button-state, the radiating bands of white on the red- dish-grey ground, with the globular form, give a primdr facie resemblance to an Echinus, denuded of its spines, which is very striking. In their native pools the specimens are often partially enveloped in gravel, from which, if closed, their six-fold star appears prettily conspicuous; while if expanded, the brilliant pencilled disk, and white- spotted tentacles, are even more attractive. The Gem is detached with ease, and becomes reconciled to captivity without difficulty, where it preserves its cha- racteristic habit of stationing itself on some exposed spot, whence it is little given to wander. It is prolific, bringing forth living and well-formed young, which are produced one, two, or three in twenty-four hours, and not scores or hundreds in a night, as are those of S. hellts. The Gem, however, will often continue to breed at this rate for weeks. The new-bom young immediately attach themselves, and display the characteristic colour and markings : they have twelve tentacles ; that is to say, the primary and secondary series are developed before birth. In this condition they greedily devour food when presented. Miss Loddiges, of Hackney, who has been very successful in breeding and preserving this, as well as other species of Anemones, has favoured me with some particulars of her treatment, which may be useful to others. Speaking of the young, this lady observes : — " I feed them from their first appearance, — rather a delicate operation, — and they steadily grow, though rather slowly Oyster seems the best food for them, but I give them lobster, and even meat. ... I am satisfied sea-weed is not necessary in the tank : I have discarded it for some time, and only admit one small piece of red for an ornament. I syringe the water daily." o 194 BUNODID^. The voracity of the species I have already alluded to. From my friend Mr. F. H. West, I learn that it is even ot cannibal propensities. A Sag. troglodytes, var. ^, he suddenly missed, and suspected gemmacea of murder. His suspicions were confirmed, for the lost wretch was disgorged in two portions, of which the first came away on the second day, the second and larger on the fourth. The result of diges- tion was manifest, in the squeezed and shapeless appearance of the masses, the dissolution of the interior, and the flaky sloughing of the exterior. In the published descriptions, often imperfect and vague, of foreign species, we can sometimes find indications of probable affinities. The Act. tuberculosa of Bass's Strait (Quoy et Gaim.), A. hicolor of St. Vincent (Lesueur), A. xanthogrammica of Kamtchatka (Brandt), A. cruentata of Tierra del Fuego (Dana), and A. Macloviana of the Malouines (Lesson), — are doubtless true Bunodes, indi- cated not only by their warty surface, but also by the white spotting of their tentacles. Of these, the first two seem closely allied to our gemmacea, the third to thalUa, while the last two deviate .more from the type, and appear parallel with Ballii. The following are the recognised British localities of the species : — Guernsey, E. W. H. H. : Jersey, G. O. : Weymouth, W. T. (w.) : Torquay, P. H. G.: Paignton, P. H. G. : Falmouth, W. P. C. : Ilfracombe, P. H. G. : Douglas, F. H. W. : Youghal, /. B. G. : Cork, /. B. G. : Mizen Head, E. P. W. : Valentia, J. M. Jones. GEMMACEA. [tuberculosa], [bicolor], thallia. ASTILEACEA. BVNOBIDjB. THE GLAUCOUS PIMPLET. Bunodes thallia. Plate IV. Figs. 5, 6. Spteific Character. Warta sub- equal, vertically remote, uniccloroua Bunodes thallia. Gosse, Annals X. H, Ser. 2, xiv. 283 : Tenby, 361 ; pi. xxiii. fig. c: Linn. Trans, xxi, 274. Annals N. H. Ser. 3, L 417. Cereut Thalia. iLxxE Edwards, Hist. CoralL i. 266. GENERAL DESCRIPTION. FOBJI. Sase. Adherent to rocks ; considerably exceeding column. Column. A rounded button in contraction, pillar-like in extension, rising to full twice the diameter. Surface covered with numerous (about thirty -six) vertical rows of sub equal prominent warts, which are separated, in moderate extension, both laterally and vertically, by interspaces of about equal width, in which the skin is irregularly corrugated. The warts are about twenty -five in each row, and reach from the base to the margin, which is serrated with the elongated topmost warts of all the rows. They are strongly adhesive, and are occasionally drawn out to the length of a line, before they yield their hold. Substance firmly fleshy. Disk. Flat, or slightly concave ; radii indistinct. Tentadcs. Sub-marginal, set in four rows; 6, 6, 12, 24^48 : — the first three rows are, however, so nearly equidistant from the centre that, on a cursory inspection, there appear but two rows altogether. They are sub- equal, thick, obtuse, about half as long as the diameter of the column ; and are commonly spread horizontally, or overarching outwards. Mouth. Set on a prominent cone. COLOCB. Column. Pale bluish or greyish green, with dark warts. Bisk. A many-rayed star of yellow rays on a blackish grovmd, produced in the following manner. The radii are blackish, each marked with a central spindle-shaped line of yellow ; in the primary and secondary radii, 2 196 BUN0D1D.E. the yellow mark is broader and near the mouth ; in the others, it is more slender, longer, and reaches to the tentacular region. Tentacles. Pellucid grey, with the front face olive, on which are scattered numerous spots of opaque white : these spots are gene- rally roundish, or polyhedral, and large and TENTACLE small ones are crowded together. (lateral view). Mouth. Blackish, with the gonidial tuber- cles of a more intense hue. Size, Button an inch and a quarter in diameter, elongating to a height of two inches ; expanse of flower two inches. Locality. Both sides of the Bristol Channel ; rocks within tide-marks. Varieties. o. Hygroxyla. The green condition described above. /8. Xeroxyla. Column dingy brown, with slightly darker warts ; disk of the same tint ; marked as in o. 7. Caustoxyla. Column reddish chocolate, with darker warts ; disk dark olive ; marked as in a ; the central half sometimes white. I first discovered this species at Lidstep, on the coast of Pembroke, in 1854, and described and figured it in " Tenby ; a Seaside Holiday." Very little has been added to its recorded history since that time ; not more than four speci- mens having occurred, so far as I am aware, to subsequent researches, all of which were obtained near Ilfracombe. Though manifestly a rare species, I was so fortunate as to light upon a numerous colony at its discovery. About a dozen individuals of different sizes were associated in the dark angles and pools of a little insular rock exposed at spring-tide, that lies just off the cove called the Droch, near Lidstep. They were not troglodyte in habit, but adherent to the open rock, and therefore easily detached. The species seems social ; clustering together in groups, mutually pressing each other's sides. The habits of the Glaucous Piraplet in captivity are THE GLAUCOUS PIMPLET. 197 closely like tliose of the Gem. Like the latter, it expands under the stimulus of the light, rather than in darkness, indicating a habitually exposed mode of life. Like gem- macea, it frequently erects itself •when closed, in the form of a pillar ; and throws off successive rings of mucus from its body, which accumulate around its base, if not removed. The action of the waves would wash these away in a state of freedom ; in a tank they should be detached by means of a stick or hair-pencil. I have never seen the warts of gemmacea used as suckers ; but in specimens of the present species, I observed this function exercised by them very signally ; not in the way of attaching extraneous fragments to the body, like ^S. hellis and T. crassicomts, but in taking hold of a firm support, like S. troglodytes. The suckers of the column adhered with force to the side of the glass vessel, and by contrac- tion were stretched as above described. The specific name " thallia'' (not Thalia, as M. Milne Edwards misquotes it) I adopted in allusion to the elon- gated form and glaucous colour, from OaWia, an olive- shoot. The same idea recurs in the epithets which distin- guish the varieties, — as if the glaucous, the dull brown, and the chocolate, were the twig as green, dry, and scorched. It is possible that the immature specimens, found by Templeton in Belfast Lough, and named by him Act. mantle, were the young of this species ; though they have been generally attributed to gemmacea. gemmacea. THALLIA. [xanth ogram mica}. [Artemisia], T. crassicomis. • Loudon's Mag, N. H. ii. 303 ; fig. 49. ASTR^ACEA. • BUNODIDjE. THE RED-SPECKED PLMPLET. Bunodes Ballii. Plate IV. Fig. 4. Specific Character. Warts sub-equal, vertically contiguous, red-spotted. Actinia Ballii. Cocks, Rep. Com. Soc. 1849, 94; Ibid. 1851, 9; pi. ii. figs. 9, 17, 18. clavata. Thompson (w.), Zoologist, 1851, App. cxxvii. GossK, Ann. N. H. Ser. 2, vol. xii. 127. Aquarium, 35. TuGWELL, Man, Sea Anem. 100, pi. iv. Jordan, Ann. N. H. Ser. 2, xv. 88. Bunodes clavata. Qosse, Linn. Trans, xxi. 274. Ann. N. H. Ser. 3, i. 417. Cereus clavata. Milne Edwards, Hist. Corall. i. 267. GENERAL DESCRIPTION. Form. Base. Adherent to rocks ; considerably exceeding tbe column ; generally lengthened-ovate in outline. Column. Low and broad, scarcely rising to a pillar-form. Surface covered with warts about equal in size, arranged in forty-eight longitu- dinal rows, of which the alternate rows are traceable from the margin only about half-way down the column ; the warts are contiguous vertically, but the rows are separated laterally, by interspaces of equal width, of corru- gated skin. The primary rows consist of about twenty-four warts, becoming indistinct towards the base; the uppermost individuals of all the rows crowning the margin as blunt teeth. Dish. Flat ; the outline nearly circular, often much overlapping the Tentacles. Nearly marginal, set in five rows ; 6, 6, 12, 24, 24 := 72 : the first three rows nearly equidistant from the centre. They are longer and more slender than in gemmacea, conical, obtuse ; decreasing in size from the first row outwards ; and are usually carried horizontally spread, with a very constant tendency to curl upward at the tips. Mouth. Raised on a cone ; often gaping ; throat membranous, protru- sile : gonidial tubercles usually prominent, often inflated. Colour. Base. Red, sometimes rich crimson. THE EED-SPECKED PIMPLET. 199 Column. Pale yellow : each wart crowned with a well-defined crimson speck, the interspaces irregularly freckled with crimson. In some instances, the pale yellow predominates on the upper half of the column, the crimson on the lower. 2>/t. Pellucid-grey, covered or dusted with opaque white specks, varying in size and shape, as if sprinkled with flour. Tentacles. Yery pellucid, pale yellow, but some or all frequently tinged with a lovely rose-colour : always sprinkled, on all sides, with minute irregularly shaped specks of opaque white. Mouth. Lip and gonidial tubercles some- times crimson or rose-pink; but sometimes whitish or pale yellow. Size. Ordinary specimens are an inch in diameter and half an inch in height, with an expanse of two inches. Mr. Tugwell figures one two inches in diameter, and three in expanse ; and Mr. Brodrick writes me that one, which has been in his possession nearly three years, measures, after feeding, four inches in expanse. LOCALITT. The southern and south-western shores of England ; on the under sur- faces of stones, and in crevices between tide-marks, and in deep water. Yabieties. o. Rosea. The most lovely condition above described. $. Dealbata. The roseate hue wanting; the tentacles cream white; in other respects as a. y. Funesta. Tentacles dark umber or wood-brown, with little trans- lucency. Disk smoke-black. Both dusted with yellowish-white specks as usual. Column as a ; but tinged with brown. Usually of large size. 8. Livida. Tentacles and disk tinged in various degrees with bluish-grey or livid green, often in a sort of changeable lustre, like that of putrescent flesh ; with the characteristic specks. Chiefly from deep water. Mr. TVilliam Thompson, of AYejmouth, described tliis species bj tbe name of Actinia davata, in the Appendix to the Zoologist for 1851. But Mr. W. P. Cocks had abready described and figured, under the title of ^. BaU{i,t]xe same 200 BUNODID^. species, in his admirable memoir " On the Actiniss of Falmouth," which was read before the Cornwall Pol jtechnic Society, in the autumn of the same year. lie had been acquainted with the species ever since 1847 ; and had pub- lished the name in the Society's Report for 1849. To Mr. Cocks's appellation, therefore, belongs the claim of priority; but even were it otherwise, Mr. Thompson's name must be rejected, not only because it had been previously applied to another species, but, according to a canon which I have already had occasion to apply to one of my own names,t hecause it conveys a false idea. The name clavata origi- nated in a misconception. In the single specimen known to Mr. Thompson at that time, he mistook the curling of the tips of the tentacles for a cluhhing^ whence the name " clavata " — clubbed. These organs have not the slightest tendency to such a form as the term implies. The name which I adopt was given, I believe, in honour of the late Robert Ball, LL.D., an eminent marine zoologist. I found the species not uncommon at Weymouth in 1853, especially on the ledges that are exposed at the recess of the tide, under Byng Cliff. Its habit is to lurk in narrow fissures in the cavities of the under side of large flat stones, and not unfrequently in the deserted holes of Pholas or Saxicava. The disk is very wide and flat ; and, as it is also very expansile, it spreads itself to a consider- able distance around the margin of its hole. So essential is it to its comfort, however, that it should have a retirement, that if it be put into an aquarium, though it may at first affix itself to a flat stone or to the surface of a shell, it will creep away, by means of its base, till it find some loose stone, under which it will insinuate itself till it is quite * M. Rathke had named clavata an Actinia, which he found on the coast of Norway, in 1843. t See ante, p. 76. THE BED-SPECKED PIMPLET. 201 concealed ; or a narrow crevice, as l)etween two contiguous stones, into which it may thrust its body. The variety h'vida, which is not rare in Weymoutli Bay, in deep water, manifests the same habit, for it is usually found to have ensconced itself in one of the angular cells or cham- bers formed by the coral-like plates of Eschara foliacea^ which afford retreat to so many and so various creatures. A remarkable peculiarity of this species is the degree to which it becomes transparent by distension with water. The effect of this is not the general swelling of the body, as in T. crassicornis, which is remarkable for the same habit effected in another way, but a great dilatation of the disk and tentacles, which then expand to an extraordinary degree, becoming so diaphanous as to be almost destitute of colour, and showing with absolute clearness the craspeda in the intersepts of the visceral cavity. The species is hardy in captivity, and the varieties a and /9 are very beautiful, especially the former. The variety 7 has not unfrequently beguiled me, on a hasty examination, into the notion that S. hellis was before me ; and I tliink that these two species form links by which the families Bunodidce and Sagartiadce are connected. There is also a remote aflSnity between this species and Aipt. Couchii. My friend, Mr. F. H. West, has received B. Ballii from the French coast of the Channel. On our own side it ranges in tolerable abundance from the Hampshire coast to the Lizard, as the following list will indicate : — Selsey ; Ventnor, G. G. : Freshwater Bay, F. N. B. : Weymouth ; Torquay, P. H. G. : Falmouth, W. P. C. thallia. Sag. bellis. Ballii. Aip. Couchii. [cruentata]. [Macloviana]. ASTR^ACEA. . BUNODIDJi. THE DIADEM PIMPLET. Bunodes coronata. Plate VII. Fig. 4. Specific Character. "Warts almost confined to upper half of column, in lines and irregularly scattered ; sub-equal, small. Bunodes coronata, GossE, Annals N. H. Ser. 3, ii. 194. GENERAL DESCRIPTION. Form. Base. Adherent to shells, scarcely exceeding column. Column. Cylindrical in expansion, much higher than wide ; covered on the upper two-thirds with moderately numerous small warts, neither per- forate nor excavate ; they are arranged in twelve longitudinal rows, with irregularly scattered ones between ; and are generally wanting towards the base. Skin between the warts smooth, and when distended having a satiny lustre. Whole column invested with a thin drab epidermis, deciduous in ragged shreds, but adhering pretty firmly. A distinct parapet, with a smooth sharp edge, but no appreciable fosse. Dislc. Circular, flat, but often protruded so as to be convex, or to form a low cone ; radii distinct. Tentacles. In five rows; 6, 6, 12, 24, 48=96. They are sub-marginal, the first row springing at about three-quarter radius ; they are shorter than radius, diminishing outwardly, conical, sub-acute. Mouth,, Large, protrusile : lip sharp : throat evertile, coarsely furrowed. Colour. Column. A rich orange, or orange-scarlet, with the warts either paler or darker than the ground-colour. Edge of parapet cream-white, immediately below which the margin is marked alternately with square patches of dark pui^plish chocolate, and narrower spaces of whitish (twelve marks of each colour in adults, six of each in young) ; these, from the fine contrasts of colour, when the button is not quite closed, have a very striking and characteristic effect, as if the animal were surmounted by an elegant coronet. Dish. Red, varying from pellucid scarlet to a reddish chocolate ; each radius bearing a longitudinal central streak of white, wliich does not reach THE DIADEM PIMPLET. 203 either tentacle or lip, and bounded by a very fine ■white line on each side ; thus is produced a pat- tern of fine radiating lines of white on red. Some- times the lines are irregularly blotched and dilated, with ragged edges. Tentacles. Pellucid, nearly colourless, crossed by three dim sub-opaque white bars, of which the middle one is most distinct ; near the base are two chocolate bars, generally divided by a central longitudinal line of pellucid white, giving the appearance of four dark spots set in square. Sometimes one bar is nearly or quite obliterated. Mouth. Lip whitish. Throat rich orange-scarlet ; te>"tacle the furrows darker than the ridges. {front rieic). Size. Diameter of column in button, one and a quarter inch ; height two inches expanse of flower one inch. LOCALITT. The south coast of Devon ; moderately deep water. Varieties. a. PcUricia. The rich orange-scarlet condition just described. /3, Pld>eia. The column of a dirty light brown ; the markings of the marginal coronet distinct, but duller. The usually red groimd of the disk replaced by deep brown, and the white lines by pellucid drab ; the whole interrupted by four or five broad irregtalar radial bands of p\u:e white. The bars of the tentacles obsolete. This fine species first occurred to myself when dredging off Berry Head, in about twenty fathoms, in August, 1858. Three or four specimens came up in about the same number of hauls. In eveiy case the animal was adherent to the shell of the living Turritella terehra, a moUusk which is so abundant there that the dredge comes up half- filled with it. The base of the Bunodes clasps the long turreted shell, nearly enveloping it when adult, only the apex and the mouth of the shell being exposed. Other specimens have occurred since in similar circum- stances ; and Mr. Densham, a collector of Torquay, informs me that in October he obtained a group of eight or ten adhering to a mass of oysters. 204 BUNODID^. It is manifest that this species departs considerably from the type of Bunodes. The irregularity of the warting, the conical form of the tentacles, and their style of colouring, in alternate undefined rings, and the occasional eversion of the walls of the throat, indicate a sensible approach to the following genus. It is always to aberrant species that we look for cross affinities ; and therefore I was more gratified than surprised to see in this animal evident marks of connexion, both in appearance and habit, with the Bagartiadm. Before I had seen it expand, I suspected it to be 8. 'parasitica, especially when in the act of unfolding. It has much resemblance to that species, as well as to 8. coccinea, with which it was associated ; for a number of this little species occurred in the same dredge-hauls ; these also adherent to the shells of the Turritellce. The whole aspect of the Diadem Pimplet, including the colouring, is that of a Sagartia, though the preponderance of its characters deter- mines it to Bunodes. It is interesting, in this relation, to notice, that one specimen in my possession protruded from the mouth a bundle of what appeared to be true acontia. The species lives well in a tank ; where it readily deserts its shell, and attaches itself to stones, or the vessel. It is lively, opening freely, frequently constricting its column, and changing its form with considerable rapidity; its vivacity and brilliant colour render it an acquisition to the aquarium. Both the scientific and the English appellations by which I distinguish the species, allude to the coronet of purple spots which surround the margin. Berry Head, P. H. G.: Torbay, E. W. H. H.: off Teignmouth, G. H. King. Ballii. Sag. parasitica. coeonata. Sag. coccinea. T. crassicornis. 205 GENUS III. TEALIA (Gosse). Actinia (Lrss.). Cribrina (Ehrenb.). Cereus (Milse Edwabds). Bunodes (Gosse). Base exceeding the column. Column not pillar-like ; the diameter usuaDy much exceeding the height. Surface studded with per- manent rounded warts, which are hollow, and have a strong adhesive power, irregularly scattered, or not set in vertical lines. Margin denticulate. Substance cartilaginous. Dis^ flat, circular in outline, considerably over- lapping the column. Radii inconspicuous. Tentacles not very numerous, arranged in several rows, sub-marginal ; short, thick, and conical ; uni- colorous, or marked with undefined rings or bands of alternate colours ; perfectly retractile. Mouth raised on a cone ; stomach habitually pro- truded to a great extent. Muscular system highly developed ; very dense, and of a cartilaginous firmness. ANALYSIS OF BRITISH SPECIES. T Warts unequal : stomach and warts red ; tentacles un- handed digitata. Warts equal : stomach and warts grey ; tentacles banded . croisicomis. k ASTR^ACEA. BVNODIDjE. THE MAEIGOLD WARTLET. Tealia digitata. Plate VI. Fig. 10. Specific Character. Warts unequal ; stomach and warts red ; tentacles not banded. Actinia digitata. Mdlleb, Zool. Dan. iv. 16; pi. cxxxiii. Alder, Zooph. of Northumberland and Durham, 44. Cereus digitatua. Milne Edwards, Corall. i. 272. Tealiqi digitata. Gossb, Ann. Nat. Hist. Ser. 3, i. 417. GENERAL DESCRIPTION. Form. Base. Adhering to shells, often exceeding the column ; outline undulate. Column, Cylindrical, about as high as wide, sometimes dilated and overarching above. Margin smooth, parapeted. Surface studded with large wai-ts, having a tendency to form transverse i-ows, but with no perpendicular arrangement. " A row of larger warts is usually found on the upper part, v^hich, when the tentacles are withdrawn, form a tuber- culated margin to the aperture." (J. A.) Bisi:. Flat, often partly everted and overarching. Radii strongly marked. Tentacles. Numerous, in three or four rows, stoutly conical, bluntly pointed, the first row largest, diminishing to the outmost, which are papil- lary : carried arching outwards. Mouth, Throat evertile, strongly ribbed. Colour. Column. Scarlet-oi"ange, with paler warts. Bisk. Dull red. Tentacles. Dull red, unhanded, a little deeper towards the tip. Mouth. Ribs of throat brownish-orange. ACTINIA MESEMBRYANTHEMUM A. CHIOCOCCA 3ACARTIA CHRYSOSPLENIirV ;» COLOURS Sr H.DICKCS. 9 ANTHEA CEREUS 0. TEALIA DICITATA. ' SAGARTIA VIDUATA. THE MARIGOLD WAETLET. 207 Size. Column one and a half inch high, and the same wide. Expanse about two inches. LOCALITT. Coast of Northumberland and Cornwall. Deep water. The name by wliich I have distinguished this genus is given as a tribute to the skill and acumen of Mr. Thomas Pridgin Teale, of Leeds, who published an elaborate and excellent Memoir on the anatomy of the following species. The English appellation is sufficiently obvious. The specific term digitata, " fingered," doubtless alludes to the thick conical form and dull reddish hue of the tentacles, in which the Danish zoologist saw a resemblance to fingers, — those of a ploughman or a scullery-maid, surely ! I distinguish this species from crassicorm's on the autho- rity of Mr. Joshua Alder, of Newcastle, who first mentioned it as British, in his Catalogue of the Zoophytes of that coast. The same gentleman has kindly favoured me with several drawings of the species, executed with his well- known beauty and precision (one of which is reproduced in my Plate), as well as with his MS. notes, from aU of which combined I have compiled the foregoing diagnosis. Mr. Alder entertains no doubt of its specific distinctness ; and his numerous opportunities of seeing it alive and comparing it with the more common kind, render his opinion valuable. He says, " It is the most coriaceous and warty species that I am acquainted with." And again, " It is always much smaller than crassicorm's, more tough and coriaceous, with larger warts, and constantly of a pale red colour." " It is not uncommon," adds the same excellent natu- ralist, " in deep water on our coast ; and as the cod-fishing boats are coming into port frequently at this season [April], 208 BUNODIDiE. I may be able to get you a specimen, though not in a lively condition." Among the numerous drawings of Actinoids for which I am indebted to Mr. W. P. Cocks, there are two which he has not named, but which are evidently identical with the Northumbrian species. Thus I am able to assign it to the Cornish coast. These are the only British localities I yet know for it. DIGITATA. crassicornis. ASTILSACEA. BirXODWJ-. THE DAHLIA WARTLET. Tealta cras»icomis. Specific Character. generally banded- Plate IV, Fig. 1. Warts equal; stomach and warts grey; tentacles Actinia felina et A. senilis, crassicmtiis. Holsatica. ? fiscella. ? himaculata, coriacea. gemmacea. Cribrina coriacea. laacmaa papulosa. Bunodea crassicomis. Tealta crassicomis. Lixs. Syst. Nat. 1088. MiJLLEB, Prod. Zool. Dan. 231. Fabr. Fatm. Groenl. 348. Johnston, Br. Zoophu i. 226 ; pi. xl. GossE, Devonsb. Coast, 34. Cocks, Rep. Com. Soc. 1851, 7 ; pi. il fig. 1. MuLLER, ZooL Dan. iv. 23, pi. cxxxix. ' Ibid. Ibid. iiL 3, pi. IxxiiL figs. 5, 6 (Juv. .'). Gbube, Actinien, 4, fig. 4. CcTTEB, Tabl. ^l^m. 653 ; R^e Anim. ed. 1, iv. 51. Rapp, Polypen, 51, pi. L fig. 3. Teale, Trans. Leeds Soc. i. 91, pis. ix. — xl Johnston, Br. Zooph. i. 224 ; pL xrxix. figs. 1, 2. Cocks, Rep. Com. Soc. 1851, 7; pL ii fig. 2. TcQWELL, Man. Sea Anem. 54, pi. iiL Dalyell, Rem. Anim. ScotL 223 ; pL xlriii. figs. 1, 2. JoHSST. Br. Zooph., Ed. L 213. Couch, Com. Fauna, iiL 76. EHRExa CoralL Roth. Meeres, 40. Ibid. Ibid. 33. GossE, Trans. Ldnn. Soc. ttj , 27^ ; Man. Mar. Zool. i. 29, fig. 42. Ibid. Ann, N. H. Ser. 3, L 417. I GENERAL DESCRIPTION. Form. Base. Adherent to rocks and stones. In general not much exceeding the column. Column. Rarely pillar-like. In expansion, the diameter greatly ex- ceeding the height. Surface covered with small hollow adhesive wart"", P 210 BUNODID^. Bometimea having a tendency to run in longitudinal lines, but more generally irregularly scattered, leaving intervals of three or four times their diameter in ordinary states of distension, and these intervals have often a silky lustre. Substance firm and even cartilaginous. Margin entire, but roughened with the scattered warts, forming a thick parapet, separated from the tentacles by a broad fosse. In freedom, the column is generally more or less disguised by fragments of stone and shell adhering to the suckers. Bisk. Flat, circular in outline, plane but overarching. Radii con- spicuous chiefly by colour. Tentacles, Arranged in five rows, the first set at about half radius, — 5, 5, 10, 20, 40 = 80; the first and second so nearly equidistant from the centre as to seem but one. Their form is conical, thick at the foot, regularly tapering to a point, which is sometimes slightly inflated. The animals appear to have the power of changing the shape of these organs at will ; for I have had individuals, in which the tentacles, after having for a while borne the ordinaiy conical form, suddenly became nearly cylin- drical, with truncate extremities, and maintained this form for a long time. These organs are nearly equal among themselves, and their length is about equal to one-third of the diameter of the disk. They are capable of little flexure, and are generally spread in a regular star-like manner, the outer rows deflected, the inner erect, and the intermediate ones horizontal. They are powerfully adhesive. Mouth, Frequently elevated on an eminence of varying form and dimensions. Throat and stomach often protruded to such an extent as to conceal the whole disk. Gonidial tubercles two pairs, small. Colour. Column, Dull green, streaked and flaked with crimson, with pale grey warts. Dish, Glaucous-olive, with conspicuous radial bands proceeding from each outer tentacle, in pairs, which curve around the foot of each tentacle of the higher rows, and are lost at varying distances from the centre ; those pairs which enclose the inner tentacles extend farthest and are most conspicuous. The colour of these bands is scarlet, often edged with white, and they are highly characteristic of the species. Tentacles. Pellucid light brown, with a band of opaque white across the foot, which frequently stretches a little way up each side : a broad band of crimson surrounds the middle, bounded below, and sometimes above, by a narrower band of sub-opaque white. All these bands are undefined, and are often rendered sub-pellucid by* distension. Mouth. Generally tinged with crimson. Gonidial tubercles crimson. Throat and stomach light grey. I THE DAHLIA WARTLET. 211 bIZE. Diameter of colaom frequently tiiree inches ; expanse of flower five ; height two. Specimens from deep water are occasionally much larger than this. < LOCAUTT. The Atlantic coasts of Europe, uniTcrsally distributed ; in tide-pools, and crevices and aqgles of rocks, near low-wat€r mark ; and in deep water. I am not certain whether it extends to the Medit«rranean. Vahibties. The colours of this species are very sportive, and scarcely two specimens can be found exactly alike ; but all these modifications may be traced to different degrees of predominance of the hues above mentioned. This variety, from its resemblance to a streaked apple, may be named, — o. Meloide, 0. Purpurea. Column wholly dull crimson ; disk crimson, with the radial bands and sometimes the central region more brilliant than the rest. Tentacles pellucid crimson, with purplish bands. y. Imiynii. As /3, but the tentacles pellucid white, with broad and con- spicuous bands of opaque white. (PL iv. fig, 1.) S. A urea. Column yellow, from a light straw or brimstone colour to the hue of a ripe apricot. «. VUis. AH colour lost in a semi-pellucid dusky grey. (Deep-water specimens generally very large.) In my " Devonshire Coast " (p. 36), I stated, with the reasons which led me to it, mj firm conviction that what had hitherto been considered as two species, under the names of A. crassi'comis and A. cortacea, were one and the same. Seven years' additional experience has only added to the strength of that conviction, and I have not been able to find a single stable character on which their separation could be grounded. It is equally clear which of the two specific names must stand. Rejecting Linnaeus's as out of the question, we find that crass icornis was applied to the species by Miiller, twenty-one years before Cuvier called it coriacea. With regard to significance, both appellations are good, perhaps equally good ; the former indicating the P 2 212 BUKODID^. thick horn-like form of the tentacles, the latter the tough and leathery consistence of the flesh. The law of priority, however, must be obeyed. Scarcely less abundant than Act. mesembryantJiemum, this magnificent species is sown broadcast upon all our shores, and seems everywhere to be equally common. In its habits, however, it is widely different from that favour- courting species. Somebody has illustrated the character of two peoples by saying, that if an Englishman retires from business and builds a box, he raises a high wall, and plants a shrubbery before it, to keep off the eye of the profanum vulgus ; but a Frenchman under similar cir- cumstances builds his house on the very edge of the high- . way, and takes his meals in the verandah. If this be true, the Actinia is a Frenchman, the Tealia an Englishman. You may hunt among the rocks till the rising tide covers them, and, finding hundreds of Beadlets, but not a single Dahlia, go away with the conviction, that the latter is a scarce species ; but to-morrow, an initiated friend accom- panies you to the same spot, and, pointing with his toe to an angle, says, " Here they are! and here! and here! — three, four, half-a-dozen in a group !" and you are tired of collecting before the profusion fails. It is in the angles formed by some great boulder with the beach, that the crassicornis delights to dwell ; and here, according to his recluse habits, he chooses to conceal his showy person from intruding eyes, by covering himself with a coat of gravel and fragments of shell, which he has attached to his adhesive suckers, till only the experienced eye can detect the difference between the animal and the surrounding rubbish. Not seldom, however, do we meet with a colony in some persistent rock-pool, in whose never-ebbing fulness the gorgeous creatures remain almost permanently ex- THE DAHLIA WARTLET. 213 panded, despising, or not needing, the precaution of con- cealment practised by their tide-deserted brethren of the beach. It is a remarkable example of the economy of creation, that these tide-pool specimens, as well as those which are brought up from deep water, rarely, if ever, indue their bodies with an extraneous covering. In such pools crassicomis makes a noble appearance. His great size, the wide expanse of the flower, the thick tentacles so symmetrically disposed, and the rich hues often finely contrasted, — make it by far the most showy of our native species. By some of our fair collectors it has been named the Dahlia; a comparison which the size, symmetry, and varying hues of that favourite flower render not inapt. I have accordingly adopted it ; designating the preceding orange-hued species by the appellation of the. Marigold. The resemblance has been acknowledged by one more conversant with flowers than even the ladies. "On one occasion," observes Mr. Jonathan Couch, " while watching a specimen that was covered merely by a rim of water, a Bee, wandering near, darted through the water to the mouth of the animal, evidently mistaking the creature for- a flower ; and though it struggled a great deal to get free, was retained till it was drowned, and was then swallowed." Mr. E. L. Williams, who has enjoyed unusual opportu- nities of acquaintance with the deep sea, writes me con- cerning this species as follows : — " When diving in bells at Dover, at the Admiralty Pier, in eight to ten fathoms^ water, I have often seen it, generally on the tops or sides of lumps of rock. The -^sop Prawn [Pandalus annuli- cornis f] was very common there, and seemed its food. I never saw a closed crassicomis in deep water, except while catching its prey." * In Johnston's Brit. Zooph. i. 225 ; et in litt. prir. 214 BUNODIDiE. My esteemed friend, Professor E. P. Wright, of Dublin, hai favoured me with one of his vivid pictures, in which this species forms a prominent feature. It will be read with interest : — ' There is a very fine cave here, [Crookhaven, county Cork,] entered at either high or low water by a boat, whose entrance is guarded on both sides by a long low reef of rocks, and of a depth at low water of about ten or twelve feet. The sea-floor is shaped somewhat like a Spanish hulk, i.e. rather flat at the bottom, and then rising up gradually and wideningly ' to a distance far above our heads, and then ending in an arch formed of sharp-pointed icicles of the by-me-never-to-be-forgotten Devonian slates. To this cave all the fat and fair anemones of the county seem to be sent, when once they have reached a good bodily condition. The cavern is of ample dimensions, so they don't crush each other for room ; and the regular manner in which they dispose of themselves is worthy of note. Actinia mesem- hryanthemum — the green, scarlet, and strawbeny varieties — occupied the highest row, some of them partly out of the water; they had eyes, and kept a 'look-out' for the rest. Then came Sag. venusta and Sag. nivea, lovingly inter- mixed, and in a large broad band some four feet deep. Then there came an empty row of benches, necessary to keep the tenants of the galleries from the aldermen in the pit, for it was filled with T. crassicornis. 1 verily believe the biggest of the big were here ; and the commonest variety was the one with the white tentacles and red disk — a splendid show for size of specimens and magnificence of colour. This cave of Anemones never can be surpassed, and seldom will the wild gi-andeur of the cliffs, a hundred feet and more high, with the Atlantic waves rolling in to fill up the picture, — be equalled." The voracity of this fine creature is remarkable. The THE DAHLIA WARTLET. 216 Shore-crab [Carcmus] is its ordinary prey, but it feeds on limpets, and other Mollusca. Dr. Johnston tells of one , that had swallowed a valve of the Great Scallop, and of the strange result;* Dr. E. P. Wright had one which discharged as the remains of his evening's meal, " a mode- rate sized Fusus, and a mass of Nereids and Shrimps, that exhaled such a fearful smell as killed all mj tank-full;" and one in Mr. F. H. West's possession actually made a honne houche of an Echinus miltaris, as large as a shilling, making no bones of the spines. Two days afterwards the shell of the Urchin was disgorged, perfectly empty, denuded of its spines, the oral plates crushed in, and partly ■wanting. The common Blenny and other fishes frequently fall victims to the rapacity of this gourmand, which spares not its own kindred. The tentacles are very adhesive, as is sufficiently mani- fest to our fingers, when we touch them ; and contact with these organs is amply sufficient to resist the most vigorous attempts to escape of the animals above-mentioned. Beautiful as is the Dahlia, it is not a very frequent tenant of our aquariums ; as it is one of the most difficult to keep. I have, however, kept specimens for four and five months ; and Mr. West still longer ; for the epicure whose urchin-diet is recorded above, had been then nine months in captivity. It appears to be little able to sustain extremes of temperature. The heat of summer is generally fatal to our captive specimens ; and a severe winter makes havoc among those which are in the enjoyment of freedom. After the intense and protracted frost of February, 1855, the shores of South Devon were strewn with dead and dying Anemones, principally of this species, which were rolled helplessly on the beach, their bodies almost concealed by the protruding craspeda. This symptom is almost the * Brit. Zooph. L 235. I 216 BUNODIDiE. invariable accompaniment of disease and death in crassi- cornis ; these organs are present in unusual profusion, and are forced out at ruptures of the integument, bj the con- tractions of the animal. The mesenteric membrane by which they are united to the septa is capable of great expansion: Sir John Dalyell has seen it protruded and spread up the side of a glass vessel, to the breadth of an inch. I have seen a similar phenomenon, but not quite to the same extent, in Peachia hastata. As in the case of A. mesembryanthemum, the ubiquity of this species renders a catalogue of its localities unnecessary: it is distributed everywhere on the British coasts. Of foreign species, so far as may be conjectured from published figures and descriptions (often imperfect), the following may belong to this genus : Artemisia (Dana) from N. W. America ; pluvia (Dana) from Peru ; gemma (Dana) from Cape Verd Isles ; papillosa and ocellata (Lesson) both from Peru; ajid fusco-rubra (Quoy et Gaim.) from the Tonga Isles. Of these the first-named seems intermediate between the present species and B. thallia. B. thallia. [Artemisia]. CRASSICORNIS. H. Margaritas, St. Churchige. Sagartia. Anthea. Bolocera. [Phymactis]. Actinia. [Echinactis]. [Cystiactis]. Tealia Greenei (Wright). Dr. E. P. Wright finds on the Irish coast a Tealia, which he thinks new, and for which he proposes the name of T. Greenei. The parapet is much smoother than in THE DAHLIA WARTLET. 217 crasstcomts, the tentacles mucli longer and more slender, the warts fewer and of a purplish hue. He has favoured me with a spirited drawing of it, but I cannot satisfy myself that it is anything more than T. crassicornis. Tealia tubercttlata (Cocks). In the Report of the Comwali Polytechnic Society for 1851, Mr. W. P. Cocks has described and figured a species, which he names Actinia tuberculata. " Body globular, light- brown, densely covered with large greyish-white tubercles, the apex of each tubercle depressed; disk white; mouth large ; lips thick, corrugated, and everted ; tentacula nume- rous, large, obtuse, some bifurcated, others trifurcated. Diameter three and a half inches when contracted." By private communication I learn further particulars. It was obtained thirteen miles south-west from Falmouth, attached to a valve of Pecten maximus ; it lived with 3Ir. Cocks for some months. " Bulky, rather loose in texture, when ftdly expanded covering the bottom of a large pan, — it had the appearance of a mammoth hellis. It appeared to be ex- tremely irritable, and upon the slightest provocation would throw oflF from its body a large quantity of thick glaire, which, if allowed to remain, produced a disagreeable smell. When contracted it had the appearance of a half-boiled sago pudding." I ventured to suggest that it might have been a great colourless deep-water specimen of crassicornis ; but Mr. Cocks repudiates the identification, while he admits the relationship. The tendency of the tentacles to a monstrous fission seems to me its most marked peculiarity. It may be distinct. 218 GENUS IV. HORMATHIA (Gosse). Base adherent ; greatly expanded. Column pillar-like, much corrugated, surrounded by a single horizontal row of warts. Disk slightly concave ; scarcely exceeding the column. Tentacles moderately long and slender; perfectly retractile. There is but a single known species, H. Margarita. ASTRJKACEA. BINODID^. \ THE NECKLET. Hormathia Margaritas. Plate Vni. Fig. 1. Specific Character. "White, with purple tentacles. Hormathia Margaritce. Gosse, Annals Nat. Hist. Ser, 3, iii. 47. ? Actinia nodota. Fabkicics, Faun. GroenL p. 350 ; No. 841. GENERAL DESCRIPTION. Form. Base. Yery closely adherent to a living Funis antiquus; far exceeding the column, and clasping the shelL Column. Skin delicate, much corrugated transversely ; below the margin a horizontal row of large well-defined warts, about ten in number ; summit extremely corrugated, and falling into radiating folds in incipient retracta- tion. A slight but distinct margin. Bisi. Slightly concave ; outline almost circular. Tentacles. Arranged in two or three rows, rather long, sub-equal, but the inner row somewhat longer than the outer; when fully expanded, curving over the margin. Mouth. Not raised on a cone, slightly corrugated. CJOLOUB. Column. White. Bisk. White, streaked with very light brown. Tentacles. Dark reddish pxirple, without any markings. Mouth. Lip slightly yellow. Size. Diameter two inches ; height two inches. LOCALITT. Moray Firth, near Banff; deep water. 220 THE NECKLET. For tliis magnificent species I am indebted to the kindness of the Rev. Walter Gregor, who obtained it in October last, from the lines of a deep-sea fishing-boat, and forwarded it to me. It was dead, however, when it reached me ; but his own careful notes and sketches, made while it was alive, have enabled me, in combination with my own imperfect observations, to characterize it as above. As he had never seen another specimen, I can add no more parti- culars of its history. The name of the genus I have formed from opfiado^;, a necklace of pearls, and the English appellation perpetuates the same allusion. The specific name is given at the discoverer's request, in honour of a lady, one of his most esteemed friends. The unsullied pearly whiteness of the animal, as well as its necklace, gives a peculiar propriety to this name, — margarita signifying a pearl. The genus is aberrant in this family; the paucity of warts, and the soft and thin texture of the skin, departing manifestly from the typical forms. It approaches the Sagartiadoe through Adamsia ^alliata and Sagartia para- sitica, with both of which it has obvious relations. T. crassicornis. Margarita. Sag. parasitica. St. Churchise. Ad. palliata. Sag. miniata. 221 GENUS V. STOMPHIA (Gosse). Base adherent, expanded. Column pillar-like; without warts or suckers, im- perforate (?) ; skin much corrugated ; substance not at all cartilaginous, but soft and lax. Disk very protrusile. Tentacles perfectly retractile. Acontia not present. Only one species has been yet recognised, S. Churchicp. ASTR^ACEA. BUNODID^. THE GAPELET. StompMa ChurcMce,. Plate VIII. Fig. 5. Specific Character. Body dashed with scarlet on white or yellow ; ten- tacles white, with scarlet bands. « Stomphia Churchice. Gosse, Ann. Nat. Hist. Ser. 3, iii. 48. GENERAL DESCRIPTION. FOBM. Base. Adherent to rocks in deep water, expansile considerably beyond the column. * Column. Very protean in shape, generally a short thick pillar, sometimes constricted hoiir-glass fashion or like a dice-box; the base sometimes detaches itself, and becomes very concave with sharp edges, or, on the other hand, protrudes as a low cone. Skin much and irregulai-ly cor- rugated transversely, and also longitudinally from the margin a little way downwards, thus giving a decussate appearance to the upper portion. Margin distinct, but without parapet or fosse, the outer tentacles springing from the very edge. Substance pulpy, or softly fleshy, very lax. IHsk. Flat, but often protruded as a low cone ; radii well marked. Tentacles. About 60, arranged in four rows, viz. 6, 6, 12, 36 ; sub-equal, the inner slightly longer than the outer, conical, much corrugated in con- traction ; when expanded, about equal in length to half the diameter of the disk ; generally carried horizontally spreading, or descending with the tips slightly up-curving. Mouth. Often widely opened. Lip sharp, protrusile, forming a nan'ow, low, circular wall. Colour. Column. Cream-white deepening to positive yellow, most irregularly sprinkled with dashes and streaks of rich scarlet, very much like a flaked carnation. THE GAPELET. 223 JHtk. White or yellowish white, pellucid. Teniaclts. "WTiite or yellowish white, pellucid, marked with three remote rings of scarlet, and, on the lower half of their front face, with two parallel stripes of the same hue, running longitudinally to the foot, sometimes confluent throughout or in part. These lateral stripes vary much in distinctness and size even in the tentacles of the same indi- vidual ; occasionally they run in upon the radii, and at times they are quite obsolete. Mouth. Edge of lip rich scarlet, " like the nectary of the Hoop-petti- coat Narcissus;" the colour sharply defined without, but within blending oflF quickly into the throat, which is white and strongly furrowed. Interior of gonidial tubercles scarlet. SiZB. Column. Two inches and a half in height, and the same in diameter ; flower about three inches in expanse. LOCAUTT. All roimd the Scottish coasts, in deep water. Varikties. o. Lychnucha. The condition just described. J3. Incensa. The red of the column predominant and almoet wholly confluent, interrupted merely by a few yellow flakes. y. Extincta. Column and disk pure white ; lip faintly tinged with red ; tentacles having the usual scarlet bars and the scarlet foot-Unea : the latter faint but distinct, and running in far upon the radii. 5. Pyriglotta. Colours nearly as a ; but remarkable for its large size, and the short thick-set form of the tentacles, which give it a considerable resemblance to Teaiia eramcomis. In the month of January, 1857, I was favoured with a communication from Miss Church of Glasgow, containing descriptions and figures of this showy and undescribed species, a specimen of Avhich she had procured in Loch Long, in the previous summer. It had been brought up in the meshes of a turbot net. Its brilliant hues, and their flaked arrangement, the protean variability of its shape, and its vivacity, attracted her notice, as did also the fact that it discharged a multitude of globular ova, of the size of mustard-seed, and of a rich scarlet hue. 224 BUNODIDiE. Last May, Mr. C. W. Peach, of Wick, sent me numerous sketches, some of which were coloured, of an Anemone which he had obtained at Peterhead, in April, 1850, and again in December, 1851 ; on each occasion from the hook of a fisherman's deep-sea line. These were manifestly- identical with Miss Church's specimen. It was not, however, until October, 1858, that I became, through the kind zeal of the Rev. W. Gregor, of Macdufi", personally acquainted with this fine species. Within three months he has sent me, on difierent occasions, half-a-dozen individuals, including all the varieties distinguished above, wliich argues its variability of character. This gentleman has been familiar with it for several years, as a not un- common inhabitant of the deep water of the Moray Frith. It is observable that all the specimens on record have been obtained by means of the deep-sea fishing boats. The generic name I have formed from <7T6fJb^o<i, wide- mouthed ; and the English appellation alludes to the same peculiarity, which is highly characteristic. The specific name is in honour of the kind correspondent to whom I am indebted for my first knowledge of the animal. More aberrant even than Ilormathia from the typical Bunodidce, and about equally intermediate between this family and the Sagartiadce, the genus might with equal propriety be placed in either. In its general aspect it rather inclines to the present family, especially by the intervention of Ilormathia, with which it has much in common. I have not been able to find any acontia, but fragments of craspeda issue from ruptures in the skin, and have much the appearance of acontia.* * On two occaslous I have seen protruded what looked like acontia. On one, it was very slender, streaming from the mouth to ueiirly an inch in length, so that I felt sura it was an acontiwm, till I put it under the microscope, when I found throughout the entire length, the ragged edge < of the mesentery from which it had been torn. It was hut a craspeduui,. THE GAPELET. 225 The Gapelet is rather difficult of domestication. In general, it attaches itself (usually to the perpendicular side of the vessel) for a short time, hut soon relinquishes its hold, and, after rolling ahout a few days on the bottom, dies. The approach of death seems to be always symptomed by spontaneous rupture and sloughing of the skin, and protrusion of the viscera. One, however, of the variety pyrtglotta, the gift of my kind friend, Mr. Gregor, esta- blished itself in my largest tank, and survived three months. My friend Mr. West has had a specimen from the Yorkshire coast a still longer time. In health, StompMa is remarkable for its extreme ver- satility of form. The column is sometimes cylindrical, sometimes shaped like a dice-box, sometimes like an hour- glass, while frequently successive constrictions chase one another along the extent. The base, when the animal is free, is sometimes concave, at others convex, and occa- sionally conical, while not unfrequently these forms are combined, the centre being conical while the rest is concave, — a cone in a crater. The disk is sometimes a deep bell, like a Convolvulus ; then a low cone, with the widely- gaping mouth crowning the summit. My first consignment fi-om Macduff consisted of two individuals, whicli on dissection proved to be of opposite sexes. They showed no external diversity of form or colour, but of one the pale salmon-colom'ed reproductive organs, which were very plump and full, were found under the compressorium to be filled with an infinite multitude of spermatozoa ; each of which consisted of a long-oval body 00015 inch in length, and a vibratile tail about thrice as long. In the other example the mesenteries were loaded with grape-like ova of a brilliant scarlet hue, varying in dimensions; — one of the largest measured '03 inch in diameter. Tliese consisted of an opaque scarlet yelk in a Q 226 BUNODID^. colourless chorion, which was perfectly globular, '0027 inch in thickness. By flattening some, 1 could discern the segmentation at the edges, which appeared to be well- advanced. When ruptured, the yelk escaped from the larger ones, — a mass of oil-globules of various sizes. The recognised localities of this species may be tabulated thus: — Loch Long, A. B. C: Peterhead, C. W. P.: Moray Frith, W. G. : Redcar, Scarborough, D. F. T. crassicornis H. Margaritse Churchi^. Bolocera Sagartia. Stomphia? SPECTA.B1LIS (Fabr.). Mr. Gregor has a strong conviction that there exists, in the same locality, an Anemone closely allied to the above, in which the colours are blue and green, arranged in a flaked or splashed manner, like the scarlet and yellow of ^S'. Churchice. This statement reminds me of the Actinia spectahilis of Greenland, which " has the body smooth, blue or green, striped longitudinally with rows of white points, and thick tentacles paler than the body, and spotted with white." From the locality of this species, it would be not unlikely to occur on the northern coasts of Scotland. * Fabricius, Fauna Groenl. p. 351, No. 342, h. 227 FAMILY VI.— ILYANTHID^. When Johnston published his second edition of the " British Zoophytes," a single Free Anemone alone was recognised : I shall have to include in the family at least a dozen, knowTi to inhabit our seas, with two or three others as yet obscurely indicated ; a number considerably greater than M. Milne Edwards assigns to the whole world, in his " Histoire Naturelle des Coralliaires," published little more than a year ago. The IlyantMdm form a very natural group, readily dis- tinguished by the important character, that, they possess no adherent base ; the column, which is generally length- ened, terminating below in a rounded, often more or less retractile, extremity. Hence they are characteristically unattached ; but many of the species, perhaps all, possess an adherent power in the entire surface of the column, by means of which they can readily crawl over a solid body. Most of them inhabit tubes, which may be membranous and free, as in Cerianthus ; membranous and investing epidermically, as in Edwardsia ; or mere burrows in the sand or mud, as in Halcampa, PeacTiia, and Hyanthus, Most of them have the habit of distending the hinder part of the column with water, assuming the form of a blown bladder. A remarkably vigorous and spasmodic contractility in this family indicates a more intense muscular force, and points to a higher physiological rank, than the preceding families possess. Q 2 228 ANxlLYSIS OF THE GENERA. Tentacles of one kind, marginal. Column tliick, pear-shaped. Mouth with a papillate gonidial tube PeacMa. Moutli simple Ilyanthus. Column slender, long, worm-shaped. Invested with an epidermis Edwardsia. Without an epidermis Halcamjta. Tentacles of two kinds, marginal and giilar. Naked; freely swimming Arachiactis. Dwelling in a membranous tube ; sedentary. Column inferiorly perforate Cerianthus. Column inferiorly imperforate {Not British) . . Saccanthus. 229 GENUS I. ILYANTHUS (Forbes). Column pear-shaped, tapering to a blunt point at the inferior extremity, which is probably perforated."* Surface smooth, without suckers, warts, or loopholes. Tentacles of one kind only, marginal, numerous {i. e. exceeding thirty). Mouth of the ordinary form, with no prominent gonidial development. • There is no evidence on this point with respect to our two British species. Dr. Kelaart, in his " Description of Ceylon Zoophytes," speaking of a species, which he has done me the honour to name Peachia Gossei, but which is evidently an Ilyanthiu, says that it has " an inferior orifice, large enough to admit a moderate sized probe, which gives passage to ova and escremeutitious matter." {Trans. Roy. Asiatic Soc. j Ceylon Branch.) ANALYSIS OF BRITISH SPECIES. Tentacles slender, filiform, long ; lined Seotietu. Tentacles thick, conical, short ; banded MUchdlii. ASTRJEACEA. ILYANTHILM. THE SCOTTISH PEAELET. Ilyanthus Scoftcus. Specific Character. Tentacles slender, filiform, long; marked with a dark line. Iluanthos Scoticus. Forbes, Ann. N. H. Ser. 1. v. 183. pi. iii. figs. 2, 3. W. Thompson, Ann. N. H. Ser. 1. xv. 322. John- ston, Brit. Zooph. Ed. 2. i. 243. pi. xlv. figs. 1, 2. M. Edwards, Hist. Nat. des CoraUiaires, i. 284. Ilyanthus Scoticus. GossE, Man. Mar. Zool. i. 30 ; Ann. N. H. Ser. 3. i. 417. GENERAL DESCRIPTION. Form. Colv/nvn, Pear-shaped, large above, tapering to a point at its lower extremity. DisJc. (" Mouth," Forbes ; but probably the Disk is meant.) Round, and rather small. Tentacles. Numerous (44, according to Forbes's figure), long (more than half as long as the body), slender, filiform, of nearly equal thickness throughout (apparently set in two or three rows). Colour. Column. Pink, with regular distant longitudinal white stripes. Tentacles. Greenish, with a dark line down the middle of each ; very nearly resembling those of Rapp's Act. filiformis. Size. Length about an inch and a half. Locality. The west coast of Scotland, and the east of Ireland : deep water. This genus was instituted by the late E. Forbes, to receive a " remarkable zoophyte," which he had dredged THE SCOTTISH PEARLET. 231 among Corbulce and other inhabitants of mud, in four fathoms, in Loch Ryan, on the west coast of Scotland, in 1839. The name of the genus is formed from iXu?, mud, and dvdo<i, a flower, and was originally written Uuanthos; but, as the Greek used in science is in a Latinized form, the correct orthography is certainly Eyanthus. The English appellation refers to the pear-like form. ILTANTHU3 SCOT1CU8 {from, Forbes). When we add that a specimen, presumed to be of this species, was found on the beach at Balbriggan, in Ireland, after a storm, in March, 1843, its whole known history is recorded. ? ScoTicus. S. viduata., Mitchellii. ASTRJiAOEA. ILYANTHIDJS, THE SCARLET PEAELET. Hyanthus Mitchellii. Plate VIII. Fig. 6. Specific Character. Tentacles thick, conical, short, marked with trana- Tcrse bands. JluantTioa Mitchellii. Gosse, Ann. N. H. Ser. 2. xii. 128. M. Edwakis, Hist. Nat. das Corall. i. 284. Hyanthus Mitchellii. Gosse, Man. Mar. Zool. i. 30, fig. 44 ; Ann. N. H Ser. 3. i. 418. GENERAL DESCRIPTION. Form. Colmnn. Stout, somewhat pear-shaped, thickening from the summit for about three-fourths of an inch, whence it gradually tapers to a blunt point, " in the centre of which is a minute wrinkled disk, which the animal does not appear to use as an adhesive sucker." * Disk. Veiy protrusile ; not so wide as the body ; radii distinct. Tentacles. About 36, set in two comjilete rows; thick, short, conical, and usually curled. The bases of the two rows are in contact, but the outer is fully one-sixth of an inch from the margin, and the inner about ae far from the base of the oral cone. MovUh. Prominent, seated on a cone. Lip thick, coarsely furrowecL COLOUR. Column. Upper parts pale scarlet ; lower two-thirds flesh-whitev blotched with scarlet ; lower extremity scarlet. Dish. A ring of purplish-black surrounds the mouth, which is suc- ceeded by a wider circle of white; and the remainder of the disk is pale red. Tentacles. Pellucid white, marked on their front faces with nnmerons alternate bands of opaque white and purple, sometimes taking a diagonal * I quote the words of my orif;inal description ; but I suspect that this appearance was only the retractation of the terminal point. THE SCARLET PEARLET. 233 direction. The tentacle that is opposite the mouth-angle on each side is wholly dull pvirple, with pale bands almost obsolescent. Mouth. Lip rich scarlet. Size. Length about two inches ; greatest diameter one inch. Locality. The coast of Dorset ; deep wat«r. This very fine species came into my possession in the spring of 1853, when I was engaged in collecting marine animals for the tanks of the Zoological Society of London. It was obtained by one of the Weymouth trawlers, who fish chiefly off the west side of Portland, As it remained with me hut a few hours, and was then forwarded to its destination, the above description and the figure were all I could contribute to its history. The species has not been met with since. I associated it with Forbes's Hyanthus (naming it in honour of D. W. Mitchell, Esq. the Secretary of the Zoological Society) ; but it appears to approach nearer to Peach ia than that species. Fuller observations are much needed on both. Scoticus. Mitchellii. p. hastata. 234 GENUS II. PEACHIA (Gosse). Siphonactinia (Dan, et Kob.). Column cylindrical, pear-shaped, or swelling in the middle, rounded at the posterior extremity, where there is an orifice ; margin entire, forming an indis- tinct parapet. Surface smooth, without loopholes, but studded in every part with very minute and very numerous suckers. Disk flat, or very slightly conical, smooth. Tentacles of one kind, twelve, thick, short, obtusely pointed ; marginal ; imperfectly retractile. Mouth not elevated on a cone ; lip thin, abrupt, pro- trusile, sometimes lobed. A single gonidial groove, the edges of which are soldered together so as to form a tube, which terminates above in a thickened, expanded rim {conchula), the margin of which is more or less divided. Acontia wanting. ANALYSIS OF BRITISH SPECIES. Column lengthened. Conchula with from 12 to 20 lobes hastata. Conchula with 3 ovate lobes tnphyUa. Column short. Conchula with 5 shallow lobes imdata. ASTJRjEACEA. ILYANTHIDjE. THE ARROW MUZZLET. Peachia hastnta. Platb VIII. Fig. 3. Specific character. Column lengthened ; conchula bearing from 12 to 20 lobes, which are mostly bifid ; tentacles marked with arrow-heads. Peachia hastata. GossE, Linn, Trans, xxi. 267, pi. xxviiL ; Man. Mar. ZooL i. 31, fig. 46 ; Ann. N. H. Ser. 3. L 418. GENERAL DESCRIPTION. Form. Colv,mn. Club-, pear-, or spindle-shaped, or cylindrical, the same indi- vidual assuming all these forms ; lower extremity rounded, with a minute central orifice, distinct, but generally closed, and apparently furnished with a sphincter. Surface smooth, but covered with microscopically minute suckers, which have the power of strong adhesion to foreign bodies. Substance fleshy, becoming more membrauous below, where, when in- flated, it resembles a blown bladder. Disk. Flat, but protrusile, as a low cone ; radii distinct. Tentacles. Twelve, in one circle, marginal; short, thick, and some- what flattened at the foot, tapering to a point ; generally carried hori- zontally expanded ; sometimes they are considerably lengthened and attenuated. Mouth. Prominent, with a pro- trusile cushion-like lip, deeply fur- rowed. Conchula. There is but one go- nidial groove, the edges of which are united, the suture marked by a depressed line, on each side of which the wall is plump. The apical edge of the tube rises into a con- spicuous organ {conchula), and is cut into papillary lobes, placed in single series, but generally so crowded as to overlap each other. They are from 12 to 20 in number, but are not perfectly regular either in form or order. Most COSCHCLA ASD MOUTH OF P. HASTATA {magnified). 236 ILYANTHTD^. of them are bifid ; tlie back lobes have a temlency to be simple, except the central back one, which is large, and composed of two bifid ones united on a single stem ; this compound one is generally bent over as a protection to the orifice of the gonidial tube. The papillae resemble tentacles in that they are hollow, with thick walls, the internal surface of which is lined with brown pigment, deepening at the tijis ; they are very moveable. Colour. Column. Pale red or flesh-colour, through which the edges of the septa appear as twelve white lines : the fore half of the column is' fre- quently marked with irregular splashes of chocolate-brown, which are sometimes confluent. Disk. Pale red or bufi", each radius marked with two \/s of deep brown, one within the other, the points of which are outwards ; the point of the outer one meets the tentacle, and sends off a branch on each side, encompassing its foot. Tentacles. Pellucid, each marked on its front face with arrow-heads cf deep brown, arranged in two longitudinal rows, the points downwards ; there are about six in each row, but near the tip they become indistinct. Each arrow-head is separated from its successor by one of opaque cream colour or pale sulphur-yellow. Mouth. White, with the fun-ows deep brown. Conchula. Pale salmon-colour; the lobes pellucid, with an opaque white core, which is crossed by a brown bar near the tip. SiZK. About four inches in length, and one in greatest diameter. I have seen the body lengthened to eight inches, without any signal attenuation. LOCALITT. Torbay, at extreme low water, and thence downward, buried in sand. In a paper read before the Linnean Society on the 20th of March, 1855, I characterised this genus and species from specimens presented to me by the Kev. Charles Kingsley. I named it after Mr. Charles W. Peach, who was the discoverer of the first British Ilyanthidan known, which I at that time referred to the same genus. In June, 1856, MM. Danielssen and Koren founded, on a species occurring on the coast of Norway, their genus Siphon- actinia, whicli is evidently identical with this, though they appear to have mistaken the conchula for the mouth. THE ARROW MUZZLET. 237 The hea,\j easterly gales of last autumn, coinciding with the October spring tides, must have disturbed the PeachicB in their burrows ; for the species suddenly became common, as many as fifty ha^dng found their way into the possession of the Torquay dealers about that time. A few of these fell to my lot, and enabled me to correct and amplify the history of the species. These specimens were very lively, ever bending their columns, and rapidly changing their forms. While under examination, they frequently adhered by various points of the column, and when lying on the side would, gradually but quickly, bring the hinder extremity round, under the body, nearly to the front, and then applying it to the bottom of tlie vessel, adhere, not by the orifice, but by the swollen surface around it. Constrictions were constantly j)assing along, commencing about the middle of the column, and passing off downwards, the effect of which was to throw out the translucent posterior extremity, like a clear dis- tended bladder, within which the septa could be very distinctly defined. One only of the specimens survived, the others I dis- sected. The former I put into a vase of sea-water with a bottom of sand. This was at night ; in the morning it was just beginning to insert the hinder extremity into the sand, and thence the process of burrowing went on regu- larly. In two hours it elevated the fore parts, and assumed a perpendicular position, continuing to descend.* By * Mr. Holdaworth, who obtained another of the Torquay specimens, has made an interesting observation on this process. " After it had selected a suitable place for burrowing, in the darkest part of the vase, the posterior extremity of the body became tapered to a fine point by a partial expulsion of the contained water, and at the same time turned downwards and pressed slightly into the ground ; the fluid contents of the animal were then forced back until the base was completely distended, and by this means a shallow depression in the sand produced ; the tail then resumed its conical shape, was again thrust into the ground, and swelled out ; and these proceedings were continued until a hole was made sufficiently large to admit the animal. Its first efiforts in burrowing had but little effect. II 238 ILYANTHID^. eleven A.M. only about an inch in length of the fore parts remained above the level of the sand, when it expanded, and seemed satisfied. At night, however, it came out of its burrow, and remained wallowing on the surface ; and for a week after this it continued to go in and out once or twice a day, grovelling and stretching awhile, and then burrowing comfortably almost to the tentacles. This individual still survives in the same vase, after six months' captivity ; it frequently remains for days completely hidden, sometimes shows only the tips of the expanded tentacles, and rarely more than the disk, above the sand. It is perfectly domiciliated. Another of the individuals referred to gave birth, while under my observation, to some half-dozen or more embryos, of oblong or ovate form, which appeared like little Peachias, but I could not see any trace of disk or tentacles in any. They were discharged one by one through the gonidial tube, as the animal lay on its side. . This one was ruptured in two places ; and as it lay in a small tank, the craspedal mesenteries were protruded, and spread in large irregular areas on the glass bottom, per- fectly flat and adherent, the membrane being pellucid and very delicate, and the craspedum bounding the outline like a white thread. The conchula is generally protruded, even when the tentacles and disk are wholly retracted. Perhaps it is the seat of some sensation. I. Mitchellii. Halcampa. hastata. Cerianthus. triphylla. undata. and it was only after an hour's labour, when the cavity had become large enough to allow the polype to work in an upright position, and with the assistance of its whole weight, that rapid progress was made." (Annals N.H, for Jan. 1859, p. 78.) ASTILEACSA. ILTANTHIDJl. THE WAVED MUZZLET. PeacMa undata. Plate VIII. Fig. 4. Specific Character. Column cylindrical, short ; conchula cut into five shallow lobes ; tentacles crossed by dark wavy bands. Peachia undata. GossE, AnnaiR "S. H. Ser. 3. i. 418. GENERAL DESCRIPTION. Form. Column. Cylindrical, rounded below, slightly fluted, about twice as long as the diameter of the disk ; terminating below in a central perforate depression, around which the skin is much puckered, and minutely cor- rugated. Surface wrinkled, both transversely and longitudinally, espe- cially when contracted. Margin distinctly angtdar, sometimes forming a very low parapet. Disk. Smooth, flat, or rising with an even and very gentle elevation from the foot of the tentacles to the edge of the mouth ; marked with twelve radii forming so many fine lines. Tentacles. Twelve, in one circle, marginal ; thick and rounded at foot, tapering regularly to the tip, which is obtusely pointed ; transverse section sub-ovate, the diameter from side to side exceeding that from back to front. By irregular contraction, they sometimes become slender and cylindrical, often with the tip clubbed or knobbed. They are generally carried widely expanded horizontally, with the tips arching downwards like a twelve-rayed star. , Mouth. Descends abruptly from the disk with a sharp angle, but which can scarcely be called a lip, as it is not thickened. It is protrusile at the will of the animal, when ordinarily it embraces the eiserted gonidial groove, and displays a number of plicae at its edge. Conchula. The groove is greatly de- veloped ; its edges are in contact until coxchula op p. undata about one-sixth of an inch from the tip, {magnified). where they separate, and turn over with a scroll-like expansion, the margin of which is cut into five shallow teeth, as follows: — one terminal and two 240 ILYANTIIID^. lateral, all of whicli are bluntly triangulai', or sub-square, two others still further removed from the terminal one, which are rounded and merge into the smooth descending edges. The mouth is sometimes widely re- tracted, and the groove exposed for the greater part of its length ; but usually the conchula only is protruded from the almost closed mouth. Colour. Column. Very pale yellow, marked with irregular longitudinal splashes and stripes, of dull red, more or less confluent at the lower extremity. Margin pellucid, with alternating spots of opaque white. Disk. Creamy white : each radius marked with a minute brown speck at the foot of each tentacle ; except that radius which is opposite (not corresiiondeni) to the gonidial groove, in which the speck is wanting. Tentacles. White, crossed by seven waved bands of deep brown, each band strong and well defined at its upper edge, but ill defined and fainter at its lower edge : the fourth band (the central one) ia broader and fainter than the rest. The lowest two bands are rather of a deep bluish-black. On the tentacle which corresponds to the groove, the lowest two bands are wanting, as are the lowest three on the tentacle opposite, leaving the face of this part of the tentacle pure white. The bands in all cases extend only across the front face and sides, disappearing on the back. Mouth. Whole interior of throat and stomach, and exterior of the lower parts of the groove, a rich red buff or salmon-colour. Conchula, Both without and within pure cream white. Size. Length about an inch and a quarter ; diameter of disk about seven- eighths of an inch ; expanded flower an inch and three-quarters : thickness of column one inch. Locality. The Channel Islands. The only individual of this species that I have seen was one which I owed to the kindness of my friend Dr. Hilton, of Guernsey, who obtained it on the island bf Herm, lying on the sand at very low water, in April, 1858. When it arrived, after just thirty-six liours' confinement, it looked much exhausted, and lay flaccid, with the mouth veiy widely gaping, displaying the thickly folded stomach, of a salmon-buff hue, and the gonidial tube greatly exposed and protruded. The tentacles were collapsed. VYhen put THE WAVED MUZZLET. 241 into sea- water, no immediate change appeared, but after an hour or two the tentacles began slowly to move one by one backward and forward, and slightly to swell and to lengthen, while the mouth partly contracted. Next morning it had quite recovered health and beauty. The tentacles were very versatile, constantly changing their form. The mouth also was perpetually opening or closing, but slowly. The animal appears unable to enclose the disk, but the tentacles contract individually, when touched, or spon- taneously, shortening to mere warts. I have seen the animal when several of its tentacles could scarcely be distinguished from the general level of the disk-edge, except by the coloured rings. It would lie rolling about on the sand in a vase, with constrictions successively passing up its body, and throwing off clear mucus. When put into a hole in the sand it would not remain ; being very buoyant, it was soon on the surface, the hole gradually filling beneath it. It remained in health for a few days, at which period the mouth gaped widely, and the lax corrugated stomach was exposed ; the tentacles contracted to warts, and, the animal being manifestly feeble and dying, I dissected it. Mr. Whitchurch, of Guernsey, reports having found a \JPeac7iia, which he supposes to be this species, on re- peated occasions; it may, however, have been the following. He mentions the interesting fact that the tentacles are luminous. The Siphonactinia (= Peachta) Boeckii has so close a resemblance to this species, that I am not certain whether my specific appellation will not have to be merged in that of the Norwegian zoologists, I rely, however, on the figure in Faun. Litt. Norv., ii., in which the lobes of the —^conchula are distinctly three in number, and are square in I 242 ILYANTHIDiE. form. The manner in which the mouth is represented as pursed out, and closely investing the gonidial tube, with the gular furrows looking like rudimentary tentacles, I have observed both in this and the foregoing species. P. BoecMi is assigned to a depth of 80 to 200 fathoms in the fjords of Norway. The posterior orifice in this genus cannot always be observed; I have, however, satisfactorily demonstrated it by dissection in both hastata and undata. When the inte- gument is cut away from the whole vicinity, it appears as a circular foramen, about half a line in diameter. It does not appear to be an anus, but probably admits water for respiration. The specific name, undata, indicates the waved pattern of colouring on the tentacles. The term Muzzlet, which I have assigned to the genus, alludes to its most prominent characteristic, — the protrusion of the gonidial tube, like a proboscis or muzzle. hastata. triphylla. UNDATA. ASTJLEACEA. ILYAHTHID^. THE TREFOIL MUZZLET. Peachia triphylla. (Sp. nov.) Plate X. Fig. 2. Srpeeific Character. Column pear-shaped, moderately long; conchula bearing three ovate or leaf-like lobes ; tentacles marked with arrow-heads, and based with brown. GENERAL DESCRIPTION. FOBK. Column. Pear-shaped ; lower extremity rounded, with a distinct central orifice, around which the skin is puckered. Surface covered with fine and close-set transverse wrinkles, and with minute suckers, which have a strong adhesive power. IHsk. Flat, but very protrusile ; radii distinct. Teniacles. Twelve, in one circle, marginal ; thick at foot, and tapering to a point. Mouth. About one-fourth of an inch wide at the margin, shelving downward funnel-like ; lip rugose and erectile. Conchula. Cut into three ovate, leaf-like lobes. CONCHULA AKD MOUTH OF P. TKIPHTLLA (magnified). COLOXTB. Column. Opaque pale reddish-brown, or bay, with numerous irr^ular longitudinal splashes of rich red-brown. No pale lines indicate the septa. b2 244 ILYANTHIDiE. Dis^. Reddish buff ; each radius marked with a minute brown speck in its centre : the gonidial radius, however, and the opposite one, are pure white, without spots. Tentacles. Pellucid, each marked with a double row of brown arrow- heads, exactly as P. hastata, but the foot is crossed by a band of deep brown, the discal edge of which is perfectly defined; the confluence of these bands forms a broad circle of brown bounding the disk. In the gonidial tentacle, however, and in the opposite one, the band is wanting, as well as the lower arrows, the opaque white of the radius running up the front of each of these tentacles half-way to the tip. Mouth. Dark brown. Conchula. Pure opaque white ; the lobes without spot or core. Size. Length three inches; greatest diameter one inch and a half. Disk three-quarters ; tentacles about one inch, j LOOALITT. The Channel Islands. I have had no opportunity of seeing the animal to which the above description applies. It was taken at Guernsey, in Decemher, 1858, and came into the possession of Dr. Gr. C. Wallich, who has kindly drawn out for my use copious notes, and furnished me with beautiful coloured drawings. It appears intermediate between hastata and undata, the species already recognised ; but I cannot satis- factorily assign it to either, as it differs from both in the form and number of the conchular lobes. I have there- fore given it a name expressive of these peculiarities. " The suctorial processes," remarks Dr. Wallich, " ap- pear to consist of simple depressions of the integument, each of which exhibits an oblong muscular body at its base, whereby a vacuum may be formed, and adhesion accordingly secured. On examining these muscular bodies under a power of 250 diameters, longitudinal as well as transverse striae are distinguished. The nature of these suckers was strikingly manifest on attempting to turn the animal in the glass, when they exhibited the appearance of THE TBEFOIL MUZZLET. 245 a number of pointed papillae, the apices of -wliieh clung forcibly to the glass, whenever a strain was put upon the creature to disengage it." hastata. TKIPHYLLA. undata. Peachia ctlihdbioa (Reid). In the Annals of N. H. for January, 1848, Dr. Keid described and figured, under the name of Actinia ci/lvidrwa, an Actinoid, which was washed ashore at St. Andrews. It must have certainly been a Peachia, and may possibly have been an immature P. hastata. The points in which it disagreed with such specimens of the latter as I have seen were the following: — 1. It was but one and a quarter inch long. 2. The conchular lobes were twelve, six of which were very minute; triangular, orange, with trans- lucent edges. 3. Twelve bands of faint reddish-brown radiated across the disk. 246 GENUS III. HALCAMPA (Gosse). Actinia (Peach). Peachia (Gosse). Column long, slender, cylindrical, or swollen at the inferior extremity, which appears to be imper- forate : no distinct margin. Surface without loop- holes, but studded with minute suckers. Dis^ fiat. Radii distinct. Tentacles of one kind, few (less than twenty), marginal or sub-marginal, cylindrical, obtuse; per- fectly retractile. Mouth simple. No obvious gonidial development. ANALYSIS OF BRITISH SPECIES. Tentacles 12, banded ; lives in sand chrysanthdlnm. Tentacles 16, white ; lives in eroded rocks .... microps. ASTJLEACFA. ILYANTHID^. THE SAJ^D PDsTLET. Halcampa chrysanthellum. Plate VII. Figi, 9, 10. Specific Character. Tentacles twelve, in one row, as long as the diameter of the column, banded. Actinia chrysanthellum. Pkach, in Johnston's Brit. Zooph. Ed. 2, i 220 ; pi. xxxvii. figs. 10 — 15. Cocks, Rep. Comw. Soc. 1851, 6; pL L figs. 20, 21. Peachia (?) chrysanthellum. GrOSSE, Linn. Trans. xxL 271 ; Man. Mar. ZooL L 31. Halcampa chrysanthellum. Ibid. Annals 2f. H. Ser. 3, i. 418. GEISTERAL DESCRIPTION. FOBM. Column. Cylindrical, lengthened, worm- like (extending to ten times its diameter or more) ; slightly invected ; terminating below in a rounded extremity, which is generally distended into a bladder-like form and translucent thinness, and is incapable of being retracted ; merging above into the tentacles without a parapet. Surface studded with excessively numerous, minute, sucking warts. IHslc. Plane. Radii twelve, distinct. I HALF-DISK OF H. CHBYSAKTUELLUH {magnified). Tentacles. Twelve, strictly marginal, set in a single row, their feet in contact. Xearly cylindrical, with roimded extremities, about as long as the general diameter of the column, usually carried pointing upwards and outwards, slightly arched ; perfectly retractile by the ordinary process of version. 248 ILYANTHID^. Mouth. A line without disjiinct lip ; not elevated on a cone. Furrowed within. COLOUB. Column. Drab or dirty white ; the septa distinct as white longitudinal lines ; the swollen bladder-like extremity translucent, and almost colour- less, except for the septa. Disk. Marked with a pretty star-like pattern, consisting of a pale blue area, inclosed in a pale line, and surrounded by twelve triangular rays of a dark brown hue; each triangle surmounted by a pale W-like figure, which incloses a dark brown area, according to the accompanying pattern. Tentacles. Pellucid brown, the front crossed by six semi-rings of opaque white, of which the second, the fourth, and the fifth (counting from the foot upward) are angular, the second pointing downward, the fourth and fifth upward. The pellucid interspaces are tinged with brown, deepest on the first, second, and fourth ; and the first white ring, surrounding the foot, is sometimes tinged with sulphur- yellow. Mouth. - Yellowish-white. Size. Specimens reach to an inch and three-quarters in length, and one-eighth of an inch in avei-age diameter ; the extremity is frequently inflated to ene-fourth. LocALiTr. Coast of Cornwall : buried in sand at low water, and in tide-pools. This is a very interesting little zoophyte, which was first made known by Mr. C. W. Peach, who has faithfully described its person and manners. Its lack of an expanded base of course removes it from the genus Actinia / and when I formed the genus PeacMaj it was under the sup- position that the present little species was to be therein included. Subsequent personal acquaintance with it, however, induced me to constitute a new genus for its reception, to which I have since added a second species. The name of this genus, Halcam^a, formed from aX?, the sea, and KafiTTT), a maggot, alludes to the grub-like form of the animal; a form which I commemorate also in the English name, pintlet, from pintle, an iron pin. The THE SAND PINTLET. 249 specific appellation must be accepted, I suppose, as ex- pressing the general resemblance of the painted disk to a flower. In May, 1858, hj the kind courtesj of J. Scott, Esq. of Her Majesty's Customs, I was favoured with two consign- ments of this pretty little species, including upwards of a dozen specimens. They were procured at Fowey, in Corn- wall When turned out of the package in which they had travelled, they looked like little earthworms. Some of them I dropped into holes which I had made with a stick in damp sand, carefully pouring the sea-water in afterwards. These maintained their place, and soon protruded and expanded their disks from the surface of the sand. Others I simply laid on the sand when covered with water; these presently began to bore with the in- ferior extremity, and soon descended as far as the level iof the disks, which then expanded, as if at home. Several of those specimens I still possess in health, after about eleven months' captivity; and I have reason to think that in the meantime they have produced living young. After they had been domiciled for a time in a wineglass nearly filled with sand, and covered with a shallow layer ■of water, I wished to remove them to a larger vase. On washing out the sand, I found the animals firmly adhering to the glass by the lower parts of their bodies. When removed, they would take instant hold of the smooth glass, ■with the suckers on any part of the body, four or five of these drawing out to a considerable length when force was applied. On examination of these suckers, we see that the skin is covered with very minute and close-set, irre- gularly shaped, rounded warts, which have a firmly adhering function. They are best seen on the distended skin of the hinder extremity, where, under a power of 150 diameters, 250 ILYANTHID^. they prove to be granular nuclei in the substance of the skin, dense in the centre, and gradually thinning to an undefined circumference, elevating the surface with a smooth rounded outline to a height about equal to their diameter ; viz. about .002 inch. Many of them certainly have a shallow pit on the summit, and I am persuaded that their adhesion is a sucking. In the middle part of the body, these warts are elongated transversely, and have a ten- dency to run in close-set annular lines. I have not been able to satisfy myself of the character of the inferior extremity. It often appears as if it were distinctly perforate; but I believe this is an illusion, produced by the following phenomenon. As the animal lies on its side, it is continually being constringed, the constriction gradually moving downward till it passes off at the extremity. The parts above and below being in- flated, and being as transparent as glass, one sees, looking directly at the extremity, the inner edge of the constriction, through the transparent integument, exactly like a ter- minal orifice, at the moment before it passes off. The manners of the species are lively and pleasing : it is very susceptible of alarm, when it closes and disappears in its burrow with great quickness ; it is, however, soon full- blown again. Under irritation, as when fine clay is mixed with the water, the tips of the tentacles are jerked from side to side with a suddenness and force that contrast with the languor common to the tribe, and which seem to indicate both a higher nervous sensibility, and also a greater development of the muscular system. My experience, as well as that of Mr. Peach, shows that it is a species well adapted for an aquarium, and that no special treatment is needful beyond a layer of sand equal in depth to the length of the column. The stomach is sometimes protruded, and inflated so as THE SAND PINTLET. 251 to form an ovate bladder as wide as the diameter of the colmnn. This occurs as well when comfortably ensconced and expanding, as when exhausted by lying out of water. Mr. Peach has favoured me with notes of a singular example of the reproduction of organs in this species. A specimen in his possession displayed a transverse cut, apparently the result of accident, which extending almost quite across the column just below the disk, caused the fore part to fall over, hanging only by a fragment of skin. The tentacles, which now of course drooped from the bottom of this hanging part, presently disappeared by absorption, while at the same time from each of the severed surfaces a new disk with new tentacles was developed. Thus the old stump became pretty much as before, only slightly shorter, but the severed piece lost the tentacles at one end, and acquired new ones at the other. Halcampa chrysanthellum has been found as yet only in Cornwall, but in the following spots : — Fowey, G. W.P. : Gwyllyn Yase, Pennance, &c., W.P. C. P. hastata. CHETSAXTHELLUM. microps. Edwardsia. A ST R^ ACE A. ILTANTHIDjE. THE KOCK PINTLET. Halcampa microps, Plate VII. Fig. 11 : XII. Fig. 6 (magn.). Specific Character. Tentacles sixteen, in two rows, very short, without markings. Halcampa microps. Gosse, Annals Nat. Hist. Ser. 3. ii. 195. GENERAL DESCRIPTION. FOBM. Column. Cylindrical ; 8-invected, the tegumental insertions of the septa being the boundaries of the swellings ; hinder extremity inflatable, pro- truaile, adhesive : skin minutely granular, enveloped in a thin mucus, which entangles foreign matters ; ordinarily covered with minute, close- set, transverse wrinkles. Disk. The rounded anterior extremity of the column, around which the tentacles are planted in two contiguous circles (though those of each row are remote inter se). Sometimes this rounded form is not observed, and then the disk is flat. Tentacles. In two rows ; the first of eight, about .014 inch long, and .0045 inch in medium diameter ; the second also of eight, marginal, remote, alternate with the former, papilliform, their length not exceeding their diameter, or .005 inch. When expanded, those of the first row either stand erect, or arch slightly outward : their movements are rather sudden ; their form quite cylindrical, with round ends; their walls thick, apparently imperforate ; a few cnidse scattered in their substance. Mouth. Elevated on a small abrupt papilla. Colour. Pellucid yellowish white, positive in the ratio of opacity of the parts without markings. Ovaries tinged with flesh-colour. Size. Column when moderately extended about .025 inch in diameter, to a point about halfway down its length ; diameter of posterior inflation at th« same time .065 inch. Total length in this condition .3 inch. Locality. ; South Devon ; rocks between tide-marks. THE BOCK PINTLET. 253 I found this tiny species in mnch eroded limestone from a cavern at Oddicombe, Devon, associated -svith Edwardsia camea, in Jnne, 1858. Having chiselled off many frag- ments of the rock, I put them into glass jars of sea-water ; and in a day or two found Halcampa mto'ops crawling up the side of the jar, adhering hy its inflated skin. In the course of a day or two more, another and another appeared, until five or six had come under my notice, most of them adhering to the glass. They were active and locomotive, moving along the surface with ease and comparative quickness (at least ten times their length in a night), adhering hy any part of the hinder moiety of the column. Very frequently they threw the anterior portion suddenly round, like an irritated caterpillar ; and almost continually constrictions were passing down in succession from head to tail. They are very coy and very sensitive, retracting forcibly and suddenly when alarmed. I attempted to feed them, hut only frightened them. The specific name is from fiiKpb<i, small, and w-^, the face. chrysanthellum . MICROPS. Edwardsia, 254 GENUS IV. EDWARDSIA (Quatrefages). Scolanthus (Gosse). Column long, slender, cylindrical, divided into three distinct regions, of which the two terminal are retractile within the central one. Anterior region forming a short thick pillar {capitulum) of less diameter than the central, and more delicate. Central region {scapus) covered by a skin {epidermis) more or less thick and opaque. Posterior region {physa) thin, pellucid, inflatable like a bladder; imperforate (?). JDish sometimes flat, sometimes conical. Tentacles of one kind, few (less than thirty), mar- ginal, arranged in one or two rows j slender, mode- rately long, pointed ; perfectly retractile. Mouth simple. No obvious gonidial development. ANALYSIS OF BRITISH SPECIES. Tentacles sixteen, transversely dashed with white; capitulvm ptirple brown, with white markings ; lives in sand callimorpha. Tentacles twenty-eight, pellucid crimson ; capitulwm pellucid cameous ; lives in eroded rocks .... camea. ASTRJBACEA. ILTANTHID^. THE PAINTED PUFFLET. Edwardsia calUmorpha. Plate VII. Fig. 7. Specific Character. Tentacles sixteen, transversely dashed with white : capitulum chocolate-brown, painted with white. Scolanthus callimorphus. GossE, Annals N. H. Ser, 2. xii. 157 ; pi. x. EdwarcUia calUmorpha. Ibid. Linn. Trans, xxi. 271 : Man. Mar. Zk)oL i. 31 ; fig. 45 : Ann. N. H. Ser. 8. L 418. GENERAL DESCRIPTION. FOBSL Column. Nearly cylindrical, slightly enlarging posteriorly, worm-like, the length in extension being to the diameter as 10 : 1. Capitulum a short piUar, slightly contracted above and below the middle, and most expanded at the margin ; mailed with eight invections, each of which is divided towards the summit into two ; the surface smooth and delicate. Scapus opaque, leathery, rough and minutely corrugated. Physa (not observed). Disk. Plane ; radii distinct. Tentacles. Sixteen, marginal, set apparently in a single row, but yet slightly alternating, corresponding to the invections and semi-invections ; long (nearly thrice the diameter of the disk), slender, slightly tapering, obtusely pointed. They radiate horizontally or diagonally, and are fre- quently intro- or retro-verted. Mouth. Set on a prominent cone. Colour. Column. Capitulum rich chocolate-brown, irregularly dashed with white and black, each invection bearing a conspicuous lozenge-shaped spot of cream-white at its foot, and each semi-invection a triangular spot of white at the summit. These marks are well defined, and their effect is very beautiful. Scapus a deep orange-yellow, somewhat tarnished. Disk. White, marked with a star of pointed arches of deep sienna- brown, each arch having a radial stria for its centre, and a circle sur- rounding the mouth for its base. The two gonidial radii dark brown. 256 ILYANTHID^. Tentacles. Transparent and colourless, marked with spots and dashes of opaque white, arranged in irregular transverse rows and rings, which increase in number and size until they become confluent towards the tips, which are thus pure white. The glassy translucency of the tentacles throws out these opaque markings with beautiful effect, especially as the foot of each is girded by a broad circle of white. Size. Column about three-quarters of an inch long when contracted, but extending to two and a half inches, with a diameter of one-fourth : disk one-fifth of an inch ; expanse of flower about one inch. Locality. The eouth-weetem coasts of England ; deep water. In the summer of 1853 I obtained, from about five fathoms in Weymouth Bay, a specimen of this species, which I described and figured in the Annals of Nat. Hist, under the name of Scolanfhus, as I supposed it to be an unrecognised form. M. de Quatrefages had, however, pub- lished an able and elaborate Memoir* on a form which he had named Edwardsia, in well-merited honour of the eminent French zoologist, M. Milne Edwards. On mature consideration, I was convinced that my Weymouth spe- cimen ought to be placed in this genus ; for though I had described a posterior orifice, which is wanting in Edwardsiay it is probable that I mistook, for such, the depression at which the physa, which I did not see, was retracted. The animal appears to be quite distinct from all of the three French species described by M. de Quatrefages, and to be well marked by its beautiful painting, which, resembling the inlayings of veneer-work, or the figures of the kalei- doscope, suggested to me a name derived from /caXo9, beau- tiful, and fiop(f>r}, form. The English term commemorates * Annales des Sci. Nat. 1842, Ser. 2, xviii. 65. THE PAINTED PUPFLET. 257 the habit of the genus, of puffing out the bladder-like termination of the column. The habit of the species, judging from what I have seen of it in captivity, is to burrow in fine gravel or sand at such a depth as allows it to protrude the coloured capitulum. from the surface. Here it expands its tentacled disk for passing prey : I fed it with fragments of a shrimp, and found that it ate with the same avidity, and in exactly the same manner, as its cousins, the Sea- Anemones ; the tentacles catching and moving to and fro the morsel, and disposing its position and direction so as to faciKtate the mouth's grasping it ; this latter organ expanding its flexible lips to an apparently indefinite width, and gradually en- veloping the presented food. If rudely touched, the disk was suddenly withdrawn ; the capitulum, and then the upper two-thirds of the scapus, disappearing in rapid succession by a process of intro- version, exactly like that by which the earthworm with- draws its fore parts, or, to use a homely simile, like the turning of a stocking. The extent to which the intro- version proceeds depends on the degree of annoyance to which the animal has been subjected, or on its wayward will. It is capable of crawling along in its subterraneous abode, while contracted ; pushing aside the gravel with the front of its body. It proceeded in this way two or three inches in as many hours, while I was watching it, before it turned upwards and thrust out its head ; the evolution of the capitulum not beginning until the surface was reached. A second specimen of this species was dredged by the Rev. Charles Kingsley, off Brixham, in January, 1854. He informed me that the form and colours agreed with my description, except that the hues of the capitulum. were more brilliant, and those of the disk less so. " He broke off his tail in disgust two days ago, but has now thought S 258 ILYANTHID^. better of it, and has begun wisely to grow a new tail, wliicli is at present transparent, hut with a well-defined orifice. He lies lialf-buried in sand, and lias several times temporarily attached himself by his new tail."* Since this page was in type, Dr. Hilton has taken a specimen at Bordeaux Harbour, Guernsey, which he has kindly transmitted to me. In its general characters and markings, it agrees with the specimen described above ; it is, however, much larger, being at least five inches long, and three-eighths in diameter. The scapus is more spindle- shaped, and more coarsely invected and corrugated ; the physa I have seen inflated, but slightly. The tentacles which correspond to the gonidial radii, and the pair at right angles to these, are much shorter than the rest. The dark gonidial radii have a flush of rich green. ]\Iany points in the form and anatomy of this genus indicate, as has been ably shown by Quatrefages, a decided approach to the EcMnodermata, through such forms as Syrinx and Svpunculus. Weymouth, P. H. O. : Brixham, C. K. : Guernsey, T. D. H. [Beautempsii.] ECHINODEEMATA. CALLIMOEPHA. ' [Harassi.] caraea. H. chrysanthellum. * Kingsley in litt. ASTR^ACEA. ILYANTHIDJS. THE CRIMSON PUFFLET. Edwardsia carnea. Plate VII. Figs. 5, 6 : XII. Fig. 3 {magn.). Specific Character. Tentacles twenty-eight, pellucid crimson ; eapitalum pellucid flesh-pink. Edwardsia carnea. GosSE, Annals N, H. Ser. 2. xviii. 219 ; pi. ii. figs. 1—4. Ibid Ser. 3. i. 418. GENERAL DESCRIPTION. FOEM. Column. Generally cylindrical, sub-equal in diameter throughout, worm- like, length to diameter as 10 : 1. Capitulunn cylindrical, or slightly barrel- shaped, marked with eight invections and eight semi-invections^ like the preceding ; margin tentaculate. Scapus slightly more coriaceous than the other regions, but clothed with a very rough epidermis, so slightly adherent that it frequently forms a partially free tube. Physa thin, membranous, globose, transpaient, revealing the septa ; imperforate. Disk. Plane ; radii distinct. Tentacles. Twenty-eight, sub-marginal, arranged in three rows, — 8, 8, 12 := 28 (perhaps the ultimate number of the third row may be 16) ; versatile in shape, being sometimes very short and fusiform, at others elongated to thrice the diameter of the disk, tapering and very slender. They generally radiate diagonally, arching outwards. Mouth. Set on a low cone; lip furrowed. COLOUB. Column. CapittUuni translucent, delicately tinted with pink, each in- vection bounded by a fine line of opaque white or brilliant pale yellow, and marked with a longitudinal dash of the same near its foot. The stomach is plainly visible, as a thick axis of rich scarlet. Scapus and physa of the same rose-tinged translucency, but the epidermis of the former is of a brownish-yellow hue. Bisk. A star of cream-white raya on a translucent ground. Tentacles. Lovely pellucid pink, sometimes with alternate bands of less s2 260 ILTANTHIDJE, and more positive colour ; frequently becoming a pale opaque yellow at foot, whicli hue runs up in a point on each aide. Mouth. Scarlet, leading to a stomach of the same rich hue. Size. Column, in extension, reaches to nearly an inch in length, with a general diameter of one-tenth ; capitulum one-sixth in length, one twenty-fourth in diameter ; expanse of flower one-fourth. Locality. The south-western coasts of England ; eroded rocks. This beautiful and interesting little species was first made known by myself in the Annals of Nat. Hist, for September, 1856, from a specimen kindly forwarded to me by Miss Pinchard, who obtained it from the rocky islet called the Orestone, ofi* Torquay. In May, 1858, three specimens were forwarded to me by my friend, Mr. F. D. Dyster, out of some hundred and fifty that were found by a collector on rocks, between tide- marks, near Tenby ; and a few weeks after this I was so fortunate as to discover a populous home of the species, in the neighbourhood of Torquay. On the south side of the promontory, called Petit Tor, on the coast of South Devon, there is a low-roofed cavern, whose orifice is left bare at the lowest water of spring-tides. The interior parts of the floor are covered with the common limestone shingle, and, being more elevated than the mouth, afibrd an opportunity of working within, whenever one can gain admittance. The roof and sides of this cave are studded with the pretty little Crimson Pufflet, as well as with many other Anemones. The tide having receded, they are very readily discovered by their crimson columns projecting an eighth of an inch from the dark floccose rock, ti The limestone is much eroded by Saxicavce; and it is in the old burrows of these Mollusca that the Edwardsia THE CRIMSON PUFFLET. 261 dwells, clmging to the sides or bottom of the hole by the suckers on its- skin, the column and disk now protruding, where formerly the siphons of the Mollusk projected. It has forcibly reminded me of Ossian's beautiful image of the fox looking out of the window in the desolate dwelling of Moiua. In captivity the animal is able to roam about the glass by means of its adhesive suckers. Under high magnification the epidermis is seen to be a film of condensed mucus, evidently composed of disin- tegrated cells, in which are entangled a few cnidce, some threads and many spores of Confervce, and multitudes of Diatomacece, of many species. I carefully removed piece- meal the whole epidermis from one, exposing the skin of the entire scapus, which then was seen to be fleshy, pel- lucid, pink, and in all respects like that of the terminal regions, except that it was slightly more dense. In a few days the scapus was again encased in an epidermic tube, thin and semi-transparent, but, instead of being yellowish or brown, it was quite grass-green. This I found to be owing to the entanglement of conferva-spores in the mucus, the water having been exposed for some days in a shallow saucer. After having been kept some days in stale water, the animal is found much contracted and retired to the middle part of the epidermic case. This may be then readily removed, the adhesion having ceased. The organic con- nexion between the epidermis and the scapus thus appears to be less in this species than in others of the genus, and approximates it to Phellia in the Sagartiadce. This pretty Pufflet is easily kept in the aquarium, but it appears to require a considerable volume of water in a state of purity. It sometimes floats at the surface, extended at foil length. It will feed readily on minute atoms of raw 262 ILYANTHID^. meat, like the common Anemones. All its movements are rapid, sudden, and spasmodic. Torquay, P. H. Q. : Tenby, F. D. Dyster. [Harass!.] Piiellia. caknea. H. clirjsanthellum. IEdwardsia Beautempsii (Quatref.). About the same time that Mr. Kingsley dredged E. cal- limorpha at Brixham, he found at Torquay, washed up after an easterly gale, an individual of the same genus, but manifestly distinct in species. While generally agreeing with E. callimorpha^ in size and form, it differed in tlie following points: — 1. The scapus was less opaque, more smooth and lubricous, and studded with longitudinal rows of minute warts between the invections. 2. The capituh/m was clavate, proportionally longer, and of the same colour as the scapus, a pale pinkish-buff, or light orange. 3. The tentacles were fourteen in number, slightly uncinate or incurved, banded with dark buff. 4. The disk was trans- parent and colourless, with a dark protruded mouth. From these characters I think it probable that the animal in question was referrible to the E. Beautempsii of M. de Quatrefages. 263 GENUS V. ARACHNACTIS (Sars). Colmmi moderately long, cylindrical, rounded at the inferior extremity, but not swollen, imperforate. Surface capable of temporary adhesion, and therefore probably studded with minute suckers. Dish ? Tentacles of two kinds, the one marginal, very long, slender ; the other gular, short ; few in each series, not retractile. Mouth, a simple slit. Hah it : freely swimming in the sea. There is but one British species, A. albida. ASTRJSACEA. ILYANTHID^. THE SPKAWLET. Arachnactis alhida. Specific Character. Marginal tentacles longer than the column, gular tentacles about one-fourth of the length of the column. AracHmactis alhida. Sabs, Fauna Litt. Norveg. i. 28; pi. iv. fig. 1 — 6. Forbes and Goodsir, Trans. Roy. See. Edinb. xx. 310. GENERAL DESCRIPTION Form. Column. Shortly cylindrical [pear-shaped, E. F.], sub-globular in contraction, becoming gradually smaller and rounded at the inferior extremity, where no orifice has been observed. Surface smooth [but with the power of adhering, at least by the inferior extremity (E. F.), which implies the existence of suckers]. Substance softly fleshy. Disk. [Undescribed.] Tentacles. Of two kinds. First series marginal, twelve to fourteen in number, filiform, tapering, very long, slender and pointed : of these eleven are about equal in length and thickness, while one or two are veiy much shorter and smaller, and un- equal inter se. Some individuals show traces of the budding forth of another tentacle. These smaller and apparently sprouting tentacles always occur at that part of the ARACHNACTIS ALBIDA. °^'"''^® ^^^""^ Corresponds to one angle of the mouth. Second series springing immediately around the mouth-slit, eight to twelve in number [sixteen, E. F.], conical, pointed, scarcely one- tenth as long as those of the first series ; some smaller than the rest, and apparently budding, and these correspond in position with the budding ones of the first series. Mouth. A simple slit. COLOUB. Column. Pellucid whitish, displaying the dark brown stomach through its translucency [dusky white, tinged with tawny, E. F.]. Tentacles. First series whitish with dark brown tips [tawny and white, E. F.]. Second series dark brown on the front face. THE SPRAWtET. 265 SiZK. I Length of colmnn about one-third of an inch [one inch, E. F.] ; diameter one-eighth ; length of marginal tentacles one and a half inch [three or four inches, K F.]. LOCAUTT. The Hebridean and Norwegian Seas. This very interesting form, the only British example of a natatory Anemone, lias occurred on two occasions, both in the month of August, and both in the Minch, the strait that divides the Isle of Lewis from Scotland : — first by Dr. Balfour in 1841, who obtained a number of specimens, but all in a mutilated condition, and subsequently by Messrs. E. Forbes and Goodsir in 1850. In the interim, the Rev. Mr. Sars, of Bergen, had described and figured it in an elaborate memoir in the '' Fauna Littoralis Xorvegiae " (1846) ; and it is from this that we derive our chief know- ledge of the species, Forbes's account being exceedingly meagre. It appears in the vicinity of the Isle of Iloroe, on the coast of Norway, in autumn and winter, swimming on the smooth sea, sometimes in dense shoals, sometimes singly, borne on the northward current. Comparing the periods of its occurrence in the Hebridean and Norwegian seas, we may infer that it comes up from the warmer parts of the Atlantic ; and it might be hopefully looked for on the west coasts of Ireland in the earlier summer. As it swims it carries the marginal tentacles horizontally spread, when it looks not unlike a long-legged spider: hence the generic name from apd^(yrj, a spider, and uktUj a ray, and hence also the English term I have assigned to it. The superior or the inferior extremity is indifierently carried uppermost. It swims by a languid undulation of the long 266 ILYANTHID^. tentacles ; but it has a certain power of crawling also ; for these organs are strongly adhesive throughout, and the animal, attaching itself bj these means to foreign bodies, slowly draws itself forward. The gular tentacles are usually projected, and clasped together, but sometimes they are horizontally spread. In the latter case, if touched, they are instantly drawn to- gether, and slightly contracted, but never retracted ; they have no adhesive power. The appearance and situation of these organs have suggested to my mind the thought that possibly they may be the lobes of a conchula, in which case the animal would be a swimming PeacJda: if, however, they are true gular tentacles, then the alliance is obvious with the following genus Cerianthus. May it not possibly be the immature condition of this latter? There are discrepancies in form and colour, and especially in size, between the specimens seen on our own coast, and those described by Mr. Sars, which make it possible that these may constitute two species. We trust other speci- mens may clear up . this and other questions of interest. Forbes found a species of the same genus abundant in the Grecian Seas, but whether identical with this, we are not informed. The internal structure, which, from the transparency of the integuments is clearly seen, presents nothing peculiar. ? Peachia. AcALEPHA. ALBiDA. Anthea. ? Cerianthus. * See M. Haime's observations on the free-swimming young of Cerian- thus, infra, p. 273. 267 GENUS VI. CERIANTHUS (Della Chiaje). Tubularia (Gmeli.v). Moschata (Blainville). Edwardsia (Forbes). Column lengthened, cylindrical, swollen and bulb- like at the inferior extremity, which is perforated with a distinct orifice ; expanding trumpet - like at the margin, which merges into the tentacles, without parapet or fosse. Surface smooth, without loop-holes, or (apparent) suckers. Usually enveloped in a loose, non-adherent tube, closed at the lower end, of tough, membranous texture, and ragged exterior. Dish wider than column, but not over-arching : funnel-shaped, with conspicuous radii. Tentacles of two kinds ; the one marginal, the other gular ; both in perfect circles, those of each equal inter se, moderately numerous, slender; absolutely incapable of retraction. There is but one British species as yet certainly assigned to this genus, C. Lloydii. ASTR^ACEA. ILYANTHIDJE. THE VESTLET. Cerianthus Lloydii. Plate VI, Fig. 8. Specific Character. Inferior orifice excentric : septa regularly graduated. Edwardsia vestita. Gosse, Ann. N. H. Ser. 2. xviii. 73. Cerianthus membranaceus. Ibid. Ibid. Ser. 3. i, 418. Lloydii. Ibid. Ibid. Ser. 3. iii. 50. GENERAL DESCRIPTION. Form. ' Column. Greatly lengtbened, cylindrical for the most part, but gene- rally swollen at the inferior end into an elliptical bulb, and gradually expanding into a trumpet-shaped summit to about twice the median diameter. No distinct margin, the summit of the column itself dividing into the tentacles, the ridges of which are apparent for some distance below the point where they separate. Inferior extremity pierced with a round orifice, which is placed at one side of the axial line. Mesenteric prolongations of the visceral septa twenty-four, of which one pair are very minute, while the opposite pair extend to the immediate vicinity of the inferior orifice. From the one to the other of these conditions there is a regular gradation in length, but from the longest to the middle pair the diminution is slight, while from the middle pair to the shortest it is great and rapid. Dish A deep funnel-shaped cavity, about twice as wide as the column, entire, circular, not overarching. Tentacles. Of two kinds. First series strictly marginal, sixty -four, set in two rows, alternating, but with their bases in mutual cbntact. They are equal, slender, conical, sharp-pointed, divided more or less conspicu- ously into knobs, by some half-dozen constrictions. Their contour is some- what stiff, and they are generally carried arching upward and outward ; but some of the inner row are frequently erect, and others inclined to a point over the disk. Second series remote from the first, crowded, in four irregular circles, springing immediately around the mouth ; filiform, obtuse, sub-equal, not half as long or thick as those of the first series. THE VESTLET. 269 Mouth, A. tranBveree slit ; lip minutely furrowed, not projecting. Investing Tube. Cylindrical, much wider than the animal, which is loosely invested by it without attachment in any part, papery or felty in texture, thick and soft, composed of many layers, the outer of which pre- sent ragged foliations. The tube can easily be detached, when the ani- mal immediately begins to form a new one, by throwing off the material from the entire surface of the column ; this at first is adhesive, tena- cious, and very tough, pellucid, but gradually becomes milky, and finally opaque, entangling mud and sand in its substance- It is wholly composed of cnidcE, the discharged ecthorcea of which, in in- calculable numbers and of great length, inter- twine and form a sort of felt. Colour. Column. Pale buff or whitish, gradually becom- ing rich chestnut brown at the summit. Disk. Pellucid white. Tentacles. First series maronne or chocolate-brown at the foot, above which pellucid whitish, with chestnut bands. Second series dark maronne. CEmXSTSTJB mthout its tube. Size. Length seven inches, under strong irritation contracting to two ; general diameter of column one-fourth of an inch ; of disk half an inch ; expanse of flower one inch and a half. 270 ^LYANTHID^. Locality. The Menai Strait, in North Wales, and the Channel Islands ; between tide-marks. Varieties. Specimens differ considerably in the depth and extent of the brown tints of the upper parts. In some the maronne or red-bi"own hue extends across the disk; in others it is scarcely discernible on the tentacles. The present species has generally been supposed to be identical with that of the Mediterranean, of which M. Jules Haime has given an elaborate memoir (Ann. d. Sci. Nat. Ser. 4, i. 341). But in that species the arrangement of the mesenteric septa, — which M. Milne Edwards (Hist. Nat. des Coralliaires, i. 308) gives as generic, differs so importantly from what obtains in ours, as to demand a revision of the generic characters. I have therefore con- stituted it a new species, naming it after Mr. W. Alford Lloyd, to whose intelligent enterprise the study of Actino- logy is so greatly indebted, and to whom we owe our acquaintance with this very animal. In the summer of 1856, this gentleman first obtained specimens from the Menai Strait, a fact which I noticed in the " Annals N. H." for July of that year, assigning the species to the Edwardsia vestita of E. Forbes. Mr. Lloyd also himself about the same time communicated two notes on the animal to the " Zoologist," in one of which he stated that he had then obtained eighteen specimens. Since that period he has procured many more, but, as I believe, only from the same locality. Some of these specimens he has courteously presented to me, and has thus enabled me to become personally familiar with the habits of the species. THE VESTLET. 271 The animal is hardy in the aquarium, bearing even the confinement of trarel with more impunity than many commoner species. It is large and handsome, with a striking and noble aspect, and as it lives habitually expanded, and manifests considerable vivacity, it is a very desirable acquisition. The appearance of its felty tube is, however, repulsive ; but this I have found by no means essential to its comfort, and have managed to dispense with it, by the following device. Having prepared a glass tube of suit- able size, by cementing it perpendicularly to a stone of sufficient weight to maintain its stability in an upright position, I carefully removed the animaPs own case, and dropped the denuded body into the new lodging. The Cerianthus, in every instance, became immediately at home, presently lengthened itself, and expanded at the margin of its new abode ; and, as if the protection hereby afforded were sufficient, it threw off a new natural coat, only to such an extent as did not interfere with the sight of the body through the glass. Another advantage is secured by this treatment ; for whereas naturally the animal burrows in the mud, so that only the expanded flower is visible, and when put into a tank sprawls uncouthly along the bottom, the upright glass tube exposes the entire animal to observation, while it is protected from injury. I have specimens now which have been kept for many months in these circumstances, and are still in the highest condition. In handling the animal during the process of stripping off the coat, it contracts by strong, sudden, and repeated jerks, at each becoming shorter. In these contractions the water in the visceral cavity is forcibly ejected from the terminal pore. This ia not placed at the extreme point, which is marked by a depression, and by the convergence of lines, but is considerably excentric. I have also seen water 272 ILYANTHID^. ejected at intervals by the same orifice, when undisturbed, and that so forcibly as to hurl the floating atoms to the distance of two inches. I am pretty sure that I have also seen an inflowing current ; but this is more gradual, and therefore less conspicuous. The orifice must be considered as only a provision for respiration, and not as a termination to tKe alimentary canal : the half-digested food is, as usual, discharged from the mouth. The Vestlet feeds freely in captivity, greedily accepting fragments of raw flesh, and also skilfully catering for itself. One evening I amused myself with observing it capture its prey. It was one of those mentioned above, set in an upright test-tube^ in an old-established tank, close to the side. The water contained a large number of minute Entomostraca, which, when the candle was placed near the tank, flocked from all parts to the light. I thus was able to direct the migrant crowd to any point that I pleased ; and so brought them, when pretty well assembled, to the quarter which the expanded tentacles of the Cerianihus occupied. One and another were continually coming into contact with the tentacles ; and it was highly interesting to mark the unerring certainty with which each was arrested the instant it touched a tentacle. No matter whether the foot, middle, or tip of the organ were touched, the little intruder inevitably adhered as if birdliraed, and apparently without a struggle ; when immediately, with the most beautiful ease and precision, the fortunate tentacle jerked inward, — all the rest remaining as they were, — and, deliver- ing the prey to the grasp of the gular tentacles, in a moment resumed its expectant position. So numerous was the giddy throng, that this manoeuvre was every moment in practice, with some or other of the tentacles ; so that scores, certainly, of the Water-fleas were captured while I was observing. THE VESTLET. 273 Mr. E. Edwards, of Menai Bridge, who has politely sent me a peculiarly fine specimen, has also favoured me with the following interesting note of the haunts and habits of the species. " The only account I can give of the Cerianthus is, that I have found it in the Menai Straits in two distinct places, about five miles apart. " The ground is a mixture of stones, gravel, and mad. The disk (some of a light and some of a dark colour) when first seen is on a level with the surface of the ground, but on approaching instantly disappears into its sac. " The operation of taking it is difficult, as on the least disturbing of the ground it slips through the sac and is lost. The plan I adopt is to surround it with two or three spades, and each to act at the same moment, so as to undermine it in an instant, and press the ground, which causes its escape to be more difficult." Mr. Holdsworth informs me that he found a specimen of this species * at the island of Herm, near Guernsey. " It was close to low-water mark, buried among mud and stones, with a large piece of granite covering it. Not more than half an inch of the tube was exposed when the stone was removed; and I found the rest winding about the irregularities of the ground in a most tortuous manner, turning sharp comers in its course downwards." M. Haime ( Op. cit.) furnishes us with some interesting details of the development of C. membranacens, which doubtless apply equally well to the present species. " The young," he observes, " which I obtained, all died in the course of a few days. I never found any young advanced, within the parent, as is so common with Actintce ; but the eggs, which float freely there, had already passed the first * It is right, however, to observe that the distinction between this species and C. meinbranaceiu was not then suspected. 274 ILYANTHIDJE. period, and I had no opportunity of seeing their segmen- tation. All were strongly ciliated, and tlierefore were already larvae. They were oval in form, § millim. iu length. One end becomes concave, the other conical. In the centre of the former an opening forms, through which granules escape, and this becomes the mouth ; the escape of the granules leaving the visceral cavity. Soon around the mouth four minute tubercles bud, which become tentacles ; then two other tubercles nearer the mouth form lips ; meanwhile the body becomes smooth, and cylindro- conical. " The young lived in this state ten or twelve days ; and attained one or one and a half millimetre in length. The body continued entirely ciliated, and was become very con- tractile. They swam freely in the manner of a Medusa. mouth downward, by means of elongations and shortenings of the trunk, and by openings and closings of the ten- tacles. Sometimes they would oscillate, or revolve on themselves." Arachnactis. Lloydii. Cyathophylliada3. [membranaceus.] '^Cbbianthus (?) VERMicuLARis (E. Forbes). Dr. Johnston, in his "Brit. Zooph." Ed. 2, p. 222; pi. xxxviii. figs. 2 — 5, has described and figured, on the authority of E. Forbes, under the name of Act. vermicu- larts, what seems either the young of the preceding- species, after it has become stationary, or else a near ally to it. It is described as " 0^ long," and the larger THE VESTLET. 275 teutacles "02^;" but what the integer is to which these fractions refer we are not informed. There is doubtless some error, as in the description these organs are called '* long; " and the figures, which are rude enough, are said to be " of the natural size," and these represent the animal as 1^ inch in length, with the tentacles, both marginal and gular, about ^ of an inch. A slender cylindrical column, Avith a trumpet-shaped margin, a funnel-shaped disk, two kinds of tentacles, and a slit-like mouth, — this animal possesses in common with the Cerianthus. It is repre- sented, indeed, as standing erect, with the base attached in the manner of an Actinia ; but this was probably drawn om assumption, and the attachment may have been similar to that which I have described in other Ilyanthidoe. Professor Forbes sajs the base was "not expanded," which vours this supposition. No tube or case is alluded to, but it may be that this is developed only at a later period of life. The specimens were dredged in fifty fathoms in the Shetland Seas ; the column was greyish pink; the disk and gular tentacles white ; the marginal tentacles fulvous. It gave out a vivid phosphorescent light when irritated in the dark. T 2 276 TRIBE IL— CARYOPHYLLIACEA. The large number of tentacles in the polypes of this tribe allies them to the Astejeacea, and at the same time separates them from the Madreporacea and Antipa- THACEA. Moreover, while the mode of increase in the compound species, by gemmation of the sides or base, removes them from the former, it affiliates them to the latter tribes. The majority of species deposit a corallum of lime, the calices of which are many-rayed. In compound species, the interstices between the corallites are not occupied by prolongations of the septal plates, but are granulous or porous, or sometimes faintly channeled. The stony plates (septa) are nearly or quite entire, rarely denticulate. Within the corallum the septa are connected laterally only by very distant dissepiments, if at all, never by series ot transverse plates. The stars, in a transverse section, are simple ; the chambers being rarely crossed by dissepi- ments : the calices are very commonly cylindrical, with narrow plates, arranged neatly around, and have often a broad bottom, generally porous and convex (Dana). The vast majority of Caryophylliacea are coralli- genous ; but this statement will not apply to those which belong to the British seas : for of the seventeen species presently to be described, seven are destitute of a corallum. So far as I am acquainted with them, the tentacles of our native species (with the exception of Zoanthus) diflfer from those of our Astr^ACEA, in having the cnida not lodged in the substance of the walls, but aggregated into masses which form warts on the siu'face. Most of them have, moreover, these organs terminated with globose heads, destitute of cnidcB, but studded with minute hairs {palpocils). 277 ANALYSIS OF THE BRITISH FAMILIES. Without a corallum. Simple , . . Capneadce. Compound Zoanikidce, With a corallum. Substance of corallum solid. Interseptal chambers free TurhinoUada. Interseptal chambers crossed by dissepiments . Cavity gradually filling up Oculinada. Cavity permanently open Angiadce. Substance of coraUum porous Eupsammiadce. 278 FAMILY L— CAPNEAD^. The members of this Family do not, at any period of their existence, so far as is known, deposit a corallum, or any trace of calcareous matter. They are, moreover, per- manently simple ; for though there is reason to believe that they increase by budding, the polypes so formed quickly sever their connexion with the parent, and become inde- pendent though associated individuals. Thus they are essentially Anemones, such as we have already considered ; yet there is something in their aspect which at once betokens their affinity with the Corals. In particular, the tentacles have the singular structure and knobbed form already noticed as peculiar to this tribe : and, contrary to the universal rule in the Astrceacea, they increase in size outwardly, — the outer row containing the largest. The body, adherent by a broad base, is fleshy or pulpy, copiously lubricated with mucus, and sometimes separating the outer skin into a deciduous epidermis. The surface is not furnished with suckers, nor pierced with loopholes. There are no acontia, but the craspeda are numerous and large, and their contained cnidcB are remarkably developed. ANALYSIS OF THE GENERA, Tentacles truncate Capnea. Tentacles crowned with bilobed heads A ureliania. Tentacles crowned with globose heads Cor}fnacHt, 279 GENUS I. CAPNEA (Forbes). Base expanded, swollen, adherent. Column cylindrical, pillar-like ; the margin forming a thick parapet, with a fosse. Surface smooth, without loopholes, invested with a woolly epidermis. Bisk circular, entire. Tentacles very thort, truncate, retractile. But one species is known, C. mnguinea CARYOPHYLLIACEA. CAPNEADjE. THE CROCK. Ccipnea sanguinea. Plate IX. Fig. 13. Specific Character. Body scarlet ; epidermis brown, 8-lobed. Kapnea sanguinea. Forbes, Ann. N. H. Ser 1. vii. 82 ; pi. i. fig. 1. Capnea sanguinea. Johnston, Brit. Zooph. Ed. 2. i. 203 ; fig. 43. Cocks, Rep. Cornw. Soc. 1851, 1 ; pi. i. figs. 1, 2. GENERAL DESCRIPTION. Form. Base. Greatly expanded, irregularly inflated and lobe-like ; its outline irregularly undulate ; adherent. Column. Cylindrical, pillar-like, higher than broad ; the margin form- ing, when fully expanded, a thick and prominent granulate parapet, or collar, with a deep fosse. Surface smooth, without loopholes, invested on the lower two-thirds with a woolly epidermis, the upper edge of which is regularly 8-lobed. Disk. Circular, entire. Tentacles. Extremely short, truncate, having the aspect of squared tubercles ; arranged in three rows of sixteen each, those of the outermost row the largest. Disk and tentacles perfectly retractile. Mouth. Round, slightly puckered. Colour. Column. Vivid vermilion, or dull brownish scarlet, with darker longitudinal stripes on the inflated basal portion. Epidermis brown. Disk. Yellowish flesh-colour. Tentacles. Orange-scarlet, paler than the column. Size. Height of column one inch ; diameter of disk one-fourth. THE CROCK. 281 LOCALITT. Deep water, off Isle of ilan, on nullipore beds : deep water, four leagnes west of Falmouth, on a valve of Pecten maximui. The late E. Forbes first obtained this interesting form in August, 1840, and assigned to it its generic and specific names; the former from Kdirvrj, a chimney, from its re- semblance to a chimney-crock, of which suggestion I have availed myself to make an English appellation. He tells us little of its history beyond what I have embodied above ; except that it is an active creature, changing its form often, but always presenting more or less of a tubular shape ; and that the upper part of the body can be retracted within the column as low as the commencement of the epidermis. Mr. W. P. Cocks has since obtained a second specimen. This was considerably smaller than Forbes's, but agreed with it in essential points. Mr. Cocks has kindly put into my possession some notes of his specimen, which have enabled me to add a few details to Forbes's diagnosis ; and also a coloured drawing made from the living animal, which I have copied in my Plate IX. Phellia. SAXGUINEA. Aureliania. 282 GENUS II. AURELIANIA (Gosse). (Gen. nov.) Corynactis (Thompson). Base expanded, adherent. Column conico-cylindrical, low, the margin forming a thick parapet with a fosse. Surface smooth, without suckers or loopholes : invested with a deciduous epi- dermis. Substance firm and coriaceous, opaque. DisA flat, entire ; radii distinct. Tentacles in several rows, very short, knobbed ; the heads more or less bilobate, and differing in form in the different rows ; perfectly retractile. Mouth slit-like, furrowed : stomach-wall protrusile. ANALYSIS OF BRITISH SPECIES. Base greatly expanded : crimson augusta. Base not exceeding column : yellow heterocera. I'LATK . rX - OOSSc D[L 1 CORYNACTi s VI Rl DIS BOLOCERA EQUES ZOANTHU S SULCATUS Z . ALDER I 10 9. 10. ZOANTHU S COUCHII 11. AURELIANIA AUGUSTA . 12. A . HETEROCERA 13- CAPNEA SANGUINEA. CAnTOPHTLLIACEA. ' C APNEA DM THE CRIMSON IMPERIAL. Aureliania augusta. (Sp. noY.) Platb IX. Pig. 11. Specie CharacUr. Column lising from a widely expanded base : crimson. GEIfERAL DESCRIPTION. FOBH. Bate. Adherent to rocks ; greatly expanded ; the outline nndolate. Colitmn. A low, thick pillar, springing gradually from the broad base like the trunk of a tree; the margin forrring a thick and prominent parapet, the inner edge of which ia crenate ; aud separated from the ten- tacles by a narrow and shallow fosse. Surface smooth, entirely invested with a soft, wooUy, firm, thin epidermis (which fell off in patches soon after capture, and was not renewed). Substance firmly fleshy ; opaque. DuJc. Somewhat elliptical, entire, flat or slightly convex ; radii fin© but distinct. Tentacles. In four rows, the outer row containing 42 ; very short, knobbed ; the knobs agreeing in form with those of the following species. Disk and tentacles freely and completely retractile. Mouth. Slit-like, slightly furrowed. COLOUB. Column. Rich crimson, splashed with deeper crimson, and with pale yellowish. Epidermis dark olive-brown. i>i«it. Light crimson. Tentaclet. Rosy white, with opaque white tips. Mouth. Deep crimson. SiZB. Diameter of base two inches and three-quarters : of disk rather more ihan an inch ; height from one to one and three-quarters. LOCALITT. North Devon : low water. 284 CAPNEAD^. In August, 1856, the Kev. J. P. Greenly, being on a visit to Ilfracombe, found in a crevice of the slaty rock at Bull Point, at extreme low water, this magnificent species, which lived in his possession till the following April. To his courtesy I owe copious descriptions and drawings made from the animal while in life and health ; by which I am enabled to draw up the foregoing diagnosis. I forwarded to him for comparison some drawings which I had by me of Mr. Thompson's Gorynactis heterocera ; and the agree- ment of the two forms in all essentials, and especially in the singular shapes of the diverse tentacles, showed that they were of one and the same genus, which was thus proved to have characters that called for its separation from Gorynactis. At the period last named my kind corre- spondent forwarded the specimen to me : but it was already dead; and while it retained its form and colour, I was precluded from adding anything to my knowledge from personal observation. In captivity the animal was lively and extremely sensi- tive, retracting its disk with remarkable suddenness and rapidity on alarm. It early crawled from the piece of slate on which it was captured, and took up a position on the side of a finger-glass in which it was kept. The tentacles were observed to vary the shape of their knobs, within slight limits: one here and there in the outer row occa- sionally approaching the hastate form of the next row. sanguinea, AUGUSTA. heterocera. CARYOPHYLLTA CEA . CAPNEADjE. THE YELLOW IMPERIAL. Aureliania heterocera. PuiTE IX. Fig. 12. Specific Character. Base scarcely exceeding column : yellow. Corynactis heterocera. W. Thompson (w.), Proc. ZooLSoc. 1853. GossK, Man. Mar. ZooL i. 28. E. P. Wbight, Nat. Hist. Review, April, 1859, p. 122. GENERAL DESCRIPTION. FOBM. Bcue. Adherent to rocks : scarcely exceeding the column in width ; Tery slightly undulate. Column. A stout cylindrical pillar, about as wide as high, but often con- stricted below the margin, when the lower portion becomes nearly hemi- spherical : margin forming a thick parapet, the inner edge of which is crenate, and separated from the tentacles by a narrow fosse. Surface smooth, entirely invested with a thin slimy epidermis, which is easily rubbed ofiT. and quickly renewed. Substance firm and coriaceous ; perfectly opaque. Disk. Nearly circular, entire, ample, membranous, flat or slightly convex : radii fine but distinct. Tentacles. About 120, set in four rows, of which the outermost con- tains 32 ; the others one or two less : they are short, thick, cylindrical, with knobbed tips, diverse among themselves. The knobs of the outermost row are little wider than fthe stems, they are sub-conical, or kidney-shaped, [seemingly formed of two lobes, with a round tubercle seated on the inner face just below the knob. Of the second row the knobs consist of two swellings divided by a constriction, each swelling oomposed of two globose lobes placed side by side, with a mucro terminating the whole. Of the two innermost rows the knobs are nearly sessile j they are rondo-quadrangular, or shaped somewhat like a loaf of bread. In the expanded state all these organs lie nearly horizontal, pointing outwards, and slightly overlapping ihe parapet. TENTACLES OF A. HETE- BOCKRA. 286 CAPNEADJE. Mouth. Silt-like, coarsely furrowed. Stomach-wall capable of protru- sion, so as to conceal the whole disk. Colour. Column. A rich apricot-yellow, which here resides in the epidermis, for when this is rubbed off, the colour is white, but when renewed the colour loturns. Disk. Pellucid white, with fine opaque white radii. Tentacles. Pellucid white, faintly tinged with red, and tipped with opaque white. Mouth. Lips deep buff. Siza. Diameter and height of column about an inch ; expanse the same. Locality. The south of England and south-west of Ireland : deep water. This fine species, only inferior in beauty to the one just described, was dredged by Mr. W. Thompson in Wey- mouth Bay, — eight fathoms, gravel,— in September, 1853. As I was at Weymouth at the time, he kindly showed it to me, and I thus had the opportunity of making careful drawings and notes from the life. We considered it as more nearly allied to Corynactis than to any other recognised form ; and, the species augusta being then unknown, I was induced to suggest the specific name heterocera, which Mr. Thompson adopted^ from €T€po<;, diverse, and Kepa^, a horn. In confinement, the species appeared hardy. When detached it readily adhered again ; soon expanded after having been provoked to close ; often passing from one condition to the other many times in quick succession. It is subject to very little change of shape, in this respect contrasting with Corynactis, which is most protean. Mr. Thompson observed that it opened slowly, exserting the tentacles of one-fourth of the periphery, while the rest remained closed. These organs were nearly motionless. THE YELLOW IMPERIAL. 287 When a piece of meat was dropped on the open disk, it remained awhile apparently unnoticed; at length the animal slowly bent itself on one side, and the unwelcome morsel rolled across the tentacles and fell to the bottom. When Dr. E. P. Wright was on the south-west coast of Ireland, in July, 1858, he found, at Crookhaven, a small number of specimens of this species, agreeing with Mr. Thompson's description in every particular, except their smaller size. He kindly sent me tliree, but they all died xn transitu, from the length of the journey. Dr. Wright says " it can assume an almost transparent appearance," — which was not the case with the Weymouth specimens ; but which assimilates it to Corynactis. He observed also that the outer tentacles were reverted, so as actually to touch the rock, which gave it a strange aspect. The circles of tentacles resemble a coronet of pearls ; and searching for a name by which to distinguish the genus, I was reminded, by this peculiarity, of the diadem which was the distinctive badge of the Eoman Augusti, and by the splendid colours of the animals, of the no less imperial gold and purple. I have therefore called it Aureliania, after him who of the Roman emperors first wore the diadem and the gold-embroidered purple.* The splendid appearance of the zoophytes, especially of the preceding species, must plead my apology for so presumptuous an appropriation. Weymouth, W. T. (w.)/ Crookhaven, E. P. W. augusta. HETEROCERA. Corynactis. * " Iste primus [sell. Atirelianus], apud Romanos, diadema cajjiti iuuexuit, geminiaqua et aurat4 omul veste, , . . usus est" I'Aorel. Vict.) lb 288 GENUS III. CORYNACTIS (Allman). Base expanded, adherent. Column versatile, tall; the margin forming a parapet, with no fosse. Surface smooth, without suckers or loopholes ; not invested with any separable epidermis. Substance fleshy or pulpy, pellucid. Disk flat, entire, circular. Tentacles in several rows, all of the same form ; each consisting of a conical stem and a globular head: perfectly retractile. Mouth simple, protrusile ; lip coarsely furrowed : stomach evertile. Only one British species exists, C. viridis. CA R TOPE TLLIA CEA . CA PNEA D£. THE GLOBEHORN. Corynactis vtridis. Plate IX. Figs. 1—5. Specific Character. Rarely exceeding half an inch in height ; trans- parent; tentacles very unequal. Corynactis Tiridis. Allman, Ann. Xat. Hist. Ser. 1, xvii. 417; pi. xi. Johnston, Brit. Zooph. Ed. 2, i. 205 ; pi. xxxv. figs. 10, 11. Cocks, Rep. Comw. Soc. 1851, 3; pi. i. figs. 3 — 5. M. Edwards, Hist. Corall. i, 258. AUmanni. Thompson, in Johnst. Br. Zooph. Ed. 2, i. 474 ; fig. 85. Cocks, Rep. Cornw. Soc. 1851, 4; pi. i. fig. 6. GossE, Dev. Coast, 422 ; pi. viii. figs. 8— 10, Ibid. Man. Mar. Zool. i. 28 ; fig. 39. K P. Wright Nat. Hist. Rev. vi. 122 ; pi. xiii. GENERAL DESCRIPTION. FoRif. Base. Adherent to rocks and shells ; generally broader than column ; its outline sometimes slightly undulate Column. Pillar-like, very variable in height and shape ; the margin forming a distinct parapet or terrace, crenated within, but not separated from the tentacles by a fosse. Surface smooth, or sUghtly furrowed, lubricous. Substance pulpy, transparent. Disk. Circular, never waved, often greatly exceeding the column, flat or slightly concave ; smooth, with the radii marked, but no gonidial distinction. Tentacles. Upwards of 100, set in four rows,— 16, 24, 32, 32, = 104 ; the outer rows largest ; each composed of a more or less pillar- like or conical stem, and a globxilar head : in the inner rows, the stem is very short, and the head nearly sessile. The outer rows usually diverge upward and outward, projecting over the margin, and not seldom hang downward. Mouth. Protrusile at pleasure into a truncate cone or cylinder, sur- U 290 CAPNEAD^. rounded by a thick lip strongly furrowetl, like the mouth of a cowry-sheli. No trace of gonidial tubercles, or grooves. Colour. Column. A yellow emerald-green, becoming far richer and more opaque at the margin. Disk. Transparent, with the radii bi'illiant emerald-green. Tentacles. Stems with dark umber-brown warts on a transparent colourless ground : heads rich rose-pink. Mouth. Emerald-green. Size. Seldom exceeding half an inch in height, and three-eighths of an inch iu diameter of disk. LOCALITT. The south-west coasts of England, Scotland, and Ireland : deep water, and between tide-marks. Varieties. a. Smaragdina. The condition above detailed, which was the one first described, and is by far the most abundant. (PI. ix. fig. 5.) ;3. Hhodoprasina. Column and disk rosy-lilac ; margin emerald-green ; tentacles, stem umber, head pearl-white. (Fig. 1 .) y. Tephrina. Column and disk pearl-grey ; margin faint emerald ; tentacle-stem and head dull wood-brown. (Fig. 3.) 5. Chrysochlorina. Column pale yellow-gieen below, blending above into orange ; margin rich orange ; disk emerald-green ; tentacle-stem maronne (or white), head pearl-white ; lip scarlet-orange. (Fig. 2.) c. Prasococcina. Column and disk pellucid pearl-grey, flushed with scarlet ; margin emerald ; tentacle-stem and head pale scarlet. f. Corallina. Column brownish scarlet, margin orange-scarlet ; disk scarlet; tentacle-stem and head pearl-white ; lip scai'let( or white). (Fig. 4.) 7}. Coina. Wholly pure white, translucent; the margin, lip, and tentacle-heads opaque. This is one of the most exquisitely lovely little gems of the aquarium ; and fortunately it is abundant on our south- western shores, and very easily preserved in confinement for an indefinite period. In the Channel Islands ; from Torquay around the promontory of Cornwall to Ilfracombe THE GLOBEHORN. 291 in the Bristol Channel ; and again on the indented coast of Cork, it occurs in profusion on the perpendicular and over- lianging rocks ; while it has been dredged in deep water oflf various points of the same shores, and even in the land- locked gulfs on the north-east of Ireland. It is almost invariably found in close-set clusters of a dozen to fifty ; and though several distinct varieties frequently occur in the same immediate vicinity, yet the individuals of the same group are invariably found to agree in their tints. Hence I incline to believe that these groups are produced either by the spontaneous fission, or the gemmation of a primitive polype. I have seen some which were evidently connected together by the base, the process of separation being incomplete. It is somewhat difficult to detach the animals ; their bodies are excessively pulpy and tender, and under irrita- tion they excrete a vast quantity of mucus, so dense as almost to equal in consistency the substance of their own bodies, and which might sometimes assume the form of an epidermis. If carefully detached, however, they will re-adhere ; and I have known individuals even crawl off from a fragment of rock to the sides of the tank. In general, however, they are stationary, and even sluggish ; allowing their tentacles to be handled without contracting. They open very freely, and ordinarily remain expanded. In a well established aquarium, they will live for a long period ; I have some which have lived in captivity fifteen months. They seem in other respects tenacious of life ; for I have seen the tentacles and margin of one side appa- rently healthy and contractile, while the whole opposite side had become a putrescent mucus, sloughing away. AU the varieties are charming ; perhaps none more so than the translucent white one which I have named Coina. * Several in the possession of Mr. Holdsworth spontaneously divided, u 2 292 . CAPNEAD^. The expanded disk, with the opaque white tentacle-heads scattered over it, looks like what the ladies call " spotted muslin ; " Vvrhile, under a lens, the tentacle-stems resemble lace, or figured blonde. Around Torquay the species exists in much variety. On the shadowed sides of the perpendicular wall-like rocks, near Meadfoot, I have seen, at extreme low water, countless groups, displaying their lovely little coronets within reach of my hand, as I was pushed in a small boat through the narrow passes of the islets. Dr. E. P. Wright finds it in amazing profusion, " covering whole rock-pools," at Crook- haven. He says that some of these expanded to nearly an inch in diameter, — dimensions which far exceed those of such as I have seen. They have been occasionally found on roots of Laminaria, and Mr. Cocks has taken a number, half-digested, from the stomach of a Plaice. They feed readily on minute morsels of raw meat; which, however, must be laid on the disk with great caution, or the animal will close. In taking-in the morsel the Cory- nactis does not protrude the lips to embrace it, nor close the tentacles over it, like the Actimcc, but dilates the mouth slowly and uniformly, until the lips form a circle of great width, nearly as wide, indeed, as the disk, within which the visceral cavity, like a broad saucer, is seen, with the coiled craspeda lining its sides and bottom. Into this gaping cavity the morsel is drawn, and then the lips gradually contract and embrace it, finally protruding in a pouting cone. This is exactly the manner of CaryopJiyllia Smithii. There is much in the appearance of this animal which agrees with CaryopJiyllia: tlie colours and their distribu- tion, the general translucency of tlie tissues, the form and crenation of the mouth, and, in particular, the shape, arrangement, and minute structure of the tentacles, are Tit& GLOBEHOEN 293 SO exactly those of the Coral, that I have often more than half suspected that the former is the immature condition of the latter. Both are found in the same localities, in the same haunts, and often in close proximity, which helps the conjecture. No trace of calcareous deposit is found in the tissues of Corynactis when crushed between plates of glass ; but the observations of Mrs. Thynne* have shown that the young of Caryophyllia attain a large size without depo- siting a corallum. But the results of this lady's experi- ments, — so far as they go, — tend to negative the identity of the two animals J though I must still consider the species as in near affinity. Under the microscope the tentacle is seen to consist of a transparent thick-walled tubular stejii, in which longi- tudinal fibres are conspicuous, and a globose head. The stem is studded with large oval warts, varying in shape and size, and without orderly arrangement, but set trans- versely on the whole, very close together in contraction, but separated by wide spaces when the tentacle is elongated. Both the head and the warts are pellucid in themselves, but are sub-opaque from their contents : both are thickly covered with palpocils, while the trans- parent portions of the stem are clothed with cilia. In conformity with the great predominance of the longi- tudinal over the annular muscular fibres in the tentacle- wall, the contraction of these organs is in length rather than in diameter ; or at least that of the diameter is only the result of elongation. The globose head seems non- contractile ; and hence, when the stem is much elongated, we see a spheryile at the tip of a narrow foot-stalk, while, when the form is much contracted, the head remaining unchanged, we have the " corrugated cup " of jVIt. Peach, with the sphere seated as it were in it. * Annals Nat. Hist, for June. 1859. 294 CAPNEAD^. The cnidce in this species attain a higher development than in any other zoophyte that I am acquainted with, and hence they afford peculiar facilities for the study of these interesting organs. No one familiar -with this beautiful little creature can for a moment doubt that the two supposed species, viridis and Allmanni, are in truth but one. The former name must of course be retained, as having the claim of priority. It was given by the discoverer, Professor AUman, who found it, where since it has been so abundantly met with by Dr. E. P. Wright. The name Corynactis is formed from Kopvvr), a club, and uktU, a ray. There are several exotic species, whose tentacles are tipped with globose knobs; — as Act. glohulosa (Quoy et Gaim.), A. ghhulifera (Ehrenb.), and A, clavigera (Dana) ; but I know too little of their structure to pronounce upon their degree of affinity with the present. The clavigera, a species of large size from the Pacific Islands, may perhaps be a link of connexion between C-orynactis and Sagartia. Guernsey, T. D. H. : Torquay, P. H. G. : Dartmouth, E. W. H. H. : Plymouth, O. D. : Fowey, Polruau, Goram Haven, C, W. P. : Falmouth, W. P. C. : Lundy, C. K. : Ilfracombe, P. H. G. : Cumbrae, D. B. : Crookhaven, G. J. AUman : Bantry Bay, Ventry, E. P. W. : Strangford Lough, W. T. : Belfast Bay, G, G. Hyndman. Aureliania. VIKIDIS. [clavigera.] Caryophyllia. Sagartia. 295 FAMILY IL— ZOANTHID.E. The polypes in this family are persistently fixed, and aggregated : the adherent base extending itself laterally, and sending up new polypes at inten-als, which remain permanently united to each other, and to the primary polype. The extension may be in irregular lines, carrying the polypes in single file; in broad bands, supporting several abreast ; or in all directions, producing large clustered masses, incrusting the foreign body to which they happen to be adherent. This variation in the manner of base-extension has been hitherto considered as so important, that genera have been constituted on thLs character alone, — Zoanthus, including those whose base runs in lines ; Palythoa^ such as form carpet-like surfaces. But e\'idence will presently be adduced to show that these variations may occur in the same species. Again, the genera Mammilifera and Corti- cifera, of Lesueur, have been formed for clustered species ; the fonner being fleshy, with a mucous surface, not en- veloped in sand ; the latter " inclosed iu cellules of sand, agglutinated ; the cellules themselves agglutinated for their whole length, and forming a corticiferous expansion." It appears, however, from Lesueur's own description, that what he considered " cellules," inhabited by the animals, was simply the integument of each polype, in which sand was imbedded. The presence or absence of sand, however, can in no wise be allowed to constitute a generic distinc- tion. I cannot, therefore, recognise in the family more than the single genus, Zoanthus. 296 GENUS I. ZOANTHUS (Cuvier). Actinia (Ellis). Zoantha ) Zoanthus ^ Mammilifera > (Lesueue). Corticifera ) Sidisia (J. E. Gray). Base permanently attached ; spreading over rocks, stones, or shells, in either a linear or incrusting manner. Column pillar-like, higher than wide ; margin cut into strongly marked teeth, which are united by a thin membrane. Surface smooth, excreting a mucus, in which occasionally grains of sand become imbedded, constituting an adventitious epidermis. Disk slightly concave ; radii inconspicuous. Tentacles conical, pointed, similar in structure to those of AsTR^ACEA : wholly retractile. Mouth more or less protrusile, simple. ANALYSIS OF BRITISH SPECIES. Invested with sand ; extension various Couchii. Without sand — Polypes cylindrical, olive ; several abreast sidcatus. Polypes obconic, pellucid white ; in single file , .... A Ideri, CARYOPHYLLIACEA. ANT HID JE. THE SANDY CEEEPLET. Zoanthus Couchii. Plate IX. Figi. 9, 10 ; X. Fig. 5. Specific Character. Basal band extending variously ; polypes invested with a sandy coating ; tentacles in two rows. Zoanthus Couchii. Johnston, Brit. Zooph. Ed. 2, i. 202 ; pi. xxxv. fig. 9. Couch, Com. Faun. iii. 73 ; pL xv. fig. 3. Holds- worth, Proc. Zool. Soc. 1858 ; pi. x. figs. 3 — 7. Di/sidea Q) papulosa. Johnston, Brit. Sponges, 190, fig. 18; pi. xvL figs. 6, 7. Sidima Barleei. J. E. Gray, Ann. X. H. Ser. 3, ii. 489 ; Proc. Zool. Soc 1858 ; pi. x. fig. 8. GENERAL DESCRIPTION. Form. Basal band. Narrow, irregtilariy creeping, soft, elastic, fleshy to the feel, very sensitive ; invested with sand, like the column. Column. Cylindrical, rising to about three or four times its diameter ; smooth, transparent. Margin cut into twelve or fourteen (generally the latter number) large fleshy triangular teeth, which are connected by a thin web of transparent membrane, the inner layer of which is composed of transverse fibres, the outer is gianular and cutaneous. In a state of semi- contraction, these teeth form strongly-marked converging ridges on the flat summit of the column. Investment. Fine sand, evidently not a secretion, but extraneous, imbedded in the epidermis, — the fragments (in Torquay specimens) being of different colours, some being of white limestone, others of red sandstone. When the column is much distended, the grains of sand become considerably separated, and we can distinctly see through the transparent and smooth integuments into the visceral cavity. Thus the sand forms manifestly only a single layer. Only very minute grains are used, and there is very little difference in their size. Dtsl: Generally flat or slightly concave, but protrusile in a conical form. Radii apparently di.stinct, but only because the upper edges of the a^ta appear through the perfectly transparent disk. 298 ZOANTHID^. Tentacles. Twenty-eight (twenty-four in less mature specimens), arranged in two rows, fourteen in each : those of the inner row correspond to the marginal teeth, those of the outer are intermediate. They are sub-equal, taper, bluntly pointed, and, when extended, about equal in length to the diameter of the column, hollow, not warted, with thick walls, which, in contraction, fall into transverse or annular corrugations. They are pro- truded in a brash, but, when fully expanded, spread out horizontally. Month. Lip shai'p, much crenated, protruded after feeding. COLOUE. Investment of root-hand and column. Pale brown, the hue of the sand. Column. Beneath the investment, transparent and colourless. Disk. Pellucid reddish-grey, dusted with excessively minute white specks. Tentacles. Translucent, nearly colourless ; but each has a small mass of opaque white pigment on the internal surface, just at the tip : the aggre- gation of white points has a pretty effect. Mouth. Lip opaque white. Size. None that I have seen alive exceeded one-eighth of an inch in diameter, and about thrice that height in extension. In contraction the button is usually about a line in height, Mr. Holdsworth has obtained specimens much larger than these. LocALixr. The extreme northern and southern points of the British Islands, North- umberland, and various other points of our coast; deep water; on stones and shells, and free on the sea-bottom. Varieties. a- Linearis. The condition above described, in which the root-baud creeps in a narrow ribbon over stones and shells. Cornwall and Devon. (Plate X. fig. 5.) /3. Diffusus. The root-baud spread over the surface of a shell as a continuous carpet, whence the polypes spring, irregularly crowded together. Northumberland. (Plate ix. fig. 10.) y. Liher. Unattached. The root-band forming a free cylinder, exactly resembling the column of the polype, and of the same diameter. The polypes in this case branch irregularly from the cylinder, and terminate both its extremities. Shetland. (Plate ix. fig. 9.) If wc selected a single specimen of each of these varieties, THE SANDY CEEEPLET. 299 and compared them, without any other information, nothing would be more manifest than that we must assign them not only to distinct species, but even to distinct genera. ]Mr. Alder has favoured me with many specimens, obtained by Mr. Barlee, at Shetland, some of which, each consisting of several full-grown polypes, are perfectly independent and compact, showing not the slightest trace of adhesion to any foreign body, nor of any part that can be distinguished as a root-band. Thus, in the specimen figured in Plate ix. fig. 9, three polypes diverge from a common centre ; others are similarly formed, sometimes with a triangular dilatation of the point of divergence, which thus becomes flat, but still with both surfaces equally entire. I have not seen more than three polypes on any free specimen. But among these, we see specimens at first sight hardly distinguishable from them, except by a slight globosity at the point of divergence : when we turn these over, we dis- cover that the globosity has been moulded on a minute shell, evidently that of a Xatica. Then others occur, in which the shell, almost always a Natica, is larger, and there is a distinct basal carpet uniformly spread over it, of the sand-covered flesh, from which spring four or more polypes : these are manifestly identical with tlie free ones. But on larger shells the colony of polypes is made up of more individuals ; in one specimen before me, in which the shell is about the size of Xatica Alderi, there are nineteen polypes. In every case the basal carpet has spread in imiform thickness over the entire shell, following the form accurately, and extending to the edge of the outer lip, and clothing the rotundity of the inner lip as far as the eye can follow it. Strange to say, in every example, the shell itself has wholly disappeared, and all that is left is the exact model of it in the sand-clothed membrane, or basal carpet, of the polype. 30ft ZOASTHIDM. In this condition, the zoophyte was mistaken by Dr. G. Johnston for a Sponge, and he has accordingly figured and described it in his " British Sponges," under the name of Dysidea papulosa. I do not see in what single particular such specimens diifer from the genus Palythoa of Lamouroux, as this is characterized by M. Milne Edwards : — " Poly- pUro'ides cylindriques, naissant sur une expansion hasilaire memhraniforme, lihres latiralement, ou sondes entre eux, et formant des masses encroHtantes / " * and thus we find the same species in some circumstances a Zoanthus, in others a Palythoa. Nay, more, as if to increase the confusion. Dr. J. E. Gray has actually made a new genus for the inter- mediate free condition, which he calls " Sidista."'f The only way in which I can account for the free condi- tion is by supposing that the germ was, in those cases, deposited on a fragment of shell or stone so minute as to be completely overspread and enveloped by the increasing base.^: The unvarying disappearance of the shell in the diffuse variety is more remarkable, and seems to imply a corrosive or absorbent power in the base. That the Shetland and Northumberland specimens are identical with ours in Torbay seems pretty certain ; for Mr. Alder, who has had opportunities of seeing both in the living state (some from the north having been sent him alive by Mr. Barlee, and some from the south by myself), can see no specific diversity between them. But that they are the same species as the Zoanthus Couchn of the Cornish coast, 1 assume rather than prove. It is unlikely that there should * Hist, des CoralHaires, i. 301, t Annals Nat. Hist. Dec. 1858. J Mr. Alder remarks on these varying conditions as follows : — " I have come to the conclusion that when the zoophyte has free space on a stone it runs over it as Zoanthus ; but when the base is confined to a shell, it spreads into an uniform crust, as Palythoa. The loose branched speci- mens, I conclude, having affixed themselves to some minute object not affording a proper base of attachment, take a tubular form until they terminate in polypes."— f/w litt.J THE SANDY CREEPLET. ^01 he two species of the same uncommon genus, having so many points in common, found in so close proximity as the Devon and Cornwall coasts, and yet there are glaring dis- crepancies between Mr. E. Q. Couch's published descrip- tions and the characters of our animal. He describes the surface as " glandular," the form as frequently "contracted to an hour-glass shape," and as being very versatile ; the habit as sluggish, and slow to change ; the tentacles as " darker at the extremities than at the base ; " not one of which particulars do our specimens confirm. My first personal acquaintance with the species I owed to Mr. Holds worth, who dredged several colonies in twelve fathoms, off the Ore Stone, near Torquay, in October, 1858, where further researches show it to be quite common. They were of the variety linearis, aflaxed to fragments of slate and old valves of Cardium rusticum, twenty or thirty polypes on each, nmning in sinuous bands from half a line to three lines apart in the series. The colonies meandered over both surfaces of the fragments. One of these colonies my friend kindly gave to me, and it has lived now ten months with me. The pol^-pes are by no means sluggish, but are continually opening and closing with considerable vivacity. AVhen completely con- tracted, each polype is a cylindrical button, with the summit round and depressed in the centre. As expansion proceeds, the centre evolves, and the summit becomes nearly flat, with the twelve or fourteen strongly marked marginal ridges radiating from the central orifice. The central aper- ture enlarges, and the white tips of the tentacles are seen protruding, and presently the tentacles themselves, blunt and pellucid white, which soon arch outwardly. They feed readily on raw flesh or earthworms, but will take only very minute fragments. These, however light their contact, cause the tentacles to retract ; but if the 302 ZOANTHID^. morsel be laid gently on the truncate summit of the closed column, the converging teeth appearing, it will remain there until the animal seizes it. The tentacles are protruded one hy one so cautiously that the meat is not disturbed, and soon we discern that it is environed by a wall of tentacles, and that the mouth is gaping widely to embrace it. After feeding, or when food which has been resting on the disk is suddenly taken away, the whole disk is protruded as a cone, on the summit of which the open throat forms a wide valley, coarsely furrowed. The creeping-band is very sensitive ; when touched with a needle-point, all the polypes suddenly contract, yet not quite simultaneously, but in the order of succession cor- responding to their proximity to the point of attack. Mr. Holdsworth tells me that " the polypes live very well when detached from their support." The generic name is formed from ^wov, an animal, and dv6o<;, a flower ; the Englisli term is meant to express its peculiar habit. Shetland, G. Barlee : Northumberland, J. A.: Guernsey, J. A.: Torquay,^. W.H.H.: Cornwall (throughout), B. Q. a : Strangford Lough, TF. T. aste^acea. zoanthus. Caryophylliacea. CA n YOPHYLLIA CEA . ZOA NTHID^ THE FURROWED CREEPLET. Zoanthtts sulcatus. (Sp. nov.) Plate IX. Fig. 7. Specific Character. Upper half of column free from sand, and indented with longitudinal furrows, GENERAL DESCRIPTIOX. Form. BcLsal band. Broad, with an iiregularly ainuous outline, and offshoots, often bearing three polypes abreast ; loosely invested with coarse sand. Coliimn. Generally cylindrical, but versatile, sometimes hour-glass shaped, springing out of a membranous epidermis, which tightly invests it, and holds a few gi-ains of very fine sand imbedded in it. When ex- tended, the column rises free and smooth out of this, which then reaches to about one-third of the height. Surface marked with twenty-two (in immature specimens twenty) longitudinal sulci, most conspicuous towards the summit : in the button state this is rounded, with a central depression, where the sulci meet. Each alternate intersulcus forms a marginal tooth. jyUk, Saucer-8hai)ed ; radii not conspicuous. Tentacles. Equal in number with the intersulci, with which they cor- respond, in two rows, the inner row to the marginal teeth, the outer inter- mediate. Sub-equal, conical, pointed, usually radiating horizontally. Mouth . Not raised on a cone. COLOUB- Cohunn. Dull uniform olive : each intersulcus having a blackish spot near its summit ; and each tooth is silvery white. Bisl: Yellow-olive ; but invariably more or less studded with very minute grains of white sand, which seem fixed, and look like silver-filings. Aggregations of these grains specially occnr at the bases of the secondary tentacles, omitting the primary ones. Tentacles. Perfectly colourless and transpai-ent, with spherical granules of yellow-brown pigment, set like pavement on the interior surface of the wall, generally in contact, yet here and there leaving large spaces alto- gether unoccupied. The colour of the column and disk ia evidently formed by similar granules, but in uninterrupted contact. 304 ZOANTHIDiE. Size. Column about one-eighth of an inch high, and one-twelfth wide. Locality. Torbay j on rock, between tide-marks. This very distinct and interesting little Zoanthus occurred in a large colony at Broadsands, near Brixham, in March, 1859. They were spread on a rock of soft red sandstone, and so numerously, that, in the fragment which came into my possession, I counted sixty polypes in a space of one-and-a-half inch square. At first their character was much disguised by the crowded sand- tubes of a very minute Terebella, out of the tangled masses of which the Zoanthi were peeping. When these were cleared away by the careful application of a needle-point and a hair-pencil, the basal expansion was apparent, an irregular broad band, with several polypes abreast, as described above. The texture of the band appears less compact than in the preceding species, with which I com- pared it, having a more cellular appearance ; the grains of sand too are coarser. The species is hardy, my specimens being healthy at the present time, after three months' captivity. They are evidently diurnal in their habits and predilections, gene- rally expanding under the stimulus of sunlight, but always closing at night. When the polype is irritated it shrinks nearly to the epidermis, and from the whole summit throws off a mucus, which presently becomes membranous, and seems identical with the epidermis, Couchii. [Solanderi.] SULCATUS. Phellia. TOPHYLLIACEA. ZOANTHID^. THE WRINKLED CREEPLET. Zoanthus Alderi. (Sp. nov.) Plate IX, Fig. 8. Specific Character. Polypes free from sand ; set in single file, obconic, transversely wrinkled. GENERAL DESCRIPTION. Form. Basal band. Narrow, smooth, irregularly branching, free from sand. Column. Inversely conical, the summit being two or more times as broad as the base ; summit (in the button state) swelling, flat, depressed in the centre, with many (about twenty ?) radiating stricR, indicating the marginal teeth. Siu-face smooth, without any investment of sand, bat marked throughout with close-set transverse or annular wrinkles. Z>i«i and Tentacles. Unknown. Colour. Basal bomd and column. Opaque milk-white. Size Height of column about two lines ; greatest diameter about half a line. Locality. Northumberland : on a stone, at extreme low- water. The slight acquaintance that I possess with this species I owe to Mr. Joshua Alder, who has sent me a drawing and description of a specimen found by him at Cullercoats, X 306 ZOANTHID^. at a very low spring-tide, in the summer of 1857. My friend favours me with the following note of the capture: — "It was soft and fleshy, without trace of corallum ; the individuals connected by a creeping fibre running over the under sur- face of the stone. I chipped a piece of it off, which fell face-downwards, and I fancy got injured in consequence ; as it never showed any signs of life after I put it into my bottle. I kept it two or three days in expectation that it might recover, but, as it began to decay, I secured the remainder by putting it into spirit." There were about a dozen polypes in the colony, all of the same size, which seems to be good evidence that they had attained adult dimensions. Couchii. Alderi. ? Sarcodictvon. 307 FAMILY III.— TURBINOLIAD^. In this, and all the families which have now to come under consideration, the tissues secrete calcareous matter, which unites into a solid internal skeleton of stone, known as the CORALLUM. The stony substance is chiefly deposited — 1. in the integuments of the base and column, forming the WALL (mtiru^) ; 2. in the septa, forming a series of perpendicular plates (lamellce), which radiate inward from the wall ; and, in some cases, another circle, or circles, of similar plates, palules (pali), which do not reach the wall ; and 3. (as I believe) in the ovarian mesenteries, form- ing a series of plates, generally twisted, in the bottom of the cavity, called the columella. The hoUow centre, formed by the upper edges of the plates, is called the CALICE {calyx). Sometimes the exterior of the wall is fm-nished with longitudinal ribs (costce), which correspond to the plates. The plates are arranged in cycles : those of the Jirst cycle project furthest inwards ; those of the second bisect the interspaces ; those of the third bisect the interspaces thus formed, and so on. The whole of the plates developed in one primary interspace constitutes a SYSTEM. In the TuRBlNOLiADJE the corallum is solid (not porous), simple, with the lamellar interspaces reaching to the bottom of the cavity, and perfectly free. The plates are highly developed, simple, and generally have a granular surface. The ribs are well-marked. X 2 308 I ANALYSIS OF BRITISH GENERA. ^ With palules : adhei-ent. Palules in a single circle : columella of many slender twisted plates Caryophyllia. Palules in several circles : columella broad and irre- gular in form Paracyathus. Without j)alules : free. Columella a single plate Sphenotroclms. Columella absent Ulocyathua. :} F>LAT£ . X". I LOPHELIA PROLirtftA . 6 PEACHIA TRIPHYLLA 6. SPHtMOTROCHUS WRICHTM . 7 . S. UAC ANOREWANUS. 8. ZOAHTHUS COUCHIl . PARACHATHUS TAXILIANUS . P PftROPUS P THULENSIS 9. PHVLlANCI* AHfHiCANA . ».ll.BALANOPHrUlA RECrA. 309 GENUS I. CARYOPHYLLIA (Lamarck). Cyathina (Ehbknb.). Corallum simple, generally obconic, often with an expanded base, permanently adherent ; outline ovate or circular. Columella composed of several thin, narrow, twisted, vertical plates. Palules broad, entire, in a single circle. Plates straight, broad, projecting, and forming six systems. Bibs straight, developed only towards the summit, granulated. The animal (for so we may conventionally term the soft tissues, though it is to be remembered that the corallum is an essential part of the living body) is, so far as we know it, translucent, the column very exten- sile, the disk protrusile, the tentacles set in several rows, diminishing in size from the outer row inward, each consisting of a stem with a globular head. I know but one British species, C. Smithii. OA R YOPHYLLIA CEA . TURBINOLIA D^. THE DEVONSHIEE CUP-CORAL. CaryophylUa Smithii. Plate X. Figs. 12, 13.* Specific Character. Plates in five cycles ; base broad j outline generally ovate ; height not exceeding the long diameter. Caryophyllia Smithii. cyathus. sessilis. t Turhinolia borealis. Cyaihina Smithii. Stokes, Zool. Journ. iii. 481 ; pi. xiii. figs. 1—6. BucKLAND, Bridgew. Tr. ii. 90 ; pi. liv. figs. 9 — 11. Johnston, Br. Zooph. Ed. 2, 198 ; pi. xxxv. figs. 4 — 8. Couch, Corn. Fauna, iii. 72 ; pi. xii. fig. 3. GossE, Dev. Coast, 108 ; pi. v. figs. 1—6. M. Edwards, Hist. Corall. ii. 14. Fleming, Brit Anim. 508. Bellamy, So. Devon, 267 ; pi. xviii. Fleming, Brit. Anim. 509.J Dana, Zooph. 371. M. Edwards and Haime, Ann. Sci. Nat. Ser. 3, ix. 288. GossE, Man. Mar. Zool. i. 33 ; fig. 50. GENERAL DESCRIPTION. caryophyllia smithii (slightly magnified). Section of corallum. CORALLUM. Corallum. Simple, constricted in various degrees ; the base generally wider than the summit, and the central region being often less than half the diameter of the latter. Outline sometimes circular, but generally more or less elliptical. Height in general less than the long diameter. Ribs. Well-marked on the upper half, leas distinct on the lower, studded with fine granules. Plates. Forming five cycles, and six systems, but the plates of the fifth cycle * Marked in the Plate " Cyaihina Smithii." THE DEVONSHIBE CUP-COEAL. 311 are wanting in some of the systems. They are broad, granular on both surfaces, with the upper edge very salient and rounded in outline. Those of the third and fourth cycles subequal between themselves, and much smaller than the first and second, which also are mutually subequal. Columella. From twelve to twenty thin plates much twisted, with sinuous edges ; the summits much lower than the pahiUs. Palules. Well -developed, more fleiuous than the septa, of which they correspond to the third cycle. Colour. In general pure white, but in some specimens tinged with a lovely permanent rose-tint ANIMAL. Form. Column. Cylindrical, very extensile, smooth, membranous, invected towards the summit, each invection becoming a tentacle, without any distinct margin. Disk. Flat, but readily assuming a conical form. No trace of gonidial radii, tubercles, or groove. Tentacles. About fifty in number, arranged in three subequal rows : stem conical, membranous, translucent, studded with transverse oblong warts ; head globose, opaque, covered with palpocils. (Plate xii. fig. 4.) Mouth. A lengthened ellipse or a slit. Lip coarsely furrowed, like the lips of a cowry-sheU. Stomach flat when empty, as in Anemones. All the tissues can be enormously distended with water. Colour. Column. A very faint bay or fawn colour, with longitudinal lines of chestnut. Dish. Transparent white, with a broad Vandyked circle of rich chestnut surrounding the mouth. Tentacles. Stem-wall colourless, with the warts deep chestnut; head opaque, pearl-white, sometimes slightly tinged with rose. Mouth, Pure white. Size. CorcUlum. Fine specimens attain a diameter of three-fourths of an inch, and a height nearly as great. Animal. The column when distended frequently stands an inch above the corallum, and exceeds it in breadth by a sixth of an inch on every side ; the tentacles augment the height still further by nearly half an inch. Locality. On various parts of our coast in deep water, attached to stones and shells : Devon and Cornwall, on rocks between tide-marks. 312 TURBINOLIAD^. Varieties, o. Castanea. As above described. )3. Esmeralda. The chestnut here replaced by vivid green in like intensity, except the border of the mouth, which is pale red. 7. Clara. Translucent white. On the perpendicular surfaces of cliffs with a northern aspect, in narrow wall-sided fissures, and on the under sides of fallen fragments of rock forming natural arches, and in dark overhung tide-pools, I have found this beau- tiful Coral in abundance on the coast of both North and South Devon. It is only at the great recesses of the equinoctial spring-tides that it is exposed, though in per- manent pools of ample dimensions it occasionally occurs at the half-tide level. For the most part gregarious in habit, it occurs more in colonies than singly, and twenty, thirty, and even more, are occasionally taken by the collectors from a single pool. It is deservedly a favourite with aquarians; for if removed from the rock with care by a proper use of the chisel, scarcely any species is more hardy, more beautiful, or more changeable in its aspects. I have been informed of a specimen which had been preserved two and a half years, and was then in health. It is free in expanding in captivity ; perhaps its most common condition being that in which the mouth is somewhat open, and the tentacle- heads just peeping from beneath the half-closed margin of the column ; but occasionally, and especially at night, the animal expands to the full, and rears its lovely form far above the level of its stony walls. This condition may, however, at any time be induced by a proffer of food ; an atom of raw flesh cautiously laid on the half-exposed disk is a temptation too great to be resisted. The protrusile lip slowly but evenly expands to embrace the food, and then closes over it, meeting in a puckered knot in the THE DEVONSHIEE CUP-COEAL. 313 centre. The unyielding stony margin of the omachal cavit)' preventing the morsel from being drawn down, as it would be in an Actinia, the whole disk projects pei-pen- dicularly, like a thick pillar, from amidst the tentacles, displaying the dark mass through the pellucid walls. Now presently a great change takes place : the whole of the soft tissues become distended with water, and take on an exquisite translucency and delicacy ; the colmnn swells out to twice the width of the corallum, the tentacles are like transparent bladders full of water, each crowned by its little white globule, and the whole appearance is most beautiful. I have seen under these circumstances the animal extended to more than an inch and a half above the level of the plates. The lip often projects like a thin oval wall, or like the brickwork surrounding a well ; marked with thick perpendicular ridges of opaque white, distinctly defined, separated by interspaces of equal width. This is well expressed in the figures (5 and 6) given by Johnston, after Alder, which are very accurate : figs. 7 and 8 of the same plate, like too many of the zoophytic deli- neations of Forbes, I can only call caricatures. I have elsewhere* given many details of the structure and economy of this Coral, to which I can here only refer the reader. Among them will be found some curious examples of reproductive power; one, in the formation of a new disk, mouth, and tentacles, at the lower end of the corallum, which had been broken firom its base ; and another, of the replacement of a large number of the septa, which had been broken away. Of the generation and development of the species I can say nothing fiom personal observation ; the smallest I have seen having been about one-sixth of an inch in diameter, with a well-formed corallum of half a line in height. • Devonshire Coast, pp. 108—127. 314 TURBINOLIAD^. Mr. R. Q. Couch, however, says, " In the youngest state the animal is naked, and measures about the fifteenth of an inch in diameter, and about the thirty-second of an inch in height. In the earliest state in which I have seen the calcareous polypidom there were four small rays, which were free or unconnected [i.e. without any wall] down to the base ; in others I have noticed six primary rays, but in every case they were unconnected with each other. Other rays soon make their appearance between those first formed ; they are mere calcareous specks at first, but after- wards increase in size. The first union of the rays is observed as a small calcareous rim at the base of the polype, which afterwards increases both in height and diameter with the age of tlie animal." From a valuable series of observations made by Mrs. Thynne,-]- it would appear that the Caryophyllia discharges its ova in spring, which in about two days become rotating infusorioid animalcules. In a week or two these afSx themselves, and develop tentacles and a disk, and gradually grow to the size, and even far more than the size, of the parent, with all the characteristic colours and marks, hut without the least trace of a corallum. During the progress of this condition, the individuals increase rapidly by spontaneous fission, the separated portions immediately becoming independent animals. It is difficult to suggest any flaw in the evidence of identity ; but it is to be regretted that the experiments terminated without any sign of the development of a corallum. Double and even triple specimens are not uncommon; and I have seen at least two examples (one of which I now possess) that are fourfold. J The appearance of such speci- mens is exactly that of a branching coral ; and, strange to * Quoted in Johnston's Br. Zooph. i. 199. t Ann. N. H. for June, 1859. . t Such a specimen I have figured in my Dev. Coast, pi. v. fig. 5. THE DEVONSHIEE CUP-COEAL. 315 say, if one alone of the disks be fed, the rest will presently become equally distended, as if partaking of a common life. On breaking one of these double skeletons, however, no communication is found to exist between the cavities ; and hence we must conclude that such instances are due to the accidental fixation of two or more geramules in close proximity to each other, and the coalescence of the cal- careous walls in process of growth. The name Caryophyllia is formed of Kcipvov, a nut, and (f)vWov, a leaf, — q.d. " a nut of leaves" = plates. The specific name is in honour of Thomas Smith, who appears to have first observed it on the south coast of Devon. A curious little Barnacle {Pyrgovia Anglicum) is para- sitic on this species, affixing itself to the outer edge of the plates ; two are sometimes found on the same coral. The corallum is very hard. An hour's rubbing of one on a slab of marble rough from the saw, with a view to a longitudinal section, produced little efiect on the coral, though it effectually polished the marble. The following list of habitats show that the species is widely scattered around our coasts. Shetland (deep-water), Fleming: Moray Firth (d. w.), W. G. : Guernsey (low- water), T. D. H. : Torquay (1. w. abimdant, d. w. rare), P. H. G. : Dartmouth (1. w.), E. W. H. H. : CornwaU (1- w. abundant), R. Q. C. : Ilfracombe (1. w. abundant), P. H. G. : Oban, J. A.: Lame (d. w.), G. D. (B.) .• Lambay, R. Ball: Dalkey Sound (1. w.) R. B. : Wexford Bay, W. WCalla : Nymph Bank (d. w.), W. T. : Youghal, R. B. : Bantry Bay (1. w. common), E.P. W.: Connemara, W.M'C: Bundoran, R. B.: Lough Swilly (d. w.), G. D. (B.) : Lough Foyle (d. w.) G. D. (b.) Corynactis. Smithii. [cyathus]. 316 GENUS 11. PARACYATHUS (M. Edw. & Haime). Corallum simple, siibturbinate or cylindrical, with an expanded base, permanently adherent. Columella very broad, terminated by a papillous surface, and formed by processes that appear to arise from the lower part of the inner edge of the septa. Palules of divers orders, forming two or more circles ; in general lobed at the summit, narrow, tall, and appearing also to arise from the lower part of the inner edge of the septa, their size diminishing as they approach the columella. JPlates nearly equal, very slightly salient, and closely set ; their lateral surfaces strongly granulated, and sometimes presenting traces of imperfect dis- sepiments. They form four or five cycles, and the systems are equally developed. Bibs nearly equal, straight, closely set, projecting very little, and delicately granulated. ANALYSIS OF BRITISH SPECIES. Plates forming five imperfect cycles : cup elliptical . . . Taxilianug, Plates forming four imperfect cycles : cup circular . , . Ribs obsolete below Thulensis. Ribs very salient below ptervpus. CARYOPHTLLIACEA. TURBINOLIAD.E. THE MORAY CUP-CORAL. ParacT/athus Taxilianus. (Spu DOT.) Plate X. Fig. 6. Spteific Character. Plates in five imperfect cycles ; calice elliptical ; ribs notched above, granulons below. GE^^alAL DESCRIPTION. Corallam. Slightly turbinate, adhering by a base broader than any other part, a little diminishing towards mid-height, and widening gently above and below. Wall thin. Height about equal to the medium diameter. Ribs. Distinct from base to margin; on the tipper half prominent, thin, with a rather sharp, but in-egularly notched edge, separated by inter- costal furrows of about twice their width ; on the lower half forming low rounded ridges, crowned with conical granules, set in two or three irregular longitudinal rows ; all are nearly alike in every respect. Calice. Elliptical ; the axes as 24 : 31. ir. i^i^ii^^^ua ■^ ' ( magnijiea J. Plates. Forming five cycles and six systems ; 4 portion cut away to but those of the fifth cycle are wholly wanting show the plates. in three systems, and present in both halves of the other three. Not very close-set, not very salient, thin, very little thick- ened externally, the highest point of their edge a little within the margin, whence it slopes very slightly inward and downward, in an imdulate line, ending with an abrupt angle, whence the inner edge descends perpendicu- larly : the entire edge rises into irregulM: eminences and blunt points, and both surfaces are roughened with coarse granules. Columella. Formed of two or three much twisted lamellse, with broad rounded lobes, rising from the tmited palules. Palule^. Thin, waved, lobed and granulate, like the septa; those of the tertiary septa large ; the others inconspicuous, and only here and there discernible ; united in the centre into an irregularly waved and perforated horizontal plate. p. TAXILIAXUS 318 TUEBINOLIAD^, Size. Diameter of long axis, 'SI inch ; of short axis, '24 ; height "21 to '14, unequal because the corallum is built partly on a shell and partly on a Serpula tube adhering to it. Animal. Unknown. Locality, The Moray Firth ; deep water. It is with some doubt that I refer this and the two following species to the genus Paracyathus. Generally agreeing with its characters, they all have the peculiarity of the union of the palules into a horizontal perforate platform, which does not appear to be the case with any of the hitherto described species. The single specimen on which the above description is founded was forwarded to me by my kind friend, Mr. Gregor, of Macduff, who obtained it from deep water. It is affixed to the inside of an old valve of Cyprtna Islandica, and has the appearance of being recent. The only species of Paracyathus with which this is likely to be confounded is the fossil P. crassus of the London Clay ; but from this it may be distinguished by the union of the palules, by the ribs being proportionally thinner and more remote, and by the diversity of their upper and lower portions. Paracyathus is derived from irapa, near, and Kva6o<;, a cup (the element of Gyathina). I have assigned a specific name from Taxilium, the ancient appellation of the pro- montory now called Kinnaird's Head, off which the specimen was taken. Caryophyllia. Taxilianus. crassus. CARYOPHYLLIA CEA. TURBINOLIADjB. THE SHETLAND CUP-COEAL. Paracyathus Thiclensis. (Sp. nov.) Plate X. Fig. 8. Specific Character. Plates in four imperfect cycles ; calice circular ; height equal to half the diameter. GENERAL DESCRIPTION. Corallum. Slightly turbinate, adhering by a base, which, though broad, is the narrowest part. Height about half the diameter. mbs. Prominent on upper half, becoming ob- solete below ; their edges set with tooth-like coni- cal tubercles ; separated by intercostal furrows, which on the whole equal the ribs in width, but both are irregular. Calice. Circular, shallow. Plates. Forming four cycles and six systems ; those of the fourth cycle wanting in the halves of four systems, and present in both halves of the other two. Rather wide apart, moderately salient, rather thick, scarcely thickened externally ; out- line of their upper edge forming a flattened arch, but not uniformly, in some the highest point being at the margin, in others far within ; inner edge nearly perpendicular : entire edge set with irregular eminences and blvmt points : both surfaces studded with coarse granules. Columella. A single flexuous plate with a somewhat tri-radiate summit, united below to the palules. Palulea. Indistinct, being confluent, and sending off horijtontal traverses to the septa, so as to form an irregular perforated horizontal lamina, whence the columella rises. p. THULEXSIS (magnified). Vertical aspect of corallum. SiZB. Diameter '19 inch ; height "1. Animal. Unknown. 320 TURBINOLIAD^. Locality. Shetland Isles ; Moray Firth ; deep water. Looking over the cabinet of Dr. Howden, of Montrose, last winter, my eye fell on this little Coral, which seemed new to me. Its owner was so kind as to transfer it to my possession, when, on careful examination, it proved to be an unrecognised species, with the characters above enumerated. It may be distinguished from P. caryophyllus by the relative proportion of the height to the diameter, and from all other described species by the number of septal cycles. Dr. Howden dredged the specimen off Ord Head in Bressai Sound, Shetland, in thirty or forty fathoms, on a bottom of small stones, to one of which it is attached. In March of the present year Mr. Gregor sent me, on a valve of Lutraria, a specimen, which appears to be of the same species, but of younger age. It is not more than half the size of the former, but in other particulars agrees sufficiently. On my putting it into sea-water on its arrival, the pellucid flesh came up and filled the intersepts, giving satisfactory evidence of its freshness. Unfortunately it had been sent through the post, packed dry ; it was probably alive when despatched. The whole corallum in this speci- men is of the purest translucent whiteness. It came up on a fisherman's line from the Moray Firth, in about forty fathoms, hard bottom. The specific name is from Thule, the ancient designa- tion, as presumed, of the Shetland Isles. Taxilianus. Thulensis. pteropus. CA R TOPff TLLIA CEA . TURBINOLIADjS. THE WINGED CUP-CORAL. Paracyathus pteropus. (Sp. nov.) Plate X. Fig. 7. Specific Character, Plates in foiir imperfect cyclea ; calice circular ; ribs very salient, dilating into wings below ; height less than half the diameter. GENERAL DESCRIPTION. Corallum. Cylindrical, adhering by the entire breadth ; height less than half the diameter. Bibs. Thin, nearly straight, sub-equal, separated by intercostal spaces about thrice their width, very salient throughout, but from the middle downward developing into triangular buttresses, the long lower edges of which are adherent to the support, so that the area inclosed by their points is far wider than that inclosed by the wall : their whole surface, as well as that of the intercostal spaces, has a slightly carious, but glossy appearance, not exactly granular. Calice. Circular, shallow ; the margin in the same plane. Plate. Forming four cycles and six equal systems, those of the fourth cycle wanting in half of each system. They are wide apart, being separated by twice or thrice their own thickness, thin, salient, but unequally so, some of the primaries and secondaries rising to twice the height, above the wall, of the tertiaries, but others are more nearly equal ; their planes are more or less waved, and their surfaces set with scattered blunt eminences : upper edge truncate, nearly horizontal, but slightly declining inwards, and rising with an abrupt blunt point at the inner edge, which then descends perpendicularly. ColnmeUa. A single flexuons plate, united below to the palules. Palulet. Distinct, united to the inner edges of the primary and secondary plates, and to some (not all) of the tertiary : they are thick, very sinuous, their surfaces set with rounded eminences, and their upper edges much p pxEROPCS lobed ; they are imited by their inner edges (corallum magnified). into an irregular horizontal platform, out of the centre of which rises the eolumelki. 322 TURBINOLIADJi. Size. Diameter from wall to wall "13 inch : height "05. Ahimal. Unknown. Locality. The Moray Firth, deep water. For this veiy distiuct and remarkable little Coral I am indebted to Mr. James Macdonald, of Elgin, who obtained it from Lossiemouth, in October, 1858, attached to a valve of Gyprina, from the deepest part of the Moray Firth. There is no other species with which it can possibly be confounded, the expansions of the ribs presenting a very striking character. They remind me of the immense but- tresses which surround the base of the giant Ceiba of the Jamaican forests. To this feature I have alluded in tlie specific name, which is formed from Trrepdv, a wing, and TToO?, a foot. My friends, Messrs. Macdonald and Gregor, speak of other Corals having at various times come under their notice, but they had always been set down, like these now recorded, as Caryojjhyllia ^mithii. It is by no means improbable that further research may considerably aug- ment the list of our living Corals. I'liulensis. PTEROPUS. 323 GENUS III. SPPIENOTROCHUS (M. Edw. & Haime). Turbinolia (Lamabck). Corallum simple, free, with no trace of adherence, wedge-shaped, the superior extremity wider in all directions than the inferior ; transversely elhptical. Columella, a single lamina, occupying the greater axis of the calice : its upper margin flexuous and bilobate. Palules entirely wanting. Plates extending to the columella, or meeting in the centre of the visceral chamber ; broad, slightly salient, forming three cycles, and six systems. Bibs broad, not very prominent, in general crisped, or represented by a series of papillous tubercles. ANALYSIS OF BRITISH SPECIES. Corallum uniformly diminiBhing downward ; ribs smooth . Afacandrewanui. Corallum pedicellate, with awelling nodes ; ribs crisped . Wrightii. T 2 CARYOPHTLLIACEA. TURBlNOLIADuE. THE SMOOTH-RIBBED WEDGE-CORAL. Sphenotrochus Macandrewanus. Plate X. Fig. 4. Specific Character. Corallum uniformly diminishing downward; ribs smooth, not salient ; edge of calice plane. Turbinolia milletiana. Thompson, Annals N. H. Ser. 1. xviii. 394. Johnston, Br. Zooph. Ed. 2, i. 196 ; pi. XXXV. figs. 1—3. E. P. Wright, N. H. Rev. vi. 122, GossE, Man. Mar, Zool. i. 32 ; fig. 49. Sphenotrochus Andrewia/nus. M. Edwards and Haime, Ann. d. Sci. Nat. Ser. 3. ix. 243 ; pi. vii. fig. 4, Macandrewanus. M. Edwards, Hist, des Corall. ii. 70. • GENERAL DESCRIPTION. Corallum. An inverted cone, compressed, lengthened, straight, with the inferior extremity forming a wedge-like blunt point. Ribs. Perfectly straight, smooth, nearly equal throughout, or slightly enlarged above, separated by intercostal spaces about twice as wide as themselves, moderately prominent, continued round the edge of the scar where the corallum was originally attached, Calice. The edges on the same horizontal plane ; outline elliptical, in the ratio of 100 : 120. Plates. Twenty -four; in three complete and well-developed cycles, close-set, straight, thick at the margin, and gradually thinning towai'ds the centre of the calice ; salient, arched at their upper edge, with a surface very slightly granulose. The primaries and secondaries are subequal and similar and hence the appearance of twelve systems ; each of these is united with the columella by two diverging laminte, as if the plate were split atits iimer edge, and the two halves sepai'ated. Columella. A single, thin, vertical lamina. Size. Height half an inch ; diameter of calice one-fourth of an inch by one-fifth. THE SMOOTH-RIBBED WEDGE-COB AL. 325: Anihal. Undescribed. Locality. The coaats of Cornwall and Galway : deep water. I am sorry that I can give no information about this species additional to what is already known, viz., that it exists in a living state on our coasts, and that the skeleton is preserved in cabinets. That in the British Museum is the only one that I have seen. As long as naturalists con- tent themselves with merely preserving the skeletons of the animals they meet with, but little progress can be made in a knowledge of their history.* The present species is said to have been dredged alive off Scilly, by Mr. MacAndrew, after whom it has been named, and off Arran, on the west coast of Ireland, by Mr. Barlee. The generic name is from acfyrjv, a wedge, and rpoxo^) a top, in allusion to the form of the corallum. S. milletianus, with which this has been confounded, is a fossU of the miocene period, with a thicker point, and a more elliptical calice. intermedins (Joss.). Macandrewanus. [Roemeri {foss.).] * M. Milne Edwards has fallen (Hist. Corall. ii. 70) into the strange inadvertence of supposing that the figure given by Johnston (Br. Zooph. Ed. 2, pi. XXXV. fig. 7), of the living animal, belongs to this species ; though the t«xt distinctly says it is a Caryophyllia Smithii. The figure is poor enough, it is true. CA R YOPB YLLIA CEA . TURBINOLIA DJ'J. THE KNOTTED WEDGE-COKAL. Sphenotrochus Wrlghtii. Plate X. Fig. 3. Specific Character. Corallum pedicellate, -with swelling nodes; libs papilliferous on the body, and crossed with zig-zag folds on the pediceL SpJienotrochus Wrightii. Gosse, Nat. Hist. Review, vi. 161 ; pi. xvii. figs. 1—L GENERAL DESCRIPTION. Corallum. Simple, straight (or else with the base considerably curved laterally), compressed above (the axes of the disk being 60 : 42 in general ; in one example, however, 60 : 50), but rounded in the lower two-thirds, pedicellate; the body and the pedicel varying exceedingly in their rela- tive proportions, the former being to the latter as 1 : o in one example ; in another, as 1 : 1; in another, as 1 : 1'2, — no two of the four specimens in my possession being alike in this respect. The pedicel is surrounded by four to six constrictions, varying gi-eatly in their relative distance : these separate nodes are more or less swollen, of which one, a little above the base, is usually more ventricose than the rest; the pedicel generally enlarges upwards, but its distinction from the body is marked by an abrupt shoulder. Bibs. About as wide as the interspacss, distinctly traceable only as far down as the termination of the body ; their course is irregularly angular ; the primaries and secondaries terminate at the shoulder in prominent knobs. On the pedicel only the six primaries are distinguishable, and these arc then crossed by numerous strongly indented zig-zag folds, of which the higher angle is on the rib, the lower in the interspace. All the ribs of the body-region aro'studded with irregularly projecting points or papillaiy eminences. Base. A small but distinct circular cavity, into which the extremities of the six primary ribs project. WRIGHTII C(i-lice. Considerably arched, the short axis being much (magnified), the higher; rather deep. Plates. Twenty -foui-, in three cycles ; the lateral primaries and secondaries more developed than the terminal ones ; moderately close- set, irregularly bent in their planes, thick exteriorly, suddenly diminishing THE KNOTTED WEDGE-CORAL. 327 just within the wall, and thence gradually becoming thinner. The primaries and secondaries equal in height and breadth ; the tertiaries much lower ; all salient, the upper edge obliquely truncate, sloping down from the margin inward. The two plate? which form the short axis are united to the columella by diverging laminae ; but this structure appears to be wanting in the others. The surfaces of all the plates are rough, with i-cattered papiUary points. Columella. Bent at each end towards one (the same) side ; ite upper edge thickened in irregular swellings. In some specimens it is not visible from above. Size (of four examples). so. LONG AXIS. SHORT AXIS. HEIGHT. 1 . . . 0-08 inch . . . 0-062 . . . . 0155 2 . . . 0-06 „ . . . 0-042 . . . . 0-140 3 . . . 006 „ . . . C-050 . . . . 0110 4 . . . 006 „ . . 0.042 . . . . 0-144 AsiMAL. Unknown. LOCALITT. Korth-cast coast of Ireland : deep wat«r. This species resembles S. crispus in its zig-zag folds, but has more agreement with S. mixtus in its general characters. In its tendency to a curved form, howcTcr, as well as in its pedicellate character, and especially in the presence of a well-formed basal area, which appears to have been a point of adhesion, it displays so much aflSnity with Ceratotrochus (according to the diagnosis of M. Milne Edwards) that I was at first disposed, to assign it to that genus. The four specimens that I have above described have been entrusted to me by my kind fiiend, Dr. E. Perceval Wright, of the Dublin University, with whose name I have honoured the species. They were dredged by G. C. Hynd- man, Esq., among shell sand, from a turbot bank off the coast of Antrim, in 1852. I have introduced the tiny form into this work, believing it to be an existing, and not a fossil species. Professor 328 TURBINOLIAD^. Milne Edwards, indeed, considers the SjpJie^iotrocId with papillate and crisped ribs to be in no case later than the eocene deposits ; while those with smooth ribs he looks upon as invariably belonging to higher strata, and reaching to the present period: but this is a canon which a new species may at any moment overturn, if it be not already subverted by the 8. nanus (Lea) of the eocene of Alabama. Dr. E. P. Wright mentions, as a suspicious circumstance, that many pleistocene shells do exist in the bed of shelly sand, where these specimens were found. But this does not confirm Professor Milne Edwards's rule ; for, so far as that could decide the question, it would prove not only that this crisped Coral is not recent, but that it is certainly as old as the miocene. Dr. Wright says : — " I have reason to think, however, that they are not fossil;" and the same is my own impres- sion, though I can scarcely assign any definite grounds for it, except the fresh appearance of one or two of the speci- mens. Some of them are rubbed, and one is polished externally. The uniformity in size of the individuals, and the full development of the plates, indicate a probability that, minute as they are, they have attained adult age. [mixtus {foss.).'\ [crispus {foss.).'] Weightii. [Ceratotrochus {foss.).'\ 329 GENUS IV. ULOCYATHUS (Sars). Flabellum (Qbat). Cordlum simple, free, turbinate, with traces of adherence (in the young state) on a very short wedge- shaped crooked pointed base. Columella and palules entirely wanting. Ribs not at all prominent, sometimes obscure. Plates very thin, high, very salient above the margin of the cup, distinct throughout their length. Calice very deep ; the margin sinuous and crisped. Animal resembling that of Caryopkyllia. Only one species has been recognised, TJ. arcticus. ULOCTATHCS ARCTICUS (after Sars) slightly magnified. CARYOPHYLLIACEA. TURBINOLIAD^E. TPIE SCAELET CRISP-COEAL. Ulocyathus arcttcus. Specific CJiaracter. Base triangular and flat, bounded by a sharp edge : calice round. Ulocyathus arcticus. Sars, Fauna Litt. Norv. ii. 73 ; pi. x. figs. 18—27. Halellum MacAndreici. J. E. Gray, Pi'oc. Zool. Soc. May, 1849 : pi. ii. figs. 10, 11. GENEllAL DESCRIPTION. CORALLUM. Corallum. Simple, free, but with traces of having been adherent iu infancy : the base with a great inferior surface, triangular, flat, often concave, separated from the superior surface, which is equally triangular and convex, by a sharp edge on each side. Ribs. Large, often indistinct, unequal; the primaries sometimes armed with minute tubercles. Calice. Very wide and deej) ; the edge almost circular, crisped with minute sinuosities. Plates. These are so irregular that it is difficult to count the cycles, but they are at least four. Those of the first and second are more than twice as high as the rest, and reach to the centre of the cup, where they unite, but irregularly : the others are lower and shorter in gradation, the lowest projecting little within the margin. All are perfectly separate throughout, extremely thin, sharp-edged, the surfaces set with minute granules often running in curved lines : the free edge of all is arched, and their greatest width is one-third from the summit. The primaries and secondaries ai'e very salient, and the edge of the calice seen in profile forms eleven or twelve triangular lobes. Columella and palules wholly wanting. ANIMAL. Form. Column. Actinia-like, without any trace of gemmae. Disk. Radii fine, distinct. Tentacles. About 140, in four rows, close-set, iri'egular; the innermost three or four times as large as the outermost : stem cylindro-conical, THE SCARLET CRISP-CORAL. 331 covered with large round prominent warts ; head globose, smooth, imper- forate ; very contiactile, but not retractile. Mouth. A wide slit in the direction of the long axis : lip crenate, with forty to sixty-five deep furrows. Colour. A brilliant orange-scarlet ; a little lighter on the inner tentacles : the furrows of the lip intense blood-red. Size. Corallum. About one and a half inch in diameter, and a little less in height. LOCALITT. The coasts of Norway and Shetland : deep water. Of this species, bj far the largest and noblest of the simple European Corals, a specimen was dredged bj Mr. MacAndrew about twenty-five miles off East Shetland, in ninety fathoms. The individual was broken by the dredge, and only a portion of the corallum was secured, which is now in the British Museum. There can be no doubt, however, of its identity. A considerable number of examples have been obtained by Mr. Sars at Oxfjord, close to North Cape, the extreme northern point of Europe. It lives at an amazing depth, even from 150 to 200 fathoms, where the pressure of the superincumbent water must be immense. Clear as are the waters of the northern seas, so vast a volume of water must surely absorb nearly the wh> le of the rays of light, and the rich hues of the animal arc therefore the more remarkable. It lies free on the mud or clay, never having occurred with evidence of recent attachment. The generic name is formed from ou\o9, crisped, and Kva6o<i, a cup. [Desmophyllum.] ARCTICUS. [Flabellum.] 332 FAMILY IV.— OCULINAD^. The corallum in this family is solid (not porous), com- pound, increasing by gemmation so as to take a form more or less branching and tree-like. The stony tissue is very compact, the surface smooth, delicately striate near the calices, or but slightly granular. The walls of the corallites (or stony skeletons of the individual polypes) are not per- forate, not distinct -from the common tissue {comenchyma) , and increase by their inner surface, so as gradually to fill up the cavity from below upwards. The interseptal chambers are only imperfectly divided by a few dissepiments, or horizontal projections of stony matter shot across. The plates {septa) are entire, or have the upper edge slightly divided ; they are well developed, and are few in number. We have but one native representative of this family, the genus Lophohelia. 333 GENUS I. LOPHOHELIA (M. Edw. & Haime). Madrepora (Lisn.). Oculina (Lahabck). Liihiode»dran (Schweiqgeb). Corallum tree-like, or formiug a branching thicket, the branches coalescing; the form results from a gemmation irregularly alternate and sub-terminal. There is no true ccenenchi/ma, but the walls are very thick, scarcely ribbed. Calices having a deep cavity, with a reverted lamellar edge. Columella and palides wanting. Plates entire, salient, unequal, the principal ones vunited towards the lower part of their inner edges, at the bottom of the visceral cavity. There is but one known British species, L. prolifera. CA B TOPH YLLIA OEA . OCVLINADJE. THE TUFT-CORAL. LoplioJielia prolifera. Plate X. Fig. 1 {reduced). Specific Character. Corallites cylindrical. Madrepora prolifera. Linn. Syst. Nat. Ed. 12, 1281. Ellis and SoLANDEB, Zooph. pi. xxxil. figs. 2 — 5, EsPER, Pflanz. i. 104; Madr. pi. xi. Lithodendron proliferum. Schweigger, Handb. der Nat. 416. Ooulina prolifera. Lamarck, An. s. verteb. ii. 286. Lamoue. Exp. mdtli. G4 ; pi. xxxii. figs, 2 — 5. Dana, Zooph. 393, Lophohelia prolifera. M. Edw. and Haime, Ann. des Sci. Nat. Ser. 3. xiii, 81. M. Edw. Hist. Corall, ii. 117. GENERAL DESCRIPTION. Corallum. Forming a massive, compact, many-branched tree, rising from a slender base, permanently attached to rocks. Corallites. Free laterally, in general budding only once or twice, cylindrical, or but slightly expanding at the summit, moderately long. Exterior surface covered with very minute close-set granules, without ribs, except very faint marginal traces. The margin is often surrounded by a thin lamellar expansion. Plates. Systems generally unequal and irregular, being formed of seven, or five, or three derived plates, but easily recognisable by following the development of tlie primaries, which are far greater than the others. The plates themselves are thick in the centre and towards the margin, but .ire thinned off to a shai'p edge, which is irregular in outline, but not notched ; their surfaces covered with minute granules. The principal ones, from eight to twelve in number, are stouter and far more salient than the rest. Walls. Very thick and dense, gradually filling up the bottom of the cavities. SrzH. The individual corallites are from one-fourth to half an inch in height and diameter. The dimensions of the compound mass vary according to THE TUFT-CORAL. 335 age : the specimen figiired is about ten inches in height, and seven in diameter. Animal. Undescribed. LOCALITT. The north-western coasts of Europe : deep water. The figure in Plate X. is taken from a noble specimen, imdoubtedlj British, reduced to half the natural size. I am indebted for the opportunity of delineating it to the kindness of Professor Dickie, of Belfast, -^ho was at the pains of having several photographs taken from it for my use, and favoured me also with many fragments including perfect corallites. Dr. Dickie informs me that it was obtained from deep water off Skye, in 1852, by means of the deep-sea lines of a fisherman, who presented it to him. He mentions having seen another British example, in the possession of Professor Fleming, the same that the latter exhibited before the Royal Society of Edinburgh in 1846, and which had been taken in the previous summer, by fishermen whose lines had become entangled with it in the .sea between the islands of Rum and Eig. This specimen, which weighs six pounds, is preserved in the Museum of King's College, Aberdeen. A third example is alluded to by Johnston, who was informed by E. Forbes that certain published figures of the species " had recalled to his mind a very large specimen in the possession of Dr. Edmonstone of Orkney." It is to be regretted that we possess no information of the living animal of so fine a Coral, the only British example of the truly dendroid species. The name Lophoheh'a is formed from \6(fio<;, a tuft, and ^Xto9, the sun : q. d. " a tuft of suns," alluding to the radiating plates of the corallites. [Acrohelia.] LOPHOHELIA. [Amphihelia.] 336 FAMILY v.— ANGIAD^. The visceral cavity of the coralluu. in this family is not obliterated, nor even subdivided ; the interseptal dis- sepiments being merely rudimentary. There is no cmnen- chyma, and the wall is imperforate. The plates have notched edges, but not very conspi- cuously. The corallum is massive. It increases by gemmation ; the buds being developed on stolons, or on basal membrani- form expansions. The corallites are not united by their sides, except accidentally by means of their walls, and they remain short. But one British genus is known, Hoplangia. 337 GENUS I. HOPLANGIA (Gosse). Phyllangia (Gossb). Corallum incrusting foreign bodies. Corallites rather short, formed by buds uhich spring from an expansion around the base of the parent, permanently united to it (but not to each other) by the inferior portions of their walls. Wall surrounded by a thin porcellanous coat (^z- ilieca), which permits the ribs to be traced through it ; granulate. Bibs thin, sharp, low, very unequally distinct. Columella a broad surface of rough papillae, merging into the plates. Falules wanting. Flafes thin, scarcely salient, unequal, straight, granulose, toothed on the edges, except the upper edges of the primaries, which are nearly entire. There is but one species, H. Durotrix. CARYOPHYLLIACEA. ANQIADjE. THE WEYMOUTH CARPET-CORAL. Hoplangia Durotrix. (Sp. nov.) Plate X. Fig. 9.* Specific Character. Plates in four imperfect cycles. Phyllangia Americana. Qosse, Annals N. H. Ser. 3. ii. 349. GENERAL DESCRIPTION. Corallum. Compound, increasing laterally on all sides ; low, not rising above the height of the individual corallites ; incrusting rocks. Corallites, Formed by budding from a permanent, thin, calcareous, carpet-like expansion, which spreads around the base of the parent, to which each is permanently united by the inferior portion of the wall. (In the specimen in my possession, four corallites of sub-equal size sire grouped around a parent, which has been long dead, for the inner portions of its plates have been worn away.) They are cylindrical, deep, about twice as high as wide, slightly inclining outwards from the common centre. Wall. Invested by a thin porcelain-like coat of calcareous matter, which appears identical with the basal carpet. It terminates above with a perfectly defined, slightly everted edge, above which the wall is beau- tifully white and clean, while the epitheca is dirty white, and coated with a minute sponge. The epitheca shows traces of periodic growth, by a succession of such everted edges not totally obliterated ; and while in one corallite the edge is level with the summits of the plates, in another there is at least one-fourth of the total height above the epitheca. Hence I infer that the wall with the septa makes a periodic growth above the last level of the epitheca, while the latter remains dormant, and that theii» the epitheca is deposited at once around the new growth ; the wall and the epitheca thus growing alternately. The wall is covered with minute scattered granules, and these as well as the ribs can be discerned through the thin epitheca. Bibs. Thin, sharp, low, in some places discernible only at the very summit of the wall, in others nearly throughout : in the former case they appear again from the edge of the epitheca a little way downward. * Marked " Phyllangia Americana " in some copies. THE WEYMOUTH CARPET-CORAL. 339 Columella. The floor of the cavity is covered with papillary emi- nences, which are very rough, with irregular points, and are identical with the lower edges of the principal plates, by the convergence of which they seem to be formed. Plutes. Thin above, but increasing in thickness below, scarcely salient, unequal, straight, the surfaces set with irregular granular tubercles, which become increasingly rough and prominent below. The edges are strongly but irregularly notched and toothed, especially below ; but the upper edge of the primaries is for the most part sub- entire; the form of the outline varies much. There are normally four cycles in six systems : but the fourth cycle is always wanting either in the whole or in half of some of the systems ; the amovmt of defection varying much in dif- ferent coraDites. The development is very un- hopla.>'gia equal, and the plates of the third or fourth {magnified). cycle are occasionally larger than those of higher rank, even in the tame system. Size. Individual corallites one-eighth of an inch in diameter, and nearly one fourth in height. AsniAL. Undescribed. LOCALITT. Weymouth Bay : deep water. When this neat and interesting little Coral first came into my hands, I thought, notwithstanding some peculiarities, that it must be referred to the PhyUangia Americana, a native of the West Indian seas, and so announced it. But I see that there are incongruities which prevent its identification with that or any other recognised genus, and I have therefore founded a new one to receive it. It has much in common with Angia, as well as PTiyUangia, but the above diagnosis will, I think, warrant my decision. In forming a generic name, I have followed the plan of M. Milne Edwards in using a common element for the genera of a given family ; though perhaps a little heterodox for stanch Linneans, it has advantages. Taking then the z 2 B40 ANGIADiE. element angia, from dyyo<i, a cup, I have completed the word from ottT^op, armour ; with a double allusion to the mail-like epttheca, and the toothing of the plates. The English name commemorates the manner of gemmation ; and the specific, the locality in which it was found ; the Durotriges having, according to Ptolemy, anciently in- habited the coast of Dorset. In September, 1858, a dealer from Torquay, dredging in Weymouth Bay, brought up a piece of the bottom, about a foot square, evidently the edge of one of the oolite ledges, torn off by the lip of the dredge. On this were from fifty to a hundred specimens of this little Coral, clustered in many groups. It was presumed to be Caryo- phyllia Smitlm, and no special notice being taken of it, the mass was broken up and dispersed ; and a small frag- ment accidentally fell under ray eye, and was secured. I was not so fortunate as to see the animal alive, my specimen, though in the flesh, being in an advanced state of decomposition ; but the discoverer, who is pretty familiar with C Smtthti, at least as to its general appearance, spoke of the Hoplangia as resembling that species, and told me that he remarked green and white hues. He observed also numerous tentacles, but did not notice whether they were knobbed. [Angia.] Hoplangia. [Phyllangia.] Mi FAMILY VI.— EUPSAMMIADiE. • The stony tissue is here deposited in such a manner that tlie corallum, instead of being compact, is porous, but not so open as to have a spongy texture. The wall is thick, and constitutes the chief part of the whole ; it is perforate, and either almost or quite naked, with a granulate ver- miculate surface. The plates are numerous ; those of the last cycle always deviate from the radius of the calice, their planes approach- ing the bisection of their system, so that the whole septal arrangement assumes the form of a six- or twelve- rayed star; by which very remarkable peculiarity this family may be infallibly recognised. The plates are perforate. The interseptal chambers are completely open to the bottom, or divided only by a few incomplete partitions. There is only one British genus known, Balanophi/lh'a. 342 GENUS I. BALANOPHYLLIA (Wood). Corallum simple, adherent, sub-pedicellate, cylin- drical, or sub-conical. Columella well-developed, but not projecting at the bottom of the calice ; of a sponge-like appearance. Plates thin, close-set ; those of the last cycle well- developed. Bibs distinct, narrow, nearly equal, crowded. The Animal is actinia-like, richly coloured, with a protrusile mouth, not conspicuously furrowed, and bluntly-pointed, warted tentacles, without terminal knobs. There is only one British species, B. regia. CARYOPHYLLIACEA. EUPSAMMIAD^. THE SCARLET AND GOLD STAR-CORAL. BalanophyUia regia. Plate X. Fiys. 10, 11. Specific Character. Corallum sub-conical, circular : epitheca extending to margin : plates in five imperfect cycles. BalanophyUia rtgia. Gosse, Dev. Coast, 399; pi. xxvi figs. 1—6. Ibid. Man. Mar. Zool. i 33 ; fig. 51. GENERAL DESCRIPTION. CORALLUM. Corallum. Conico-cylindrical, rising like the trunk of a tree from a base much broader than the column ; height rarely exceeding, often not equal- ling, the diameter. Calice. Circular or nearly so : varying much in depth. Wall. Rather thick, porous, but scarcely spongy, invested with an epitheca, which in general extends to the margin, but not always, and occa- sionally (as in a specimen in my possession) seems wholly wanting. Ribs. Continuous (not formed of separate granules) but very sinuous, and in some part^ branching, the branches so confluent as to form a rough network : they are often distinct through the epitheca. Columella. Much developed, forming a large spongiose mass (or more like the crumb of well-raised bread), often rising almost to the level of the margin, but more commonly to about half that height. Plates. Well develojjed, thick, here and there perforate, with a frosted surface and minutely toothed edges, not salient, the upper edge sloping downward and inward. The star is six-rayed, and is always distinctly formed, and generally symmetrical. There are five cycles, but some of the fourth and fifth are wanting in each system. The gradation in deve- lopment is pretty regular downward from the first to the fourth ; but the fifth are exceedingly irregular and unequal. The two plates of the fifth cycle in each system, which stand next to the primaries (that is, those of the sixth order), are developed to an extent much exceeding even the Hist, des Coi-aU. i. 45. 344 EUPSAMMIAD^. primaries themselves, from which they diverge at such an angle that they mutually meet and coalesce at a point about midway between the origin of the secondary of that system and the axis of the calice, but at a level much lower than the margin ; the two united plates thence pro- ceed in the intermediate line to join the columella. In many examples, however, this continuation of the united quinaries is obsolete in each ^alternate system. The quinaries that are contiguous to the secondaries (the 7th order) are also much developed, but not so as to equaUthe secondaries, with which they often cohere. ANIMAL. Form. Column. Cylindrical, extensile, smooth, or somewhat invected. Dish Protrusile, in the form of a high truncate cone, on the summit of which is the mouth, without any thickened or furrowed lip. No trace of gonidial radii, tubercles, or grooves. 1 Tentacles. About fifty in number, large, conical, obtusely-pointed, with- out terminal knobs : their walls are translucent, and studded with opaque transversely-oblong warts, which become confluent towards the tip. Colour. Column and Dish. Yivid scarlet in adults, orange in young individuals, opaque. Tentacles. Gamboge yellow : the hue residing only in the warts. Size. IDiameter of corallum one-fourth of an inch at margin, and occasionally twice as much at base ; height from one-sixth to one-fourth. The animal in full expansion may reach one-third of an inch in diameter, and one-half in height. Locality. The coast of North Devon : on rocks at extreme low water. This showy little Coral, interesting not merely for its Ijcauty while alive, but for its peculiar structure when dead, was discovered by myself in 1852. I had been spending a THE SCARLET AND GOLD STAR-CORAL. 345 summer at Ilfracombe, and the chills and storms of autumn were already warning the migrant inhabitants away. It was a spring-tide in September, and the water had receded lower than I had seen it since I had been at the place. I was searching among the extremely rugged rocks that run out from the Tunnels, forming walls and pinnacles of dan- gerous abruptness, with deep, almost inaccessible cavities between. Into one of these, at the very verge of the water, I managed to scramble down ; and found round a comer a sort of oblong basin, about ten feet long, in which the water remained, a tide-pool of three feet depth in the middle. The whole concavity of the interior was so smooth tliat I could find no resting-place for my foot in order to examine it; though the sides, all covered with the pink lichen-like Coralline, and bristling with Laminariae and Zoophytes, looked so tempting that I walked round and round, reluctant to leave it. At length I fairly stripped, though it was blowing very cold, and jumped in. I had examined a good many things, of which the only novelty was the pretty narrow fronds of Flustra cliartacea in some abundance, and was just about to come out, when my eye rested on what I at once saw to be a Madrepore, but of an imusual colour, a most refulgent orange. It was detached by means of the hammer, as were several more, which were associated with it. Not suspecting, however, that it was anything more than a variation in colour of that very vari- able species, CaryophyUia Smitkn, I left a good many remaining, for which I was afterwards sorry, since they proved to belong to this new and interesting form before us. All were affixed to the perpendicular side of the pool, above the permanent water-mark : and there were some of the common CaryophylUce associated with them. I afterwards found the same species in considerable number, especially during the very low springs of the 346 EUPSAMMIAD^. October new moon, among the rocks off the Tunnels, all in the vicinity of the spot where I found the first. They were always in the same circumstances, crowded in colonies; one cavity, just large enough to turn in, containing perhaps a hundred, speckling the walls with their little scarlet disks, near extreme low water. Not one that I took presented the least variation from the characters I had jotted down already ; but one specimen had adhering to its base two very young ones, one about a line in diameter, the other not more than one-third of a line. Examination with a lens revealed no difference either in form or colour between these and the adult ; the condition of their skeleton is un- known, as I did not choose to destroy the infant specimen, much to my present regret. Since that time it has been found in considerable abund- ance along the same line of coast; and it has become common in our aquariums. It is always attractive from its brilliancy, and is moderately hardy, though it appears rather more difficult to keep than Garyophyllia. The integuments are opaque, even when distended; indeed they never become filled with water to anything like the extent which makes the species just named so beautiful. The plates are never visible, during life, in any degree of contraction, the red flesh lying as an opaque cushion over them even when all the tentacles are withdrawn. I am not sure that the disk is ever wholly covered by the inver- sion of the column ; even when the tentacles are quite con- cealed beneath the margin, the large mouth-cone still pro- trudes from the central orifice. Sometimes the tentacles sink to very low warts or minute yellow eminences on the scarlet plain that constitutes the disk. I have said that the epitheca is not unvarying ; and I think that the flesh does not extend externally below its edge. One in my possession, however, had the exterior of THE SCAELET AND GOLD STAR-CORAL. 347- the corallum wholly clothed with the scarlet integument, even down to the base. The covering was exceedingly thin, for with a needle-point I could feel the stony corallum without any sensible indentation of the surface, and the points at the margin were projecting. I have no information about the reproduction of the species, except such as may be gathered from the following observation. In the month of September, in a vase in which several specimens were kept, and which contained nothing else to which I could reasonably attribute the phenomenon, I found several clusters of ova. Each cluster consisted of about a dozen, loosely aggregated, and all con- nected by a kind of twisted cord, which formed a footstalk for each. The eggs were perfectly globular, Jgth of an inch in diameter, of a pellucid orange-yellow hue. One of them under the microscope showed the contents granular, and receding from the chorion, with a definite outline. None of them developed the embryo to my knowledge. The genus was established by ]Mr. Wood in 1844, to receive a fossil species from the Red Crag of Sutton. It now contains eleven species, most of them fossil, but one exists in the Italian seas, and two others elsewhere. There is none with which B. regia can be confounded. The generic name is derived from ^dXavo'j, an acorn or nut, and <f>vWov, a leaf, and the specific alludes to the royal colours in which the animal is arrayed. Ilfracombe, F. H. G. ; Lundy, C. K. REGIA. [cylindrica {foss.).'\ 348 ' LUCERNARIAD^. (?) PociLLoroRA INTERSTINCTA (Miiller). At a meeting of the Eoyal Society of Edinburgh (Trans. March, 1846), Dr. Fleming exhibited a characteristic draw- ing of a Pocillopora presumed to be of this species, which was obtained by Dr. Hibbert in the Shetland Seas. Dr. Fleming had expected that a detailed description of this would have been published before the appearance of his "History of British Animals," in 1828. It is, however, I believe, still a desideratum. The genus is marked by the following characters : Corallum massive or sub-tree-like, with thick, imperforate walls. Visceral chambers divided by well-developed hori- zontal partitions, or floors, in successive stages. Plates rudimentary. Calices shallow, with a thick ring at the bottom of each, forming a sort of columella. lucer:n'akiad^. Contrary to my original intention, I have determined to exclude this family from my work. Their true affinities are with the Hydrozoa and Meduscp. The gelatinous tex- ture, the expanded umbrella, the ovaries in the substa::ce of the umbrella, the four-lipped mouth placed at the end of a free peduncle,* and the quadripartite arrangement, are all Medusan characters. The tentacles in marginal groups are found in Bougainvillma, and their form, — knobs at the tip of long footstalks, — agrees more with Slahheria than with Corynactis and Caryojpliyllia. '* See my fig. of Campanularia, in Devonsh. Coast, p. 296, pi. xviii. PLATE Xr ANATOMICAL DETAILS If ontt;. sc I APPENDIX. I. SPECIES DISCOVEEED TOO LATE FOR DESCRIPTION IN THEIR PROPER PLACES IN THIS VOLUME, ASTR^ACEA. SAGARTIAD^. THE LATTICED COKKLET. Phellia Brodricii. V-LXTY. Tin. Fig. 2. Speci^c Character. Epidermis free at the margin, dense, transrersely corrugated. Tentacles marked with a latticed pattern. Phellia Brodricii. GossE, Annals N. H. Ser. 3. iii 46. GENERAL DESCRIPTIOK. Form. Rase. Adherent to rocks ; considerably exceeding the column. Column. Flat and wrinkled when completely contracted : rising to a tall, somewhat slender pillar, studded with low warts on its upper portion, but covered on its lower two-thirds with a tough, firmly adherent epi- dermis, the upper edge of which is free, with a ragged foliaceous margin, not foi-ming a tube. The surface of this is transversely corrugated, but not warted. The animal frequently expands in its low condition, when the flower occupies the summit of a very low cone, and is not half the diameter of the base. A slight margin, much wrinkled in semi-contraction, and forming a star of radiating furrows in closing. DisJ:. Flat or slightly concave ; outline circular. Tentacles. Arranged in five rows, viz. 6, 6, 12, 24, 48 = 96 ; short and slender, diminishing from the first row outwards ; in ordinary extension not longer than one-fourth the diameter of the disk ; generally carried arching over the margin, the tips occasionally turned up. Mouth. Elevated on a strongly marked cone. Acontia. Not emitted, even under strong irritation, while in my posses- sion. Mr. Brodrick, however, has seen them projected from the mouth. '^50 APPENDIX. Colour. Colvmn. Exposed part pellucid white, with the warts opaque white. Epidermis. Ochreous drab, slightly darker in some parts, with longi- tudinal white lines proceeding from the base, and vanishing a little way up. Central star of button formed of alternate whitish and blackish rays. Dish. Drab : each primary and secondary radius marked with two parallel lines of dark chocolate-brown ; each tertiary radius is similai-ly but more faintly marked, and the space inclosed is in these latter radii drab on their outer and white on their inner moiety, the divisions of the two colours being marked by a black spot. The space immediately bounding the foot of each primary tentacle dark brown. Tentacles. Pellucid whitish ; the lower half opaque white on the front, crossed by four transverse bars of dusky, the whole (except the lowest one) being connected by three longitudinal lines of the same colour, which impart a latticed or window-like pattern to the tentacle. Mouth. Lip white ; throat white, with black furrows. Size. Diameter of base nearly an inch, of extended column half an inch, of flower from one-third of an inch to an inch ; height one inch. LOCALITT. Lundy Island : on rocks at low water. My acquaintance with this species I owe to the courtesy of William Brodrick, Esq., of Ilfracombe, with whose name I have honoured it. He kindly sent rae a specimen in November, 1858, which had at that time been in his possession about sixteen months, having been taken with another individual in the summer of 1857. Its habit is to remain on an exposed stone, without any disposition to roam : it is generally closed by day, or if open the column is contracted ; but it elongates in darkness. It is very timid, and cannot on this account be fed : the slightest touch of the tentacles I found to be followed by an instant closing. The light of a candle, concentrated by a lens, presently causes it to shrink and contract. gausapata. Broduicu. troglodytes. ASTILEACEA. BUNODIBjE. THE RINGED DEEPLET. Bolocera eques. (Sp. nov.) Plate IX. Fig. 6. Specific Character. Tentacles wholly retractile ; white, encircled with a red ring. GENERAL DESCRIPTION. Form. JBase. Adherent, scarcely exceeding the column. Column. Cylindrical ; very changeable in shape ; very distensible ; surface covered with numerous slightly indented, close-set, longitudinal striae ; studded, on the upper two-thirds, with numerous minute warts, increasing in number to the margin : these are either prominent or level, at the pleasure of the animal, and they have the power of attaching frag- ments of extraneous matter, which, however, seems rarely exercised. Substance lax and pulpy, with thin integuments. Margin forming a thick parapet, the summit obtusely edged, and notched with close-set denticulations, which are not warts, but are the terminations of the striae. IHgk. Flat, smooth, with very delicate and inconspicuous radii ; outline expansile beyond the column. Tentacles. Sub-marginal, set in six rows: 6,6,12,24,48,48^144; short, thick, conical, but versatile in form, in contraction being slender, in distension often ovate, or when this is partial, ovate with a slender point {mucro) ; constricted at foot, and in contraction marked with longitudinal mlci, both of which are very readily obliterated ; the tip perforate. They are subequal, about an inch and a half in length, and when distended, . upwards of one-third of an inch in diameter ; are flexuous, and thrown in various directions ; are strongly adhesive ; they are perfectly and readily retractile, but in a peculiar mode ; the margin contracts, till its edges meet over the tentacles, but it never involves itself. Mouth. Occasionally protruded in form of a wide cone. Two gonidial grooves, each with its pair of tubercles, and its broad, though faintly marked, radius. Lips thickened. Stomach-wall capable of being pro- truded in great bladder-like lobes. COLOUB. Column. A rich light orange-scarlet, rather duller towards the base ; the striae marked by slightly paler lines ; the warts white, each inclosed in 352 APPENDIX. a ring a little deeper than the general hue ; the region below the warts studded with much more minute and more crowded whitish specks. Dish. Pale buff or dx'ab, unspotted ; pellucid. Tentacles. Pellucid white ; a broad scarlet ring, bounded below by a narrower one of opaque white, surrounds the middle of each tentacle. Mouth. Lip as the disk. Gonidial tubercles white. Stomach-wall marked with alternate lines of pellucid and opaque white. Size. Height of column, when distended, four inches, diameter nearly the same ; expanse of flower about seven inches. Locality. North Sea : deep water. The acquisition of tlie magnificent animal above de- scribed, for Avhich I am indebted to the kindness of Mr. D. Ferguson, of Coutham, not only enables me to augment the genus Bolocera, and at the same time the British Fauna, •with another species, but also makes me better satisfied with the establishment of such a genus. Equal in dimen- sions to B. Tuedice, and presenting much in common with that species, there are peculiarities in this specimen which compel me to consider it specifically distinct. These are tlie brilliant hue of the column, its striate surface, the thinness of the integuments, the much feebler sulcation and constriction of the tentacles, and the rings of positive colour which adorn them, together with their power of complete retractation. All these characters make the pre- sent species a decidedly nearer approximation to Tealia. Indeed, when fully expanded, so remarkable is the resem- blance in form, size, and colour, to a fine T. crassicotmis, tliat I have little doubt the reason of its having been hitherto overlooked, is that it has been passed over as that familiar species. Yet the minute warts, the (really though slightly) constricted and furrowed tentacles, and the non-retractility of the margin, determine its place in this genus. The nobleness of its tout ensemble, and especially the 1 APPENDIX. 3SS rings on its many fingers, suggested to me a specific appel- lation, in allusion to old Rome's coxcomb chivalry, whose gold rings were no less characteristic than their valour. My friend informs me that the specimen was procured on the 17th of December, 1858, in twenty-eight fathoms' water, about ten miles east of the mouth of the Tees. The fisherman who obtained it (a carefal collector) had never seen one like it, though he had been very familiar with T. crassicomis, fix)m the circumstance of some hundreds of specimens having been sent to 3£r. Teale, fi-om Redcar, when that gentleman was engaged in his important re- searches into its anatomy. It lived upwards of three weeks with its first possessor, and after that a fortnight with me. The greater portion of this latter period it passed in a large tank, where it attached itself, expanded and dilated most gorgeously, presenting a grandeur of beauty which all who beheld it could scarce sufficiently admire. But for a few days before its death it loosed the hold of its base, and began to rupture the integuments, displaying the cras- peda. Then the Stomach-wall protruded, at first in a vesi- cular manner, and then by the inordinate recession of the lip, so that the plicate and corrugated stomach occupied the whole place of the disk. Then the tentacles lost their power of distension, and resumed their flaccid and con- tracted condition, when the longitudinal sulci became again conspicuous. And so the illustrious stranger died. I subsequently received another specimen from Banfi", in every respect like the former. It survived but ten days. TuedisB. EQUES. T. crassicomis. A A 354 APPENDIX. II. SPECIES DESCRIBED AS BRITISH, BUT WHICH I AM NOT ABLE TO APPORTION TO THEIR TRUE PLACE, FROM THE LACK OF PERSONAL ACQUAINTANCE WITH THEM. Alderi (Cocks). " Body cylindrical, hyaline, smooth ; numerous grass- green longitudinal striae ; tentacles twelve, short, obtuse, with a continu- ation of the green line on the posterior surface of each. Disk and mouth crimson, the latter marked with eight spots of same colour, but much darker; edge of disk entire; suctorials minute, numerous, imbedded." Deep water, off Falmouth. Pellucida (Cocks). " Body cylindrical, smooth, opalescent ; numerous white longitudinal grooves ; suctorials minute ; tentacles short, filiform, transparent, plain ; mouth small ; disk circular, flat, crossed by opaque white lines ; edge entire." Falmouth. Yan-ellii (Cocks). " Body conoid, hyaline, with twenty-four longitu- dinal semi-opaque white striae ; suctorials numerous, minute, imbedded. Three rows of tentacles, short, obtus© (rather clavate), spotted all over with white. The ovarian filaments, &c. distinctly seen through the trans- parent tunics." Falmouth. Bella (Cocks). "Body cylindrical, hyaline, spotted with yellow; twelve longitudinal opaque white striae ; mouth bright orange-red ; two yellow patches extending from the angle on each side to the base of the tentacles ; tentacles twenty, long, filiform, dotted anteriorly, and tipped, with yellow." Falmouth. .ff^ci^te^a (Wright). "Base adherent to rock; not exceeding column. Column , smooth ; height about equal to breadth (one inch). Disk hollow, hardly equalling diameter of column. Tentacles numerous ; in five or six rows, set close to margin ; nearly equal ; very conical and short ; thickly crowded. Mouth set on a cone ; lip tumid, furrowed. Column and disk sienna-brown, or salmon colour. Tentacles light brown, with two white bars across the base, tip slightly white or translucent. Lips orange or brick-red." Berehaven, Co. Cork, N.B. The above five species seem all referrible to that group of the genus Sagartia, which I have provisionally named Thoe. Intestmalis (Fabric). " Body cylindrical, the upper half suddenly con- tracted and narrow." — " When contracted, the body seems like two broad rings, of nearly equal bi'eadth, and about half an inch in diameter ; when expanded to nearly two inches, the body consists of two cylindrical por- tions of difiereut dimensions, smooth, pellucid, yellowish ; a few longi- tudinal white streaks ; disk not expanded ; tentacles about eighteen, filiform, in two rows." (Fleming.) Shetland. APPENDIX. 355 m. ADDENDA. Sagartia bellis. The Act. Johrutoni of Mr. Cocks is a variety of this species ; two specimens have come under my notice. miniata. A friend {E. W. H. H.) thinks that the Act. elegans of Dalyell is this specie (see supra, p. 100). If so, my name must give place to liis. omata. I have taken this at Torquay. It has been also found at Mizen Head, and sent me from Banff. The markings are true to the description, and leave no doubt of its distinctness as a species. pallida. Sent me in some numbers from Banff. A consider- able colony has also been found at Torquay. cocdnea. Abundant in deep water, Torbay. parasitica. Found, at Jersey, between tide-marks. Pkellia gausapata. 1 have since seen numerous specimens ; the species is quite distinct from P. murocincta. A very large specimen has been taken from deep water in Torbay. picta. Other specimens have been sent me from Banff. The epi- dermis is very thin and deciduous ; and altogether the species seems inter- mediate between the true Pkellice and such Sagartia as cocdnea. Adamsia palliata. Some interesting facts concerning this species and its connexion with the Hermit-crab will be found in a paper of mine, " On the Transfer of Adamsia palliata from Shell to Shell," published jn the Zoologist for June, 1S59. Sphenotrochus Macandrewanus. This has occurred more abundantly than the text seems to imply. Both Dr. Cocks and Mr. Alder inform me of having seen numerous specimens, chiefly from tbe Cornish coast ; and the latter has kindly presented me with two specimens. Wrightii. Dr. "Wright has sent me a fifth specimen from the same bank as the other four, differing considerably in form from all. LophoTielia prolifera. I have omitted to mention a fine British specimen, preserved in the Museum of Newcastle ; and another mentioned by Lands- borough, from Barra, one of the Hebrides. Balanophyllia regia. Two living specimens have been dredged in Ply- mouth Sound, by Mr. T. H. Stewart of the Roy. Coll. Surg. 356 APPENDIX. IV. GEOGKAPHICAL DISTRIBUTION. In the following attempt to distribute our Sea- Anemones geographicailj, I divide the whole British Coast into ten provinces, thus (somewhat arbitrarily) defined. 1. The Shetland, including the Orkneys, and Scotland as far aa Kinnaird's Head. 2. The North Sea, including the coast from Kinnaird's Head to Spurn Head. 3. The Eastern ; from the Humber to the Thames, a fiat low shore. 4. The South-east; firom the Foreland to St. Alban's Head; chiefly chalk cliSa. 5. The Devonian; from St. Alban's to St. David's Head; a rugged rocky coast. 6. The Irish Sea, to the Mull of Cantyre, including Man, and the Irish shore. 7. The Hebridean, from Cantyre to the Orkneys. 8. The Sonth Irish, from Cai-usore Point to Mizen Head. 9. The Atlantic, from Mizen Head to Eathlin Island. 10. The Channel Islands. A glance at the table will show that the Devonian dis- trict is by far the richest in species, including two-thirds of the whole. Next in fecundity to this extreme south comes the extreme north, numbering, however, less than two- thirds of the Devonian total. The Irish Sea, the Atlantic coast of Ireland, and the Channel Isles, each claims about two-thirds of the Shetland total. The province of the North Sea holds about two-thirds of this last number ; and then come in succession the South-east, the Eastern, and South Irish, and finally the Hebridean. These numbers represent, of course, the state of our knowledge rather than the fact. I look for additions in the Devonian province, and far more in the Shetland and Hebridean, of which last I know almost nothing. The Atlantic province will doubtless be farther enriched, and that of the Channel Isles. But I do not look for many species to be added to tlie North Sea ; and few if any to the Eastern and South-eastern provinces ; — mud and chalk being essentially ungenial to Sea-anemones. X Z -" -■ § 5 ? - III •a i i X o 2; s £ S H 6 1 X — X 5 4! jHwithnit . . • • • 1 ! . ciassicoTnis . . . . . . . . . bdHs. ... . . . . rtnbercnlata . . '■iniaU • Margaritae . . • naea. ... • • • ChnTchiae . . . • • •niata ...» • • ! spectabilis . • idtlhystoma . • Scotunu . . • Tennsta . . . . . . • • Mitchellii . . . ■irea • • • • hastata(Pe.) . • ^lyrodeta . . • • • • nndata . . . paOUda .... • triphylla . . • f«in ... • ? cylindrica. . • • ehiysantlielL . pdbieida . . • microps . . . Tarrellii . . • callimorpha . . R«llii. . . . • eainea . . . ka«tata(8ag.). r Beantempsii. coecinea. . . • • alhida . . . . troglodytes • Uoydii . . . • • vidiuU . . . • • • • • Teimieulatis . • panntka . . • • sangninea . . • duysosplen. . • angusta . . . intestinalis . . - heterocera . . • palliata . . . • . . . • • Tindis . . . • • • • mnrocincta. . • CoQchiKZo.) . • • • • gansapata . . • • tuleatns. . . Biodricii . . • Alderi (Zo.) . • pieta. . . . • Smithii . . . • • • • • • fenestiata . . • pteropns . . • CaiicIui(Aip.). • • Taadlianiu . . Tholeosis . . Macandieiraii. • cercus ... * mesembiy. . . • • • . . . • . . . . Tnedis . . . • • • • Wrightii . . • eqaet. . . . • • arcticus . . . • gemmacea . . • • • • • prolifera . . . • • thallla . . . • DoTotrix . . • Ballii . . . regia .... • amoiiata. . . • I intentineta . • digitata ... 1 ■ Total 75 30 u 7 9 51 20 6 7 21 22 358 APPENDIX. NAMES OF AUTHORITIES EXPRESSED BY INITIALS. A. B.C. Miss Church. A. M. M. Mrs. Murray Menzies. A.R. Mr. A. Rohertson. O.K. Rev. Charles Kingsley. a w. p. Mr. Chas. AV. Peach. D.B. Miss Barnie. B.F. Mr. D. Ferguson. JD.L. Rev. David Landsborough. D.R. Mr. David Robertson. E. C. H. Mr. E. C. Holwell. E. F. Professor Edward Forbes. E. L. W. Mr. E. L. Williams, Juu. E. P. W. Dr. E. Perceval Wright. E.W.H.H. Mr. E. W. H. Holdsworth. F. H. W. Mr. F. H. West. F. L. a Rev. F. L. Currie. F. N. B. Mr. F. N. Broderick. G.B. Mr. G. Barlee. 0. a H. Dr. G. C. Hyndmau. 6.D. Dr. G. Dansey. G. B. (B.) Professor Dickie. G. G. Mr. G. Gatehouse. G. G. (F.) Mr. G. Guyon. G. H. L. Mr. G. H. Lewes. G.J. Dr. George Johnston. G. J. A. Professor Allman. Q. T. Rev. George Tiigwell. H. H. D. Rev. H. H. Dombrain. U. 0. Mr. H. Owen. J. A. Mr. Joshua Alder. J. C. Dr. John Coldstream. J. C. G. Miss Gloag. /. D. H. (A misprint for T. D. H.) /. G. Rev. James Guillemard. /. G. D. Sir John G. Dalyell. /. M. Mr. James Macdonald. /. M. J. Mr. J. M. Jones. /. P. Mr. J. Price. /. R. G. Prof. J. Reay Greene. /. R. M. Mr. J. R. Mummery. J. T. Mr. John Templeton. J. T. H. Mr. James T. Hillier. M. E. G. Miss Guille. M. Y. Miss Vigurs. P. H. G. Mr. P. H. Gosse. R. B. Dr. Robert Ball. R. C. J. Prof. R. C. Jordan. R. H. Mr. R. Howse. R P. Mr. Robert Patterson. R Q. C. Mr. Richard Q. Couch. S. H. Mr. Sydney Hodges. S. W. Mr. S. Whitchurch, T. D. H. Dr. Thos. D. Hilton. T. S. W. Dr. T. Strethill Wright, W. A. L. Mr. Wm. Alford Lloyd. W. F. S. Rev, W, F, Short, W. G. Rev. Walter Gregor. W. H. Rev. Wm. Houghton. W. M'O. Mr, W, M'Calla, W. P. C. Mr. W. P. Cocks. , W. T. Mr, Wm. Thompson (Bel- fast). W. T. ( W.) Mr. Wm. Thompson (Weyi mouth). MAGN IFI ED. PLATE XII F.fl.0O$SE.DCL * DICKCS SC 1 PHELLIA PICTA. 2 ZOANTHUS SULCATUS 3 EDWARDSIA CARNEA ■1. CARYOPHYLLIA /Of «?/«£/'. 5. ZOANTHUS ALDERI. 6 HALCAMPA MICROPS 7. GRECORIA FENESTRATA 8. PHELLIA MUROCINCTA. INDEX. N.B. The names inclosed within bracketa are such as are not adopted in thi work. Acontia, xxii. Actinia, 174. ACTINIA D^, 171. ACTIXOLOBA, 11. AcTiNOPsis, 150, 170. AlPTASIA, 151. albida, 264. Alderi, 305. ? A Ideri, 354. [Allmanni], 289. [amachd], 152. ^ Amencanal, 338. Anemone, origin of the name of, 14. Anemone, Cave-dwelling, 88. eioak, 125. Daisy, 27. Eyed, 84. Fish-mouth, 57. Gold-spangled, 119. PaUid, 78. Parasitic, 112. Plumose, 12. Orange-disked, 60. Ornate, 54. Rosy, 48. Sandalled, 73. Scarlet-fringed, 41. Snake-locked, 105. Snowy, 66. Translucent, 82. Anemones, enemies of, 168. food of, 103, 164, 193, 272. voracity of, 215. {anguicomd], 105. AXTHEA. 159. Arachnactis, 263. arcticus, 330. ASTRiEACEA, 8. augusta, 283. [aua-anliaca], 12. AuKELIA^^A, 282. [^aurora], 88. Authorities, Names of, 358. Balanophtlua, 342, £allii, 198. Bantry Bay, riches of, 64. [Barleei], 297. Base, 1. [Beautempsil], 262. Bee, mistake of, 213. ? Bella, 354. bellis, 27. IbiTnaciUata'}, 209. [biserialis], 152. BOLOCERA, 185, 351. [borealis], 310. Brodricii, S49. BUNODE.S, 189. BUNODID^, 183. callimorpka, 2o5. [^Candida], 73. • Capxea, 279. Capstone Hill, 31, 74. camea, 259. Carpet-coral, 338. Cartophtllia, 309. CARYOPHYLLIACEA, 276. Cavity, 4. [cerasurri], 175. ccreus, 160. [Cereus], 205. Ceriakthus, 267. [chiococca], 175. Chrysoela, 123. chrysanlhelltim, 247. chrysosplenium, 119. Churchiw, 222. Cinclides, xxiii. [ciow/a], 198. 360 INDEX. Cnidse, xx. xx^-ii. Cnidae, chambered, zxTiiL tangled, xxx. spiral, xxxi. globate, xxxii. eoccinea, 84. Colour, change of, 180. Column, 2. Concealment, instinct of, 212. \corallin<i\, 175. [coriacea], 209. Corklet, Walled, 135. Warted, 140. Painted, 143. Latticed, 349. eoronata, 202. CORYNACTia, 288. Couchii, 152. Couchii, 297. Crab, Hermit, 115, 128. Craspeda, xxi. crassic<yrnis, 209. Crawling, mode of, 81, 164, 253. Creeplet, Sandy, 297. Furrowed, 303. Wrinkled, 305. [Cribbina], 205. Crisp-coral, Scarlet, 330. Crock, 280. Crookhaven, cavern of, 214. Cup-Coral, Devonshire, 310. Moray, 317. Shetland, 319. Winged, 321. [Ctathina], 309. [er/athivs], 310. ?, cylhidrica 245. Cyusta, 123. Deeplet, 186. Ringed, 351. diantkus, 12. digitata, 206. Disk, 3. Division, spontaneous, 19, 46, 66, 86,110,168,291. Durotrix, 338. Ecthorseum, xxix. \eduliii\, 160. Edwardsia, 254. [^cE(a], 112. Eggs, discharge of, 97, 100, 117, 225,314,347,223. [eUgans], 88. eqv/e^ 351. [e^ina]. 175. [explorator], 88. Eyelet, 146. lfdma\, 209. fenestraia, 146. \JUcdla\, 209. \FordcaUii\ 175. [/rogrcMJea], 175. Gapelet, 222. Gardens of Anemones, 51, 62, 64, 68, 71, 164, 214. gausapata, 140. gemmacea, 190. [gemmacea], 209. Geographical disfefibtition, 356. Germs, discharge of, 101, 132, 139, 238, 273. [glandtdosa], 190. Globehom, 289. [gramiTiea], 175. Gbegoria, 145. ? Oreenei, 216. Halcamipa, 246. hastata, 235. ? hastata, 354. [hemisphcerica^, 175. heferocera, 285. [Ifohatica], 209. hoplangia, 337. Hobmathia, 218. iekthygtoma, 57. ILTANTHIDjE, 227. Ilyanthus, 229. Imperial, Crimson, 283. Yellow, 285. ? itUerstincta, 348. ? irUestinaiis, 354. [judaica], 12. \laceratd], 105. xjife, tenacity of, 96, 118. Moydii, 268. lophohelia. 333. Macandrewanus, 325. [J/oc^wdreici], 330. [macvlata], 125. MargaritfE, 219. {margaritifera], 175. [memArona^MJ) 268. INDEX. 361 [mesembryanthernvni], 88. mesembryanthemtim, 175. microps, 252. miniata, 41. Mitcheim, 232. Morecambe Bay, 93. Mouth, 4. murocincta. 135. Muzzlet, Arrow, 235. Trefoil, 2. Wared, 239. Necklet, 219. nivea, 66. Inodosa], 219. [Oculina], 333. Odour, rank, 117. Opelet, 160. Organs, reproduction of, 251. OMnata, 54. \pmatd\, 41. palliata, 125. pallida, 78. [Palyihoa], 300. I>a^i7/o«o], 209. [/)ap?7Zo«a], 297. Paractathus, 316. jjarasitica, 112. [peduTiculatd], 27. ? pellucida, 354. [ue//MCM/a], 82. Peachia, 234. Pearlet, Scottish, 230. Scarlet. 232. [pentapetala], 12. Peribola, xxxiv. Petit Tor, 31, 68, 136, 260. Pheixia, 134, 349. [PhtllaxgiaJ, 337. Pimplet, Gem, 190. Glaucous, 195. Red-specked, 198. Pintlet, Sand, 247. Rock, 252. picta, 143. [plumosa], 12. Plumose Anemone, 12. POCILLOPORA, 348. Poisoning power, sxxyL proUfera, 334. pteropus, 321. Pterygia, xxx. Pufflet, painted, 255. crimson, 259. [pulchem^ma}, 48. ,' pura, 82. [purpurea], 176. reffia, 343. rosea, 48. [rufal 175. Sagabtia, 25. subdivision of, 121. tanguinea, 280. [SCOLASTHUS], 254. Scoticus, 230. SCTPHIA, 123. Screw, zxix. [senilis], 12. [senilis], 209. Septa, xi. [sessilis], 310. [Sidisia], 300. [Siphonactixia], 236. Bmithii, 310. Species, what ? 50. ? spectabilis, 226. Spermatozoa, 99, 225. [spkceroidesl, 88. Sphesotrochus, 323. Spherules, 180. sphyrodeta, 73. Sprawlet, 264. Star-coral, Scarlet and Gold, 343. Stinging power, 136. Stomach, proti-usion of, 32. Stomphia, 221. Strawberry, 177. Strebla. xxix. [sulcata], 160. sulcatus, 303. Swimming, mode of, 165, 265. System, tegumentary, x. muscular, x. nei-vous and sensory, xii. digestive, xiiL circulatory, xvL respiratory, xvi. reproductive, xis. teliferous, xi. [labeUa], 175. Taxilianus, 317. Tkalia, 205. [Templetonii], 27. Tenby, Caves of, 61, 70, 92. Tentacles, branching of, 109, 168. B B 362 JXDEX. Tentacles, 3. elongation of, IG, 34, 44, 70,101. Terms, explanation of, 1. [Thalia], 195, thalUa, 1 95. Thoe, 122. Tkulensis, 319. Tide-pools, 31, 62, 68, 162, 344. Torquay, rocks at, 44. friphylla, 243. troglodytes, 88. Trumplet, 152. ? tiiherculata, 217. Tuedtce, 186. Tuft-coral, 334. [Turbinolia], 323. Uloctathus, 329. [undaia'\, 105. undata, 239. venusta, 60. ? vermicidaris, 274. •yej-rwcosa], 190. yestita^, 169. Vestlet, 268. [rn^MCfto], 88. viduata, 105. [vmosa], 48. viridis, 289. Wartlet, Dahlia, 209. Marigold, 206. Watcombe, 32. Wedge-coral, Smooth- ribbed, 325, Knotted, 326. "Woolhouse Rocks, 43, 51, 61. Wrightii, 326. ? Tarrellii, 354. Young, birth of, 36, 46, 71, SO, 99, 118, 193. ZOANTHID^, 295. ZOANTHUS, 296. ERRATA. Page 10, line 4 ■) Add the qualifying phrase "in general," to Page 11, line 20 \ the character that there is but a single Page 12, sec( nd line fioni bottom .) mouth-angle and pair of tubercles. Page 13, line 10 for "Always," read " generally." Page 90, line a ^^^,^^^^lou:cst part of ,^c). ier.i^c\. t^.\X 11. CI.Ay, I'lllSTEB, BUEAD STEEET HILL. CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY QL Gosse, Philip Henry 376 Actinologia Britannica .4 G7G6	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6029634475708008, "perplexity": 14035.255559964151}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721067.83/warc/CC-MAIN-20161020183841-00044-ip-10-171-6-4.ec2.internal.warc.gz"}
https://arxiv.org/abs/0904.3670	hep-th (what is this?) # Title: Topological Black Holes in Horava-Lifshitz Gravity Abstract: We find topological (charged) black holes whose horizon has an arbitrary constant scalar curvature $2k$ in Ho\v{r}ava-Lifshitz theory. Without loss of generality, one may take $k=1,0$ and -1. The black hole solution is asymptotically AdS with a nonstandard asymptotic behavior. Using the Hamiltonian approach, we define a finite mass associated with the solution. We discuss the thermodynamics of the topological black holes and find that the black hole entropy has a logarithmic term in addition to an area term. We find a duality in Hawking temperature between topological black holes in Ho\v{r}ava-Lifshitz theory and Einstein's general relativity: the temperature behaviors of black holes with $k=1, 0$ and -1 in Ho\v{r}ava-Lifshitz theory are respectively dual to those of topological black holes with $k=-1, 0$ and 1 in Einstein's general relativity. The topological black holes in Ho\v{r}ava-Lifshitz theory are thermodynamically stable. Comments: Latex, 14 pages; v2, v3: typos corrected Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc) Journal reference: Phys.Rev.D80:024003,2009 DOI: 10.1103/PhysRevD.80.024003 Report number: CAS-KITPC/ITP-106,KU-TP 031 Cite as: arXiv:0904.3670 [hep-th] (or arXiv:0904.3670v3 [hep-th] for this version) ## Submission history From: Rong-Gen Cai [view email] [v1] Thu, 23 Apr 2009 12:59:17 GMT (12kb) [v2] Sat, 25 Apr 2009 08:06:13 GMT (13kb) [v3] Thu, 30 Apr 2009 03:27:24 GMT (13kb)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25573796033859253, "perplexity": 2705.8394812300476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549429485.0/warc/CC-MAIN-20170727202516-20170727222516-00379.warc.gz"}
https://sciencing.com/isothermal-processes-definition-formula-examples-13722767.html	# Isothermal Processes: Definition, Formula & Examples Print Understanding what different thermodynamic processes are and how you use the first law of thermodynamics with each one is crucial when you start to consider heat engines and Carnot cycles. Many of the processes are idealized, so while they don’t accurately reflect how things occur in the real world, they’re useful approximations that simplify calculations and make it easier to draw conclusions. These idealized processes describe how the states of an ideal gas can undergo change. The isothermal process is just one example, and the fact that it occurs at a single temperature by definition drastically simplifies working with the first law of thermodynamics when you’re calculating things like heat-engine processes. ## What Is an Isothermal Process? An isothermal process is a thermodynamic process that occurs at a constant temperature. The benefit of working at a constant temperature and with an ideal gas is that you can use Boyle’s law and the ideal gas law to relate pressure and volume. Both of these expressions (as Boyle’s law is one of the several laws that were incorporated into the ideal gas law) show an inverse relationship between pressure and volume. Boyle’s law implies that: P_1V_1 = P_2V_2 Where the subscripts denote the pressure (P) and volume (V) at time 1 and the pressure and volume at time 2. The equation shows that if the volume doubles, for instance, the pressure has to reduce by half in order to keep the equation balanced, and vice versa. The full ideal gas law is PV=nRT where n is the number of moles of the gas, R is the universal gas constant and T is the temperature. With a fixed amount of gas and a fixed temperature, PV must take a constant value, which leads to the previous result. On a pressure-volume (PV) diagram, which is a plot of pressure vs. volume often used for thermodynamic processes, an isothermal process looks like the graph of y = 1/x, curving downwards towards its minimum value. One point that often confuses people is the distinction between isothermal vs. adiabatic, but breaking down the word into its two parts can help you remember this. “Iso” means equal and “thermal” refers to something’s heat (i.e., its temperature), so “isothermal” literally means “at an equal temperature.” Adiabatic processes don’t involve heat transfer, but the temperature of the system often changes during them. ## Isothermal Processes and the First Law of Thermodynamics The first law of thermodynamics states that the change in internal energy (∆U) for a system is equal to the heat added to the system (Q) minus the work done by the system (W), or in symbols: ∆U= Q - W When you’re dealing with an isothermal process, you can use the fact that internal energy is directly proportional to temperature alongside this law to draw a useful conclusion. The internal energy of an ideal gas is: U = \frac{3}{2} nRT This means that for a constant temperature, you have a constant internal energy. So with ∆U= 0, the first law of thermodynamics can easily be re-arranged to: Q=W Or, in words, the heat added to the system is equal to the work done by the system, meaning that the heat added is used to do the work. For example, in isothermal expansion, heat is added to the system, which causes it to expand, doing work on the environment without losing internal energy. In an isothermal compression, the environment does work on the system, and causes the system to lose this energy as heat. ## Isothermal Processes in Heat Engines Heat engines use a complete cycle of thermodynamic processes to convert heat energy into mechanical energy, usually by moving a piston as the gas in the heat engine expands. Isothermal processes are a key part of this cycle, with the added heat energy being completely converted into work without any loss. However, this is a highly idealized process, because in practice there will always be some energy lost when the heat energy is converted into work. For it to work in reality, it would need to take an infinite amount of time so that the system could remain in thermal equilibrium with its surroundings at all times. Isothermal processes are considered reversible processes, because if you’ve completed a process (for example, an isothermal expansion) you could run the same process in reverse (an isothermal compression) and return the system to its original state. In essence, you can run the same process forwards or backwards in time without breaking any laws of physics. However, if you attempted this in real life, the second law of thermodynamics would mean there was an increase in entropy during the “forwards” process, so the “backwards” one wouldn’t completely return the system to its original state. If you plot an isothermal process on a PV diagram, the work done during the process is equal to the area under the curve. While you can calculate the work done isothermally in this way, it’s often easier to just use the first law of thermodynamics and the fact that the work done is equal to the heat added to the system. ## Other Expressions for Work Done in Isothermal Processes If you’re doing calculations for an isothermal process, there are a couple of other equations you can use to find the work done. The first of these is: W = nRT \ln \bigg(\frac{V_f}{V_i}\bigg) Where Vf is the final volume and Vi is the initial volume. Using the ideal gas law, you can substitute the initial pressure and volume (Pi and Vi) for the nRT in this equation to get: W = P_iV_i \ln \bigg(\frac{V_f}{V_i}\bigg) It may be easier in most cases to the work through the heat added, but if you only have information about the pressure, volume or temperature, one of these equations could simplify the problem. Since work is a form of energy, its unit is the joule (J). ## Other Thermodynamic Processes There are many other thermodynamic processes, and many of these can be classified in a similar way to isothermal processes, except that quantities other than temperature are constant throughout. An isobaric process is one that occurs at a constant pressure, and because of this, the force exerted on the walls of the container is constant, and the work done is given by W = P∆V. For gas undergoing isobaric expansion, there needs to be heat transfer in order to keep the pressure constant, and this heat changes the internal energy of the system as well as doing work. An isochoric process takes place at a constant volume. This allows you to make a simplification in the first law of thermodynamics, because if the volume is constant, the system can’t do work on the environment. As a result, the change in internal energy of the system is entirely due to the heat transferred. An adiabatic process is one that occurs without heat exchange between the system and the environment. This doesn’t mean that there is no change in temperature in the system, though, because the process could lead to an increase or a decrease in temperature without direct heat transfer. However, with no heat transfer, the first law shows that any change in internal energy must be due to work done on the system or by the system, since it sets Q = 0 in the equation.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8958890438079834, "perplexity": 238.04837828890305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703496947.2/warc/CC-MAIN-20210115194851-20210115224851-00339.warc.gz"}
https://www.physicsforums.com/threads/is-this-separable.413520/	# Homework Help: Is this separable 1. Jul 1, 2010 ### seand How do you find a solution for: 2tv' - v = 0 The text says it's separable but I'm not seeing it. I'm just learning so extra details are appreciated. Thanks. (this should have been posted in the homework section - but I can't seem to move it there, sorry) Last edited: Jul 1, 2010 2. Jul 1, 2010 ### Staff: Mentor 2t dv/dt = v ==> 2t dv = v dt Can you continue? 3. Jul 1, 2010 ### seand well I think I want to switch things around so v and t are separated 1/v dv = 1/2t dt integrating both sides ln(v) = 1/2 ln(t)+C e^x both sides: v = k*sqrt(t) Which looks right! Is that how I was meant to do it? If so, thanks - I got stalled before when I got the ln() on both sides. 4. Jul 1, 2010 ### Staff: Mentor Looking right might not be good enough. You can check by taking the derivative and verifying that tv' - v = 0. 5. Jul 3, 2010 ### Unit Verifying that 2tv' - v = 0	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.880341649055481, "perplexity": 3738.5917183042134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159744.52/warc/CC-MAIN-20180923193039-20180923213439-00465.warc.gz"}
https://mathoverflow.net/questions/180704/the-space-of-varieties-between-two-given-varieties	# The space of varieties between two given varieties Let $\mathbf{P} = \mathbf{P}^n(k)$ be the $n$-dimensional projective space over a field $k$, let $A, B$ be projective varieties in $\mathbf{P}$ such that $A \subset B$. Now define $V(A,B)$ to be the set of all projective varieties $C$ such that $A \subseteq C \subseteq B$. What can be said about the structure of $V(A,B)$ ? What about the same question where we allow "quasi-projective'' instead of ''projective" ? Can one see $V(A,B)$ as a variety (and what is its dimension) ? This is more reasonable if you insist that $C$ have a given Hilbert polynomial. Otherwise, consider the case $A = \emptyset$, $B = \bf P = \bf P^2$. Then you have curves of every degree, so your $V(A,B)$ will be of infinite type. With $C$'s Hilbert polynomial fixed, the natural thing to look at is the subschemes $C$ between $A$ and $B$. This defines a closed subscheme $X$ of the Hilbert scheme of $\bf P$ with this chosen Hilbert polynomial, so in particular is projective. (I get the impression from your question that you need to learn about Hilbert schemes, from e.g. Eisenbud and Harris.) If you only want varieties, then you're thinking about an open subscheme $V(A,B)$ of that $X$. Given how badly behaved Hilbert schemes are, I doubt it $V(A,B)$ will be irreducible, and I suspect figuring out the dimensions of its components will be quite intractable in general. • Hey - thanks for the nice answer ! Supposing that one considers quasi-projective varieties $A, B, C$ (so that, as you state, it is possible to obtain a thing of infinite type), is there a formal statement somewhere that $V(A,B)$ iself is always a variety, or a scheme ?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.977124035358429, "perplexity": 107.33294280664738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573029.81/warc/CC-MAIN-20220817153027-20220817183027-00395.warc.gz"}
http://mathhelpforum.com/advanced-statistics/171603-crossover-design.html	## Crossover design The compund symmetry model can be written: $Y_{ij}=\beta_0+\beta_1time_{ij}+\beta_2trt_{ij}+\b eta_3CO_{ij}+b_i+w_{ij}$ Say after inputting the SAS code with PROC mixed, I have the following: Row Col1 Col2 1 12.5250 8.4750 2 8.4750 12.5250 So which is $var(b_i), var(w_{ij})$?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6609976887702942, "perplexity": 22317.629901037988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128321497.77/warc/CC-MAIN-20170627170831-20170627190831-00585.warc.gz"}
https://mathematica.stackexchange.com/questions/208315/how-to-solve-transient-3d-heat-equation-with-robin-boundary-conditions	# How to solve transient 3D heat equation with robin boundary conditions Good afternoon! I'm trying to solve the following heat equation: with the following boundary conditions and initial value: Nut I'm getting error while solving it with NDSolve: s = NDSolve[{(1.0/alpha) D[T[x, y, z, t], t] == D[T[x, y, z, t], {x, 2}] + D[T[x, y, z, t], {y, 2}] + D[T[x, y, z, t], {z, 2}],T[x, y, z, 0] == T0, -lambda D[T[x, y, 0, t], z] ==Piecewise[{{qmax Exp[-c ((x - xs0 - vx t)^2 + y^2)/r0^2], (x - xs0 - vx t)^2 + y^2 <=r0^2}, {-h (T[x, y, 0, t] - Tinf), (x - xs0 - vx t)^2 + y^2 >r0^2}}],lambda D[T[0, y, z, t], x] == h (T[0, y, z, t] - Tinf), -lambda D[T[L, y, z, t], x] == h (T[L, y, z, t] - Tinf), D[T[x, 0, z, t], y] == 0, -lambda D[T[x, W/2, z, t], y] == h (T[x, W/2, z, t] - Tinf), -lambda D[T[x, y, H, t], z] == h (T[x, y, H, t] - Tinf)}, T, {x, 0, L}, {y, 0, W}, {z, 0, H}, {t, 0, tf}] Could someone help? Full code here: https://pastebin.com/U0qhNSeJ • You need to define all parameters tf, alpha,lamda,... before NDSolve is applied. – Ulrich Neumann Oct 22 '19 at 14:20 • You might increase the chance to get a helpful answer if you provide complete executable code! Together with some information about the errors. – Ulrich Neumann Oct 22 '19 at 14:37 • Full code here: pastebin.com/U0qhNSeJ – João Vitor Oct 22 '19 at 14:48 We must change everything to meters. I updated the code and added a step function f[x], changed the boundary condition for y = 0 to Automatic (=NeumannValue[0, y==0]). Now all the pictures are in real time (sec) and in meters. Needs["NDSolveFEM"] L = 1000(scale); (Plate dimensions)Ls = 250/L; W = 200/L; H = 30/L; reg = Cuboid[{0, 0, 0}, {Ls, W, H}]; mesh = ToElementMesh[reg, MaxCellMeasure -> .0000005]; mesh["Wireframe"] (Material properties) rhocp = 4898556; lambda = 36; alpha = 1; ts = (lambda/rhocp)^(-1)(time scale); T0 = 293(Initial temperature); Tinf = 293(Ambient temperature); Q \ = 4256(Source power); xs0 = 6/L(Initial position of moving source); xsf = (Ls - xs0)(final position of moving source); r0 = 3/L(radius of source); c = 1(source constant parameter); vx = ts 1.61/L(moving source velocity x direction. Velocity ); qmax = c Q/(Pi r0^2)/lambda(source term); h = 10/lambda(heat convection coeffcient); tf = (Ls - 2 xs0)/ vx(Final time of calculation);(eq={(1.0/alpha) D[T[x,y,z,t],t]\ \[Equal]D[T[x,y,z,t],{x,2}]+D[T[x,y,z,t],{y,2}]+D[T[x,y,z,t],{z,2}],,-\ lambda D[T[x,y,0,t],z]\[Equal]Piecewise[{{qmax Exp[-c ((x-xs0-vx \ t)^2+y^2)/r0^2],(x-xs0-vx t)^2+y^2\[LessEqual]r0^2},{-h \ (T[x,y,0,t]-Tinf),(x-xs0-vx t)^2+y^2>r0^2}}],lambda D[T[0,y,z,t],x]\ \[Equal]h (T[0,y,z,t]-Tinf),-lambda D[T[L,y,z,t],x]\[Equal]h \ (T[L,y,z,t]-Tinf),D[T[x,0,z,t],y]\[Equal]0,-lambda D[T[x,W/2,z,t],y]\ \[Equal]h (T[x,W/2,z,t]-Tinf),-lambda D[T[x,y,H,t],z]\[Equal]h \ (T[x,y,H,t]-Tinf)}) f[x_] := (1 + Tanh[10000 x])/2 eq = D[T[x, y, z, t], t] - (D[T[x, y, z, t], {x, 2}] + D[T[x, y, z, t], {y, 2}] + D[T[x, y, z, t], {z, 2}]); ic = T[x, y, z, 0] == T0; bc = NeumannValue[ qmax Exp[-c ((x - xs0 - vx t)^2 + y^2)/r0^2] f[ r0^2 - (x - xs0 - vx t)^2 - y^2] - h (T[x, y, z, t] - Tinf) f[((x - xs0 - vx t)^2 + y^2) - r0^2], z == 0] + NeumannValue[-h (T[x, y, z, t] - Tinf), x == 0 \|\| x == Ls \|\| y == W \|\| z == H]; sol = NDSolve[{eq == bc, ic}, T, {t, 0, tf}, {x, y, z} \[Element] mesh] Table[DensityPlot[ Evaluate[T[x, y, 0, t] /. sol], {x, 0, Ls}, {y, 0, W}, PlotLegends -> Automatic, ColorFunction -> "Rainbow", FrameLabel -> Automatic, PlotLabel -> Row[{"t =", t ts}], PlotRange -> All], {t, .2 tf, tf, .2 tf}] Table[DensityPlot[ Evaluate[T[x, 0, z, t] /. sol], {x, 0, Ls}, {z, 0, H}, PlotLegends -> Automatic, PlotRange -> All, ColorFunction -> "Rainbow", FrameLabel -> Automatic], {t, .2 tf, tf, .2 tf}] We have nice pictures at z=H Table[DensityPlot[ Evaluate[T[x, y, H, t] /. sol], {x, 0, Ls}, {y, 0, W}, PlotLegends -> Automatic, ColorFunction -> "Rainbow", FrameLabel -> Automatic, PlotLabel -> Row[{"t =", t ts}], PlotRange -> All], {t, .2 tf, tf, .2 tf}] • Dear Alex, thank you very much for the solution! I'm new with Mathematica and I think I haven't understood all the steps you did. For example, on eq there is only one minus sign on the x derivative, should it also be negative for the y and z derivatives? I don't understand the boundary conditions... does one NeumannValue[h (T[x, y, z, t] - Tinf), True] accounts the convection on all surfaces? For the heat source term I couldn't follow how you did. Could you please help me? Thank you! – João Vitor Oct 23 '19 at 10:44 • 1. eq has a form D[T[x, y, z, t]-(…), and bracket () has the same meaning as in standard calculus. 2. I usually use True in such problems. But you can replace it with bc=NeumannValue[…,z==0]+NeumannValue[-h (T[x, y, z, t] - Tinf), x==0\|\|y==0\|\|x==Ls\|\|y==W\|\|z==H] and compare. 3. There is a thin source of radius r0 and a mesh with a mesh size of r0. Effectively, the source heats one cell. On the next cell we have Exp[-8]=0.000335463, there the cooling is turned on as (1-Exp[-8]). – Alex Trounev Oct 23 '19 at 11:48 • Is it possible to use the Piecewise[...] function to define the moving source as I did or NDSolve will raise error with it? Another problem is the symmetry bc at y=0, since the moving source is located at the center, i.e. at position W/2, for this case is it possible to also set .NeumannValue[0, y==0]? This is a Welding simulation and I'm also trying to solve it with finite difference method using ADI Douglas-Gunn method + TDMA. – João Vitor Oct 23 '19 at 14:49 • @JoãoVitor If this is welding, then explain what material it is and in what units the parameters are expressed? – Alex Trounev Oct 23 '19 at 16:23 • L, W, H, xs0, xsf and r0 in mm. tf in seconds. vx in mm/s. T0 and Tinf in °C. qmax in W. Q in J. c is dimensionless. lambda in W/m.K h in W/m².K rhocp in J/m³.K – João Vitor Oct 23 '19 at 17:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3683018982410431, "perplexity": 8748.172848016431}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371858664.82/warc/CC-MAIN-20200409122719-20200409153219-00400.warc.gz"}
https://www.wikiplanet.click/enciclopedia/tab/en/Electrical_resistance_and_conductance/740d26f7a67cb83e38ad190d1ce73358dabe54b6	# Electrical resistance and conductance \| dependence of resistance on other conditions ## Dependence of resistance on other conditions ### Temperature dependence Near room temperature, the resistivity of metals typically increases as temperature is increased, while the resistivity of semiconductors typically decreases as temperature is increased. The resistivity of insulators and electrolytes may increase or decrease depending on the system. For the detailed behavior and explanation, see Electrical resistivity and conductivity. As a consequence, the resistance of wires, resistors, and other components often change with temperature. This effect may be undesired, causing an electronic circuit to malfunction at extreme temperatures. In some cases, however, the effect is put to good use. When temperature-dependent resistance of a component is used purposefully, the component is called a resistance thermometer or thermistor. (A resistance thermometer is made of metal, usually platinum, while a thermistor is made of ceramic or polymer.) Resistance thermometers and thermistors are generally used in two ways. First, they can be used as thermometers: By measuring the resistance, the temperature of the environment can be inferred. Second, they can be used in conjunction with Joule heating (also called self-heating): If a large current is running through the resistor, the resistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in a circuit-protection role similar to fuses, or for feedback in circuits, or for many other purposes. In general, self-heating can turn a resistor into a nonlinear and hysteretic circuit element. For more details see Thermistor#Self-heating effects. If the temperature T does not vary too much, a linear approximation is typically used: ${\displaystyle R(T)=R_{0}[1+\alpha (T-T_{0})]}$ where ${\displaystyle \alpha }$ is called the temperature coefficient of resistance, ${\displaystyle T_{0}}$ is a fixed reference temperature (usually room temperature), and ${\displaystyle R_{0}}$ is the resistance at temperature ${\displaystyle T_{0}}$. The parameter ${\displaystyle \alpha }$ is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, ${\displaystyle \alpha }$ is different for different reference temperatures. For this reason it is usual to specify the temperature that ${\displaystyle \alpha }$ was measured at with a suffix, such as ${\displaystyle \alpha _{15}}$, and the relationship only holds in a range of temperatures around the reference.[9] The temperature coefficient ${\displaystyle \alpha }$ is typically +3×10−3 K−1 to +6×10−3 K−1 for metals near room temperature. It is usually negative for semiconductors and insulators, with highly variable magnitude.[10] ### Strain dependence Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon strain. By placing a conductor under tension (a form of stress that leads to strain in the form of stretching of the conductor), the length of the section of conductor under tension increases and its cross-sectional area decreases. Both these effects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on strain gauges for details about devices constructed to take advantage of this effect. ### Light illumination dependence Some resistors, particularly those made from semiconductors, exhibit photoconductivity, meaning that their resistance changes when light is shining on them. Therefore, they are called photoresistors (or light dependent resistors). These are a common type of light detector. Other Languages azərbaycanca: Elektrik müqaviməti বাংলা: রোধ Bân-lâm-gú: Tiān-chó͘ беларуская (тарашкевіца)‎: Супор chiShona: Mukweso dansk: Resistans eesti: Takistus 한국어: 전기저항 Bahasa Indonesia: Hambatan listrik íslenska: Rafmótstaða Kiswahili: Ukinzani Kreyòl ayisyen: Rezistans (kouran) македонски: Електричен отпор Bahasa Melayu: Rintangan elektrik norsk nynorsk: Elektrisk motstand oʻzbekcha/ўзбекча: Elektr qarshilik Plattdüütsch: Elektrisch Wedderstand polski: Rezystancja Seeltersk: Wierstand Simple English: Electrical resistance slovenčina: Elektrický odpor slovenščina: Električni upor srpskohrvatski / српскохрватски: Električni otpor svenska: Resistans Tagalog: Resistensiya தமிழ்: மின்தடை татарча/tatarça: Электр каршылыгы українська: Електричний опір ئۇيغۇرچە / Uyghurche: قارشىلىق Tiếng Việt: Điện trở	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9365173578262329, "perplexity": 2212.8788576456263}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201812.2/warc/CC-MAIN-20190318232014-20190319014014-00195.warc.gz"}
https://cris.bgu.ac.il/en/publications/on-r-simple-k-path-and-related-problems-parameterized-by-kr	On R-simple k-path and related problems parameterized by k/r Gregory Gutin, Magnus Wahlström, Meirav Zehavi Research output: Contribution to conferencePaperpeer-review Abstract Abasi et al. (2014) introduced the following two problems. In the r-Simple k-Path problem, given a digraph G on n vertices and positive integers r, k, decide whether G has an r-simple k-path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k. In the (r, k)Monomial Detection problem, given an arithmetic circuit that succinctly encodes some polynomial P on n variables and positive integers k, r, decide whether P has a monomial of total degree k where the degree of each variable is at most r. Abasi et al. obtained randomized algorithms of running time 4(k/r) log r·nO(1) for both problems. Gabizon et al. (2015) designed deterministic 2O((k/r) log r) · nO(1)-time algorithms for both problems (however, for the (r, k)-Monomial Detection problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. In the p-Set (r, q)-Packing problem, given a universe V , positive integers p, q, r, and a collection H of sets of size p whose elements belong to V , decide whether there exists a subcollection H0 of H of size q where each element occurs in at most r sets of H0. Gabizon et al. obtained a deterministic 2O((pq/r) log r) ·nO(1)time algorithm for p-Set (r, q)-Packing. The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by the product of two parameters, that is, k/r and log r, where k = pq for p-Set (r, q)-Packing. Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (r, k)-Monomial Detection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2o((k/r) log r) · (n + log k)O(1)-time algorithm for (r, k)-Monomial Detection, i.e. (r, k)-Monomial Detection is highly unlikely to be single-exponentially FPT when parameterized by k/r alone. The question remains open for r-Simple k-Path and p-Set (r, q)Packing. We consider the question from a wider perspective: are the above problems FPT when parameterized by k/r only, i.e. whether there exists a computable function f such that the problems admit a f(k/r)(n + log k)O(1)time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by k/r alone. We resolve the wider question by (a) obtaining a 2O((k/r)2 log(k/r)) · (n + log k)O(1)-time algorithm for r-Simple k-Path on digraphs and a 2O(k/r)·(n+log k)O(1)-time algorithm for r-Simple k-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that p-Set (r, q)-Packing is FPT (in contrast, we prove that p-Multiset (r, q)-Packing is W[1]-hard), and (c) proving that (r, k)-Monomial Detection is para-NP-hard even if only two distinct variables are in polynomial P and the circuit is non-canceling. For the special case of (r, k)-Monomial Detection where k is polynomially bounded by the input size (which is in XP), we show W[1]-hardness. Along the way to solve p-Set (r, q)-Packing, we obtain a polynomial kernel for any fixed p, which resolves a question posed by Gabizon et al. regarding the existence of polynomial kernels for problems with relaxed disjointness constraints. All our algorithms are deterministic. Original language English 1750-1769 20 https://doi.org/10.1137/1.9781611975482.105 Published - 1 Jan 2019 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United StatesDuration: 6 Jan 2019 → 9 Jan 2019 Conference Conference 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 United States San Diego 6/01/19 → 9/01/19 ASJC Scopus subject areas • Software • Mathematics (all) Fingerprint Dive into the research topics of 'On R-simple k-path and related problems parameterized by k/r'. Together they form a unique fingerprint.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8763522505760193, "perplexity": 2171.7828847771093}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104293758.72/warc/CC-MAIN-20220704015700-20220704045700-00311.warc.gz"}
http://gmatclub.com/forum/until-the-1980s-most-scientists-76327.html	Find all School-related info fast with the new School-Specific MBA Forum It is currently 13 Feb 2016, 08:09 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # =================== Until the 1980s, most scientists Author Message TAGS: Director Joined: 04 Jan 2008 Posts: 914 Followers: 59 Kudos [?]: 368 [0], given: 17 =================== Until the 1980s, most scientists [#permalink] 06 Mar 2009, 10:46 =================== Until the 1980s, most scientists believed that noncatastrophic geological processes caused the extinction of dinosaurs that occurred approximately 66 million years ago, at the end of the Cretaceous period. Geologists argued that a dramatic drop in sea level coincided with the extinction of the dinosaurs and could have caused the climatic changes that resulted in this extinction as well as the extinction of many ocean species. This view was seriously challenged in the 1980s by the discovery of large amounts of iridium in a layer of clay deposited at the end of the Cretaceous period. Because iridium is extremely rare in rocks on the Earth’s surface but common in meteorites, researchers theorized that it was the impact of a large meteorite that dramatically changed the earth’s climate and thus triggered the extinction of the dinosaurs. Currently available evidence, however, offers more support for a new theory, the volcanic-eruption theory. A vast eruption of lava in India coincided with the extinctions that occurred at the end of the Cretaceous period, and the release of carbon dioxide from this episode of volcanism could have caused the climatic change responsible for the demise of the dinosaurs. Such outpourings of lava are caused by instability in the lowest layer of the Earth’s mantle, located just above the Earth’s core. As the rock that constitutes this layer is heated by the Earth’s core, it becomes less dense and portions of it eventually escape upward as blobs or molten rock, called “diapirs,” that can, under certain circumstances, erupt violently through the Earth’s crust. Moreover, the volcanic-eruption theory, like the impact theory, accounts for the presence of iridium in sedimentary deposits; it also explains matters that the meteorite-impact theory does not. Although iridium is extremely rare on the Earth’s surface, the lower regions of the Earth’s mantle have roughly the same composition as meteorites and contain large amounts of iridium, which in the case of a diapir eruption would probably be emitted as iridium hexafluoride, a gas that would disperse more uniformly in the atmosphere than the iridium-containing matter thrown out from a meteorite impact. In addition, the volcanic-eruption theory may explain why the end of the Cretaceous period was marked by a gradual change in sea level. Fossil records indicate that for several hundred thousand years prior to the relatively sudden disappearance of the dinosaurs, the level of the sea gradually fell, causing many marine organisms to die out. This change in sea level might well have been the result of a distortion in the Earth’s surface that resulted from the movement of diapirs upward toward the Earth’s crust, and the more cataclysmic extinction of the dinosaurs could have resulted from the explosive volcanism that occurred as material from the diapirs erupted onto the Earth’s surface. 1. The passage suggests that during the 1980s researchers found meteorite impact a convincing explanation for the extinction of dinosaurs, in part because (A) earlier theories had failed to account for the gradual extinction of many ocean species at the end of the Cretaceous period (B) geologists had, up until that time, underestimated the amount of carbon dioxide that would be released during an episode of explosive volcanism (C) a meteorite could have served as a source of the iridium found in a layer of clay deposited at the end of the Cretaceous period (D) no theory relying on purely geological processes had, up until that time, explained the cause of the precipitous drop in sea level that occurred at the end of the Cretaceous period (E) the impact of a large meteorite could have resulted in the release of enough carbon dioxide to cause global climatic change 2. According to the passage, the lower regions of the Earth’s mantle are characterized by (A) a composition similar to that of meteorites (B) the absence of elements found in rocks on the Earth’s crust (C) a greater stability than that of the upper regions (D) the presence of large amounts of carbon dioxide (E) a uniformly lower density than that of the upper regions 3. It can be inferred from the passage that which one of the following was true of the lava that erupted in India at the end of the Cretaceous period? (A) It contained less carbon dioxide than did the meteorites that were striking the Earth’s surface during that period. (B) It was more dense than the molten rock, located just above the Earth’s core. (C) It released enough iridium hexafluoride into the atmosphere to change the Earth’s climate dramatically. (D) It was richer in iridium than rocks usually found on the Earth’s surface. (E) It was richer in iridium than were the meteorites that were striking the Earth’s surface during that period. 4. In the passage, the author is primarily concerned with doing which one of the following? (A) describing three theories and explaining why the latest of these appears to be the best of the three (B) attacking the assumptions inherent in theories that until the 1980s had been largely accepted by geologists (C) outlining the inadequacies of three different explanations of the same phenomenon (D) providing concrete examples in support of the more general assertion that theories must often be revised in light of new evidence (E) citing evidence that appears to confirm the skepticism of geologists regarding a view held prior to the 1980s 5. The author implies that if the theory described in the third paragraph is true, which one of the following would have been true of iridium in the atmosphere at the end of the Cretaceous period? (A) Its level of concentration in the Earth’s atmosphere would have been high due to a slow but steady increase in the atmospheric iridium that began in the early Cretaceous period. (B) Its concentration in the Earth’s atmosphere would have increased due to the dramatic decrease in sea level that occurred during the Cretaceous period. (C) It would have been directly responsible for the extinction of many ocean species. (D) It would have been more uniformly dispersed than iridium whose source had been the impact of a meteorite on the Earth’s surface. (E) It would have been more uniformly dispersed than indium released into the atmosphere as a result of normal geological processes that occur on Earth. 6. The passage supports which one of the following claims about the volcanic-eruption theory? (A) It does not rely on assumptions concerning the temperature of molten rock at the lowest pan of the Earth’s mantle. (B) It may explain what caused the gradual fall in sea level that occurred for hundreds of thousands of years prior to the more sudden disappearance of the dinosaurs. (C) It bases its explanation on the occurrence of periods of increased volcanic activity similar to those shown to have caused earlier mass extinctions. (D) It may explain the relative scarcity of iridium in rocks on the Earth’s surface compared to its abundance in meteorites. (E) It accounts for the relatively uneven distribution of iridium in the layer of clay deposited at the end of the Cretaceous period. 7. Which one of the following, if true, would cast the most doubt on the theory described in the last paragraph of the passage? (A) Fragments of meteorites that have struck the Earth are examined and found to have only minuscule amounts of iridium hexafluoride trapped inside of them. (B) Most diapir eruptions in the geological history of the Earth have been similar in size to the one that occurred in India at the end of the Cretaceous period and have not been succeeded by periods of climatic change. (C) There have been several periods in the geological history of the Earth, before and after the Cretaceous period, during which large numbers of marine species have perished. (D) The frequency with which meteorites struck the Earth was higher at the end of the Cretaceous period than at the beginning of the period. (E) Marine species tend to be much more vulnerable to extinction when exposed to a dramatic and relatively sudden change in sea level than when they are exposed to a gradual change in sea level similar to the one that preceded the extinction of the dinosaurs. ======================================================= _________________ SVP Joined: 30 Apr 2008 Posts: 1888 Location: Oklahoma City Schools: Hard Knocks Followers: 38 Kudos [?]: 506 [0], given: 32 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 06 Mar 2009, 10:57 7th March (Saturday) means you're taking this thing tomorrow? _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a. GMAT Club Premium Membership - big benefits and savings Director Joined: 04 Jan 2008 Posts: 914 Followers: 59 Kudos [?]: 368 [0], given: 17 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 06 Mar 2009, 11:05 I normally post it midnight(Its Indian time-Saturday already ) _________________ VP Joined: 18 May 2008 Posts: 1287 Followers: 14 Kudos [?]: 264 [0], given: 0 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 07 Mar 2009, 00:32 My ans are: 1.C 2 A 3 B 4 A 5 E 6 A/E 7 B VP Joined: 05 Jul 2008 Posts: 1431 Followers: 37 Kudos [?]: 293 [0], given: 1 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 07 Mar 2009, 10:34 Tough Q's C,A,C,A,C,B,E Director Joined: 04 Jan 2008 Posts: 914 Followers: 59 Kudos [?]: 368 [0], given: 17 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 08 Mar 2009, 03:10 Very interesting the passage was but Qs were tricky what you all say? My choices are C A(why not E?) C A D B B OAs to be posted now icandy wrote: Tough Q's C,A,C,A,C,B,E _________________ Director Joined: 04 Jan 2008 Posts: 914 Followers: 59 Kudos [?]: 368 [0], given: 17 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 08 Mar 2009, 03:30 OAs are .......... Scroll down _________________ VP Joined: 18 May 2008 Posts: 1287 Followers: 14 Kudos [?]: 264 [0], given: 0 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 08 Mar 2009, 06:42 yes passage was interesting one. Though i answered some questions a bit carelessly , it was an easy one as compared to medical passages. Manager Joined: 27 Feb 2009 Posts: 62 Followers: 1 Kudos [?]: 11 [0], given: 0 Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 08 Mar 2009, 16:15 can some 1 explain no 7 for me please Re: RC No 1-extinction of dinosaurs-7th March(Saturday) [#permalink] 08 Mar 2009, 16:15 Similar topics Replies Last post Similar Topics: 1 Until the 1980s, most scientists believed that 14 10 Mar 2010, 02:24 2 In the early to mid-1980s, a business practice known as a 16 13 May 2008, 21:15 2 Until recently most astronomers believed that the space 9 16 Jan 2008, 21:39 Linda Kerber argued in the mid- 1980's that after the 0 14 Jan 2008, 15:33 Linda Kerber argued in the mid- 1980's that after the 19 08 Oct 2007, 16:24 Display posts from previous: Sort by # =================== Until the 1980s, most scientists Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5486246347427368, "perplexity": 2964.6522807003885}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701166739.77/warc/CC-MAIN-20160205193926-00301-ip-10-236-182-209.ec2.internal.warc.gz"}
https://email.esm.psu.edu/pipermail/macosx-tex/2005-October/018244.html	# [OS X TeX] Re: MacOSX-TeX Digest #1524 - 10/14/05 John Vokey vokey at uleth.ca Sat Oct 15 01:41:45 EDT 2005 Nope. If I don't check keep keep postscript file, I get this (and I have both the latest TeXShop, and thr latest of your (Gerben's) i- installer installations), which is even worse: ### This is /usr/local/teTeX/bin/powerpc-apple-darwin-current/ simpdftex, Version \$Revision: 2.18 \$ ### /usr/local/teTeX/bin/powerpc-apple-darwin-current/tex fpe.tex This is TeX, Version 3.141592 (Web2C 7.5.5) (./fpe.tex ! Undefined control sequence. l.2 \documentclass [doc,fignum,nobf]{apa} So, what? Reinstall everything? How do I clear out what is installed? And, oddly, this has never happened before, and, it has affected every computer I have (too many to list) (all of which have been updated to the latest of both i-installer files and TeXShop--- before I discovered the problem). I have no desire to spend the many hours on each reinstalling. Something is seriously toasted. On 14-Oct-05, at 6:00 PM, TeX on Mac OS X Mailing List wrote: > I think you might be using an older TeXShop and you have the "keep PS > file" checkbox checked. Uncheck it and it may work. > > Secondly, try the command line > > simpdftex latex --maxpfb yourfilename.tex > > If that works, the problem lies in your frontend. > > G > -- Please avoid sending me Word or PowerPoint attachments. See <http://www.gnu.org/philosophy/no-word-attachments.html> -Dr. John R. Vokey ------------------------- Info -------------------------- Mac-TeX Website: http://www.esm.psu.edu/mac-tex/ & FAQ: http://latex.yauh.de/faq/ TeX FAQ: http://www.tex.ac.uk/faq List Archive: http://tug.org/pipermail/macostex-archives/	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9239674806594849, "perplexity": 29898.621228336964}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402124756.81/warc/CC-MAIN-20201001062039-20201001092039-00283.warc.gz"}
https://forum.swissmicros.com/viewtopic.php?f=23&t=1877&p=16305	## [DM16] bit field manipulation Contributed software for the DM10, DM11, DM12, DM15 and DM16 goes here. Please prefix the subject of your post with the model of the calculator that your program is for. RobFisher Posts: 3 Joined: Fri Mar 16, 2018 10:35 am ### [DM16] bit field manipulation There isn't a "software library" section in this part of the forum, so I hope this is welcome. I have been playing with my DM16C and finding it useful. Often I am working with data formats that encode values in bit fields with odd numbers of bits. For example, MPEG 2 transport streams are used for sending video over satellites. I can look at a hex dump of a stream but there are fields that don't line up with the hex digits. The DM16C is useful for this. My program lets me type in the hex digits and pull out individual fields to registers, or else plug in the values and see the resulting hex. I'll put the raw mnemonics below so that you can paste them into http://www.swissmicros.com/nut_decoder/ but for the full explanation and commented code see https://github.com/RobFisher/hp_calcula ... encode.txt Aside: What I find quite fun about the 16C is manipulating the stack so as to avoid using registers for intermediate values. It might make the code harder to understand, though. At some point I might see if I can document a toolkit of stack manipulations. Anyway, briefly, to use this you program in the sizes of the fields in bits into registers .1 to .F, using 0 to indicate there are no more fields. The values for the fields are (left to right) read from or written to registers 1 to F. You can GSB A to parse X into fields stored in the registers, or GSB E to encode the fields in the registers and store them in X. Example: To read an MPEG2 transport packet header, set the word length to 32 and the complement to unsigned. Store the number of bits in each field in registers as follows: .1 = 8 (for the sync byte) .2 = 1 (for the transport error indicator) .3 = 1 (for the payload unit start indicator) .4 = 1 (for the transport priority) .5 = 13 (for the PID) .6 = 2 (for the transport scrambling control) .7 = 2 (for the adaptaion field control) .8 = 4 (for the continuity counter) .9 = 0 to indicate that there are no more fields. Set the window size to 32 bits by pressing DEC 32 [f] WSIZE. Store the transport stream header in X, e.g. 47ABCDEF h Run the program with GSB A When it terminates, the sync byte (0x47) will be stored in register 1, the transport error indicator in register 2, and so on. You can convert the other way with GSB E. It's quite handy to parse the field values, change one of the fields by changing a register, then re-encode. Here is the program for pasting, but see the above link for a readable version: Code: Select all LBL A HEX 0 STO I Rv LBL B x<>I 11 + x<>I RCL (i) x=0 RTN x<>I 10 - x<>I GSB F STO (i) Rv GTO B LBL F x<>y ENTER ENTER R^ RLn LST X x<>y ENTER Rv x<>y AND x<>y Rv RTN LBL E HEX 11 STO I 0 LBL C RCL (i) x<>y ENTER ENTER R^ ENTER LST X x<>I 10 - x<>I x<>y RCL (i) AND R^ OR x<>I 11 + x<>I RCL (i) x=0 GTO D RLn GTO C LBL D Rv RTN Mr_Plant Posts: 13 Joined: Fri Mar 06, 2020 4:27 am ### Re: [DM16] bit field manipulation Very interesting .love the implementation.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3474903106689453, "perplexity": 3621.7415044913137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703538082.57/warc/CC-MAIN-20210123125715-20210123155715-00302.warc.gz"}
https://mitpress.mit.edu/books/evolution-gender-and-rape	Hardcover \| Out of Print \| ISBN: 9780262201438 \| 472 pp. \| 6 x 9 in \| January 2003 Paperback \| $7.75 Short \| £5.95 \| ISBN: 9780262700900 \| 472 pp. \| 6 x 9 in \| January 2003 eBook \|$21.00 Short \| ISBN: 9780262253802 \| 472 pp. \| January 2003 ## Overview Are women and men biologically destined to be in perpetual conflict? Does evolutionary genetics adequately explain sexual aggression? Such questions have been much debated in both the media and academia. In particular, the notion that rape is an evolutionary adaptation, put forth by Randy Thornhill and Craig T. Palmer in their book A Natural History of Rape (MIT Press, 2000), vaulted the debate into national prominence. This book assesses Thornhill and Palmer's ideas, as well as the critical responses to their work. Drawing on theory and data from anthropology, behavioral ecology, evolutionary biology, primatology, psychology, and sociology, the essays explain the flaws and limitations of a strictly biological model of rape. They argue that traditionally stereotyped gender roles are grounded more in culture than in differing biological reproductive roles.The book is divided into three parts. The first part, "Evolutionary Models and Gender," addresses broad theoretical and methodological issues of evolutionary theory and sociobiology. Part 2, "Critiquing Evolutionary Models of Rape," addresses specific propositions of Thornhill and Palmer, making explicit their unexamined assumptions and challenging the scientific bases for their conclusions. It also considers other studies on biological gender differences. Part 3, "Integrative Cultural Models of Gender and Rape," offers alternative models of rape, which incorporate psychology and cultural systems, as well as a broader interpretation of evolutionary theory.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20140206813812256, "perplexity": 6531.069456976561}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257836397.31/warc/CC-MAIN-20160723071036-00205-ip-10-185-27-174.ec2.internal.warc.gz"}
http://electronics.stackexchange.com/questions/52160/can-daq-devices-be-used-as-oscilloscopes	# Can DAQ devices be used as Oscilloscopes? Recently, I took computer architecture course, and got interested in analog electronics. At present moment, I can't afford good USB oscilloscope such as Agilent or Cleverscope (though, they are not that terribly expensive - $1500) I am curious if USB Digital Acquisition Devices(DAQ) can be used as inexpensive temporary substitution for oscilloscope ? What would be drawbacks (apart from not having probes and having to record the data to PC) ? Thanks ! Edit: some of the DAQ devices I have looked at: http://www.ni.com/products/usb-6008/ (NI make an array of different USB DAQ) http://www.keithley.com/products/data/multifunction/usb/?mn=KUSB-3100 (though it looks like this DAQ is better suited for power electronics) Some USB oscilloscopes I am interested in: http://www.cleverscope.com/products/CS320A http://www.home.agilent.com/en/pc-1418982/usb-modular-oscilloscope?nid=-34492.0&cc=US&lc=eng - I'm interested in the differences between a DAQ vs. an Oscilloscope, too, but what about investing into a bench-top scope? There are some pretty decent ones for around$400 or so, and they usually have a USB connection for hooking up to the PC. – helloworld922 Dec 25 '12 at 7:09 Probably 99% of all the available sub-$2K digital oscilloscopes are going to be only 8-bit. – Connor Wolf Dec 25 '12 at 8:34 The DAQ's yo are referring to have very limited bandwidth, barely enough to sample telephone quality audio. – jippie Dec 25 '12 at 10:06 The biggest disadvantage for USB scope's, in my opinion, is that they are often supported for the current Windows versions available. Once you want to upgrade you Windows to a newer release, the scope hardware is often rendered useless because of driver and post-processing software issues. – jippie Dec 25 '12 at 10:08 @newprint I am unfamiliar with Cleverscope, but most of the time this sort of devices have closed protocols and closed source software. Don't forget about the USB/.dll driver module here! Also writing your own software from scratch is not an easy task for many people. I've written software for a DVM, which is a lot simpler than a scope, and it took me several weeks to come to a useful tool (then again, I'm not much of a programmer). – jippie Dec 25 '12 at 10:44 ## 4 Answers DAQ systems can make very functional low-speed oscilloscopes, with a number of caveats: • You're not going to get a very broad voltage range. Most of them will maybe do ±10V input range. • Probably won't support offset subtraction on the inputs, like scopes do. • DC-coupled only, unless you supply the series cap. • Inputs can be low(ish)-impedance (some may have buffer amps, on cheap ones the input may literally just connect to the ADC pin). Not the 1MΩ standard that scopes have. • Most importantly: • PC based oscilloscope interfaces suck • Also, it's likely a DAQ won't even have a traditional oscilloscope-like software tool. You may have to write your own. Anyways, if you have a situation where you have fixed or low voltages, and don't mind doing a bunch of work on the PC end, a DAQ could be used as rather pokey oscilloscope. They're really different tools, though, and while they do share some characteristics, they have very different intended uses, and this tends to show in their approach and the software design considerations. It's also worth noting that most DAQ systems are designed for continuous, rather then triggered data acquisition. This means you're maximum sample rate is largely limited by the interface the DAQ uses. For example, USB2 only has 480 Mbps(more like 400 Mbps real-world) bandwidth. As such the best sample rate that could ever be achieved would be 50 Msps(million samples per second) at 8 bits resolution, and very few implementations will even approach that. Somewhere in the range of 1-10 Msps at 8 or 16 bits is more realistic. Extracting all the available bandwidth from USB is very challenging. Another consideration is what you're going to do with all the data. 1 Msps is a lot of data. If the 1 Msps stream is 16 bits, that's 2 Megabytes of data per second, or a gigabyte every 8 minutes. I don't know what you're intending to do with this pseudo-oscilloscope, but you can't just take samples willy-nilly, unless you're just displaying them and then immediately discarding them. I've actually written a minimal real-time visualization tool for some IOtech-branded DAQ systems at work. It's kind of an oscilloscope. It works well, but I've also designed all the PCBs that interface with the DAQ system, so I could design them to work to the DAQ system's input specifications. - Thank you for a long and detailed response. I have accepted your response as the answer. Somehow, I forgot all about the USB bandwidth. If Agilent claims that it's USB scope has sampling rate of up to 1GSa/s, how they manage to get all this date to PC ?? – newprint Dec 26 '12 at 2:04 @newprint the samples are all taken and stored on the device before they get transferred to the PC. – helloworld922 Dec 26 '12 at 3:23 @newprint - the critical thing is that the Agilent scope is non-continuous. It samples for a little while, and then spends a lot longer transfering that data to the PC, the duration of which it's not sampling the inputs. DAQ devices, pretty much universally, are going to be designed for continuous sampling. As such, you then need to worry about bandwidth concerns. – Connor Wolf Dec 26 '12 at 18:01 You can get DAQ devices that take more then 480 Mbps of data per second, but they're going to use a different interface, such as PCI-e, or cardbus. – Connor Wolf Dec 26 '12 at 18:01 Check ebay for low cost second hand equipment. HP and others brands has very long life and they are generally still very good even second hand. The 2 DAQ models are lower in resolution and "high frequency" bandwidth than PC sound card. They can go down to 0 Hz DC whereas sound card typically cut off at 20Hz. - Likely many DAQ ADC cards come with such software. Some (not a lot) software use PC parallel port as very low speed logic analyzer. You can buy adaptor to make "parallel port' out of latest USB-only PC. However, this may limited the speed further. Many software use PC sound card as oscilloscope/spectrum analyzer as well as signal generator. Apparently, these may more suit to EE engineering than computer architecture course. Allow on screen 'experiments' too. Like, http://www.qsl.net/dl4yhf/spectra1.html - Personally, I wouldn't go there. You didn't specify a piece of equipment, but you would likely have speed issues. I would look at something like this. It's not very fast, but with a 25 MHz sampling frequency, you can easily look at 5 MHz signals, and theoretically up to 12.5 MHz. The transient recorder, bode plotter, and spectrum analyzer are nice features. The function generator is an added bonus, and most importantly, it doesn't cost$1500. - Personally I am not really impressed with Velleman's software and its support over time. See also my comment at top of the page. – jippie Dec 25 '12 at 10:19	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5055156350135803, "perplexity": 2209.754433288743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931012703.33/warc/CC-MAIN-20141125155652-00125-ip-10-235-23-156.ec2.internal.warc.gz"}
https://open.kattis.com/problems/sixdegrees	Kattis # Six Degrees Picture by: Fo0bar For years and years, the ICT Senior Service Desk (ISSD) of the university has been confronted with a slow wired network that gives unexpected time-outs and seemingly random slow connection speeds. A new manager has been hired to solve these problems once and for all. The manager does not have any computer science or IT knowledge, but he does happen to have a strong background in sociology. He quickly finds that the network problems only affect devices of old professors with an office in some distant corner of the building. Obsessed by the idea of six degrees of separation, the new manager proposes a rule to counter the network problems. This rule says that any two devices in the network should be connected via at most 5 intermediary devices. So, given the current lay-out of the university’s wired computer network, he decides to prepare a list of all devices of peripheral professors that are currently not able to connect to all other devices within 6 steps. The manager’s solution to the network problems is then to disconnect all devices on this list from the wired network at once. He explicitly ignores the fact that, possibly, then disconnecting these devices in a particular order may lead to a network structure such that some devices on the list actually no longer have to be disconnected, or that afterwards additional devices may have to be disconnected or connected to reach the actual desired result. The board of the university, not having a background in computer science, IT or sociology is also not bothered by whether or not the proposed solution is correct, but will instead only base its decision on whether or not the prepared list of devices to be disconnected is not too long, so that not too many professors would be affected. The board will therefore only approve the plan if no more than 5% of the wired network devices is on the list. Given the lay-out of the network, in the form of a list of pairs of IP addresses or hostnames representing directly connected devices, determine whether or not the university board will allow the new manager to execute his plan. ## Input The input starts with a line containing an integer $T$ ($1 \leq T \leq 2$), the number of test cases. Then for each test case: • One line containing an integer $1 \leq M \leq 30\, 000$ denoting the number of (directly) connected pairs of devices (with at most $3\, 000$ unique devices). • $M$ lines, each line containing two IP addresses or hostnames of (directly) connected devices, represented by two strings of printable ASCII characters (of length $\leq 64$) without whitespace. Each pair of connected devices is included once in the input file. All connections are bidirectional. You may assume that all devices in the university network are in the same connected component of devices. ## Output For each test case, output one line containing either YES if the plan is allowed to be executed or NO if the plan is not allowed to be executed. Sample Input 1 Sample Output 1 2 6 132.229.123.1 132.229.123.2 132.229.123.2 132.229.123.3 132.229.123.3 132.229.123.4 132.229.123.4 132.229.123.5 132.229.123.5 132.229.123.6 132.229.123.6 xxxxx 7 a b b c c d d e e f f g g h YES NO	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2787652313709259, "perplexity": 904.058030242424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998084.36/warc/CC-MAIN-20190616082703-20190616104703-00488.warc.gz"}
https://freevcenotes.com/methods/notes/discrete-probability	# Sample Spaces and Events The outcome of a random experiment is uncertain, but there exists a set of possible outcomes, $$\varepsilon$$, known as the sample space. The sum of the probabilities of all the outcomes in, $$\varepsilon$$ is 1. For example, the sample space for rolling a six-sided dice is: $\varepsilon = \{1,2,3,4,5,6\}$ An event is a subset of the sample space denoted by a capital letter. For example, if the event A is defined as the odd numbers when rolling a sixes sided dice, then we have: $A = \{1,3,5\}$ If the event B is impossible, $$\text{Pr}$$(B) = 0. If the event B is certain, $$\text{Pr}$$(B) = 1. So, for any event B, 0 $$\leqslant$$ $$\text{Pr}$$(B) $$\leqslant$$ 1. # Determining Probabilities When the sample space is finite, the probability of an event is the sum of the probabilities of the outcomes in that event. For example, if A is defined as the odd numbers when rolling a six sided dice, then: \begin{aligned} \Pr(A) &= \Pr(\text{Roll 1}) + \Pr(\text{Roll 3}) + \Pr(\text{Roll 5})\\ &= \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\\ &= \frac{3}{6}\\ &= \frac{1}{2}\\ \end{aligned}\\ When dealing with area questions, assume that it is equally likely to hit any region of the define area. So, the probability of hitting a certain region A is: $\text{Pr}(A) = \frac{\text{Area of A}}{\text{Total area}}$ When an experiment has only two possible outcomes (events), they are said to be complementary. The complement of the event A is denoted by A'. In the example with the six sided dice above, A' would represent everything in the sample space, except the odd numbers. So, $A' = \{2,4,6\}$ Because the sum of the probabilities of events A and A' must be 1, we have: $\text{Pr}(A') = 1 - \text{Pr}(A)$ # The Addition Rule and Mutual Exclusivity The addition rule is generally used to calculate $$\Pr(A \cap B)$$ or $$\Pr(A \cup B)$$ $\Pr(A) + \Pr(B) - \Pr(A \cap B) = \Pr(A \cup B)$ We say that two events are mutually exclusive if: $\Pr(A \cap B) = 0$ That is the two events will never occur at the same time. # Probability Tables A very powerful table which isn't emphasised enough! $A$ $A'$ $B$ $\Pr(A \cap B)$ $\Pr(A' \cap B)$ $\Pr(B)$ $B’$ $\Pr(A \cap B’)$ $\Pr(A' \cap B’)$ $\Pr(B’)$ $\Pr(A)$ $\Pr(A')$ $1$ For appropriate questions, place the probabilities given in their corresponding box. The sum of each column and row is the last entry. For example: $\Pr(A \cap B) + \Pr(A \cap B') = \Pr(A)$ $\text{Example 9.1: John has lost his class timetable. The probability that}\\\text{ he will have Methods period one is 0.35.}\\ \text{The probability that he has PE on a given day is 0.1 and the probability}\\ \text{ that he will have Methods period one and PE on the same day is 0.05.}\\ \text{Find the probability that John does not have Methods period one and PE on the same day.}\\ \text{ }\\ \text{Let } M \text{ represent methods period one and } P \text{ represent PE. From the information given we have:}\\$ $M$ $M’$ $P$ $0.05$ $\Pr(M’ \cap P)$ $0.1$ $P’$ $\Pr(M \cap P’)$ $\Pr(M’ \cap P’)$ $\Pr(P’)$ $0.35$ $\Pr(M’)$ $1$ \text{Looking at the second row}\\ \begin{aligned} 0.05 + \Pr(M’ \cap P) &= 0.1\\ \Pr(M’ \cap P) &= 0.05\\ \text{} \\ \text{Looking at the last row}\\ 0.35 + \Pr(M’) &= 1\\ \Pr(M’) &= 0.65\\ \end{aligned} $M$ $M’$ $P$ $0.05$ $0.05$ $0.1$ $P’$ $\Pr(M \cap P’)$ $\Pr(M’ \cap P’)$ $\Pr(P’)$ $0.35$ $0.65$ $1$ \text{Looking at the third column}\\ \begin{aligned} 0.05 + \Pr(M’ \cap P’) &= 0.65\\ \Pr(M’ \cap P’) &= 0.6\\ \end{aligned}\\ \text{So, the probability that John does not have Methods period one and PE on a given day is } 0.6\\ # Conditional Probability The probability that event A happens when we know that event B has already occured: $\Pr(A \mid B) = \frac{\Pr(A \cap B)}{\Pr(B)}$ It is often difficult to recognise when we are being asked a conditional probability question. However, generally speaking, if the question includes “if” or “given that”, you can be almost certain that you are dealing with conditional probability. We have written the same question below twice using the two different phrases. \text{Example 9.2 Find the probability that a six is rolled with a} \\\text{ six sided die given that an even number has been rolled. }\\ \text{Or equivelantly, If an even number has been rolled on a six sided die}\\\text{ find the probability that a six is rolled. }\\ \text{ }\\ \begin{aligned} \Pr(\text{Even}) &= \frac{3}{6} \\ \Pr(\text{Even and Six}) &= \Pr(\text{Six}) \\ &= \frac{1}{6} \\ \Pr(\text{Six if Even}) &= \frac{\Pr(\text{Even and Six})}{\Pr(\text{Even})} \\ &= \frac{\frac{1}{6}}{\frac{3}{6}}\\ &= \frac{1}{3}\\ \end{aligned}\\ # Independence If knowing that event B has happened does not change the probability of event A from happening, then we say that events A and B are independent. For example, it raining outside and you going to school is independent as you will go to school regardless of whether it is raining or not. However, you being in class and having lunch is not independent as you will (probably) not be able to have lunch during class. Mathematically two events are independent if: \begin{aligned} \Pr(A \cap B) &= \Pr(A) \cdot \Pr(B)\\ \Pr(A) \neq 0 &\text{ and } \Pr(B) \neq 0 \end{aligned} \text{Example 9.3: John has lost his class timetable.}\\\text{ The probability that he will have Methods period one is 0.35.}\\ \text{The probability that he has PE on a given day is 0.1 and the probability} \\\text{that he will have Methods period one and PE on the same day is 0.035.}\\ \text{Is John having Methods period one independent to him having PE on the same day?}\\ \text{ } \\ \text{A. Let } M \text{ represent methods period one and } P \text{ represent PE. From the information given we have:}\\ \begin{aligned} \Pr(M) &= 0.35\\ \Pr(P) &= 0.1\\ \Pr(M) \cdot \Pr(P) &= 0.035 \\ &= \Pr(M \cap P) \\ \end{aligned}\\ \text{So, John having Methods period one and PE on the same day are independent events} \\ # Discrete Random Variables A random variable is a function that assigns a number to each outcome of an experiment. A discrete random variable can take one of a countable number of possible outcomes. Continuous random variables will be considered in the next section. For example, the number of free throws John can score when taking two is a discrete random variable which may take one of the values 0,1 or 2. More on this below. # Discrete Probability Distributions The probability distribution for a random variable consists of all the values the variable can take along with the associated probabilities. The general format is: $\text{}$ $x_1$ $x_2$ $...$ $x_n$ $\Pr(X=x)$ $\Pr(X=x_1)$ $\Pr(X=x_2)$ $...$ $\Pr(X=x_n)$ The table allows us to easily find probabilities such as: $$\text{Pr}(X>1)$$ and $$\text{Pr}(X<2)$$ by summing the relevant probabilities in the table Note: the bottom row must sum to 1 and each probability must be at least zero and at most one. A discrete probability function, also called a probability mass function describes the distributions of a discrete random variable. An example of a graph for a discrete probability function is given below. \text{Example 9.4: James scores 80\% of all free throws he takes.}\\ \text{Create a probability distribution table and graph the probability mass function if James has two free throws.}\\ \text{ }\\ \text{Let } X \text{ be the number of free throws James scores}\\ \begin{aligned} \varepsilon &= {0,1,2}\\ Pr(X = 2) &= 0.80 \cdot 0.80 \\ &= 0.64\\ Pr(X = 1) &= \binom{2}{1} \cdot 0.8 \cdot 0.2\\ &= 0.32\\ Pr(X = 0) &= 0.2 \cdot 0.2 \\ &=0.04\\ \text{ }\\ \end{aligned}\\ \text{Bear with us on how we calculated } \Pr(X = 1). \\ \text{We will explain how we calculated this in the next chapter - binomial distribution}\\ $x$ $0$ $1$ $2$ $\Pr(X=x)$ $0.04$ $0.32$ $0.64$ # Mean, Variance and Standard Deviation The expected value, or mean, is the average value of a discrete random variable. To calculate it, we sum the products of each value of X and its associated probability. That is, we first find the product of each column of the probability distribution table and then sum them up. Mathematically: $E(X) = \mu = \sum_{x} x \cdot \Pr(x)$ Variance and standard deviation are a measure of spread. Standard deviation is more relevant to us as it is in the same units as those which we are measuring. The formula provided by VCAA involves a very large number of calculations, as such we recommend you memorising: $\text{Var}(X) = E(X^2) - [E(X)]^2$ To calculate the first term, simply square all of the x values in the top row of the probability distribution table and then find the product of each column of the probability distribution table before summing them up. To get standard deviation we simply square root the variance. $\text{sd}(X) = \sigma = \sqrt{\text{Var}(X)}$ \text{Note:}\\ \begin{aligned} E(aX+b) &= a \cdot E(X) + b\\ \text{Var}(aX+b) &= a^2 \cdot Var(X)\\ \end{aligned} For many random variables, there is a 95% chance of obtaining an outcome within two standard deviations either side of the mean. That is, $\Pr(\mu - 2\sigma \leqslant X \leqslant \mu + 2\sigma) \approx 0.95$ \text{Example 9.5: James scores 80\% of all free throws he takes. James takes two shots.}\\ \text{Find the expected value, variance and standard deviation for this scenario.}\\ \text{Correct your answers to 2 decimal points where appropriate}\\ \text{ }\\ \text{The probabilities for each outcome was calculated in example 9.3.}\\ \text{Let } X \text{ be the number of free throws James scores.}\\ \text{ }\\ \begin{aligned} E(X) &= 0 \times 0.04 + 1 \times 0.32 + 2 \times 0.64\\ &= 0 + 0.32 + 1.28\\ &= 1.6\\ \text{ }\\ E(X^2) &= 0^2 \times 0.04 + 1^2 \times 0.32 + 2^2 \times 0.64\\ &= 2.88\\ \text{ }\\ \text{Var}(X) &= E(x^2) - [E(X)]^2\\ &= 2.88 - 1.6^2\\ &= 0.32\\ \text{ }\\ \text{sd}(X) &= \sqrt{Var(X)}\\ \text{sd}(X) &= 0.57\\ \end{aligned}	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999585151672363, "perplexity": 662.8091400310481}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571989.67/warc/CC-MAIN-20220813232744-20220814022744-00771.warc.gz"}
https://www.physicsforums.com/threads/intrinsic-carrier-concentration-formula.597231/	# Intrinsic carrier concentration formula 1. Apr 16, 2012 ### aarnes In the ni=$\sqrt{NcNv}e^{\frac{-Eg}{2kT}}$ formula for intrinsic carrier concentration, what values Nc,Nv and Eg take in relation to temperature? Are they each calculated for that temperature or, for example, is Eg always taken for 300K? My guess is they should all be calculated for given temperature but that would mean some solved problems I got my hands on are wrong. In them, Eg is always taken to be 1.196eV. 2. Apr 20, 2012 ### reddvoid Yes, Nc ,Nv and Eg are functions of temperature, you should take into account changes in Nc Nv when temperature is not 300k but I think you can neglect change in Eg, you can consider it too if you want more accurate answer	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8364067673683167, "perplexity": 1409.8779669596624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661296.34/warc/CC-MAIN-20160924173741-00169-ip-10-143-35-109.ec2.internal.warc.gz"}
https://www.wisdom.weizmann.ac.il/~oded/MC/066.html	## Property Testing Lower Bounds Via Communication Complexity by Eric Blais, Joshua Brody, and Kevin Matulef I find the general technique very appealing. I wish to emphasize that the fact that property testing lower bounds can be obtained via communication complexity problems is highly non-trivial, since doing so calls for reducing a communication problem to a testing problem that (in general) lacks any a priori input partition (nor a natural clue on how to partition the tested input so that the inputs to the communication problem can be embedded in it). Indeed, the new technique is a methodology, which as such involved a creative step. The user (i.e., lower bound prover) has to come up with a decomposition of the input that allows local reduction of the individual inputs (of the communication problem) into two parts that when composed back yield an input for the testing problem. Furthermore, the composition should be locally computable (i.e., its value at any desired point has to be determined by few bits in each of the two parts). Viewing the testing problem as referring to a set of functions, denoted PI, one should present a decomposition of functions into pairs of functions. Specifically, one should suggest a communication problem in which the inputs of the two parties are function, denoted $f$ and $g$, such that $h=compose(f,g)$ is in PI iff the pair $(f,g)$ is a yes-instance of the communication problem. An example, taken from Section 1.2, is indeed useful and follows. Consider the task of proving a lower bound on the query complexity of testing $k$-linearity (i.e., whether the tested function is a linear function that depends on exactly $k$ variables). We use a reduction from the communication problem SET DISJOINTNESS, when restricted to instances in which the sets are of size $k$ (and the intersection is either empty or a singleton). The parties are given $k$-subsets, $A$ and $B$, of $[n]$, and (locally) form the linear functions $f(x)=\sum_{i\in A} x_i$ and $g(x)=\sum_{i\in B} x_i$, respectably. They use an algorithm for testing $2k$-linearity, invoked on the (imaginary) function $h=f+g$. The parties emulate the testing algorithm in the joint-randomness model so that whenever they need the value of $h$ at point $x$, they send each other the values $f(x)$ and $g(x)$. Finally, observe that $h$ is $2k$-linear iff $A$ and $B$ are disjoint (and otherwise $h$ is $(2k-2)$-linear). Indeed, the foregoing description refers to the standard model of randomized communication complexity (i.e., the joint-randomness model). Using it adaptive testers yield standard two-way communication protocols, whereas non-adaptive testers yield one-way communication protocols. One-sided error probability is preserved. The point in all these comments is that they offer a variety of potential lower bound attempts while relying on different existing communication complexity lower bounds. Let me mention that the new method allows settling all three conjectures of [23] (only one is addressed in the current write-up of this work, but the other two are doable too - details will appear in a forthcoming revision). However, this is not the reason for my excitment of the new method, but rather an excellent demonstration to its power and versatility. P.S.: I found the paper HERE. Update (4/4/11): Posted as ECCC TR11-045. #### The original abstract We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Omega(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich [23]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions, and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds. Back to list of Oded's choices.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9196661114692688, "perplexity": 438.32772882040365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571222.74/warc/CC-MAIN-20220810222056-20220811012056-00496.warc.gz"}
https://www.koreascience.or.kr/search.page?keywords=continuity&pageSize=10&pageNo=1	• Title, Summary, Keyword: continuity ### A Study on the Continuity Expressed in Modern Fashion Design - Focusing on the Continuance Theory of H. Bergson and the Continuity theory of J. Deleuze - (현대 패션 디자인에 나타난 연속성 연구 - 베르그송의 지속 이론과 들뢰즈의 연속성 이론을 중심으로 -) • Yang, Hee-Young;Yang, Sook-Hi • Journal of the Korean Society of Costume • / • v.58 no.2 • / • pp.15-33 • / • 2008 • Continuity and discontinuity is a relative concept, and there are various categories of the continuity and discontinuity in our circumference. Generally, characteristics of postmodernism including between the difference and the variety have being regarded as a discontinuity. Concept of the continuity includes between the quantitative continuity and the qualitative continuity qualitative continuity has organic characteristic, which encourages creating something permanently through the flowing of the time. Therefore, this thesis has studied like this complex social condition and various relationships expressed in modern fashion focusing on permanently creative movements and behaviors equal to the 'continuance' theory of Herni Bergson and 'continuity' theory of Jill Deleuze. This thesis classifies characteristics of the qualitative continuity into spatiotemporal and spatial continuity, and subdivides into 3 sets: perceptual continuity, spatial continuity, transferring continuity of physical experience, immaterial informational continuity, and fluid continuity with environment. Continuous viewpoint, which accepts the existing elements and allows them to flow liberally, should be present more appropriative thinking direction in explaining the complex situation expressed in the modern fashion, rather than discontinuous viewpoints focused on the only changing moment. ### COMPARISON BETWEEN DIGITAL CONTINUITY AND COMPUTER CONTINUITY • HAN, SANG-EON • Honam Mathematical Journal • / • v.26 no.3 • / • pp.331-339 • / • 2004 • The aim of this paper is to show the difference between the notion of digital continuity and that of computer continuity. More precisely, for digital images $(X,\;k_0){\subset}Z^{n_0}$ and $(Y,\;k_1){\subset}Z^{n_1}$, $if(k_0,\;k_1)=(3^{n_0}-1,\;3^{n_1}-1)$, then the equivalence between digital continuity and computer continuity is proved. Meanwhile, if $(k_0,\;k_1){\neq}(3^{n_0}-1,\;3^{n_1}-1)$, then the difference between them is shown in terms of the uniform continuity property. ### FROM STRONG CONTINUITY TO WEAK CONTINUITY • Kim, Jae-Woon • Journal of the Chungcheong Mathematical Society • / • v.14 no.1 • / • pp.29-40 • / • 2001 • In this note, we get the conditions such that strong continuity ${\Rightarrow}$ weak continuity plus interiority condition( wc+ic), and continuity ${\Rightarrow}$ wc+ic are true. And we investigate some equivalent conditions with weak continuity, some properties of weak continuity. And we show that almost compactness is preserved by weakly continuous function, and we improve some known results with respect to strong continuity. ### A Study on the Factors Influencing the Number of Network Organizations, Network Continuity and Service Continuity of Community Child Centers - Focusing on Chonbuk Province - (지역아동센터의 네트워크 연계기관수 및 네트워크 지속성, 서비스 지속성 영향요인 연구 - 전북지역을 중심으로 -) • Kim, Hyeon Suk;Seo, Yoon • Korean Journal of Child Studies • / • v.34 no.6 • / • pp.159-175 • / • 2013 • The purpose of this study was to explore how the number of network organizations, network continuity and service continuity of community child centers were affected by the charge factors, organizational factors and environmental factors of community child centers. To better understand this, we researched 107 community child centers in Chonbuk Province. The key results are as follows. The interpersonal connections of the charge factors had a strong effect on the number of network organizations, but organizational factors and environmental factors appeared to have no effect on the number of network organizations. The operational type(religious group), organizational support level of the organizational factors and public-private cooperation, regular consultative group operation(no) of the environmental factors appeared to have an effect on the level of network continuity. The region(local), regular consultative group operation(yes) and public-private cooperation of the environmental factors appeared to have an effect on the level of service continuity. It was hoped that the results would enable us to explore how factors related to a network had an effect on the service continuity, the number of network organizations and clients referral network continuity, and they appeared to have an effect on the service continuity. ### FUZZY D-CONTINUOUS FUNCTIONS • Akdag, Metin • East Asian mathematical journal • / • v.17 no.1 • / • pp.1-17 • / • 2001 • In this paper, fuzzy D-continuous function is defined. Some basic properties of this continuity are summarized; and sufficient conditions on domain and/or ranges implying fuzzy D-continuity of fuzzy D-continuous functions are given. Also fuzzy D-regular space is defined and by using fuzzy D-continuity, the condition which is equivalent to fuzzy D-regular space, is given. ### ON M-CONTINUITY • Min, Won Keun;Chang, Hong Soon • Korean Journal of Mathematics • / • v.6 no.2 • / • pp.323-329 • / • 1998 • In this paper, we introduce a new class of sets, called $m$-sets, and the notion of $m$-continuity. In particular, $m$-sets and $m$-continuity are used to extend known results for ${\alpha}$-continuity and semi-continuity and precontinuity. ### Continuity of Ambulatory Care among Adult Patients with Type 2 Diabetes and Its Associated Factors in Korea (우리나라 성인 2형 당뇨환자의 외래진료 지속성과 관련요인 분석) • Hong, Jae-Seok;Kim, Jai-Yong;Kang, Hee-Chung • Health Policy and Management • / • v.19 no.2 • / • pp.51-70 • / • 2009 • Background : Previous studies have reported that enhanced continuity of care prevented a sudden worsening in progress among chronic disease patients, and as a result was favorable for efficient spending of health care funds. This study aims to estimate the continuity of care of Korean with diabetes and to identify factors affecting the continuity of care. Methods : This study used the Korean National Health Insurance Claims Database which includes E11 (ICD-10) as a primary or secondary disease as of 2006. Study population is 1,160,725 type 2 diabetics (20-84 years). Continuity of Care Index (COC), Modified, Modified Continuity Index (MMCI), and Most Frequent Provider Continuity (MFPC) were used as indexes of continuity of care. Results : The continuity of care in the study population was $0.94{\pm}0.10$ as calculated by MMCI, $0.91{\pm}0.16$ as calculated by MFPC and $0.86{\pm}0.23$ as calculated by COC. The lower continuity of care was shown in the patients who were female, 65 and over years old, Medical Aid recipients, 13 times or more visitors, hospital users as main attending medical institution, patients experienced hospitalizations or comorbidities. Conclusion : The continuity of care for adult patients with type 2 diabetes was high in Korea, and showed variation according to patients' characteristics. This result provides empirical evidence for policymakers to develop or strengthen programs for managing patients showing low continuity of care. ### R(g, g')-CONTINUITY ON GENERALIZED TOPOLOGICAL SPACES • Kim, Young-Key;Min, Won-Keun • Communications of the Korean Mathematical Society • / • v.27 no.4 • / • pp.809-813 • / • 2012 • We introduce the notion of R($g$, $g^{\prime}$)-continuity on generalized topological spaces, which is a strong form of ($g$, $g^{\prime}$)-continuity. We investigate some properties and relationships among R($g$, $g^{\prime}$)-continuity, ($g$, $g^{\prime}$)-continuity and some strong forms of ($g$, $g^{\prime}$)-continuity. ### On Fuzzy Almost c-Continuous Mappings • Baik, B.S.;Hur, K. • International Journal of Fuzzy Logic and Intelligent Systems • / • v.2 no.2 • / • pp.153-158 • / • 2002 • In this paper we introduce the concept of a fuzzy almost c-continuity and investigate some of its properties. ### On Fuzzy Almost c-Continuous Mappings (퍼지 Almost c-연속사상에 관하여) • Baik, B.S.;Hur, K. • Proceedings of the Korean Institute of Intelligent Systems Conference • / • / • pp.285-288 • / • 2002 • In this paper, we introduce the concept of a fuzzy almost c-continuity and investigate some of its properties.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8175064325332642, "perplexity": 6651.700200320166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153391.5/warc/CC-MAIN-20210727103626-20210727133626-00581.warc.gz"}
https://beniyiyimanne.com/nonton-the-npb/viewtopic.php?tag=cramer%27s-rule-for-4-variables-42b123	Best Male Hip Hop Artist 2020 Nominees, Epic Clarity Documentation, Hangi Pit Opening Hours, Masterchef Hokey Pokey Recipe, East African Federation, Ocean Quest Oc Birthday Party, The Relationship Between Law And Social Work Practice, New Liquor Prices In Andhra Pradesh, Condos For Sale In Brooklyn For $200,000, " /> Best Male Hip Hop Artist 2020 Nominees, Epic Clarity Documentation, Hangi Pit Opening Hours, Masterchef Hokey Pokey Recipe, East African Federation, Ocean Quest Oc Birthday Party, The Relationship Between Law And Social Work Practice, New Liquor Prices In Andhra Pradesh, Condos For Sale In Brooklyn For$200,000, " /> # cramer's rule for 4 variables cx1 + dx2. Know the naming convention. Rule. would then be: Copyright Then divide this determinant by the main one - this is one part of the solution set, determined using Cramer's rule. The reason for this will become apparent as we describe the method. Cramer’s Rule is an explicit formula for the solution of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. To solve only for z, Rules for 3 by 3 systems of equations are also presented. var now = new Date(); 1x Find detD, detD x, detD y, and detD z. x … I know how to solve it … of the system with the variables (the "coefficient matrix") How to Find Unknown Variables by Cramers Rule? The number of calculations required does increase for large systems, but the procedure is exactly the same, regardless of the size of the system. Sometimes, a more compact notation is used for determinants, as it is shown below: So, using the notation above, we would get these more compact formulas for the Cramer's rule: Let us have a visual way of understanding what is happening. Cramer's rule is a mathematical trick using matrices to solve a system of equations. In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. The beauty of Cramer's rule is that it applies exactly the same procedure, whether it is a 2x2 system or if it is a 10x10 system. Cramer’s Rule can be extended to systems of four or more linear equations in the same number of variables. and Dz Cramer's Rule For Solving a Linear System Of n Equations With n Variables. It involves the use of determinants to make very straightforward a task that otherwise would be really complicated, especially for larger systems. A X = B. you'll have to use some other method (such as matrix Question 18759: I have to solve this 4x4 matrix using Cramer's Rule: 4x + 0y + 3z - 2w = 2 3x + 1y + 2z - 1w = 4 1x - 6y - 2z + 2w = 0 2x + 2y + 0z - 1w = 1 Once I get to finding the determinants of the three 3x3 matrices, I am completly lost. Choose language... Python. Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. You just pick the variable Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 × 2 matrices. equations, Cramer's Rule is a handy way to solve for just one of the variables + 1z The system may be inconsistent Given a system of linear equations, Cramer's Rule is a handy way to solve for just one of the variables without having to solve the whole system of equations. = 3. 'November','December'); Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. See the image below: Now we see that $$x$$ and $$y$$ differ in what they have in the numerator. Find detD, detD x, detD y, and detD z. x … is to it. You may assume that you will always be given the same number of equations as there are number of variables, i.e. determinant, and divide by the coefficient determinant. The concept is the same. Repeat this operation for each variable. If before the variable in equation no number then in the appropriate field, enter the number "1". we get: Cramer's Rule says that x This app allows the user to solve the variables in the equations. However, pump B can pump water in or out at the same rate. Cramer's Rule provide and unequivocal, systematic way of finding solutions to systems of linear equations, no matter the size of the system. Below is the Step by Step tutorial of solved examples, which elaborates that how to solve a complex electric circuit and network by Cramer's rule. x + 2y 2z Solved Examples on Cramer’s Rule. Step 1: Find the determinant, D, by using the x, y, and z values from the problem. = 9/3 = 3, That's all there is to Cramer's 4 6 −60 The denominator is the determinant of the coefficient matrix and the numerator is the determinant of the matrix formed by replacing the column of the variable being solved by the column representing the constants. D x y 2x + 4y – 2z = -6 6x + 2y + 2z = 8 2x – 2y + 4z = 12. Given a system of linear In a square system, you would have an #nxx(n+1)# matrix.. teach Cramer's Rule this way, but this is supposed to be the point of Guidelines", Tutoring from Purplemath In addition to providing the results, this app provides all intermediate steps and details which can be a tremendous help with your homework and understanding of the concept. The denominator determinant (dn) is created from the … See the image below. = v, where a, b, c, d, u, and v are numbers with a, b, c, and d nonzero. Using Cramer’s Rule to Solve a System of Three Equations in Three Variables. Cramer's Rule says that x = D x ÷ D, y = D y ÷ D, and z = D z ÷ D. That is: x = 3 / 3 = 1, y = –6 / 3 = –2, and z = 9 / 3 = 3. A square matrix A matrix with the same number of rows and columns. The determinant of the coefficient matrix must be non-zero. Step 1: All 2x2 linear systems can be written in the following form: So then your first step is finding these values $$a_1, b_1, c_1$$ and $$a_2, b_2, c_2$$ for the system you want to solve. Now, in this case $$c_1 = 10, c_2 = 4$$, for the determinant used to compute $$x$$, we replace the previous matrix by changing the FIRST column: For the determinant used to compute $$y$$ we replace the previous matrix by changing the SECOND column: Therefore, the solution is $$x = 3$$, $$y = 1/2$$. Solving linear equations using elimination method. In terms of notations, a matrix is an array of numbers enclosed by square brackets while determinant is an array of numbers enclosed by two vertical bars. Do you solve like a normal 3x3 and just multiply the determinent found by the number on the outside? The algebra is as follows: ∣ A ∣ = a 1 b 2 c 3 + b 1 c 2 a 3 + c 1 a 2 b 3 − a 3 b 2 c 1 − b 3 c 2 a 1 − c 3 a 2 b 1. Observe that both $$x$$ and $$y$$ have the same determinant in the denominator. I can't go It's a simple method which requires you to find three matrices to get the values of the variables. The coefficients of that common matrix used in the denominator are directly derived from the coefficients that multiply $$x$$ and $$y$$ in the system. Step 2: Once you have the coefficients $$a_1, b_1, c_1$$ and $$a_2, b_2, c_2$$ you use the following formulas to solve for $$x$$ and $$y$$: In the above formula, where it says "det", it means the determinant of the corresponding matrix. Problem 5. Find the value of variable x. Cramer's Rule For Solving a Linear System Of n Equations With n Variables. The determinant D of the coefficient matrix is . = 3 Below is the Step by Step tutorial of solved examples, which elaborates that how to solve a complex electric circuit and network by Cramer's rule. That's all there is to Cramer's Rule. However, pump B can pump water in or out at the same rate. var months = new Array( We'll assume you're ok with this, but you can opt-out if you wish. ), 2x Cramer’s rule: In linear algebra, Cramer’s rule is an explicit formula for the solution of a system of linear equations with as many equations as unknown variables. ÷ D. (Please Recall that a matrix is a rectangular array of numbers consisting of rows and columns. The Cramers Rule App will help you solve a system of equations in two or three variables using the Cramer's Rule method. confuse you; the Rule is really pretty simple. The cardinal rule for investors is to understand what they are investing in by doing their homework, the "Mad Money" host says. Following the Cramer’s Rule, first find the determinant values of all four matrices. How long would it take each pump to fill the tank by itself ? (no solution at all) or dependent (an infinite solution, which may be Cramer's Rule for Linear Circuit Analysis \| Cramer's Rule Calculator Solved Example Today, we are going to share another simple but powerful circuit analysis technique which is known as "Cramer's Rule". We will now introduce a final method for solving systems of equations that uses determinants. A #2xx2# matrix would only have the coefficients of the variables; you need to include the constants of the equations. Unfortunately it's impossible to check this out exactly using Cramer's rule. = 0, Similarly, Dy For the matrix that goes in the denominator we use. We get: Cramer's Rule has a specific role in efficiently solving systems of linear equations. ... Variables and constants. = 0, you can't use Cramer's We have the left-hand side In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us. Solution: So, in order to solve the given equation, we will make four matrices. but Cramer's Rule was so much faster than any other solution method (and 4 Cramers Rule for 2x2 System. They don't usually Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables. Question 18759: I have to solve this 4x4 matrix using Cramer's Rule: 4x + 0y + 3z - 2w = 2 3x + 1y + 2z - 1w = 4 1x - 6y - 2z + 2w = 0 2x + 2y + 0z - 1w = 1 Once I get to finding the determinants of the three 3x3 matrices, I am completly lost. without having to solve the whole system of equations. Problem 5. We can then express x x and y y as a quotient of two determinants. you want to solve for, replace that variable's column of values in the Recall that a matrix is a rectangular array of numbers consisting of rows and columns. You can't divide by zero, so what does this mean? in Order \| Print-friendly X Y = X Y = Detailed Answer Two Linear 2 Variable Cramers Rule Example Problem: Example:[Step by Step Explanation] 9x + 9y = 13; 3x + 10y = 10 ; We need to compute three determinants: D, D x, and D y. VIDEO 0:54 00:54. + y + z Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 … = 0" means that The denominator consists of the coefficients of variables (x in the first column, and y in the second column). and z 3x3 Cramers Rule Calculator - Solving system of equations using Cramer's rule in just a click 3x3 CRAMER'S RULE CALCULATOR The calculator given in this section can be used to solve the system of linear equations with three unknowns using Cramer's rule or determinant method. 3x3 and 4x4 matrix determinants and Cramer rule for 3x3.notebook 4 April 14, 2015 Homework: Pg. It is named after Gabriel Cramer (1704-1752) who published the rule for arbitrary number of unknowns in 1750. 1. Evaluating each determinant (using the method explained here), See the image below, For $$y$$, you use the SAME matrix as the one in the denominator, only that you replace the SECOND column with the coefficients $$c_1$$ and $$c_2$$. 5 5 0 100% of 2 4 raulbc777. Answer. row operations) to Use Cramer's Rule to solve each for each of the variables. Solve the following system of 3x3 linear equations using Cramer's Rule. [Date] [Month] 2016, The "Homework One straightforward way to solve for x1 and x2 is to isolate one of the variables in one of the equations and substitute the result into the other equation. Using Cramer’s Rule to Solve a System of Three Equations in Three Variables. number + 1900 : number;} Cramer’s rule is most useful for a 2-x-2 or higher system of linear equations. 'June','July','August','September','October', = Dz ÷ D. Now that we can find the determinant of a $$3 × 3$$ matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. Cramer's Rule with Questions and Solutions Cramer's rule are used to solve a systems of n linear equations with n variables using explicit formulas. x2 = ( v − cx1 )/ d. I am using Cramer's rule to solve a system of linear equations but don't know how to find the determinant of a $4\times 4$ matrix. be the determinant of the coefficient matrix of the above system, and As a way of remembering the rule, think of this: For $$x$$, you use the SAME matrix as the one in the denominator, only that you replace the FIRST column with the coefficients $$c_1$$ and $$c_2$$. Cramer's to solve for just one single variable. What if the coefficient determinant Cramer's Rule is a technique used to systematically solve systems of linear equations, based on the calculations of determinants. Python. If the main determinant is zero the system of linear equations is either inconsistent or has infinitely many solutions. Accessed A i. Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for $$2 × 2$$ matrices. Since is a matrix of integers, its determinant is an integer. var date = ((now.getDate()<10) ? z = 0 4 6 −60 Use Cramer's Rule to give a formula for the solution of a two equations/two unknowns linear system. Do you solve like a normal 3x3 and just multiply the determinent found by the number on the outside? system of equations: 2x Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. coefficient determinantwith answer-columnvalues in x-column, 2x x + 2y + z + 4z = 0 Notations The formula to find the … Cramer’s Rule with Two Variables Read More » 4-4 Determinants and Cramer’s Rule You can use the determinant of a matrix to help you solve a system of equations. God knows I needed the extra time). Python. Choose language... Python. Example 2A ContinuedStep 2 Solve for each variable by replacing the coefficients ofthat variable with the constants as shown below.The solution is (4, 2). That is: x page. 0+y-z+0=0. Using Cramer’s Rule to Solve a System of Three Equations in Three Variables. Rule. Use Cramer's Rule to give a formula for the solution of a two equations/two unknowns linear system. Cramer’s Rule for 2×2 Systems. Cramer's Rule is a technique used to systematically solve systems of linear equations, based on the calculations of determinants. Fundamentals. Cramer’s Rule; Cramer’s Rule is a method of solving systems of equations using determinants. We first start with a proof of Cramer's rule to solve a 2 by 2 systems of linear equations. Cramer’s Rule for a 3×3 System (with Three Variables) In our previous lesson, we studied how to use Cramer’s Rule with two variables.Our goal here is to expand the application of Cramer’s Rule to three variables usually in terms of \large{x}, \large{y}, and \large{z}.I will go over five (5) worked examples to help you get familiar with this concept. For the system the … For $$x_2$$ we change the second column by $$(c_1, ..., c_n)$$, for $$x_3$$ we change the third column, and so on. Holt Algebra 2 4-4 Determinants and Cramer’s Rule The coefficient matrix for a system of linear equations in standard form is the matrix formed by the coefficients for the variables in the equations. Don't let all the subscripts and stuff (Use Cramer’s rule to solve the problem). It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. expressed as a parametric solution such as "(a, That's all there Solution (4) A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. Example 1: Solve the given system of equations using Cramer’s Rule. can work many kinds of magic. determinants of two matrices 5x + 4y = 28 Find the determinant of the 3x 2y = 8 matrix where one of the variables coefficient are replaced with the answers. © Elizabeth Stapel 2004-2011 All Rights Reserved. In terms of notations, a matrix is an array of numbers enclosed by square brackets while determinant is an array of numbers enclosed by two vertical bars. document.write(accessdate); The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. Solution: So, in order to solve the given equation, we will make four matrices. Cramer's rule are used to solve a systems of n linear equations with n variables using explicit formulas. 1z I first find the coefficient determinant. If your pre-calculus teacher asks you to solve a system of equations, you can impress him or her by using Cramer’s rule instead of using a graphing calculator. by replacing the third column of values with the answer column: Then I form the quotient 3x3 CRAMER'S RULE CALCULATOR . Let \displaystyle \|A\|= {a}_ {1} {b}_ {2} {c}_ {3}+ {b}_ {1} {c}_ {2} {a}_ {3}+ {c}_ {1} {a}_ {2} {b}_ {3}- {a}_ {3} {b}_ {2} {c}_ {1}- {b}_ {3} {c}_ {2} {a}_ {1}- {c}_ {3} {a}_ {2} {b}_ {1} ∣A∣ = a. . Seki wrote about it first in 1683 with his Method of Solving the Dissimulated Problems.Seki developed the pattern for determinants for $2 \times 2$, $3 \times 3$, $4 \times 4$, and $5 \times 5$ matrices and used them to solve equations. Using Cramer’s Rule to Solve Three Equations with Three Unknowns – Notes Page 3 of 4 Example 2: Use Cramer’s Rule to solve4x −x+3y−2z=5 −y 3z= 8 2x+2y−5z=7. To find whichever variable you want (call it "ß" or "beta"), just evaluate the determinant quotient D ß ÷ D. (Please don't ask me to explain why this works. Available from is zero? You get the idea. Stapel, Elizabeth. + 2y Return to the Degrees of Freedom Calculator Paired Samples, Degrees of Freedom Calculator Two Samples. 5 Key Points. Cramer's Rule for Linear Circuit Analysis \| Cramer's Rule Calculator Solved Example Today, we are going to share another simple but powerful circuit analysis technique which is known as "Cramer's Rule". x y Cramer: My top 4 rules for owning stocks. is that you don't have to solve the whole system to get the one value return (number < 1000) ? I The concept of the matrix determinant appeared in Germany and Japan at almost identical times. For finding the value variable x, the first step is to find the determinant … (fourdigityear(now.getYear())); and the right-hand side with the answer values. a + 3, a 4)"). construct a matrix of the coefficients of the variables. The system must have the same number of equations as variables, that is, the coefficient matrix of the system must be square. Cramer’s Rule for a 2×2 System (with Two Variables) Cramer’s Rule is another method that can solve systems of linear equations using determinants. How do I use Cramer's rule to solve a system with 4 variables? Step 1: Find the determinant, D, by using the x, y, and z values from the problem. Linear Systems of Two Variables and Cramer’s Rule. = 0. Cramer's Rule For Two Linear Equations in Two Variables & Formula Calculation. [Tex]D_1 = \begin {vmatrix} d_1 & b_1 & c_1\\ d_2 & b_2 & c_2\\ d_3 & b_3 & c_3\\ \end {vmatrix} [/Tex] [Tex]D_3 = \begin {vmatrix} a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & … Using Cramer’s Rule to Solve a System of Three Equations in Three Variables. Using Cramer’s Rule to Solve a System of Two Equations in Two Variables. Typically, solving systems of linear equations can be messy for systems that are larger than 2x2, because there are many ways to go around reducing it when there are three or more variables. This website uses cookies to improve your experience. Let's use the following Cramer's Rule states that: x = y = z = Thus, to solve a system of three equations with three variables using Cramer's Rule, Arrange the system in the following form: a 1 x + b 1 y + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3; Create D, D x, D y, and D z. Solves systems of equations in 2 or 3 variables "0" : "")+ now.getDate(); Since is a matrix of integers, its determinant is an integer. How to Find Unknown Variables by Cramers Rule? 5 5 0 100% of 2 4 raulbc777. X1 X2 X3 = X1 X2 X3 = X1 X2 X3 = Detailed Answer Cramer rule for systems of three linear equations [ Cramers Rule Example Problem: Step by Step Explanation ] Example; 3x1 + 4x2 - 3x3 = 5; 3x1 - 2x2 + 4x3 = 7; 3x1 + 2x2 - x3 = 3; In matrix form Ax = b [ a1 a2 a3 ] x = b this is; Cramer Rules / Formula: Matrix Calculator 2x2 Cramers Rule. Purplemath. The value of each variable is a quotient of two determinants. Functions: What They Are and How to Deal with Them, Normal Probability Calculator for Sampling Distributions. The system is: x-y-z-w=0. As you can see, the determinant in the denominator is the same, and the one in the numerator is obtained by changing the first column with $$(c_1, ..., c_n)$$ for $$x_1$$. So, assume that $$x_1, x_2, ..., x_n$$ are the variables (the unknowns), and we want to solve the following n x n system of linear equations: In order to solve for $$x_1, x_2, ..., x_n$$, we will use the following determinant on the denominator: And so on. 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http://aircraftdata.net/view/6353804134712348691563/what-is-the-difference-between-oat-rat-tat-and-sat	# What is the difference between OAT, RAT, TAT, and SAT? Lnafziger • What is the difference between OAT, RAT, TAT, and SAT? Lnafziger When I first started flying jet aircraft, I found different instruments that measure temperature in different ways and it was quite confusing. What is the difference between Outside Air Temperature, Ram Air Temperature (RAT), Total Air Temperature (TAT), Static Air Temperature (SAT), and any that I might have missed? • As you measure temperature moving at high velocities, your outside thermometer will measure a higher temp than what is actually outside (what a non-moving thermometer would get). That's because as the air rams into your thermometer it gets a little bit compressed, and that makes it heat up a little bit. Amazingly, some smart people have even calculated how much that "ram rise" is, and you can actually compensate (well, in theory) from the indicated temperature to calculate what the actual outside air temp is: $$Ram~Rise=SAT\times0.2\times{M}^2$$ Problem is, not all instruments will compress that air in the same way, and not all of them will pick up this Ram Rise entirely. So they will publish a $K$, a ram-rise-coefficient (also called recovery factor) meaning how much of from the theoretical Ram Rise they actually pick up. (OAT/SAT) Static Air Temperature = noted as $T_S$, actual temperature of the air outside, the same your thermometer will pick up if you were not moving (say, in a balloon?) (RAT) Ram Air Temperature = noted as $T_M$, actual measured temperature by your instrument. $$RAM=SAT+k\times{Ram~Rise}$$ (TAT) Total Air Temperature = noted as $T_T$, total temperature as measured by an instrument with $K$-coefficient of $1$. $$TAT=SAT+Ram~Rise$$ Tags • When I first started flying jet aircraft, I found different instruments that measure temperature in different ways and it was quite confusing. What is the difference between Outside Air Temperature, Ram Air Temperature (RAT), Total Air Temperature (TAT), Static Air Temperature (SAT), and any that I might have missed? • My electronic E6B does calculations for both planned and actual true airspeed (yielding different results). I was told the difference has to do with assumptions about temperature, but the manual doesn't give any details. What is the difference between the two? • I'm from Brazil, and here we use the West/East rule, so we use an odd flight level when we fly between 0/360 - 179, and when we fly between 180 - 359 we fly in an even flight level. But what should you do in other countries? Where I can find those rules? I've heard that in Europe it's totally different, and that in some countries in Asia they use meters, instead of feet. Where can I find this information? • , segment, or route between the radio fixes defining the airway, segment, or route. (Both quotes taken straight from the Pilot/Controller Glossary) The only difference in language is the bit about 22 miles from a VOR. Therefore, when you're within those 22 miles, there's no practical difference between a MEA and a MOCA, right? If that's true, why is there an 1800 foot difference between...As we all know from our instrument training, the MOCA is: MINIMUM OBSTRUCTION CLEARANCE ALTITUDE (MOCA)- The lowest published altitude in effect between radio fixes on VOR airways, off-airway • Inspired by a discussion in chat. Most GA piston singles are powered by either Lycoming or Continental engines. The engine designs used by both manufacturers are broadly similar (4-cycle, horizontally-opposed, gasoline-powered, air-cooled), and they're both generally available with either carburetors or fuel injection, but I know they're not "identical products". What are some of the differences between the designs used by the two manufacturers, and what practical implications do those choices have for pilots? • Are there any LSA aircraft that are IFR certified? A LSA would be the perfect private commuter plane for an instrument rated private pilot. If not, what are the most cost effective airplanes that are IFR certified? Is there anything cheaper than a C172? Just to be clear: I am asking about aircraft that are IFR certified, meaning that they can be flown IFR in IMC conditions. I am also aware that there is a huge difference between different countries with regard to aircraft certification so my question is mainly about the U.S. • in a commercial product? For those of you who don't know what the Boeing winds are, I found this description of their software product on am informal message board (not related to Boeing): PC WindTemp..., with or without altitude changes. Calculates average enroute wind and temperature or any reliability (quantile) between 50% and 99%. Months or seasons, reliabilities, altitudes, and speeds may...There are various services that use world-wide Boeing Winds for forecast wind data which can be used to calculate an approximate flight time between two locations. They usually have best case, worst • Some aircraft come with a Pilot Operating Handbook and some come with an Aircraft Flight Manual. Why the different name, and is there a difference between them? • Non-precision instrument approaches generally have altitude restrictions which get lower when you get closer to the airport. I always figured these restrictions were AMSL using the current altimeter setting, not compensating for temperature. Some have heard the mnemonic that mountains are higher come wintertime, which basically means that colder weather make your altimeter read higher than you actually are (or, as most pilots prefer to think, you're lower than what your altimeter reads) Have a look at this VOR approach into Newark Most altitude restrictions are a minimum level, so • This question is somewhat related to this other one. I listened to this exchange between a helicopter and Newark. http://www.youtube.com/watch?v=oHNvXPbZ7WI The helicopter wants to land at Newark. The controller tells the helicopter to remain clear of the Class B. I'm aware that the controllers must give clearance to operate in certain classes of airspace, and the helicopter wasn't granted clearance to do so. Why was the helicopter denied (as far as can be deduced)? What should the pilot have done differently, either to get clearance to land at Newark or to anticipate not being able to? 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http://mathhelpforum.com/algebra/107154-fraction-help.html	1. ## fraction help In a polling booth, total number of voters is 1575, of which 0.4 part are male voters. If a candidate gets 0.6 part of male voters and 0.4 part of female voters, then find out how many votes did the candidate get? (A) 189 (B) 756 (C) 378 (D) 630 (E) 945 2. Originally Posted by sri340 In a polling booth, total number of voters is 1575, of which 0.4 part are male voters. If a candidate gets 0.6 part of male voters and 0.4 part of female voters, then find out how many votes did the candidate get? (A) 189 (B) 756 (C) 378 (D) 630 (E) 945 We have 1575 voters, out of which 40% are male and 60% are female. First we want to calculate exactly how many male and female voters there are. To calculate 40% of 1575 we will simply take 10% (which is 157.5) and multiply it by 4: $0.4 \cdot 1575 = \frac{4}{10}\cdot 1575 = 4 \cdot 157.5 = 630$ so we have 630 male voters and 945 female voters. Now, use the same way to calculate 60% of 630, 40% of 945 and add them together to get the final result. 3. Originally Posted by sri340 In a polling booth, total number of voters is 1575, of which 0.4 part are male voters. If a candidate gets 0.6 part of male voters and 0.4 part of female voters, then find out how many votes did the candidate get? (A) 189 (B) 756 (C) 378 (D) 630 (E) 945 From the male voters, the candidate received: (0.4)(0.6) of 1575 From the female voters, the candidate received: (0.6)(0.4) of 1575 The total votes received by this candidate: $0.4 \times 0.6 \times 2 \times 1575$ .	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6032259464263916, "perplexity": 2303.636069362239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718285.69/warc/CC-MAIN-20161020183838-00508-ip-10-171-6-4.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/integration-by-parts.373585/	# Integration by parts 1. Jan 28, 2010 ### Zhalfirin88 1. The problem statement, all variables and given/known data $$\int \frac{x^3e^{x^2}}{(x^2+1)^2}$$ 3. The attempt at a solution Well, this problem is hard, so I thought to use u = x3ex2 so du = x2ex2(3+2x2) dx and dv = (x2+1)-2 then v = -2(x2+1)-1 Please check v though to make sure my algebra is right. so then using the by parts formula: $$\int \frac{x^3e^{x^2}}{x^2+1} = \frac{x^3e^{2x}}{2(x^2+1)} - \int \frac{x^2e^{x^2}(3+2x^2)}{2(x^2 +1)}$$ But where do I go from here? 2. Jan 28, 2010 ### Staff: Mentor I wouldl start with u = x2/(x2 + 1)2 and dv = xex2. No guarantee that this will get you anywhere, but a useful strategy is to make dv the most complicated expression that you can integrate. 3. Jan 28, 2010 ### l'Hôpital Notice the fact that you have x^3 on the top. Try a u-sub with u = (1 + x^2). After that, use parts cleverly and you'll get your answer. 4. Jan 28, 2010 ### Dick This is one of those cases where there are several plausible alternatives for picking the parts, but only one works. You just have to mess around. Hint: if you pick u=x^2e^(x^2) then du=(2xe^(x^2)+2x^3e^(x^2))dx=2(1+x^2)e^(x^2)xdx. Notice the (1+x^2) factor in du. You need that. 5. Jan 28, 2010 ### LCKurtz You don't get that v for your choice of dv. You probably meant $$\frac{(x^2+1)^{-1}}{-1}$$ but that's still wrong because you would need a 2x in the in the numerator of dv. Similar Discussions: Integration by parts	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8414114117622375, "perplexity": 1830.0374183321164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948513784.2/warc/CC-MAIN-20171211164220-20171211184220-00785.warc.gz"}
https://en.m.wikibooks.org/wiki/Basic_Algebra/Solving_Equations/Solving_Equations_with_Variables_on_Both_Sides_of_the_Equation	# Basic Algebra/Solving Equations/Solving Equations with Variables on Both Sides of the Equation ## VocabularyEdit Variable - a letter (${\displaystyle a}$ -${\displaystyle z}$ ) that takes the place of a number. Equation - an example would be like ${\displaystyle 8y-3=1+10y}$ (The answer is ${\displaystyle y=-2}$ ) ## LessonEdit NOTE: WHAT YOU DO TO ONE SIDE YOU MUST DO TO THE OTHER SIDE! NO EXCEPTIONS! 1) do the distributive property. 2) Combine like terms on both sides. 3) add/subtract numbers next to a variable on both sides. 4) divide by the number next to the variable on both sides. 5) The answer should look like: ${\displaystyle x=20}$ or ${\displaystyle 20=x}$ . (Note: the variables and numbers may vary in your answer.) ## Example ProblemsEdit A simple problem: ${\displaystyle 2(x+5)=5(x-10)}$ <---Problem ${\displaystyle 2x+10=5x-50}$ <---Distributive Prop. ${\displaystyle 2x+10-2x=5x-2x-50}$ <---Subtract the variables with numbers next to them. ${\displaystyle 10=3x-50}$ <---This is what you're left with. ${\displaystyle 10+50=3x-50+50}$ <---get rid of the 50 by subtracting ${\displaystyle 60=3x}$ <---This is what you're left with. ${\displaystyle 60/3=3x/3}$ <---Get rid of the three by dividing by three ${\displaystyle 20=x}$ <---This is your answer. ## Practice GamesEdit Put links here to games that reinforce these skills Purplemath.com: http://www.purplemath.com/modules/index.htm ## Practice ProblemsEdit (Note: put answer in parentheses after each problem you write) Practice Problem #1: ${\displaystyle 3x-2=x+16}$ Answer: ${\displaystyle x=9}$ Practice Problem #2: ${\displaystyle 2y=15-y}$ Answer :${\displaystyle y=5}$ Practice Problem #3: ${\displaystyle {\frac {z-5}{-3}}=9+3.4z}$ Answer: ${\displaystyle z=-1{\frac {27}{28}}}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 20, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20815370976924896, "perplexity": 2552.677264329859}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948522999.27/warc/CC-MAIN-20171213104259-20171213124259-00104.warc.gz"}
https://mathandmultimedia.com/tag/slope-of-a-function/	Increasing and Decreasing a in the linear function y = ax In the previous post, we have learned the effect of the sign of a in the linear function $y = ax$. In this post, we learn the effect of increasing and decreasing the value of a. Since we have already learned that if $a = 0$, the graph is a horizontal line, we will discuss 2 cases in this post: $a > 0$, and $a < 0$. Case 1: a > 0 Let us consider several cases of the graph of $y = ax$ where $a > 0$. Let the equation of the functions be $f(x) = \frac{1}{2}x$, $g(x) = x$, and $h(x) = 2x$ making $a = \frac{1}{2}$, $1$, and $2$, respectively. As we can see from the table, for the same $x > 0$, the larger the slope, the larger its corresponding y-value. This means that for $x = 1$, the point $(1,h(1))$ is above $(1, g(1))$ and that the point $(1,g(1))$ is above $(1,f(1))$. We can say that as $x$ increases, $h$ is increasing faster than $g$, and $g$ is increasing faster than the increase in $f$» Read more GeoGebra Tutorial 17 – Functions, Tangent Lines and Derivatives This is the 17th tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, I strongly suggest that you read the GeoGebra Essentials Series. In this tutorial, we are going to use slider control a, b, c, d and e and graph the function f(x) = ax4 + bx3 + cx2 + dx + e. Figure 1 We then construct a line tangent to the function and passing through point A and trace the graph of the point whose x-coordinate is the x-coordinate of A, and whose y-coordinate as the slope of the tangent line. We compute for the derivative of f(x), and see if there is a relationship between the trace and the derivative. If you want to follow the this tutorial step by step, you can open the GeoGebra window. Before following the tutorial, you may want to see the final output. Slope Concept 2 – Slope of the Graph of a Linear Function Note: This is the second part of the the Slope Concept Series. The second and third articles are Part I – Understanding the Basic Concepts of Slope and Part III – Slopes of Vertical and Horizontal Lines. *** In the Understanding the Basic Concepts of Slope post, we have discussed that slope is described as rise over run. In this post, we are going to show that the slope of a straight line is constant. To get the slope of a straight line or a segment, we determine two points on the line, say A and B, draw a horizontal line through point A and a vertical segment through point B. We then determine the intersection of the two line segments and name it C as shown in Figure 1. Angle C is a right angle since BC is a vertical segment and AC is a horizontal segment. Figure 1 - A line containing points A and B. The slope of the line containing AB is $\frac{BC}{AC}$. If we determine two more points, say D and E on the line, and do the process mentioned above, we can come up with triangle DEF right angled at F as shown in Figure 2. In terms of DEF, the slope of the line containing points D and E is $\frac{EF}{DF}$. Since the line containing DE and AB is practically the same line, we have learned from high school mathematics that their slopes must be equal. Hence, the following relationship holds: $\frac{BC}{AC} = \frac{EF}{DF}$. Why is this so? Figure 2 - Triangle ABC and DEF with their hypotenuse contained on the line. To show that the slope of the line is constant, we must show that $\frac{BC}{AC} =\frac{EF}{DF}$. Proof That the Slope of a Straight Line is Constant From Figure 2, BC is parallel to EF, since they are both vertical segments. Similarly, DF is parallel to AC, since they are both horizontal segments. By the Parallel Postulate, we can consider AB as a transversal of the two pairs of parallel segments. We can see that angles DEF and ABC are congruent since they are corresponding angles. Angles C and F are also congruent since they are both right angles. Hence, by AA similarity, ABC is similar to DEF. Since the ratio of the corresponding sides of a similar triangles are equal, it follows that $\frac{BC}{AC} =\frac{EF}{DF}$. From the above proof, we have shown that the slope of a straight line, or the slope of the graph of a linear function, is constant.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 28, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.95329350233078, "perplexity": 199.95974262138316}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302723.60/warc/CC-MAIN-20220121040956-20220121070956-00558.warc.gz"}
http://mathhelpforum.com/advanced-algebra/122400-finite-group-question.html	1. ## Finite Group Question Let $G$ be a finite group and let $H\neq G$ be a subgroup of $G$. Show that $G\neq \cup_{a\in G} aHa^{-1}$. 2. Originally Posted by paulk Let $G$ be a finite group and let $H\neq G$ be a subgroup of $G$. Show that $G\neq \cup_{a\in G} aHa^{-1}$. Hint 2: Conjugation of a subgroup preserves order. 3. Let $[G:H]=n\geq 2$. Then we have $\|\cup_{a\in G} aHa^{-1}\|\leq n(\|H\|-1)+1.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9972517490386963, "perplexity": 207.18948050258612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280242.65/warc/CC-MAIN-20170116095120-00490-ip-10-171-10-70.ec2.internal.warc.gz"}
https://www.hepdata.net/search/?q=abstract%3A%22baryon+production%22&collaboration=WA85	Showing 1 of 1 results #### Measurement of the Omega / Xi production ratio in central S - W interactions at 200-A-GeV/c The collaboration Abatzis, S. ; Andrighetto, A. ; Antinori, F. ; et al. Phys.Lett. B347 (1995) 158-160, 1995. Inspire Record 402400 Strange baryon and in particular multi-strange baryon production is suggested to be a useful probe in the search for quark gluon plasma formation in heavy ion collisions. We have measured the (Ω − + Ω + ) (Ξ − + Ξ + ) production ratio to be 0.8±0.4 at central rapidity and ϱ T > 1.6 GeV/c. 0 data tables match query	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9928291440010071, "perplexity": 9388.571768235182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584331733.89/warc/CC-MAIN-20190123105843-20190123131843-00125.warc.gz"}
http://mathhelpforum.com/advanced-algebra/171282-isomorphism-theorems.html	# Math Help - isomorphism theorems 1. ## isomorphism theorems Show that $ \mathbb{Z}_n \backslash \mathbb{Z}_m \cong \mathbb{Z}_{n\backslash m} $ Im trying to show it using the first isomorphism theorem , but i have trouble with constructing the function (maybe i didnt understand that well the theorem D: ) thanks! 2. nevermind I found the function :P	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9718620777130127, "perplexity": 895.8512855151448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042985647.51/warc/CC-MAIN-20150728002305-00265-ip-10-236-191-2.ec2.internal.warc.gz"}
https://ww2.mathworks.cn/help/symbolic/sym.latex.html	# latex LaTeX form of symbolic expression ## Syntax chr = latex(S) ## Description example chr = latex(S) returns the LaTeX form of the symbolic expression S. ## Examples collapse all Find the LaTeX form of the symbolic expressions x^2 + 1/x and sin(pix) + phi. syms x phi chr = latex(x^2 + 1/x) chr = '\frac{1}{x}+x^2' chr = latex(sin(pix) + phi) chr = '\phi +\sin\left(\pi \,x\right)' Find the LaTeX form of the symbolic array S. syms x S = [sym(1)/3 x; exp(x) x^2] S = $\left(\begin{array}{cc}\frac{1}{3}& x\\ {\mathrm{e}}^{x}& {x}^{2}\end{array}\right)$ chr = latex(S) chr = '\left(\begin{array}{cc} \frac{1}{3} & x\\ {\mathrm{e}}^x & x^2 \end{array}\right)' Since R2021b Compute several symbolic matrix variables, and then find their LaTeX form. Create 3-by-3 and 3-by-1 symbolic matrix variables. syms A 3 matrix syms X [3 1] matrix Find the Hessian matrix of ${X}^{T}AX$. Derived equations involving symbolic matrix variables appear in typeset as they would be in textbooks. f = X.'AX f = ${X}^{\mathrm{T}} A X$ H = diff(f,X,X.') H = ${A}^{\mathrm{T}}+A$ Generate the LaTeX form of the symbolic matrix variables f and H. chrf = latex(f) chrf = '{\textbf{X}}^{\mathrm{T}}\,\textbf{A}\,\textbf{X}' chrH = latex(H) chrH = '{\textbf{A}}^{\mathrm{T}}+\textbf{A}' Modify generated LaTeX by setting symbolic preferences using the sympref function. Generate the LaTeX form of the expression $\pi$ with the default symbolic preference. sympref('default'); chr = latex(sym(pi)) chr = '\pi ' Set the 'FloatingPointOutput' preference to true to return symbolic output in floating-point format. Generate the LaTeX form of $\pi$ in floating-point format. sympref('FloatingPointOutput',true); chr = latex(sym(pi)) chr = '3.1416' Now change the output order of a symbolic polynomial. Create a symbolic polynomial and set 'PolynomialDisplayStyle' preference to 'ascend'. Generate LaTeX form of the polynomial sorted in ascending order. syms x; poly = x^2 - 2x + 1; sympref('PolynomialDisplayStyle','ascend'); chr = latex(poly) chr = '1-2\,x+x^2' The preferences you set using sympref persist through your current and future MATLAB® sessions. Restore the default values by specifying the 'default' option. sympref('default'); For $x$ and $y$ from $-2\pi$ to $2\pi$, plot the 3-D surface $y\mathrm{sin}\left(x\right)-x\mathrm{cos}\left(y\right)$. Store the axes handle in a by using gca. Display the axes box by using a.Box and set the tick label interpreter to latex. Create the x-axis ticks by spanning the x-axis limits at intervals of pi/2. Convert the axis limits to precise multiples of pi/2 using round and get the symbolic tick values in S. Display the ticks by setting the XTick property of a to S. Create the LaTeX labels for the x-axis by using arrayfun to apply latex to S and then concatenating $. Display the labels by assigning them to the XTickLabel property of a. Repeat these steps for the y-axis. Set the x- and y-axes labels and the title using the latex interpreter. syms x y f = y.sin(x)-x.cos(y); fsurf(f,[-2pi 2pi]) a = gca; a.TickLabelInterpreter = 'latex'; a.Box = 'on'; a.BoxStyle = 'full'; S = sym(a.XLim(1):pi/2:a.XLim(2)); S = sym(round(vpa(S/pi2))pi/2); a.XTick = double(S); a.XTickLabel = strcat('$',arrayfun(@latex, S, 'UniformOutput', false),'$'); S = sym(a.YLim(1):pi/2:a.YLim(2)); S = sym(round(vpa(S/pi2))*pi/2); a.YTick = double(S); a.YTickLabel = strcat('$',arrayfun(@latex, S, 'UniformOutput', false),'$'); xlabel('x','Interpreter','latex'); ylabel('y','Interpreter','latex'); zlabel('z','Interpreter','latex'); title(['$' latex(f) '$for$x$and$y$in$[-2\pi,2\pi]\$'],'Interpreter','latex') ## Input Arguments collapse all Input, specified as a symbolic number, variable, vector, array, function, expression, or symbolic matrix variable (since R2021b).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9518007040023804, "perplexity": 9456.805963701285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587963.12/warc/CC-MAIN-20211026231833-20211027021833-00705.warc.gz"}
https://brilliant.org/discussions/thread/problem-writing-party-february-29-march-13/	Problem Writing Party: February 29 - March 13 Hello Brilliant! Brilliant's third Problem Writing Party was a great success as always. Thanks to all the contributions from Rohit, Efren, Sravanth, Agnishom, Hobart, Refaat, Chan Lye, Hjalmar, Pranshu, Manmeswar, Soumava, Racchit, Lakshya, Chinmay, Worranat, Anish, Arjen, Rajdeep, Nihar, Hobart, Jason, Pranshu, Margaret,, Lee, Jason, Vignesh and the countless others of you who helped! This week, we were able to create a 5 new quizzes. Here are the quizzes that you all helped create: New Brilliant Challenge Quizzes As always, we're going to continue showing off the hard work these members of the community put in over the last week, so Brilliant will featuring these quizzes in topic listings and social media. Now that those announcements have been made, we're moving on to our fourth Problem Writing Party! How it Works The party starts right now (February 29th, 2016) and will last for the next two weeks. Throughout the two weeks, we will be focusing on writing awesome problems for the topics listed in quizzes that need your help on the publish page. To join, submit as many problems as you want to these listed topics. At the end of the party, Brilliant staff will be picking the best 5-10 problems for each topic. These problems will then be immortalized and formed into a challenge quiz. If we pick your problem, then you can brag to your friends because it will be displayed on Brilliant forever! Your problem has a better chance of being selected if you include a graphic (when appropriate) and a solution. Use this note to 1. Ask questions about the party or brainstorming ideas from Brilliant staff. 2. Share links to great relevant problems. 3. Bounce your ideas off each other to help formulate the best problem you can. To be selected, problems you submit should be fun and challenging to a student in levels 1-3 for the given topic. -------->This Party's Topic Listing<-------- The topics of problem submission for this party can be found by navigating over to the Brilliant publish page and checking under the quizzes that need your help section. Just click the contribute button next to the topic you want make a submission to. Happy writing and continue to party on! Note by Sandeep Bhardwaj 3 years, 1 month ago MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$ Sort by: $\Large \text{The topics for this party can be found in the link below}$ Topics for the problem writing party Remeber: Your problem has a better chance of being selected if you include a graphic (when appropriate) and a solution. - 3 years, 1 month ago And don't forget to use the latex (where appropriate) in the problem and the solution. - 3 years, 1 month ago Can I use Daum application to post a problem or to write solution? - 3 years, 1 month ago @Sridhar Sri Yes you can, for now. But it would be great if you post the problem/solution into latex and text. If you need any guidance on latex, Daniel's LaTex Guide will help out gently. Thanks! Waiting for you problems. $$\ddot \smile$$ - 3 years, 1 month ago the time-keeper :-a problem i made SHM physical pendulum - 3 years, 1 month ago @Rohith M.Athreya Very nice problem. Thanks for your contribution. Looking for your innovative problems of more topics. :) - 3 years, 1 month ago Hey there people :) Here is my set.I will keep adding more problems. - 3 years, 1 month ago Wow! Great, you've created a complete set which is awesome. Thanks a lot! Waiting for new problems to be added to the GREAT SET of yours. $$\ddot \smile$$ - 3 years, 1 month ago Thank you sir! :D Just a result of Fiitjee teaching me SHM :P - 3 years, 1 month ago Great. All the best for your excellence at all the subjects you're studying! - 3 years, 1 month ago Thank you so much sir ! :D - 3 years, 1 month ago Computer Science $$\rightarrow$$ Sorts - 3 years, 1 month ago I knew I could count on you for some good computer science problems Zeeshan. ;) - 3 years, 1 month ago This is a great contribution. I loved the problems. Thanks! $$\ddot \smile$$ - 3 years, 1 month ago Here are some problems on Floor Functions, hope you like it . This Floor is slippery and I love 333, but this is ... . - 3 years, 1 month ago Nice problems Anish. The first one is yummy. Thanks! @Anish Harsha - 3 years, 1 month ago You're welcome, sir ! :D - 3 years, 1 month ago My submission: Floor and ceiling functions: One, Two, Three Capacitors: One, Two, Three, Four, Five, Six Transformers: Step-up or Step-down?, Transformer DC 3D Coordinate Geometry: Three I will update this comment if I share more problems. This is the set containing all of my problems for this party - Set - 3 years, 1 month ago Here's my first one for Geometry-3D Coordinates: Box in the Air. Hope it's not too easy. ;) - 3 years, 1 month ago Cool problem with a cool name. Thanks Dr. Warm! @Worranat Pakornrat - 3 years, 1 month ago You're welcome. Surely more will come. ;) - 3 years, 1 month ago See my problem, Battery Charger on the topic Capacitors and Transformers. :) - 3 years, 1 month ago Looks great! :) - 3 years, 1 month ago I would like to contribute this problem I'm Floored for the party - 3 years, 1 month ago Nice Problem @neelesh vij Thanks :) - 3 years, 1 month ago Here are mine: ( I will keep updating) Great Circles (recursion) Only 0,1,2,3,4 (recursion) Is it Algebra at all? (3-D Geometry) Another recursion problem (recursion) Fat Subsets (recursion) Emergency to return (recursion) - 3 years, 1 month ago Cool Problem. Thanks! - 3 years, 1 month ago Thanks - 3 years, 1 month ago @Sandeep Bhardwaj Do check out my problems. :) I added 5 more of them. I will add more after my chemistry exam on the 9th. - 3 years, 1 month ago Nice problems. @Soumava Pal :) - 3 years, 1 month ago I like trees - 3 years, 1 month ago I posted a problem on recurrence relations a few days ago. - 3 years, 1 month ago Wow! This one is a bit too hard for what we're looking for in recurrence relations, but I think you've got a knack for writing some good problems. Got any fun, easy ones tucked away? - 3 years, 1 month ago Sure, will try to come up with some fun but accessible problems. I will update the post above if I manage to write some nice ones. :) - 3 years, 1 month ago Here's mine contribution on capacitors - 3 years, 1 month ago Nice one. It would be great if you post an awesome solution to your problem to get it featured. Thanks! $$\ddot \smile$$ - 3 years, 1 month ago Here is a problem on recurrence. - 3 years, 1 month ago Nice problem @Svatejas Shivakumar It would be great if you post an awesome solution to your problem to get it featured. Thanks! $$\ddot \smile$$ - 3 years, 1 month ago Here are my entries.............. Capacitors - 1 Capacitors - 2 3D - 1 3D - 2 F n C - 1 SHM - 1 SHM - 2 Enjoy Solving!!! - 3 years, 1 month ago Great problems @Manmeswar Patnaik Thanks $$\ddot \smile$$ - 3 years, 1 month ago Thanks Sir!!.....Please check out the other entries............ @Sandeep Bhardwaj - 3 years, 1 month ago Here comes my new one on Combinatorics-Recursion: Fractal Squares. Hope you'd like it. ;) - 3 years, 1 month ago I don't see any Computer Science question.!! Hurry up, you computer scientists! :) - 3 years, 1 month ago Electricity and Magnetism $$\rightarrow$$ Circuits $$\rightarrow$$ Resistors $$\rightarrow$$ Equivalent Resistance $$\rightarrow$$ FUN :) - 3 years, 1 month ago @Zeeshan Ali The circuit looks like the first letter of my name. I loved it. Thanks! :) - 3 years, 1 month ago Another level of question on Combinatorics-Recursion: Fractal Cubes. This one may be a little crazy. ;) - 3 years, 1 month ago Why are all of your problems so awesome? - 3 years, 1 month ago Thanks! Maybe because I got inspired. ;) - 3 years, 1 month ago This is my fun question on Algebra-Floor & Ceiling Functions: Which day was it?. And for Combinatorics-Variance: Who's the tallest of them all?. It's kind of straight forward, but I still like it. ;) - 3 years, 1 month ago The first problem is really fun to check your patience on calculations ;). FYI you should use "\left \lfloor ... \right \lfloor" if the expression inside the floor doesn't fit well. Thanks! - 3 years, 1 month ago Oh, OK. Probably I was kind of lazy at night. Thanks for your advice. ;) - 3 years, 1 month ago Here are the questions from my stock on Combinatorics-Introduction to Recursion: Snip! Snip! Have fun! ;) - 3 years, 1 month ago You've got a nice stock. Congratulations! - 3 years, 1 month ago Thank you. Hope to come up with new ones, too. ;) - 3 years, 1 month ago What are the topics that we need to put questions? - 3 years, 1 month ago You can go to the Publish Page to get the topics needed for contribution. Looking forward to have awesome problem from your side. Thanks! :) - 3 years, 1 month ago Thanks. - 3 years, 1 month ago See My Problem Roots Finding - 3 years, 1 month ago Nice problem @Rishabh Deep Singh I'm quite glad to see you contributing such wonderful problems. Thanks $$\ddot \smile$$ - 3 years, 1 month ago Thanks Friend. - 3 years, 1 month ago My Problem on Oscillations A classical mechanics problem by Rishabh Deep Singh - 3 years, 1 month ago Again a nice problem! - 3 years, 1 month ago I have included over 27 mind blowing problems in my set 26 mind turning problems - 3 years, 1 month ago But do they have anything to do with the topics listed in this problem writing party? - 3 years, 1 month ago First thing, why are you after my life? 2 thing 50% of the problems are related to the topics, and the last thing I did not consider making a new set for just posting here - 3 years, 1 month ago Rishabh, First of all, please be a bit polite. Andrew is staff, and he knows exactly what is good for this wiki writing party. He only wants to make this Community a better place. He is not after your life. - 3 years, 1 month ago Hmmmmm, ok - 3 years, 1 month ago I viewed your set and it has very nice problems. But sorry to say , I can't find a single problem related to the topics in this problem writing party. You might review the topics here and contribute awesome problems which would force Andrew to select! - 3 years, 1 month ago Ok!, I will do it, but I have my pre-boards and finals so I will come back to brilliant only after that - 3 years, 1 month ago Here are some problems (I will keep updating this as I post more) Floor and ceiling functions: Curve sketching: Dueling functions Where do we start? - 3 years, 1 month ago I knew I could count on you to post a handful Jonas. These look great! - 3 years, 1 month ago Thank you guys for the support! :) - 3 years, 1 month ago Nice set! I particularly enjoyed Pixelated Poke Ball, being a huge pokemon fan myself. - 3 years, 1 month ago Here's my introductory problem for Simple Harmonic Motion(Waves- Doppler Effect) Shake it to my Beats - 3 years, 1 month ago Beautiful! More Simple Harmonic Motion problems would be great if you can think of some fun ones! :) - 3 years, 1 month ago More coming. These are for floor/ceiling fxns: - 3 years, 1 month ago Now these look like some mind benders! I'm excited for these. - 3 years, 1 month ago Thanks! :) - 3 years, 1 month ago I was writing the solution for your last two problems... But then I realised it was very tough to explain in solution and tiring latex work is also required!! - 3 years, 1 month ago This is my new question of Calculus-Curve Sketching: Rise within Roots. Enjoy! ;) - 3 years, 1 month ago All right! Looks good. We could use some great curve-sketching problems. - 3 years, 1 month ago Thank you. Glad you'd like it. ;) - 3 years, 1 month ago Here are my problems(I would keep updating): - 3 years, 1 month ago Nice problem @Samarth Agarwal FYI if you want to link a problem, you can do it this way $$\to$$ [Name of the problem] (link) I'm linking your problem in this comment. Here it goes: A simple recursion for 2016 and 2018 Excited to see more problems from your side. Let me know if you've any queries. Thanks $$\ddot \smile$$ - 3 years, 1 month ago Thanx sir for creating a link to my problem, I would try to contribute more but I have exams till 18th March :( - 3 years, 1 month ago Ahh, never mind. Focus on your exams and you can contribute to the further Problem Writing Party. Thanks $$\ddot \smile$$ All the best for your exams! - 3 years, 1 month ago Thank you Sir.... :) - 3 years, 1 month ago My question on Capacitance An electricity and magnetism problem by Rishabh Deep Singh - 3 years, 1 month ago My first question. - 3 years, 1 month ago Nice Problem. But the expression needs to be made more clear. In the numerator, does "2015/2016" represents the power of "2015"? - 3 years, 1 month ago This time my contribution will be poor, but here is something for Floor and Ceiling Functions - 3 years, 1 month ago Here's one for floor and ceiling functions: - 3 years, 1 month ago Nice problem @Brandon Monsen It would be great if you post an awesome solution to your problem to get it featured. Thanks! $$\ddot \smile$$ - 3 years, 1 month ago Just posted one! I'll make some edits for coherence but the meat of it is there. - 3 years, 1 month ago Can I post the questions till 13 March - 3 years, 1 month ago Yes , of course! Looks like @Andrew Ellinor can definitely count upon your numerous contributions for many topics for this party :) - 3 years, 1 month ago @Deepanshu Dhruw I see that you post good problems. Excited to see your problems for this party. And yeah you can post the problems till 13 March. I'm waiting. :) - 3 years, 1 month ago Thanks - 3 years, 1 month ago Electricity and Magnetism $$\rightarrow$$ Circuits $$\rightarrow$$ Capacitors and Transformers $$\rightarrow$$ Never mind - 3 years, 1 month ago Try my new question Limit It is out of given topics. - 3 years, 1 month ago Hi Rishabh, this is a great question! Do you think you could post problems that are focused on the topics for this problem writing party? - 3 years, 1 month ago Ok I will. - 3 years, 1 month ago Computer Science $$\rightarrow$$ Sorts - 3 years, 1 month ago Nice ones. These will be great for a quiz. Thanks! - 3 years, 1 month ago My second question. - 3 years, 1 month ago Is the ground floor left empty for rent? - 3 years, 1 month ago Maybe XD - 3 years, 1 month ago I added a few problems in logarithm inequalities. What do you think of them? Calculus of parametric equations can be tricky. I might try just parametric equations. - 3 years, 1 month ago I loved all the problems. It would be great if you post the problems from the topic list of this party which can be seen at the Publish Page. Thanks! - 3 years, 1 month ago My contlibusion: Andlew's Dleam Sorry , I can contribute only one problem this time :( - 3 years, 1 month ago No problem, Nihar! You can contribute more problems to the following parties. Night ploblem. ;) - 3 years, 1 month ago Thyankss!! - 3 years, 1 month ago You'll definitely enjoy solving the following problems: - 3 years, 1 month ago @Zeeshan Ali I like the beautiful circuits in your problems. Thanks! - 3 years, 1 month ago Another new one on Geometry-3D coordinates: Box in the Air (Part 2). Have fun! ;) - 3 years, 1 month ago I would like to fit myself diagonally in that box which will be a great exercise. :P - 3 years, 1 month ago Haha...would you like to be a cat in the box? ;) - 3 years, 1 month ago No way man! Then I will be stuck forever there if no one comes to help me out. :P How will I use Brilliant then? - 3 years, 1 month ago By using local box network maybe. Lol... - 3 years, 1 month ago My new Question on Inductor + Capacitor - 3 years, 1 month ago Good problem. But it is recommended and it's great that you don't include the text into the latex. Use latex only for mathematical expressions. Can you please make edits to your problem accordingly? Thanks! - 3 years, 1 month ago I will do it in my upcoming Questions. - 3 years, 1 month ago Great! I will convert this problem into text and latex necessarily. Thanks! - 3 years, 1 month ago @Rishabh Deep Singh I've edited the above problem of yours. Can you please check if it's all good? - 3 years, 1 month ago Thanks Sir It is perfect - 3 years, 1 month ago Here is another problem, for 3D Coordinate Geometry - 3 years, 1 month ago @Hjalmar Orellana Soto Tangent in three dimensions is coolest. Thanks! :) - 3 years, 1 month ago This one for starters: Triangle and Line. - 3 years, 1 month ago @Arjen Vreugdenhil Yummy starter! :) - 3 years, 1 month ago the time-keeper :-a problem i made SHM physical pendulum - 3 years, 1 month ago @Rohith M.Athreya Great Great Great! Looking for more of your innovative problems for other topics of the Problem Writing Party. :) - 3 years, 1 month ago a problem on floor functions that i made :) - 3 years, 1 month ago Very nice problem. I've edit this to make the latex clearer, see if that's good? - 3 years, 1 month ago Yeah thanks! looks much more clearer now. :) - 3 years, 1 month ago My third question. - 3 years, 1 month ago @Lee Care Gene Nice building of ceilings. I enjoy your problems a lot. :) - 3 years, 1 month ago Here's my recursion problem: Vole crossing This might be too difficult for a quiz, but I think it's a pretty neat problem. - 3 years, 1 month ago @Kerry Soderdahl This problem can be defintely used for some Level 4-5 quiz. Thanks! Looking for more great problems. :) - 3 years, 1 month ago @Andrew Ellinor I just had another problem created, but I could not format it correctly. Could you please help? the problem means "the floor function of the absolute value of $$\frac {2x-3}{4}$$". What floor is it? - 3 years, 1 month ago Fixed Margaret! Check out the LaTeX to learn something new. :) Also, I like this problem so much that it's definitely going into a quiz. - 3 years, 1 month ago thanks:D - 3 years, 1 month ago A floor/ceiling problem. I'm going to try to think of some for curve sketching, but so far the only ones I have done before/thought of require you to actually sketch the curves... • Twentieth integral [For this one, I'm aware that it's listed under calc, but it is a floor function problem and doesn't really require calculus other than the geometric concept of an integral] - 3 years, 1 month ago @Hobart Pao Lovely problem. Yeah, I agree with the point you made. It involves the use of many concepts likely graph of floor function, geometrical interpretation of integral, arithmetic progression sum, finding the roots of a quadratic equation. :P - 3 years, 1 month ago Thanks for the encouragement @Kunal Verma, @Sandeep Bhardwaj! - 3 years, 1 month ago Cool problem.c: - 3 years, 1 month ago My new problem on Curve Sketching Inspired From Inspiration - 3 years, 1 month ago You post the problems which are quite helpful specifically for the guys who are preparing for JEE. That's great. Thanks! - 3 years, 1 month ago Sir i am preparing for JEE-ADV 2016 and my board exams are finished. Now i am Free. - 3 years, 1 month ago That's great. Now you can post more problems for the party. ;) - 3 years, 1 month ago I need to prepare for JEE Mains so i need to study for that . Any suggestions? - 3 years, 1 month ago You can check out my set: Jee Queries with the Best Solutions. And if you have any specific questions, let me know and I'll help you out. - 3 years, 1 month ago Thanks Sir. - 3 years, 1 month ago This is my new question on Geometry-3D coordinates: Simple Section. Enjoy! ;) - 3 years, 1 month ago Enjoyed! ;) It was enjoyable. - 3 years, 1 month ago Thanks! I enjoyed writing it, too. ;) - 3 years, 1 month ago a problem on recursions at loggerheads with logarithmic recursions - 3 years, 1 month ago Great one, thanks! - 3 years, 1 month ago 3D coordinate geometry: It's a bird! It's a plane! It's not superman... - 3 years, 1 month ago My new one on Algebra-Ceiling & Floor Functions: Hit the ceiling! Hit the floor!. Hope it's not too easy. ;) - 3 years, 1 month ago - 3 years, 1 month ago It's more likely to be chosen if you give it a solution. ;) - 3 years, 1 month ago Sorry this is a late entry please improve it PS:- I don't have solution - 3 years, 1 month ago Your problem is too general... State what kind of sentence it is! (It looks quadratic, but is it?) - 3 years, 1 month ago	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.953433632850647, "perplexity": 5777.098504440251}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578596541.52/warc/CC-MAIN-20190423074936-20190423100854-00083.warc.gz"}
https://physics.stackexchange.com/questions/250512/is-a-weyl-fermion-its-own-antiparticle	# Is a Weyl fermion its own antiparticle? Majorana fermions are their own antiparticles, and Weyl fermions are just Majorana fermions without mass. However, I haven't been able to find any source that says whether a Weyl fermion is its own antiparticle. My suspicion is that the question is meaningless. My impression is that "X is its own antiparticle" means "the mass eigenstates are mapped to themselves under charge conjugation". In the absence of a mass, we can choose any basis we want, so the question doesn't have a well-defined answer. Then again, we can unambiguously pick out particles and antiparticles for a massless complex scalar field using the $U(1)$ charge. But I'm not sure if such a charge exists for Weyl spinors. Is a Weyl fermion its own antiparticle? Generally, what does 'being your own antiparticle' mean? • Absence of mass, or even charge, does not make the notion of antiparticle meaningless, and a Weyl fermion is not its own antiparticle: the antiparticle is distinguished by opposite helicity. See neutrino vs. antineutrino, for instance physics.upenn.edu/~pgl/neutrino/now2000/node4.html – udrv Apr 20 '16 at 15:28 • On a 2nd thought, you might want to look up "Majorana neutrino", which indeed is its own antiparticle. These slides give a quick overview of the neutrino problem and the difference between Weyl and Majorana neutrinos: desy.de/~troms/teaching/SoSe12/slides/neutrinoII_b.pdf – udrv Apr 20 '16 at 15:39 • @udrv I agree with what you said, but on the level of the Lagrangian, isn't a Weyl fermion just a Majorana neutrino with zero mass? How can the property of 'being its own antiparticle' change discontinuously? – knzhou Apr 20 '16 at 16:39 • @urdv I think there are two slightly different notions of 'antiparticle' being used interchangably; this is related to my other question. It's fine for Majorana, but for Weyl the two notions give different answers. – knzhou Apr 20 '16 at 18:36 • I think you are onto a subtle point. There is a fundamental difference between a massless Weyl fermion and a massive Majorana one, and it has to do with the distinction between chirality and helicity. In essence, a massless fermion has no rest frame, its helicity cannot be flipped by a boost that reverses its direction of motion, and in this case chirality = helicity. Otoh, a massive fermion always has a rest frame and chirality $\neq$ helicity. A chirally left-handed Majorana fermion can have right-handed helicity, but remains chirally left-handed under boosts. – udrv Apr 21 '16 at 3:50 Consider a single Weyl fermion, which has equation of motion $$\sigma^\mu \partial_\mu \psi = 0.$$ Just like the Dirac equation, the Weyl equation is linear in momenta and hence has negative energy solutions. We then perform the usual procedure to regard these as positive energy solutions (e.g., historically, we might imagine them as holes in the Dirac sea). These two classes of solutions, $\psi$ and $\psi^c$, are related by charge conjugation. • A Majorana spinor has a mass term, which couples $\psi$ and $\psi^c$ together. Then it's natural to consider the mass eigenstates, which are $\psi \pm \psi^c$. These states are mapped to themselves by charge conjugation, which is why Majorana fermions are their own antiparticles. • A Weyl spinor may be coupled to a gauge field in the usual way. Then $\psi$ and $\psi^c$ have conjugate gauge charge (for example, opposite $U(1)$ charges). Then it's natural to consider eigenstates that have definite charge, in which case our basis is $\psi$ and $\psi^c$. These are mapped to each other by charge conjugation, so Weyl spinors are not their own antiparticles.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8225793242454529, "perplexity": 331.51385011970405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145533.1/warc/CC-MAIN-20200221142006-20200221172006-00540.warc.gz"}
https://weiminwang.blog/2017/08/05/credit-card-fraud-detection-2-using-restricted-boltzmann-machine-in-tensorflow/	# Credit card fraud detection 2 – using Restricted Boltzmann Machine in TensorFlow In my previous post, I have demo-ed how to use Autoencoder for credit card fraud detection and achieved an AUC score of 0.94. This time, I will be exploring another model – Restricted Boltzmann Machine – as well as its detailed implementation and results in tensorflow. RBM was one of the earliest models introduced in the world of deep learning. There have been many successful use cases of RBM in areas such as dimensionality reduction, classification, collaborative filtering, feature learning as well as anomaly detection. In this tutorial, we will also use the unsupervised way – without feeding labels to model – to train on our data, and will achieve a slightly improved result over auto-encoder! The whole post is divided into three parts below: 1. Introduction to RBM 2. Implementation in TensorFlow 3. Results and interpretation All codes can be found in github. ### Overview of the data set I will be using the exact same credit card data here. So please go to my previous post if you would like to see more about it. Also, you can download the data from Kaggle here if you want. ## RBM – a quick introduction There are many good online resources that offer either brief or in-depth explanation of RBM: 1. https://www.youtube.com/watch?v=FJ0z3Ubagt4 & https://www.youtube.com/watch?v=p4Vh_zMw-HQ – Best youtube explanations to RBM in my opinion. 2. https://deeplearning4j.org/restrictedboltzmannmachine – Gives a very intuitive and easy-to-understand ways of RBM. 3. http://deeplearning.net/tutorial/rbm.html – Theories + Theano implementation Basically, a RBM is network consists of two layers – the Visible Layer and Hidden layer. There are symmetric connections between any pair of nodes from Visible and Hidden layers, and no connections within each layer. For majority cases, both hidden and visible layers are binary-valued. There are also extensions with visible layer being Gaussian, and hidden layer being Bernoulli. The latter will be our case for fraud detection. (Since our input data will be normalized into mean of 0 and std of 1) When signals propagate from visible to hidden, the input layer (i.e. the data sample) will be multiplied by the matrix W, added with bias vector b of hidden layer, and finally go through the sigmoid function to be squashed to be within 0 and 1, which are also the probabilities for each hidden neuron to be on. However, it is very important to keep the hidden states binary (0 or 1 for each), rather than using the probabilities itself. And only during the last update of the gibbs sampling should we use probabilities for hidden layer, which we will talk about later on. During backward pass, or reconstruction, the hidden layer activation will become the input, which is multiplied by the same matrix W, added with visible biases, and then will either go through the sigmoid function (for Bernoulli visible), or being sampled from a multivariate Gaussian distribution (for Gaussian visible), as below: Intuitively, we can understand that the model is adjusting its weights, during training, such that it could best approximate the training data distribution p with its reconstruction distribution q, as below: We first define our so-called Energy Function E(x, h)as well as the joint probability p(x, h)given any pair of visible and hidden layers, as below: In the above equations, x is the visible layer, h is the hidden layer activations, W, b and c are the matrix, hidden bias and visible bias, respectively. So basically, for any pair of x and h, we are able to calculate E(x, h). Its value is a scalar. And the higher the value of Energy, the lower the p(x, h). ### 1) How to detect fraud with RBM? From the energy function, we are able to derive the equation below for so-called Free Energy of x: This Free Energy, F(x), which is also a scalar, is exactly what we will need for test data, from which we will use the distribution to detect anomalies. The higher the free energy, the higher the chance of x being a fraud. ### 2) How to update model parameters? To update our parameters W, b and c, we use below equations combined with SGD (the alpha is learning rate): The left part, $h(x^{(t)})x^{(t)^T}$, is easy to calculate, as x(t) is just the training sample, and h(x(t)) is simply below: So the outcome is simply a vector multiplication which gives same shape as W. Okay, that’s easy! But how to calculate $h(\widetilde{x})\widetilde{x}^{T}$? The answer is to use gibbs sampling: We start with a training sample x(t), and sample each hj by first calculating: Then we draw a value from a uniform distribution [0, 1], and if the drawn value is smaller than the one calculated above, we assign 1 to hj, otherwise we assign 0. And then we do this for each h Next, we sample visible layer xk, by using previously sampled hidden layer as input, with a similar equation below: Here, we directly use the results from sigmoid , without sampling it into states of 0 and 1, to get visible layer. However, we will do this step slightly different if the input data, or visible layer x, is a Gaussian distribution. If Gaussian, we will sample the x vector using Gaussian with mean μ = c + WTh as well as identity covariance matrix. This part is fully implemented in the code where you can check for verification (under equation sample_visible_from_hidden). We will do this k steps, which is referred to as Contrastive Divergence, or CDk. After last step k, we will use the sampled visible layer as $\widetilde{x}$, together with the last hidden probabilities as $h(\widetilde{x})$. Please note that what we are using here is the probabilities, not the sampled 0 and 1 states, for $h(\widetilde{x})$ In summary, the whole process looks like this: • Start with training sample x – $x^{(t)}$ • Sample h from input x – $h(x^{(t)})$ • Sample x from h • Sample h from x • Sample x from h – $\widetilde{x}$ • Sample h from x – $h(\widetilde{x})$ In practice, using k = 1 can give a good result. Until here, we have covered the whole process of model updating. ### 3) How to tune hyper-parameters? We will stick to data-driven approach. Split the data into training and validation sets, and train the model on training set, while evaluate the performance on validation. Start the hidden layer with a smaller dimension than input layer(e.g. 5, 10), set the learning rate to be a small value (such as 0.001), monitor the validation data set reconstruction error (not the actual error against labels) The reconstruction error is basically the mean squared of the difference between predicted $\widetilde{x}$ and the actual data x, averaged over the entire mini-batch. If the reconstruction error stops decreasing, that would be a sign for early-stopping. However, a comprehensive guide to tuning RBM is fully covered in Geoffrey Hinton’s notes, which you are encouraged to take a look at. ## Coding the RBM The code was modified from here, which was an excellent implementation in TensorFlow. I only added in a few changes to its implementation: 1. Implemented Momentum for faster convergence. 2. Added in L2 regularisation. 3. Added in methods for retrieving Free Energy as well as Reconstruction Error in validation data. 4. Simplified the code a bit by removing parts of tf summaries (originally not compatible with tf version 1.1 above) 5. Added in a bit utilities such as plotting training loss Basically, the code is a sklearn – style RBM class that you can directly use to train and predict. ## Training and Results ### 1) Training and validation We split our data by transaction time into training and validation by 50 – 50, and train our model on the training set. ```TEST_RATIO = 0.50 df.sort_values('Time', inplace = True) TRA_INDEX = int((1-TEST_RATIO) df.shape[0]) train_x = df.iloc[:TRA_INDEX, 1:-2].values train_y = df.iloc[:TRA_INDEX, -1].values test_x = df.iloc[TRA_INDEX:, 1:-2].values test_y = df.iloc[TRA_INDEX:, -1].values ``` After we train the model, we will calculate the Free Energy of val set, and visualize the distributions for both fraud as well as non-fraud samples. ### 2) Data pre-processing Since the data are already PCA transformed, we will only need to standardize them with z-score to get mean of 0 and standard deviation of 1, as below: ```cols_mean = [] cols_std = [] for c in range(train_x.shape[1]): cols_mean.append(train_x[:,c].mean()) cols_std.append(train_x[:,c].std()) train_x[:, c] = (train_x[:, c] - cols_mean[-1]) / cols_std[-1] test_x[:, c] = (test_x[:, c] - cols_mean[-1]) / cols_std[-1] ``` Please note that you need to calculate statistics using training set only, as I did above, instead of on the full data set (training and val combined). After that, we will fit the data using model with Gaussian visible layer. (This is the Gaussian – Bernoulli RBM, since the hidden layer is still binary-valued) ### 3) Visualization of results It is clearly seen that fraud data has Free Energy much more uniformly distributed than non-fraud data. If you calculate the AUC score (Area Under the ROC Curve) on val set, you will get a score around 0.96! ### 4) Real time application To enable it as real time fraud detector, we will need to find a threshold based on validation data set. This can be done by trading off the precision & recall curves as below (e.g. a value of 100 might give a relative good balance): ### 5) A further interpretation at the val AUC score of 0.96 There’s another way to intuitively look at the AUC score: The val data’s fraud percentage is around 0.16%; For example, if we choose top 500 transactions ranked by model’s free energy predictions, the number of fraud is around 11.82% … So, precision increases from 0.16% to 11.82% in the top 500 … Again, please find in github for details on code implementation and notebook. ## *References Exercises: Try to use Reconstruction Error in place of Free Energy as fraud score, and compare the result to using Free Energy. This is similar to what I did previously for autoencoder tutorial. ## 10 thoughts on “Credit card fraud detection 2 – using Restricted Boltzmann Machine in TensorFlow” 1. The visualization shows that normal energy and fraud energry are different. However, it does not make sense in case unsupervised learning because you make the energy distribution graph by using label data column test_y ==1. So, If we don’t use test_y, how can we decide a cut off point of fraud and normal energy? Like 1. You are supposed to collect validation data for threshold selection. In reality, you can also work with your fraud ops team to create feedback loop for the task. Like 2. Thanks for the article and sample code. Can I re-use your code? I understand that your code was derived from Gabriele Angeletti’s or https://github.com/blackecho/Deep-Learning-TensorFlow Can you confirm that it is also released under the MIT license? Like 1. sure. Feel free to reuse it Like 3. Why do you compute positive = tf.matmul(tf.transpose(visible), hidden_states) if visible_unit is binary. I think the right formula is positive=tf.matmul(tf.transpose(visible), hidden_probs). Like 4. Hey, do you know how you would code in Persistent Contrast Divergence instead of Contrast Divergence? Like 5. Just wondering – is the implementation of CD actually PCD ? Like 6. How do you calculate ROC or AUC when the test_cost only provides the free energy and not a 0 or 1 classification?? this is the exact code line “fpr, tpr, _ = roc_curve(test_y, test_cost)” where test_cost = model.getFreeEnergy(test_x).reshape(-1) can you explain to me in this case how do you calculate the FPR and such ?? Like 1. To calculate AUC you only need scores / predicted probabilities (test_cost) so that you can rank the data. And you will need ground truth binary labels (0 or 1) which is provided in test_y. You can’t convert test_cost into 0 or 1 to calculate AUC, you need the raw scores. Please refer to online / Google for AUC calculation details. Like 7. Is Free Energy as reconstruction error when I use RBM to anomaly detection? Can I use MSE(prediction values – true values) as reconstruction error when I use RBM to anomaly detection? Thank you Like	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 10, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5829635858535767, "perplexity": 1264.3568996636345}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573827.2/warc/CC-MAIN-20190920030357-20190920052357-00137.warc.gz"}
https://atarnotes.com/forum/index.php?topic=180336.msg1087291	August 24, 2019, 05:50:10 am ### AuthorTopic: Mundane life updates (Read 41650 times) Tweet Share 0 Members and 1 Guest are viewing this topic. #### technodisney • MOTM: AUG 2018 • Forum Obsessive • Posts: 350 • Master Procrastinator • Respect: +442 ##### Re: Mundane life updates « Reply #390 on: November 14, 2018, 04:12:13 pm » +7 Most of the people in my EngLang class are starting to worry because they haven't put much study in. That is making me feel a little better as I have been putting some effort in. My Informatics 3/4 SAT Guide 2018 Methods, BusMan, EngLang, Informatics, VET IT technodisney's VCE Journal 2019 Cert IV Live Production and Technical Services RMIT (City Campus) technodisney's journey into Live Theatre The Disney Nerd needs to get fit The more you like yourself, the less you are like anyone else, which makes you unique. ~ Walt Disney #### beatroot • B-RICE • National Moderator • Part of the furniture • Posts: 1340 • Respect: +1308 ##### Re: Mundane life updates « Reply #391 on: November 17, 2018, 04:40:45 pm » +7 I just watched ‘The Crimes of Grindewald’ and I honestly don’t know how I feel about it 😂 Which will hold greater rule over you? Your fear or your curiosity? #### geek123456 • Trendsetter • Posts: 141 • Respect: +20 ##### Re: Mundane life updates « Reply #392 on: November 17, 2018, 06:18:07 pm » 0 I just watched ‘The Crimes of Grindewald’ and I honestly don’t know how I feel about it like in a good way or a bad way ;[ ? It is really hard being realistic on the Atar calculator when all those 50s are just a keyboard tap away mod edit: merged posts « Last Edit: November 17, 2018, 06:37:35 pm by geek123456 » Healer at St. Mungo's Hospital #### beatroot • B-RICE • National Moderator • Part of the furniture • Posts: 1340 • Respect: +1308 ##### Re: Mundane life updates « Reply #393 on: November 17, 2018, 06:32:48 pm » +7 like in a good way or a bad way ;[ ? Bad way- plot was confusing, characters were meh compared to the first film, editing was off, cinematography made me dizzy, Johnny Depp as Grindewald. They're really milking the Fantastic Beasts series to earn more revenue- which sucks because this entire series could be easily summed up in three films. Which will hold greater rule over you? Your fear or your curiosity? #### geek123456 • Trendsetter • Posts: 141 • Respect: +20 ##### Re: Mundane life updates « Reply #394 on: November 17, 2018, 06:35:58 pm » +1 Bad way- plot was confusing, characters were meh compared to the first film, editing was off, cinematography made me dizzy, Johnny Depp as Grindewald. They're really milking the Fantastic Beasts series to earn more revenue- which sucks because this entire series could be easily summed up in three films. sad to hear... I was really looking forward to see it ;( Healer at St. Mungo's Hospital #### katie,rinos • HSC Moderator • Posts: 852 • Respect: +847 ##### Re: Mundane life updates « Reply #395 on: November 17, 2018, 06:36:06 pm » +6 I just watched ‘The Crimes of Grindewald’ and I honestly don’t know how I feel about it 😂 I just finished watching it and was so disappointed!! Class of 2017 (Year 12): Advanced English, General Maths, Legal Studies, Music 1, Ancient History, History Extension, Hospitality 2018-2022: B Music/B Education (Secondary) [UNSW] #### tomatosauce • Forum Obsessive • Posts: 246 • Masters degree in Procrastination • Respect: +113 ##### Re: Mundane life updates « Reply #396 on: November 17, 2018, 06:55:03 pm » +3 19.6° feels like 25°... perfect day! why do I have exam revision? Someone told me to grow up yesterday. I immediately banned that person from ever riding my unicorn! Class of 2020 graduate... if I ever get there! :D Year 11 - 2019: English, Math Methods, VET Business 3/4, Legal Studies and Accounting Year 12 - 2020: English, VET Accounting 3/4, Math Methods, Legal Studies and VCE Accounting 2021: Chartered Accountant #### hums_student • MOTM: SEP 18 • Forum Obsessive • Posts: 314 • "Clean your room" - JBP • Respect: +417 ##### Re: Mundane life updates « Reply #397 on: November 18, 2018, 06:29:11 pm » +10 It only took me eleven months to figure out that various forum tabs are collapsible Spoiler Gonna save me from doing a lot of scrollin' in the future. "You know what I consider the worst disability of all? Procrastination and laziness." . VCE (98.35 ATAR) – Literature, History, Politics, Chinese, Methods, Chemistry UniMelb – B Arts (History, Economics) / M Teaching #### jazcstuart • MOTM: SEP 18 • Forum Obsessive • Posts: 231 • Respect: +178 ##### Re: Mundane life updates « Reply #398 on: November 18, 2018, 08:21:50 pm » +6 It only took me eleven months to figure out that various forum tabs are collapsible Spoiler Gonna save me from doing a lot of scrollin' in the future. What, how did I not know this??! You have also saved me a lot of scrolling, thanks haha HSC 2017 - Mathematics, Music 1 HSC 2018 - English (Advanced), Maths Extension 1, Chemistry, Geography, Earth and Environmental Science 2019 - B Renewable Energy Engineering @ University of Newcastle #### technodisney • MOTM: AUG 2018 • Forum Obsessive • Posts: 350 • Master Procrastinator • Respect: +442 ##### Re: Mundane life updates « Reply #399 on: November 19, 2018, 07:56:10 am » +10 I posted a meme I made on a Facebook group last night and I woke up to it being posted on another Facebook group. I don’t know whether to be happy it’s good enough to be stolen or sad that I wasn’t credited. My Informatics 3/4 SAT Guide 2018 Methods, BusMan, EngLang, Informatics, VET IT technodisney's VCE Journal 2019 Cert IV Live Production and Technical Services RMIT (City Campus) technodisney's journey into Live Theatre The Disney Nerd needs to get fit The more you like yourself, the less you are like anyone else, which makes you unique. ~ Walt Disney #### Joseph41 • It's Over 9000!! • Posts: 9408 • Oxford comma and Avett Brothers enthusiast. • Respect: +6106 ##### Re: Mundane life updates « Reply #400 on: November 19, 2018, 09:14:14 am » +5 I posted a meme I made on a Facebook group last night and I woke up to it being posted on another Facebook group. I don’t know whether to be happy it’s good enough to be stolen or sad that I wasn’t credited. The life of a meme lord. Take it as a compliment! A noble spirit embiggens the smallest man. Yeet ahoy! Apply for a 2019/2020 TuteSmart scholarship! #### PhoenixxFire • VIC MVP - 2018 • Victorian Moderator • ATAR Notes Superstar • Posts: 2637 • Bad puns are how eye roll • Respect: +1948 ##### Re: Mundane life updates « Reply #401 on: November 19, 2018, 03:09:42 pm » +7 My new thing is listening to songs sped up. 2019: B Environment and Sustainability/B Science @ ANU #### lm21074 • MOTM: JAN 19 • Trendsetter • Posts: 103 • Respect: +99 ##### Re: Mundane life updates « Reply #402 on: November 19, 2018, 04:18:42 pm » +7 Today I was doing an English practice exam in class. The substitute teacher we had wrote, "The time is..." on the board and got up every couple of minutes or so to change it. #### S200 • Part of the furniture • Posts: 1097 • Yeah well that happened... • Respect: +238 ##### Re: Mundane life updates « Reply #403 on: November 19, 2018, 04:48:06 pm » +2 Today I was doing an English practice exam in class. The substitute teacher we had wrote, "The time is..." on the board and got up every couple of minutes or so to change it. Did you introduce them to a clock? Carpe Cerevisi $\LaTeX$ - $e^{\pi i }$ #ThanksRui! - #Rui$^2$ - #Jamon10000 5233718311 #### lm21074 • MOTM: JAN 19 • Trendsetter • Posts: 103 • Respect: +99 ##### Re: Mundane life updates « Reply #404 on: November 19, 2018, 06:20:46 pm » +5 Did you introduce them to a clock? Sadly, there was no clock to introduce them to, which is why they kept the time in such a weird way. I mean, shouldn't there be a clock in every classroom? During one part of the exam this makeshift "clock" from 12:47 to 12:50 in the spam of 30 seconds.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.16575460135936737, "perplexity": 26224.43684033687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318986.84/warc/CC-MAIN-20190823192831-20190823214831-00222.warc.gz"}
https://www.mycoursehelp.com/QA/if-the-phillips-curve-equation-is-repres/137967/1	Create an Account Home / Questions / If the Phillips curve equation is represented by equation : \pi _(t)-\pi _(t-1)=(m+z)-\alp... If the Phillips curve equation is represented by equation : \pi _(t)-\pi _(t-1)=(m+z)-\alpha u_(t) , with the change in the inflation rate equal to m plus z minus the product of alpha and u, the natur If the Phillips curve equation is represented by equation : , with the change in the inflation rate equal to m plus z minus the product of alpha and u, the natural rate of unemployment will be equal to... a. m + z. b. m + z - alpha. c. 0. d. none of the above Apr 16 2021 View more View Less	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9330264329910278, "perplexity": 2232.9253478763167}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487613380.12/warc/CC-MAIN-20210614170602-20210614200602-00539.warc.gz"}
https://grandmaster.colorado.edu/copper/2016/abstract/gu_xianming____093481/	Next: Bibliography Xian-Ming Gu Nested Krylov subspace methods based on the Hessenberg procedure for shifted linear systems Institute of Mathematics and Computing Science University of Groningen Nijenborgh 9 P O Box 407 9700 AK Groningen The Netherlands [email protected] T.-Z. Huang B. Carpentieri In many computational applications it is required to solve simultaneously shifted linear systems of the form (1) where is a non-Hermitian matrix, is the unknown vector and is the right-hand side, both of them are complex vectors of length . For instance, shifted linear systems arise in the solution of PageRank problems and the rational approximation of the action of a matrix function to a vector, i.e. , where is an arbitrary vector with length . In principle, a sequence of shifted linear systems (1) can be solved simultaneously at the cost of a single solve using shifted Krylov subspace methods (abbreviated as KSMs). These methods employ the property that Krylov subspaces are invariant under arbitrary diagonal shifts to the matrix , i.e., (2) As we know, the restarted GMRES method is a widely popular choice for solving (1), refer to [1]. The computed GMRES shifted residuals are not collinear in general after the first restart so that it loses the computational efficiency mentioned above. Consequently, certain enforcement has to be made to guarantee the computed GMRES shifted residuals collinear to each other in order to maintain the computational efficiency. Note that in this case, only the seed system satisfies the minimum residual property, the solution of the other shifted systems is not equivalent to GMRES applied to those systems, see [1] for details. Moreover, another attractive approach is to use the restarted FOM method for solving the whole sequence of shifted linear systems simultaneously, for all residuals are naturally collinear [1,2]. As a result, the computational efficiency can be maintained because the orthonormal basis and the Hessenberg matrix are required to be calculated only once each time. However, both the restarted GMRES method and the restarted FOM method for (1) are inherently derived by the Arnoldi procedure, which tends to be expensive when becomes large in aspects of memory and algorithmic cost. To alleviate these difficulties, our recent work [5] proposed a new iterative strategy with collinear residuals for solving sequence of shifted linear systems based on the Hessenberg process, which is similar but cheaper compared to the Arnoldi procedure. Our theoretical and numerical analyses indicate that iterative schemes referred to as may outperform the conventional shifted FOM method in aspects of iteration counts and CPU time. From the algorithmic property, it turns out that preserving orthonormality in the construction of the Krylov basis is sometimes not necessary for an efficient iterative solution of (1). This observation motivates us to extend the restarted Changing Minimal Residual method based on the Hessenberg process (CMRH) [3], which is similar to the restarted shifted GMRES method but much cheaper, for solving (1). In practice, preconditioning shifted systems (1) is necessary to accelerate the convergence of shifted KSMs and this remains an open question, because standard preconditioning techniques may destroy the shift-invariance property (2). In this work, we construct an inner-outer scheme based on nested KSMs, where the inner solver is a multi-shift KSM (i.e., ) that acts as a preconditioner for an outer flexible CMRH (FCMRH) method. Our numerical experiments show that the new nested iterative scheme based on the Hessenberg procedure is very competitive, and sometimes superior, compared to the early established combination (FOM-FGMRES) in [4] in terms of CPU time. In summary, in our presentation first we discuss the derivations of both the method [5] and the restarted shifted CMRH method. Then, we present their flexible variant combined with nested inner-outer iterative schemes, namely multi-shift Hessen-FCMRH. Meanwhile, extensive numerical experiments involving PageRank problems and numerical solutions of space(-time) fractional differential equations are reported to illustrate the effectiveness of the restarted shifted CMRH method and the Hessen-FCMRH method, respectively. Next: Bibliography root 2016-02-22	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8245626091957092, "perplexity": 674.7658300721916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00081.warc.gz"}
https://researchoutput.ncku.edu.tw/zh/publications/a-software-reliability-growth-model-for-imperfect-debugging	# A software reliability growth model for imperfect debugging Yeu Shiang Huang, Kuei Chen Chiu, Wan Ming Chen 6 引文斯高帕斯（Scopus） ## 摘要 In general, software reliability growth models (SRGMs) are often developed based on the assumptions of perfect debugging, single error type, and consistent testing environment. However, such the assumptions may be unrealistic for testing projects because new errors are always introduced during the debugging process, software errors are not alike, and the testing environment may change as the workforce escalates. Furthermore, the learning effect of the debugging process is taken under advisement, and assumes that it is unstable since the process of the error removal is also imperfect, which may cause a fluctuation of errors in the system. Therefore, the study is based on the Non-Homogeneous Poisson Process with considerations of the phenomenon of imperfect debugging, varieties of errors and change points during the testing period to extend the practicability of SRGMs. Furthermore, the error fluctuation rate is described by a sine cyclical function with a decreasing fluctuation breadth during the detection period, which indicates that the influence of increasing new errors during the testing period will be gradually unremarkable as the testing time elapses. Besides, the expected time of removing simple or complex errors is assumed to be different truncated exponential distributions. Finally, the optimal software release policies are proposed with considerations of the costs which occur in the testing and warranty period under an acceptable threshold of software reliability. 原文 English 111267 Journal of Systems and Software 188 https://doi.org/10.1016/j.jss.2022.111267 Published - 2022 6月 • 軟體 • 資訊系統 • 硬體和架構	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8945745229721069, "perplexity": 2235.030095591304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499790.41/warc/CC-MAIN-20230130003215-20230130033215-00059.warc.gz"}
http://gasmar.com.br/th-gxyma/93e625-graphing-horizontal-and-vertical-lines	Intercepts p. 7 . Organizing and providing relevant educational content, resources and information for students. Graphing Lines Information Packet: Table of Contents: Graphing Ordered Pairs p. 1 Slope p. 2-4. The graph is a vertical line passing through the $$x$$-axis at $$2.$$, What if the equation has $$y$$ but no $$x$$? This means the change in Y = 0, while the change in X = 1. Solutions for both equations are listed. All Rights Reserved. We're used to these: If the Then the slope is: Topic: Horizontal and Vertical Lines II. Mathematics » Introducing Graphs » Graphing Linear Equations. Just $$x$$ and no $$y,$$ or just $$y$$ without an $$x?$$ How will we make a table of values to get the points to plot? All the ordered pairs in the table have the same $$y$$-coordinate, $$-1$$. Let’s consider the equation $$x=-3.$$ The equation says that $$x$$ is always equal to $$-3,$$ so its value does not depend on $$y.$$ No matter what $$y$$ is, the value of $$x$$ is always $$-3.$$. But, to better understand this test, you need to meet its partner, the vertical line test. How Do You Write an Equation for a Horizontal Line? 1/0 = undefined. When you see To graph a horizontal line that goes through a given point, first plot that point. A full lesson to develop understanding about horizontal and vertical lines. The concept of slope simply does not work for vertical lines. Notice that the first equation has the variable $$x,$$ while the second does not. ©a F2v0r1 K14 qK xu 6t6a 4 sSAo2f9tBwIa Lrse a tL 3LWCc.y s 1A 9l1l u 1r ci5gmhtHsz Urfewse1r Pv Wevdh. A line parallel to the x-axis is called a horizontal line. The points where the graph intercepts the y-axis are called y-intercepts. Graphing a Line Using the Intercepts. is always Horizontal and Vertical Lines 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. Save my name, email, and website in this browser for the next time I comment. Then plot the points and connect them with a straight line. −1 − 1. the graph should look like! Example 8. You don't have access to this video. Follow along with this tutorial as you see how use the information given to write the equation of a horizontal line. B. MEMORY METER. This is a lesson from the tutorial, Introducing Graphs and you are encouraged to log in or register, so that you can track your progress. Graph the Equation: Horizontal / Vertical lines. Using the Slope Formula to Find the Slope of a Line … Describing Horizontal and Vertical Lines. The equations for vertical and horizontal lines look very similar to equations like $$y=4x.$$ What is the difference between the equations $$y=4x$$ and $$y=4?$$, The equation $$y=4x$$ has both $$x$$ and $$y.$$ The value of $$y$$ depends on the value of $$x.$$ The $$y\text{-coordinate}$$ changes according to the value of $$x.$$, The equation $$y=4$$ has only one variable. Observe it and locate the following points : (a) If point D is + 8, then which point is – 8? News Feed ... Graphing Horizontal & Vertical Lines. We use first party cookies on our website to enhance your browsing experience, and third party cookies to provide advertising that may be of interest to you. The change in outputs between any two points, therefore, is 0. All for only $14.95 per month. isn't there, then it They just move left to right. Verdict: vertical lines have NO SLOPE. This modified article is licensed under a CC BY-NC-SA 4.0 license. Evaluate A(x)… ... Horizontal and Vertical Line Graphs. Unless specified, this website is not in any way affiliated with any of the institutions featured. Plot the points and connect them as shown. 0/1 = 0 as a slope. say this: y Remember that vertical lines only have a 'y' value and no 'x' value. A vertical line is the graph of an equation that can be written in the form $$x=a.$$. It is always recommended to visit an institution's official website for more information. From the line's graph, I'll use the (arbitrary) points (4, 5) and (4, –3). can be anything! PowerPoint and worksheet with an interesting extension to allow pupils to discover horizontal and vertical lines for themselves. Slope Intercept Form p. 8 Writing Equations of Lines … It doesn't look like M V kM ta Id 3eb kw biotRhw sIYngf ji HnmiHt3e q bAhlDg7eobJr eaP B14. Let’s graph the equation $$y=4.$$ This time the $$y$$-value is a constant, so in this equation $$y$$ does not depend on $$x.$$, To make a table of solutions, write $$4$$ for all the $$y$$ values and then choose any values for $$x.$$. https://www.onlinemathlearning.com/graphing-horizontal-lines.html Hope this helps. Using$m=\frac {\text {rise}} {\text {run}}\$ to Find the Slope of a Line From Its Graph. Example 1 : The below vertical number line, representing integers. Students graph linear equations in standard form, + = ( or =0), that produce a horizontal or a vertical line. Graphing Horizontal and Vertical Lines; Sign Up Create an account to see this video. y = Make a Table p. 6 . −1 − 1. Author: John Sweeney. y = 4 x. Play this game to review Pre-algebra. Trying to find the equation of a horizontal line that goes through a given point? Graph the equation of the vertical line (x = k) or horizontal line (y = k) in this series of printable high school worksheets. We’ll use $$0,2,$$ and $$4$$ for the $$x$$-values. All names, acronyms, logos and trademarks displayed on this website are those of their respective owners. Where's the This GeoGebra worksheet illustrates the equations of horizontal and vertical lines. of this line? Register or login to receive notifications when there's a reply to your comment or update on this information. LO: To recognise and be able to plot Horizontal and Vertical lines on a graph KW: Horizontal, Vertical, Graph, Parallel, Value, Function Create Class; Home. Notice that the equation $$y=4x$$ gives a slanted line whereas $$y=4$$ gives a horizontal line. Rejecting cookies may impair some of our website’s functionality. Lesson designed to … Practice your knowledge of horizontal and vertical lines: their graphical presentation, their slopes, and their equations. Starting with looking at coordinates to spot the link, then identifying 7 lines, then drawing given lines onto a graph. Note that this is not a function (fails the vertical line test) but the graph intercepts the y-axis three separate times. Graph $$y=-3x$$ and $$y=-3$$ in the same rectangular coordinate system. To see this process in action, watch this tutorial! Then draw a straight line left and right that goes through the point, and you're done! The value of $$y$$ is constant. We choose $$0,3,$$ and $$-3$$ as values for $$x.$$. can be anything! So, what's the slope Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and … Rewrite the given linear equation in slope-intercept form to find the slope and y-intercept and then graph the line accordingly. Lesson Notes The goal of this lesson is for students to know that the graph of an equation in the form of = or = , where is a constant, is the graph of a vertical or horizontal line… Then, grab two points and find the slope. The There are a few special equations that also produce a straight line. Practice. 0 0. Don't want to keep filling in name and email whenever you want to comment? Vertical lines go up/down, but they never go left or right. (By the way: that's true only because the variables don't have any exponents, but that's for another day). There are two special cases of lines on a graph—horizontal and vertical lines. The line passes through the $$x$$-axis at $$\left(a,0\right)$$. C. The students will find the equation of horizontal and vertical lines. This foldable has notes and examples for graphing horizontal and vertical lines. Preview; Then choose any values for $$y.$$ Since $$x$$ does not depend on $$y,$$ you can chose any numbers you like. Using Geoboards to Model Slope. The following diagrams show horizontal and vertical transformations of functions and graphs. Some of the worksheets for this concept are Lines lines lines horizontal and vertical lines, Graphing lines, Lesson 14 the graph of a linear equation horizontal and, Parallel and perpendicular lines, 14 straight line graphs mep y8 practice book b, Work from the graphs of, Graphing lines in slope … Let us see some example problems to understand how to graph the given integers on horizontal and vertical number lines. This tells you what Finding the Slope of a Line from the Graph, Finding the Slope of a Line from Two Points, Finding the Slope of a Line from the Equation, Finding the Equation of a Line Given a Point and a Slope, Finding the Equation of a Line Given Two Points. Improve your math knowledge with free questions in "Graph a horizontal or vertical line" and thousands of other math skills. Register or login to make commenting easier. Improve your math knowledge with free questions in "Equations of horizontal and vertical lines" and thousands of other math skills. Scroll down the page for more examples and solutions on horizontal and vertical transformations. Graph the equation $$x=2.$$ What type of line does it form? -2, The graph is a horizontal line passing through the $$y$$-axis at $$–1$$ as shown. The students will understand the difference between horizontal and vertical lines. ... Note: A line parallel to the y-axis is called a vertical line. Horizontal and Vertical Lines: If we are given the equation of y equal to a number, our graph will be a horizontal line. Find three solutions for each equation. The equations for vertical and horizontal lines look very similar to equations like y = 4x. Is this equation a function? 0! there's enough there! … This indicates how strong in your memory this concept is. Can we graph an equation with only one variable? A horizontal line indicates a constant output, or y-value. Equations are written in slope-intercept form and standard form. The $$y\text{-coordinate}$$ is always $$4.$$ It does not depend on the value of $$x.$$. Goals and Objectives: A. It explains the mnemonic device HOY VUX - horizontal lines have zero slope and cross the y-axis and vertical lines have undefined slopes and cross the x-axis. Your browser seems to have Javascript disabled. Let's do the calculations to confirm the logic. The slope of a vertical line does not exist! This means the change in Y = 1, while the change in X = 0. In this video, I show how to graph horizontal and vertical lines by graphing quickly, using a table, or thinking of y=mx+b. But to fit the size of our coordinate graph, we’ll use $$1,2,$$ and $$3$$ for the $$y$$-coordinates as shown in the table. The graph of a relation of the form x = 5 is a line parallel to the y-axis because the x value never changes. Which table is correct for y=4x+3? Constructive Media, LLC. Finding the Slope of Horizontal and Vertical Lines. Displaying top 8 worksheets found for - Graph Horizontal And Vertical Lines. Horizontal/Vertical Lines p. 5 Graphing Linear Equations p. 6-8. If you believe that your own copyrighted content is on our Site without your permission, please follow this Copyright Infringement Notice procedure. Solution for Let A(x) represent the area bounded by the graph, the horizontal axis, and the vertical lines at t = 0 and t = x for the graph below. You can accept or reject cookies on our website by clicking one of the buttons below. The basic graphing technique is the horizontal line test. © 2019 Coolmath.com LLC. This graph is a horizontal line passing through the $$y\text{-axis}$$ at $$4.$$, A horizontal line is the graph of an equation that can be written in the form $$y=b.$$, The line passes through the $$y\text{-axis}$$ at $$\left(0,b\right).$$, The equation $$y=-1$$ has only variable, $$y.$$ The value of $$y$$ is constant. The students will graph the different type of lines. and x We're sorry, but in order to log in and use all the features of this website, you will need to enable JavaScript in your browser. Line y is parallel to x-axis and line x is parallel to y-axis. Graphing Linear Equations Line-Up Activity: This product contains 15 cards in which students need to match the equation with the graph. (3,−1) ( 3, − 1) The graph is a horizontal line passing through the y y -axis at –1 – 1 as shown. slope of a horizontal line is Write and graph lines of the form y = a constant value and x = a constant value. horizontal line, vertical line Get full access to over 1,300 online videos and slideshows from multiple courses ranging from Algebra 1 to Calculus. The graph of a function always passes the vertical line test. $$\overset{\underset{\mathrm{def}}{}}{=}$$, Plotting Points on a Rectangular Coordinate System, Verifying Solutions to an Equation in Two Variables, Completing a Table of Solutions to a Linear Equation, Finding Solutions to Linear Equations in Two Variables, Recognizing the Relation Between the Solutions of an Equation and its Graph, Graphing a Linear Equation by Plotting Points, Finding the Intercepts from an Equation of a Line, Choosing the Most Convenient Method to Graph a Line, Finding the Slope of a Line from its Graph, Finding the Slope of Horizontal and Vertical Lines, Using the Slope Formula to Find the Slope of a Line between Two Points, Graphing a Line Given a Point and the Slope, http://cnx.org/contents/[email protected]. Identify a function with the vertical line test. III. Plot the points and connect them, as shown in the figure below. There are also a few horizontal and vertical lines. x -2 Horizontal and Vertical Lines ~ Lesson Plan I. This GeoGebra worksheet illustrates the equations of horizontal and vertical lines. Key Concepts. To understand more about how we and our advertising partners use cookies or to change your preference and browser settings, please see our Global Privacy Policy. The equation has only variable, $$x,$$ and $$x$$ is always equal to $$2.$$ We make a table where $$x$$ is always $$2$$ and we put in any values for $$y.$$. This is … Horizontal lines do not go up/down. Plot the graph of each of the following relations: Solution: Key Terms. To make a table of solutions, we write $$-3$$ for all the $$x$$ values. Grab two points and see? Rejecting cookies may impair some of our website’s functionality. % Progress . (b) Is point G a negative integer or a positive integer? x? (0,−1) ( 0, − 1) 3 3. Graphing Horizontal & Vertical Lines We know that we can graph a linear equation with two variables, such as the famous x and y, as a straight line. Notice in the figure below that the graph is a vertical line. In , we see that the output has a value of 2 for every input value. N'T look like CC BY-NC-SA 4.0 license partner, the vertical line the different type of does! + = ( or =0 ), that produce a horizontal line that goes through a given point and... The x-axis is called a horizontal line access to over 1,300 online videos and slideshows from courses... Ranging from Algebra 1 to Calculus a ' y ' value representing integers therefore, 0!... then, grab two points, therefore, is 0 n't there, identifying... Y-Axis are called y-intercepts to equations like y = 1 \ ( )! Also produce a straight line coordinate system will graph the given integers on horizontal and vertical lines = 0 x. Its partner, the vertical line Practice your knowledge of horizontal and vertical lines this video videos slideshows. In this browser for the \ ( -1\ ) the point, first plot that.. Let us see some example problems to understand how graphing horizontal and vertical lines graph the accordingly. The given Linear equation in slope-intercept form to find the slope of this line lines for.... 0, while the second does not exist on horizontal and vertical lines following points: ( a ) point... Notice procedure a,0\right ) \ ) while the change in y =,., while the change in x = 1, while the change in x =,. Ranging from Algebra 1 to Calculus to better understand this test, you need to meet its partner the. Relevant educational content, resources and information for students bAhlDg7eobJr eaP B14 never go or... Have the same \ ( y=4\ ) gives a horizontal line given lines onto a graph the first has... 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https://socratic.org/questions/how-do-you-solve-10-2x-4-8-9x-3x	Algebra Topics # How do you solve 10(2x + 4) = - (-8 -9x) + 3x? Jan 25, 2016 x = - 4 #### Explanation: First 'distribute' the brackets : ie 20x + 40 = 8 + 9x + 3x ( collect x terms to left side and numbers to the right ) 20x - 9x - 3x = 8 - 40 hence : 8x = - 32 ( divide both sides by 8 ) so x = - 4 ##### Impact of this question 187 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42873695492744446, "perplexity": 2298.111471064204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540490743.16/warc/CC-MAIN-20191206173152-20191206201152-00158.warc.gz"}
http://www.perimeterinstitute.ca/video-library?title=&page=656	# Video Library Since 2002 Perimeter Institute has been recording seminars, conference talks, and public outreach events using video cameras installed in our lecture theatres. Perimeter now has 7 formal presentation spaces for its many scientific conferences, seminars, workshops and educational outreach activities, all with advanced audio-visual technical capabilities. Recordings of events in these areas are all available On-Demand from this Video Library and on Perimeter Institute Recorded Seminar Archive (PIRSA)PIRSA is a permanent, free, searchable, and citable archive of recorded seminars from relevant bodies in physics. This resource has been partially modelled after Cornell University's arXiv.org. ## Mathematical Physics (PHYS 624) - Lecture 10 Friday Nov 27, 2009 Speaker(s): ## The Worldline Approach to Black Hole Dynamics Thursday Nov 26, 2009 Speaker(s): TBA Collection/Series: Scientific Areas: ## Evaporating black holes in the presence of a minimal length Thursday Nov 26, 2009 Speaker(s): After implementing an effective minimal length, we will present a new class of spacetimes, describing both neutral and charged black holes. As a result, we will improve the conventional Schwarzschild and Reisner-Nordstroem spacetimes, smearing out their singularities at the origin. On the thermodynamic side, we will show how the new black holes admit a maximum temperature, followed by the SCRAM phase'', a thermodynamic stable shut down, characterized by a positive black hole heat capacity. Collection/Series: Scientific Areas: ## Non-Equilibrium Systems (PHYS 606) - Lecture 9 Thursday Nov 26, 2009 Speaker(s): ## Mathematical Physics (PHYS 624) - Lecture 9 Thursday Nov 26, 2009 Speaker(s): ## Born--Oppenheimer approximation for quantum fields on quantum spacetimes Wednesday Nov 25, 2009 Speaker(s): The relation between loop quantum gravity (LQG) and ordinary quantum field theory (QFT) on a fixed background spacetime still bears many obstacles. When looking at LQG and ordinary QFT from a mathematical perspective it turns out that the two frameworks are rather different: Although LQG is a true continuum theory its Hilbert space is defined in terms of certain embedded graphs which are labeled by irreducible representations of SU(2). Collection/Series: Scientific Areas: ## Introduction to Effective Field Theory - Lecture 10B Wednesday Nov 25, 2009 Speaker(s): ## Was Einstein Right Handed? Wednesday Nov 25, 2009 Speaker(s): Of all four forces only the weak interaction has experimentally exhibited parity violation. At the same time observations suggest that general relativity may require modification to account for dark matter and dark energy. Could it be that this modification involves gravitational parity violation? Many of the dominant approaches to quantum gravity, such as string theory and loop quantum gravity, point to an effective parity violating extension to general relativity known as Chern-Simons General Relativity (CSGR). Collection/Series: Scientific Areas: ## Non-Equilibrium Systems (PHYS 606) - Lecture 8 Wednesday Nov 25, 2009 Speaker(s): ## Introduction to Effective Field Theory - Lecture 10 Wednesday Nov 25, 2009 Speaker(s): ## Next Public Lecture Check back for details on the next lecture in Perimeter's Public Lectures Series ## RECENT PUBLIC LECTURE ### Mario Livio: Brilliant Blunders Speaker: Mario Livio	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3791300654411316, "perplexity": 3327.0165178912202}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982952852.53/warc/CC-MAIN-20160823200912-00121-ip-10-153-172-175.ec2.internal.warc.gz"}
http://www.physicsforums.com/showthread.php?p=4247791	# Basic Fourier analysis proof by Wingeer Tags: analysis, basic, fourier, proof P: 79 Hi. Just going through my notes from the last lecture I remember having some troubles understanding the proof the lecturer gave for the following theorem: Suppose that f is Riemann integrable and that all its Fourier coefficients are equal to 0, then f(x)=0 at all points of continuity. The proof is a bit tricky so I will sketch the basic gists of it. We start by defining $$\delta_n = \frac{\left(\frac{1+cos(x)}{2}\right)^n}{\alpha_n}$$ where $$\alpha_n = \int_{-\pi}^{\pi} \left(\frac{1+cos(x)}{2}\right)^n dx$$. This implies of course that $$\int_{-\pi}^{\pi} \delta_n dx = 1, \forall n$$ He then proceeds by showing that $$\frac{1}{\alpha_n} \leq \left(2\delta \left(\frac{1+cos(\delta)}{2}\right)^n\right)^{-1}$$ Which will be useful later in the proof (this part was also understandable). Now: $$\left(\frac{1+cos(x)}{2}\right)^n = \left(\frac{exp(\frac{ix}{2}) + exp(\frac{-ix}{2})}{2}\right)^{2n}$$ He then uses the binomial theorem on this expression to get: $$\sum_{k=0}^{2n} {2n \choose k} e^{i(k-n)x}$$ My first question is then: Where did the 2's from the denominator go? Now he uses this expansion to conclude that: $$\int_{-\pi}^{\pi} f(y) \left(\frac{1+cos(y-x)}{2}\right)^n dy = 0$$ By arguing that the Fourier coefficients are 0 by assumption. My second question is regarding this step. I do not understand it at all. any clarity at all will be a tremendous help. Then $$\|f(x)\| = \left\|f(x) - \alpha_n ^{-1} \int_{-\pi}^{\pi} f(y) \left(\frac{1+cos(y-x)}{2}\right)^n dy \right\| = \left\|\alpha_n ^{-1}\int_{-\pi}^{\pi} (f(x)-f(y)) \left(\frac{1+cos(y-x)}{2}\right)^n dy \right\|$$ Which follows from how the delta function is defined. He then proceeds by noting that $$\lim_{y \to x}f(y) = f(x)$$ so that for any given epsilon>0 there exists 2delta such that $$\|x-y\| < 2\delta \Leftarrow \|f(x) - f(y)\| < \epsilon$$ By definition of limits. Now by the triangle inequality for integrals we have that the expression over is less than or equal to: $$\alpha_n ^{-1}\int_{-\pi}^{\pi} \left\|(f(x)-f(y)) \left(\frac{1+cos(y-x)}{2}\right)^n \right\| dy$$ Now he chooses to split the integral by using the limits, so that we get: $$\int_{\|x-y\|<2\delta}\alpha_n ^{-1} \left\|(f(x)-f(y)) \left(\frac{1+cos(y-x)}{2}\right)^n \right\| dy + \int_{\|x-y\| \geq 2\delta} \alpha_n ^{-1} \left\|(f(x)-f(y)) \left(\frac{1+cos(y-x)}{2}\right)^n \right\| dy$$ Now the first integral is smaller than epsilon due to the limits and also by the normalisation of delta. In the second integral we pull out alpha and use the inequality mentioned earlier. Now he also claims that: $$\int_{\|x-y\| \geq 2\delta} \left\|(f(x)-f(y)) \left(\frac{1+cos(y-x)}{2}\right)^n \right\| dy \leq 2M \left(\frac{1+cos(2\delta)}{2}\right)^n$$ Which I believe to be the case since f is Riemann integrable and therefore must be bounded by, say M. Also on the right side we have the maximal value that the cosine expression takes, and therefore the inequality is justified. Is this correct thinking? Now by taking limits we get the conclusion that $$\|f(x)\| < \epsilon$$ where epsilon was arbitrarily chosen, and therefore f(x)=0 where it is continuous. I know this result probably goes a bit deeper and can be better understood with measure theory and the like. I have not had any courses with this yet, so my question is then: Is there a more general formulation of this theorem where we do not restrict the functions to be Riemann integrable? Thanks in advance. HW Helper PF Gold P: 2,913 This is quite a long problem, so I will only address part of it for now. Regarding your first question: Quote by Wingeer Now: $$\left(\frac{1+cos(x)}{2}\right)^n = \left(\frac{exp(\frac{ix}{2}) + exp(\frac{-ix}{2})}{2}\right)^{2n}$$ He then uses the binomial theorem on this expression to get: $$\sum_{k=0}^{2n} {2n \choose k} e^{i(k-n)x}$$ My first question is then: Where did the 2's from the denominator go? The 2's in the denominators of the exponentials cancel, but you are right that there is still the 1/2 in the denominator of the larger fraction which is unaccounted for. Here is what I get: \begin{align} \left(\frac{\exp\left(\frac{ix}{2}\right) + \exp\left(\frac{-ix}{2}\right)}{2}\right)^{2n} &= \frac{1}{2^{2n}}\sum_{k=0}^{2n} {2n \choose k} \left(\exp\left(\frac{ix}{2}\right)\right)^{k} \left(\exp\left(-\frac{ix}{2}\right)\right)^{2n-k} \\ &= \frac{1}{2^{2n}}\sum_{k=0}^{2n} {2n \choose k} \exp\left(\frac{ixk}{2}\right) \exp\left(\frac{-ix(2n-k)}{2}\right) \\ &= \frac{1}{2^{2n}}\sum_{k=0}^{2n} {2n \choose k} \exp\left(\frac{ixk}{2}\right) \exp\left(\frac{ixk}{2}\right) \exp\left(\frac{-i2xn}{2}\right) \\ &= \frac{1}{2^{2n}}\sum_{k=0}^{2n} {2n \choose k} \exp\left(ixk\right) \exp\left(-ixn\right) \\ &= \frac{1}{2^{2n}}\sum_{k=0}^{2n} {2n \choose k} \exp\left(ix(k-n)\right) \end{align} So this matches your lecturer's expression except for the $1/2^{2n}$ factor, but his conclusion that the result integrates to zero is true with or without this factor: ]Now he uses this expansion to conclude that: $$\int_{-\pi}^{\pi} f(y) \left(\frac{1+cos(y-x)}{2}\right)^n dy = 0$$ By arguing that the Fourier coefficients are 0 by assumption. All he has done here is to substitute the expression I just obtained into this integral: \begin{align}\int_{-\pi}^{\pi} f(y) \left(\frac{1+cos(y-x)}{2}\right)^n dy &= \frac{1}{2^{2n}} \int_{-\pi}^{\pi} f(y) \sum_{k=0}^{2n} {2n \choose k} \exp\left(i(y-x)(k-n)\right) dy \\ &= \frac{1}{2^{2n}} \sum_{k=0}^{2n} {2n \choose k} \int_{-\pi}^{\pi} f(y) \exp\left(i(y-x)(k-n)\right) dy\\ &= \frac{1}{2^{2n}} \sum_{k=0}^{2n} {2n \choose k} \exp(-ix(k-n)) \int_{-\pi}^{\pi} f(y) \exp(iy(k-n)) dy \\ \end{align} Note that $$\int_{-\pi}^{\pi} f(y) \exp\left(iy(k-n)\right) dy$$ is a Fourier coefficient of ##f##, hence zero by assumption. Therefore the whole expression reduces to zero. I will read over the rest of your post and reply again if I can offer any further insight. HW Helper PF Gold P: 2,913 Quote by Wingeer In the second integral we pull out alpha and use the inequality mentioned earlier. Now he also claims that: $$\int_{\|x-y\| \geq 2\delta} \left\|(f(x)-f(y)) \left(\frac{1+cos(y-x)}{2}\right)^n \right\| dy \leq 2M \left(\frac{1+cos(2\delta)}{2}\right)^n$$ Which I believe to be the case since f is Riemann integrable and therefore must be bounded by, say M. Also on the right side we have the maximal value that the cosine expression takes, and therefore the inequality is justified. Is this correct thinking? Yes, it sounds correct to me. Now by taking limits we get the conclusion that $$\|f(x)\| < \epsilon$$ where epsilon was arbitrarily chosen, and therefore f(x)=0 where it is continuous. Looks good. This is a fairly standard type of proof in real analysis: the $\delta_n$ family of functions are called kernels and they are used to "smooth" the integrand in order to make limiting arguments more feasible. If you're interested in reading further I would highly recommend Stein and Shakarchi's book Fourier Analysis: An Introduction. This book should be very accessible to you as it only uses the Riemann integral. In chapter 2, they lay out the properties of what they call a family of "good" kernels $K_n$: (a) For all $n \geq 1$, $$\int_{-\pi}^{\pi} K_n(x) dx = 1$$ (b) There exists $M > 0$ such that for all $n \geq 1$, $$\int_{-\pi}^{\pi} \|K_n(x)\| dx \leq M$$ (c) For every $\delta > 0$, $$\int_{\delta \leq \|x\| \leq \pi} \|K_n(x)\|dx \rightarrow 0 \textrm{ as } n \rightarrow \infty$$ and they prove some nice theorems regarding what you can do with a good family of kernels, as well as giving several examples of such families. You can check for yourself whether the kernels in your problem satisfy these conditions. I know this result probably goes a bit deeper and can be better understood with measure theory and the like. I have not had any courses with this yet, so my question is then: Is there a more general formulation of this theorem where we do not restrict the functions to be Riemann integrable? Yes, if we use the Lebesgue integral, then if $f$ satisfies $\int_{-\pi}^{\pi} \|f(x)\| dx < \infty$ and all of its Fourier coefficients are zero, it follows that $f = 0$ almost everywhere. "Almost everywhere" means except possibly on a set of measure zero. A set of measure zero is small: it can be covered by a countable family of open intervals of arbitrarily small total length. Note that this implies that $f$ must equal zero at any point of continuity, for if it were not, then we would also have $f \neq 0$ throughout some open neighborhood around the point, and any such neighborhood has measure greater than zero. P: 79 ## Basic Fourier analysis proof We are actually using the book by Stein and Shakarchi in this course. I am not sure if I like it or not; It is certainly something else from what I've read before. Also I am just recently diving into analysis, so I might need to get used to this way of thinking. Could you please elaborate on the purpose of kernels? Especially in how they make limiting arguments more feasible. Is not the integrand supposed to be squared in the last paragraph? Since that is the norm on L2(-pi,pi)? Or am I wrong? Probably a difficult question, but how small does the total length have to be? And how do the length compare with the ones of neighbourhoods of open intervals? Another question arose last week when we were talking about the basis of L2(-pi,pi). We proved that the family of function e^(inx) for integer n is a basis for the aforementioned space. Then he posed a related question: Prove that $$\frac{\sin(nx)}{\sqrt{\pi / 2}}$$ for integer n is a basis for the space L2(0,pi). First off the family of functions are definitely orthonormal by a quick computation of an integral. The completion part is however stalling me. I really want to use the fact that the exponentials are a basis to prove that this family is a basis, but I cannot really see how to go about that. Maybe a translation and scaling, or something up that alley? I really appreciate you taking your time to help! HW Helper PF Gold P: 2,913 Quote by Wingeer Thank you for your answer and time! We are actually using the book by Stein and Shakarchi in this course. I am not sure if I like it or not; It is certainly something else from what I've read before. Also I am just recently diving into analysis, so I might need to get used to this way of thinking. Could you please elaborate on the purpose of kernels? Especially in how they make limiting arguments more feasible. The operation of convolution with a kernel results in a smoother function. For example, if ##K_n## is a sequence of "good" kernels, and ##f_n## is any integrable function, then for each ##n##, $$f_n(x) = \int_{-\pi}^{\pi} f(y)K_n(x-y) dy$$ is continuous at all points even if the original function ##f## was not. Furthermore, at any point ##x## where ##f## is continuous, we have ##f_n(x) \rightarrow f(x)##. Even better, if S is the set of points where ##f## is continuous, and S is compact, then ##f_n \rightarrow f## uniformly on S. Now, why is this useful in Fourier analysis? One of the key questions is under what circumstances we can reconstruct ##f## given its Fourier coefficients. The simplest candidate way to do this is to define a sequence of partial sums: $$f_n(x) = \sum_{k=-n}^{n} \hat{f}(k) e^{ikx}$$ If you do some clever rearranging, it turns out that this can be written as a convolution with a kernel: $$f_n(x) = \int_{-\pi}^{\pi} f(y) D_n(x-y) dy$$ where ##D_n(x) = \sin((2n+1)x)/\sin(x)## is the so-called Dirichlet kernel. Unfortunately it turns out that the Dirichlet kernel is not a "good" kernel; it violates the second condition laid out by Stein and Shakarchi because there is no M such that $$\int_{-\pi}^{\pi}\|D_n(x)\| dx \leq M$$ for all ##n##. A consequence of this is that we aren't guaranteed that the partial sums of the Fourier series will converge to the original function, even at points where that function is continuous. (Indeed, explicit counterexamples have been constructed.) However, there are other ways to reconstruct the function ##f## from its Fourier coefficients. One way is to take averages of the partial sums of the Fourier series. In other words, $$g_n(x) = \frac{1}{n}\sum_{m=1}^{n} f_m(x)$$ where ##f_m(x)## is as defined above. This turns out to be equivalent to forming a sum like $$g_n(x) = \sum_{k=-n}^n \hat{f}(k) w_n(k) e^{ikx}$$ where ##w_n(k)## is a sequence of weighting factors that has a triangular profile between -n and n. It turns out that this is also equivalent to convolution with a kernel: $$g_n(x) = \int_{-\pi}^{\pi} f(y) S_n(x-y) dy$$ Here ##S_n(x)## is called a Fejer kernel, which (if I recall correctly) equals the SQUARE of ##D_n(x)## (times a scale factor like 1/n), and this does turn out to be a good kernel. Therefore the sequence of AVERAGES of the partial sums of the Fourier series does converge to the original function at all points of continuity. (By the way, there's no hope at points of discontinuity, since we can change the function's value at each of these without having any effect on its Fourier coefficients, because these are defined by an integral.) Is not the integrand supposed to be squared in the last paragraph? Since that is the norm on L2(-pi,pi)? Or am I wrong? On a finite-length interval such as ##[-\pi,\pi]##, if the L2 norm is finite then the L1 norm will also be finite. But all that is needed for the theorem I mentioned is for the L1 norm to be finite. On a finite-length interval, both norms will always be finite if the function is bounded. In the context of Riemann integration, all functions are assumed to be bounded because the integral is not defined otherwise. The Lebesgue integral can integrate unbounded functions too, so we have to require the finite norm in that case. Probably a difficult question, but how small does the total length have to be? And how do the length compare with the ones of neighbourhoods of open intervals? You mean, for a set of measure zero? Well, any countable set has measure zero (for example, the rationals have measure zero), and some uncountable sets also have measure zero, for example the Cantor set. A set of measure zero cannot contain any open intervals, no matter how small. However, this is not sufficient: the set of irrationals contains no open intervals but it does NOT have measure zero. Another question arose last week when we were talking about the basis of L2(-pi,pi). We proved that the family of function e^(inx) for integer n is a basis for the aforementioned space. Then he posed a related question: Prove that $$\frac{\sin(nx)}{\sqrt{\pi / 2}}$$ for integer n is a basis for the space L2(0,pi). I'm not sure I believe that. All of these functions have value 0 at ##x=0## and ##x=\pi##, so any linear combination of them will also be zero at these points. Now maybe ##L^2(0,\pi)## refers to the open interval ##(0,\pi)## so those two points are conveniently excluded, but I'm still not so sure... Let me think about it for a while and see if I can find an argument for or against this claim. P: 79 Wow. Great answer again. Really clarifying, but still leaves me curious. I recently bought the real analysis book by McDonald and Weiss and I really enjoy reading in it. It is indeed meant to be the open interval, yes. P: 79 I have a suggestion: Let ##f \in L^2(0, \pi)##, then we have: $$f' = \left\{ \begin{array}{l} f(x) \text{ if } x \in (0, \pi)\\ -f(-x) \text{ if } x \in (-\pi,0) \end{array} \right.$$ and define f'(0) to be the average of the limiting processes of each of the piecewise functions (or something like that). Then f' is in ##L^2( -\pi, \pi)## and it is odd and can therefore be expressed by a linear combination of sin(nx). This was what me and a friend came up with. Now that I see over it myself, I can't help but think: What's stopping us from doing the same argument with even functions? And also, will the claim imply that the only functions in L2(0,pi) is odd? That seems weird ... Sci Advisor HW Helper PF Gold P: 2,913 You know that the family ##f_m(x) = e^{imx}## is a basis for ##L^2(-\pi, \pi)##, or for any interval of length ##2\pi##, for example ##L^2(0, 2\pi)##. So scale the x axis by a factor of 2 to obtain $$g_m(x) = f_m(2x) = e^{i2mx}$$ as a basis for ##L^2(0, \pi)##. Now, can you show that each ##g_m## can be expanded in terms of the proposed basis ##\sin(nx)/\sqrt{\pi/2}##? If so, you may be able to argue that ##\sin(nx)/\sqrt{\pi/2}## is indeed a basis for ##L^2(0, \pi)##. Consider the ##n##'th Fourier coefficient of ##g_m## with respect to the proposed basis. I'm too lazy to keep track of constant scale factors so I'll just write ##K## for these: \begin{align}c_{n,m} &= K \int_0^{\pi} e^{i2mx} \sin(nx) dx \\ &= \frac{K}{2i} \int_0^{\pi} e^{i2mx} [e^{inx} - e^{-inx}] dx \\ &= \frac{K}{2i} \left[\int_0^{\pi} e^{i(2m+n)x} dx - \int_0^{\pi} e^{i(2m-n)x} dx \right] \end{align} Note that if ##k## is an integer, then $$\int_0^{\pi} e^{ikx} dx = \pi$$ if ##k## = 0, and otherwise we have $$\int_0^{\pi} e^{ikx} dx = \frac{1}{ik} [e^{ik\pi} - 1] = \frac{1}{ik} [(-1)^k - 1] = \begin{cases} 2i/k & \textrm{if }k \textrm{ is odd} \\ 0 & \textrm{if }k \textrm{ is even} \\ \end{cases}$$ It's clear that we can continue carrying out this calculation to end up with a formula for ##c_{n,m}##, which will give us $$g_m(x) \sim \sum_{n=-\infty}^{\infty} c_{n,m} \sin(nx)$$ where I wrote ~ instead of = because convergence needs to be established. Assuming that works out, then an arbitrary function ##f\in L^2(0, \pi)## can be written as $$f(x) \sim \sum_{m=-\infty}^{\infty} a_m g_m(x)$$ where ##a_m## is the ##m##'th Fourier coefficient for ##f## in terms of the basis ##g_m##. This then becomes $$f(x) \sim \sum_{m=-\infty}^{\infty} a_m\left( \sum_{n=-\infty}^{\infty} c_{n,m} \sin(nx)\right)$$ and if the sums are well-behaved, we can rearrange as follows $$f(x) \sim \sum_{n=-\infty}^{\infty} \left(\sum_{m=-\infty}^{\infty} a_m c_{n,m} \right) \sin(nx)$$ thereby giving us, in principle, a Fourier coefficient ##d_n## for ##f## in terms of the basis ##\sin(nx)## equal to $$d_n = \sum_{m=-\infty}^{\infty} a_m c_{n,m}$$ Of course, all of these manipulations will need to be justified. Absolute convergence will be required in order to move summations around like that. Related Discussions Calculus & Beyond Homework 2 Calculus & Beyond Homework 2 Calculus & Beyond Homework 9 Engineering, Comp Sci, & Technology Homework 2 Calculus 11	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9885666370391846, "perplexity": 305.17149272844955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00222-ip-10-147-4-33.ec2.internal.warc.gz"}
http://mathhelpforum.com/discrete-math/134577-solved-von-neumann-ordinals-print.html	# [SOLVED] von Neumann Ordinals • Mar 19th 2010, 10:47 AM Let $\alpha$ and $\beta$ be two von Neumann ordinals. Show that $\alpha \subset \beta$ if and only if $\alpha \in \beta$ .	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7593116164207458, "perplexity": 610.4199336226152}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816841.86/warc/CC-MAIN-20180225170106-20180225190106-00350.warc.gz"}
https://people.maths.bris.ac.uk/~matyd/GroupNames/129/S3xD12.html	Copied to clipboard ## G = S3×D12order 144 = 24·32 ### Direct product of S3 and D12 Series: Derived Chief Lower central Upper central Derived series C1 — C3×C6 — S3×D12 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×D12 Lower central C32 — C3×C6 — S3×D12 Upper central C1 — C2 — C4 Generators and relations for S3×D12 G = < a,b,c,d \| a3=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 > Subgroups: 476 in 116 conjugacy classes, 36 normal (22 characteristic) C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×D12, S3×D4, C3⋊D12, S3×C12, C3×D12, C12⋊S3, C2×S32, S3×D12 Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, S32, C2×D12, S3×D4, C2×S32, S3×D12 Character table of S3×D12 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 12G size 1 1 3 3 6 6 18 18 2 2 4 2 6 2 2 4 6 6 12 12 2 2 4 4 4 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ9 2 2 -2 -2 0 0 0 0 2 -1 -1 -2 2 2 -1 -1 1 1 0 0 1 1 1 -2 1 -1 -1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 0 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 0 0 2 -1 -1 -2 -2 2 -1 -1 -1 -1 0 0 1 1 1 -2 1 1 1 orthogonal lifted from D6 ρ12 2 -2 2 -2 0 0 0 0 2 2 2 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 0 0 2 -2 0 0 -1 2 -1 -2 0 -1 2 -1 0 0 -1 1 -2 -2 1 1 1 0 0 orthogonal lifted from D6 ρ14 2 2 2 2 0 0 0 0 2 -1 -1 2 2 2 -1 -1 -1 -1 0 0 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 0 0 2 2 0 0 -1 2 -1 2 0 -1 2 -1 0 0 -1 -1 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ16 2 2 -2 -2 0 0 0 0 2 -1 -1 2 -2 2 -1 -1 1 1 0 0 -1 -1 -1 2 -1 1 1 orthogonal lifted from D6 ρ17 2 2 0 0 -2 -2 0 0 -1 2 -1 2 0 -1 2 -1 0 0 1 1 2 2 -1 -1 -1 0 0 orthogonal lifted from D6 ρ18 2 2 0 0 -2 2 0 0 -1 2 -1 -2 0 -1 2 -1 0 0 1 -1 -2 -2 1 1 1 0 0 orthogonal lifted from D6 ρ19 2 -2 -2 2 0 0 0 0 2 -1 -1 0 0 -2 1 1 1 -1 0 0 √3 -√3 √3 0 -√3 -√3 √3 orthogonal lifted from D12 ρ20 2 -2 2 -2 0 0 0 0 2 -1 -1 0 0 -2 1 1 -1 1 0 0 √3 -√3 √3 0 -√3 √3 -√3 orthogonal lifted from D12 ρ21 2 -2 -2 2 0 0 0 0 2 -1 -1 0 0 -2 1 1 1 -1 0 0 -√3 √3 -√3 0 √3 √3 -√3 orthogonal lifted from D12 ρ22 2 -2 2 -2 0 0 0 0 2 -1 -1 0 0 -2 1 1 -1 1 0 0 -√3 √3 -√3 0 √3 -√3 √3 orthogonal lifted from D12 ρ23 4 -4 0 0 0 0 0 0 -2 4 -2 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 0 0 0 0 0 0 -2 -2 1 4 0 -2 -2 1 0 0 0 0 -2 -2 1 -2 1 0 0 orthogonal lifted from S32 ρ25 4 4 0 0 0 0 0 0 -2 -2 1 -4 0 -2 -2 1 0 0 0 0 2 2 -1 2 -1 0 0 orthogonal lifted from C2×S32 ρ26 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 2√3 -2√3 -√3 0 √3 0 0 orthogonal faithful ρ27 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 -2√3 2√3 √3 0 -√3 0 0 orthogonal faithful Permutation representations of S3×D12 On 24 points - transitive group 24T229 Generators in S24 (1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24) (1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18) (1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24) (1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13) G:=sub<Sym(24)\| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>; G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) ); G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)]]) G:=TransitiveGroup(24,229); Matrix representation of S3×D12 in GL6(ℤ) 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 , -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 , 1 2 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 , 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 1 G:=sub<GL(6,Integers())\| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >; S3×D12 in GAP, Magma, Sage, TeX S_3\times D_{12} % in TeX G:=Group("S3xD12"); // GroupNames label G:=SmallGroup(144,144); // by ID G=gap.SmallGroup(144,144); # by ID G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,50,490,3461]); // Polycyclic G:=Group<a,b,c,d\|a^3=b^2=c^12=d^2=1,bab=a^-1,ac=ca,ad=da,bc=cb,bd=db,dcd=c^-1>; // generators/relations Export ׿ × 𝔽	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6430867910385132, "perplexity": 112.53402683545438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585382.32/warc/CC-MAIN-20211021071407-20211021101407-00594.warc.gz"}

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