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https://www.physicsforums.com/threads/finding-limits-of-line-integral.315743/
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# Finding limits of line integral
1. May 22, 2009
### boneill3
1. The problem statement, all variables and given/known data
Integrate along the line segment from (0,0) to $(\pi,-1)$
The integral
$\int_{(0,1)}^{(\pi,-1)} [y sin(x) dx - (cos(x))]dy$
2. Relevant equations
3. The attempt at a solution
I have used the parameterization of $x=\pi t$ and $y= 1-2t$
To get the integral:
$\int_{(0,1)}^{(\pi,-1)} [1-2t sin(\pi t) -(cos(\pi t))]dt$
But now because it is an integral of variable t I need to change the limits .
I'm not sure if I just have to put the limits of t just from 0 to $\pi$
I suppose I'm having trouble with getting from the limit of 2 variables (x,y) to a limit of one variable t
Thanks
2. May 22, 2009
### Dick
If x=pi*t and y=1-2t, then if you put t=0 then x=0 and y=1, right? If you put t=1 then x=pi and y=(-1), also right? As you came up with that fine parametrization what's the problem with finding limits for t?
3. May 23, 2009
### boneill3
I will need to go back and study more about parametrization.
4. May 23, 2009
### boneill3
When calculating this line integral
$\int_{(0,1)}^{(\pi,-1)} [y sin(x) dx - (cos(x))]dy$
I'm using the formula
$\int_{a}^{b}[f(x(t),y(t))x'(t) + g(x(t),y(t))y'(t)]dt$
with parameterization
I have $x = \pi t$
$y = 1-2t$
so
$x' = \pi$
and
$y' = -2$
plugging into the integral I get
$\int_{(0)}^{(1)} [1-2y sin(\pi t) \pi - (cos(\pi t))-2]$
$= -1$
The question states that the integral is independant of path.
So if I integrate along the initial line segment $(0,1)$to $(0,\pi)$
I should be able to plug in the values f($(-1,\pi)$)-f($(0,1)$)
And it should equal my original integral vaue of -1.
However I get 0
Could someone please check what I've done I show me where I am going wrong ?
Similar Discussions: Finding limits of line integral
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https://www.physicsforums.com/threads/critical-radius-of-insulation.408544/
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1. ### mjki9ec3
3
In class, we were taught that the critical radius of insulation (found by differentiating thermal resistance WRT outer radius and setting to zero) gives a minimum resistance and thus and maximum heat loss- i.e. it's not always productive to insulate your pipe.
but could it not also give a maximum resistance and therefore a minimum heat loss?
differentiating again surely would tell you when it would be minimum and maximum?
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https://www.oncotarget.com/article/1845/text/
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Oncotarget
∆N-P63α and TA-P63α exhibit intrinsic differences in transactivation specificities that depend on distinct features of DNA target sites.
Oncotarget. 2014; 5:2116-2130. https://doi.org/10.18632/oncotarget.1845
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Abstract
Paola Monti1,*, Yari Ciribilli2,*, Alessandra Bisio*2, Giorgia Foggetti1, Ivan Raimondi2,3, Paola Campomenosi3,4, Paola Menichini1, Gilberto Fronza1,**, Alberto Inga2,**
1 Mutagenesis Unit, Istituto di Ricerca e Cura a Carattere Scientifico Azienda Ospedaliera Universitaria San Martino-IST-Istituto Nazionale per la Ricerca sul Cancro, Genoa, Italy;
2 Laboratory of Transcriptional Networks, Centre for Integrative Biology, CIBIO, University of Trento, Trento, Italy;
3 Department of Biotechnology and Life Sciences, DBSV, University of Insubria, Varese, Italy;
4 The Protein Factory, Centro Interuniversitario di Ricerca in Biotecnologie Proteiche, Politecnico di Milano, ICRM-CNR Milano and Università degli Studi dell’Insubria, Varese, Italy
* These authors contributed equally to this work
** These authors share last authorship
Correspondence:
Alberto Inga, email:
Gilberto Fronza, email:
Keywords: TP63; transcription; transactivation specificity; TP73; TP53; Response Element
Received: February 5, 2014 Accepted: March 21, 2014 Published: March 23, 2014
Abstract
TP63 is a member of the TP53 gene family that encodes for up to ten different TA and ∆N isoforms through alternative promoter usage and alternative splicing. Besides being a master regulator of gene expression for squamous epithelial proliferation, differentiation and maintenance, P63, through differential expression of its isoforms, plays important roles in tumorigenesis. All P63 isoforms share an immunoglobulin-like folded DNA binding domain responsible for binding to sequence-specific response elements (REs), whose overall consensus sequence is similar to that of the canonical p53 RE. Using a defined assay in yeast, where P63 isoforms and RE sequences are the only variables, and gene expression assays in human cell lines, we demonstrated that human TA- and ∆N-P63α proteins exhibited differences in transactivation specificity not observed with the corresponding P73 or P53 protein isoforms. These differences 1) were dependent on specific features of the RE sequence, 2) could be related to intrinsic differences in their oligomeric state and cooperative DNA binding, and 3) appeared to be conserved in evolution. Sicen genotoxic stress can change relative ratio of TA- and ∆N-P63α protein levels, the different transactivation specificity of each P63 isoform could potentially influence cellular responses to specific stresses.
INTRODUCTION
TP63 is a member of the TP53 gene family [1] that encodes for up to ten different TA- and ∆N- isoforms (α, β, γ, δ, ε) through differential promoter usage and alternative splicing [2]. The TA isoforms contain the N-terminal transactivation domain (TA1), whereas the ∆N isoforms are transcribed from an internal promoter (P2) and lack the entire TA1 domain. A second C-terminal transactivation domain (TA2), present in all P63α and β isoforms, has been reported [3].
P63 is a master regulator of gene expression for squamous epithelial proliferation, differentiation and maintenance; in fact, heterozygous germline TP63 mutations are causative for a subset of human ectodermal dysplasia syndromes, confirming the key role of P63 in epidermis and limb formation during development [4]. The ∆N-P63α isoform is the variant predominantly expressed in basal epithelial cells of skin, breast, prostate and urinary tract, its inactivation being associated with the developmental defects. Conversely, TA-P63 isoforms are barely detectable in the same tissues and have instead been identified in the female germ line [5-7], where they play a role in quality control in response to DNA damage through the modulation of apoptosis [8].
Since the ∆N-P63 isoforms lack the N-terminal transactivation domain, it was originally proposed that these proteins might act primarily as oncogenes through dominant-negative mechanisms [1]. Indeed, these isoforms can play an important role in the progression of triple negative breast cancers where ∆N-P63α promotes tumor survival by inhibiting TA-P73 pro-apoptotic activity [9]. However, different studies indicate that ∆N-P63 can be transcriptionally active [3, 10-13]. For example, ∆N-P63α may directly contribute to tumorigenesis by up-regulating the expression of the chaperone protein HSP70, which displays proliferative and anti-apoptotic functions [14] or by repressing pro-apoptotic genes [15]. However, ∆N-P63 isoforms may have also tumor suppression function; in fact ΔN-P63α has been shown to transcriptionally activate genes like VDR and ID-3, causing a decrease in cell invasion [16, 17].
The TA-P63 isoforms have been described to induce cell cycle arrest, senescence and apoptosis [18, 19]. Recent evidence suggested that TA-P63 also inhibits metastasis either by transcriptionally activating metastasis suppressor genes, such as BHLHE41 (Sharp1) and CCNG2 (Cyclin G2) [20] or by directly up-regulating miR-130b and the microRNA processing enzyme Dicer [21].
Regulation of target genes expression by P63 is achieved through the binding to sequence-specific REs whose consensus sequence is highly similar to that of p53 REs (RRRCWWGYYY-n-RRRCWWGYYY; R=purine; W=A/T; Y=pyrimidine; n=0-13 bp spacer; CWWG = CORE) [22]. Structures of P53, P63 and P73 DNA binding domains (DBDs) co-crystallized with DNA target sequences revealed overall conserved conformation and DNA-protein contact sites, even though subtle structural differences were observed [23-25]. In vivo DNA binding studies have demonstrated specific features in P53, P73 and P63 target sites’ recognition [26-31], supporting the view that promoter selectivity contributes to the functional divergence that is apparent by genetic studies in mice or by germ line mutations in humans diseases.
In particular, relative to the P53 consensus, the P63 consensus RE was enriched in A/G at position 5 and C/T at position 16 [27]. Recently, the observation that REs containing two half-sites with one overlapping base pair were recognized by P63, but not by P53, provided additional evidence that specific recognition of cis-elements contribute to functional divergence among P53 family proteins [32]. In particular, ChIP-sequencing data from different cellular systems indicated a high correlation among P63-bound sites, the majority of which are not bound by P53 proteins activated by DNA damage response [33].
To examine the contribution of RE sequence features to P63-dependent transactivation specificity we took advantage of a functional assay in yeast where a single P63 isoform can be expressed and its capacity to stimulate transcription from isogenic promoter-reporter constructs measured [28, 34-36]. Being the promoter landscape constant except for the RE being tested, we anticipate that differences in transactivation potential could be directly related to the nature of the interactions of P63 protein with DNA target sites. From the results obtained with P63 proteins on more than 80 different REs, those obtained with P73 and P53 proteins for a selected group of REs, and also from the correlation between yeast- and mammalian-transcription assays, we uncovered that ∆N- and TA-P63α exhibit different transactivation specificities. These changes are dependent on specific features of the target RE sequences, were not observed with corresponding P73 and P53 proteins, and are likely related to intrinsic differences in the oligomeric state and in the cooperative DNA binding between ∆N- and TA-P63α proteins.
RESULTS
∆N- and TA-P63α isoforms exhibit different transactivation specificities in a genetically defined yeast-based assay.
Initially, we compared TA-P63α and ΔN-P63α transactivation abilities towards 53 canonical P53 family REs using the yeast functional assay (Table S1A). The results clearly showed isoform-dependent transactivation capacity with the identification of groups of REs exhibiting different ∆N-P63α/TA-P63α activity ratios. Next to a group of 14 REs towards which TA-P63α was at least 1.5 times more active than ∆N-P63α (Figure 1A, gray bars; Table S1A, gray highlight), there was a group of 10 REs exhibiting quite similar responsiveness to the two isoforms (Figure 1A, white bars; Table S1A, white cells), followed by another group of 21 REs that was more responsive to ∆N-P63α (Figure 1A, blue bars; Table S1A, blue highlight). Eight REs were not active (Table S1A, green highlight). We then extended the functional analysis of P63α isoforms to 34 additional REs (canonical but containing a spacer sequence between the two decameric half-sites, or non-canonical, comprising three-quarter sites and half-sites, multimers or altered-structure REs) (Table S1B). The results confirmed the existence of groups of REs with different ∆N-P63α/TA-P63α activity ratios. As an example, results obtained with three selected REs from each of the different groups are presented in Figure 1B. The sequence features of the two groups of canonical REs (without spacer) (Table S1A) showing different ∆N-P63α/TA-P63α activity ratios were summarized using two logo views, the second of which takes into account the sequences of dinucleotide motifs flanking the conserved C and G bases of the RE consensus (Figures 1C and 1D). Overall, ∆N-P63α showed higher activity than TA-P63α towards REs containing a higher frequency of non-consensus bases and a reduced frequency of the CATG sequence at the CWWG core motif. The estimation of relative binding affinities (Log Kd) for the canonical REs (Table S1A) based on a binding predictor matrix [37], in relation to the ΔN-P63α / TA-P63α activity ratio (Figure 1E) or to the transactivation capacity of each protein (Figure 1F), strongly supports the finding that TA-P63α exhibited higher transactivation potential but limited to higher affinity REs (Figure 1F, right panel). Conversely, ∆N-P63α exhibited a generally lower transactivation potential but towards a wider range of REs, including those with predicted lower Kd (Figure 1F, left panel). Moreover, the comparison of the relative transactivation capabilities of TA-P63α and ∆N-P63α on CON A (2xGGGCATGTCC) and CON B (2xGGGCTAGTCC) supports the impact of CWWG sequences in determining isoform specificity; in fact, the ratio of ∆N-P63α/TA-P63α responsiveness changed from 0.28 for CON A to 4.74 for CON B (Table S1A). Interestingly, differences in torsional flexibility were described for P53 consensus REs containing CATG or CTAG core motifs, and shown to impact on P53 binding cooperativity [38].
Figure 1: ∆N-P63α and TA-P63α exhibit differences in transactivation capacity and specificity. A) Ratio between ∆N-P63α and TA-P63α transactivation towards 45 canonical REs. The names of the REs are listed on the left, while the sequences and feature information can be found in Table S1A. Results are plotted in a logarithmic scale. Grey or blue bars mark REs that were at least 1.5 times more responsive to TA-P63a or to ∆N-P63α, respectively (i.e. ∆N-P63α/TA-P63α ratio being lower than -0.17 or higher than 0.17 in log scale). Differences in ratio comprised in that interval were not considered as significant (white bars). The very high ratio for the top three REs is caused by extremely weak responsiveness to TA-P63α (see Table S1A). B) Example of the yeast-based transactivation assay. ∆N-P63α or TA-P63α were expressed under the GAL1 inducible promoter (8 hours at 0.128% galactose) in isogenic yeast reporter strains containing the indicated P53-family REs. Light Units were normalized to the optical density of the cultures and also to the activity measured in strains transformed with an empty expression vector (RLU). Average and standard errors of four biological replicates are plotted. C) Traditional Logo view showing sequence features for the group of 21 REs that were more responsive to ∆N-P63α than to TA-P63α (see panel A and Table S1A) or for the 14 REs that were more responsive to TA-P63α than to ∆N-P63α (see panel A and Table S1A). The logo was generated with the WebLogo 3 tool (http://weblogo.berkeley.edu/logo.cgi). D) Custom logo summary highlighting dinucleotide motifs at the CWWG CORE and the RR or YY DNA contact sites. The height of the color box reflects the percentage of consensus bases at each position. Font size of consensus bases or motifs is proportional to their frequency of occurrence. The consensus sequence motif is indicated below the figure. E) Comparison of predicted DNA binding affinity for the p53 REs listed in Table S1A and the ratio of relative transactivation measured with TA-P63α and ∆N-P63α. Each dot represents a different RE sequence and the color matches the groupings based on the ∆N-P63α/ΤΑ-P63α transactivation ratio reported in panel A and in Table S1A, plotted on a logarithmic scale. F) Relative transactivation potential of ∆N-P63α (left panel) and ΤΑ-P63α (right panel) compared to the predicted values of Log Kd for all REs in panel E.
Regarding P63α isoforms activity on non-canonical sites, both ∆N-P63α and TA-P63α were inactive with half-site REs (Table S1B). Moreover, ∆N-P63α was slightly less sensitive than TA-P63α to the negative impact of spacers between half-sites and was more active towards three-quarter sites.
Overall, TA-P63α showed a transactivation preference towards high affinity REs while ∆N-P63α revealed a preference for REs predicted to have lower affinity, but high cooperativity [38].
∆N- and TA-P63β and P73 isoforms, as wll as P53 N-ter and C-ter variants do not exhibit different transactivation specificities in yeast.
Given the unexpected differences in transactivation specificity observed with P63α variants at the N-terminal region, we extended the analysis to other P53 family proteins. Firstly, we measured TA-P63β and ∆N-P63β transactivation specificities using a selected subset of 35 REs (Table S2).
TA-P63β was more active than ∆N-P63β, with the exception of four REs (PRODH+4.7, PRODH-3.1, C1 SP2 and PRODH-0.9) (Table S2). TA-P63β also showed higher transactivation capacity than TA-P63α and appeared to be less sensitive to the deleterious impact of a spacer between half-sites, although it remained inactive on half-sites (Tables S1 and S2). On the contrary, ∆N-P63β exhibited weak activity on all tested REs, generally much lower than ∆N-P63α (Tables S1 and S2). Hence, there was no clear evidence for a change in specificity for TA- and ∆N-P63β unlike the case of the corresponding P63α isoforms (Figure 2A).
∆N- and TA-P73α isoforms, along with the corresponding β isoforms, were also analyzed on a total of 14 selected REs (Table S3). As observed for P63 proteins, TA-P73β had a higher transactivation capacity than TA-P73α (Figure 2B). TA-P73α had always a much higher transactivation capacity than ∆N-P73α, the latter being barely active in yeast. Furthermore, ∆N-P73β was completely inactive in our experimental system.
Figure 2: ∆N- and TA isoforms of P63β as well as P73α or P73β isoforms do not exhibit transactivation specificity in yeast. The indicated P63 (A) or P73 (B) proteins were expressed from stable transformants of isogenic yeast reporter strains and relative transactivation was measured as described in Figure 1B. Average and standard errors of four biological replicates are plotted. In panel A, the light gray bars indicate results obtained with the P63 proteins that were presented in Figure 1 and are included here to facilitate the comparison of results obtained with the P63β isoforms. Additional results are presented in Table S2 and S3.
In order to rule out that the observed heterogeneity in transactivation could be due to differences in protein steady-state levels, Western blot analysis was performed. Similar expression of TA-P63α and ∆N-P63α isoforms was observed in yeast (Figure S1), confirming previous results [39]. TA-P63β protein levels appeared to be lower than those of ∆N-P63β hence the higher activity of the TA-P63β is even underestimated. The inactive ∆N-P73β was expressed at fairly good levels.
To determine the relative transactivation potential of all the three P53 family gene members, the activity of P53 was also measured on a subset of 35 REs previously tested with P63 isoforms (Table S4). As expected, P53 was always a more potent transcription factor compared to P63 and P73 isoforms (TA, ∆N, α, β), and the only protein active with half-site REs in our assay. These data are in agreement with previous results and could be related, at least in part, to the higher relative expression of P53 proteins in yeast [28]. However, a group of five canonical REs (without spacer) was identified (P48, PERP, COL18A1, H2, miR-198) that exhibited comparable or even higher responsiveness to ∆N-P63α or TA-P63β than to P53 (Table S5). Considering REs where both ∆N-P63α/P53 and TA-P63β/P53 ratios were above the 1.5 threshold (P48, PERP and mir-198), we noticed an enrichment of CAAG sequences at core motif (3/5, 60%). In the case of the miR-198 RE, two single nucleotide changes from CAAG to CATG in both core motifs led to ~30 fold increase in responsiveness to P53 but only to a 2.5 fold increase to TA-P63β (Tables S2, S4 and S5).
Lastly, we examined the relative transactivation specificity of a ∆N-P53 variant i.e. ∆40-P53 (obtained from an alternative translation start site) compared to full-length P53. A sample of 10 REs, representative of different P63α isoforms’ transactivation specificities, and two different levels of expression of the GAL1 inducible promoter were used (Figures S2A and B). In all cases, ∆N-P53 exhibited reduced transactivation capacity, with no evidence for a change in transactivation specificity; a C-terminal deletion construct (∆C-P53), lacking the basic domain was also tested, revealing lower transactivation potential and no impact on specificity (Figures S2A and B).
Overall, the functional analysis in yeast allowed us to establish differences in sequence-specific transactivation for P53 family proteins: i) ∆N- and TA-P63α isoforms (but not the corresponding ∆N- and TA-P63β, P73 or P53 isoforms) exhibited an unexpected difference in relative transactivation specificity towards REs with distinct sequence features; ii) TA-β-isoforms for both P63 and P73 were usually more active than the corresponding α isoforms; iii) regarding the ∆N-β isoforms, ∆N-P63β showed very low levels of activity while ∆N-P73β was virtually inactive.
Ectopic expression of different P63 isoforms in HCT116 P53-/- cells partially supports the RE-dependent changes in transactivation specificity.
We next asked whether the observations in yeast could be confirmed in human cells. For these proof-of-principle experiments, we chose the colon cancer derived HCT116 cell clone where TP53 has been knocked out and the expression of endogenous P63 or P73 proteins is almost undetectable [39, 40]. Ectopic expression of P63 ∆N, TA, α, β quartet was achieved upon transient transfection. Gene reporter assays, endogenous gene expression (qPCR) and Western blots were used as endpoints, the latter also to control the relative expression of the P63 proteins.
With the α isoforms, results with the gene reporter assays confirmed the higher transactivation capacity of TA-P63α with respect to ∆N-P63α on the P21 and MDM2 promoter sequences and the quite similar transactivation on BAX, as highlighted in yeast (Figure 3A). Also the observation that ∆N-P63α was more active than TA-P63α towards the AIP1 reporter was consistent with the results in yeast (Figure 3A). Conversely, P63α isoforms were barely active in HCT116 P53-/- cells towards the PG13 RE cluster, in contrast with what observed in yeast, where both isoforms were active
Endogenous genes expression or Western blot analyses confirmed a higher activity of TA-P63α compared to ∆N-P63α towards P21 and MDM2 (Figure 3B, 3C). Moreover, the expression of ∆N-P63α was also associated with higher endogenous induction of AIP1 and NOXA, as observed in yeast, and the same trend was observed for KILLER (Figure 3B). The BAX gene was more responsive to TA-P63 than to ∆N-P63α, unlike what obtained in yeast and with the gene reporter assay in mammalian cells. These discrepancies could be due to the known presence of additional REs in the promoter that are not present in the reporter constructs.
Lastly, TA-P63β was the most active isoform also in the human cell line, considering the relative low levels in the transient transfection experiments (Figure 3C). Differently from the yeast results, ∆N-P63β was more active than ∆N-P63α towards the majority of reporter constructs (Figure 3A), although such a difference was not always observed when endogenous genes expression was measured (Figure 3B).
Figure 3: ∆N- and TA-P63α isoforms exhibit some differences in transactivation specificity in HCT116 P53-/- cells. A) Dual luciferase assays from HCT116 P53-/- cells transiently cotransfected with the indicated reporter plasmids and P63 expression vectors. P53 was included as control. Renilla luciferase was measured to normalize transfection efficiencies. Data are expressed as fold of induction relative to the results obtained with an empty expression vector. Presented are the average and standard deviations of at least three biological replicates. B) qPCR analysis of the indicated endogenous target genes in HCT116 P53-/- cells transfected with the indicated expression plasmids. Data are normalized to GAPDH and B2M reference genes and are shown relative to the expression measured from cells transfected with an empty vector. Plots show the average and standard deviations of three technical replicates. The experiment was repeated twice. C) Western Blot analysis of total soluble proteins extracted from HCT116 P53-/- cells transfected with the indicated plasmids. Besides P53 and P63 proteins, the target proteins P21 and MDM2 and the reference controls α-actinin and GAPDH are shown.
The same experiments were conducted after ectopic expression of P73 quartets (TA, ∆N, α, β) (Figure 4). Results with the α isoforms confirmed those in yeast, showing a higher transactivation capacity of TA-P73s with respect to ∆N-P73s on P21, MDM2 and BAX target sequences. ∆N-P73α was, in fact, inactive towards most reporter constructs (Figure 4A) and endogenous genes tested (Figure 4B). The AIP1 promoter construct, but not the endogenous gene, showed instead higher responsiveness to the ΔN-P73s compared to TA-P73s (α and β isoforms) (Figures 4 A and 4B). ∆N-P73β that was highly expressed (Figure 4C), exhibited a significant activity in the reporter assays (in contrast to what observed with yeast), but was weakly active towards most endogenous genes (Figure 4B), and did not stimulate the expression of P21 and MDM2 proteins (Figure 4C). TA-P73β was in general the most active isoform also in the human cell line.
Figure 4; ∆N- and TA-P73 isoforms exhibit differences in transactivation capacity but not in specificity in HCT116 P53-/- cells. A) Dual luciferase assays from transiently transfected HCT116 P53-/- cells with the indicated reporter plasmids and P73 isoforms. P53 was included as control. Experiments were performed and data are plotted as described in Figure 3A. B) qPCR analysis of the indicated endogenous target genes in HCT116 P53-/- cells transfected with the indicated expression plasmids. See Figure 3B. C) Western Blot analysis of total soluble proteins extracted from HCT116 P53-/- cells transfected with the indicated plasmids. Besides P53 and P73 proteins, the target proteins P21 and MDM2 and the reference controls α-actinin and GAPDH are shown.
Overall, the ectopic expression of TA-P63α and ∆N-P63α in HCT116 P53-/- cells partially supports the RE-dependent changes in transactivation specificity.
Genotoxic stress and/or P53 activating molecules influence the expression of ∆N-P63 isoforms in human cell lines.
The relative expression from the P1 and P2 promoters of the TP63 gene is well-established in the context of squamous epithelial differentiation [41]. Given the identification of a functional p53 RE in the P2 promoter of TP63 [42-44], we examined whether genotoxic stress or P53 activating molecules could affect the relative expression of P63 promoter variants (TA or ∆N) by qPCR (Figures 5A and 5B). Experiments were performed in five human cell lines that differ for P53 status: HEK293T and HepG2 (wild type), HaCat (mutated), JHU-011 and JHU-029 (null). These cells lines were chosen as they endogenously expressed P63 proteins [45].
In HEK293T and HepG2 cells, ∆N-P63 was expressed at lower levels compared to TA-P63 and its expression decreased both after genotoxic stress or Nutlin treatment, particularly in HEK293T (Figure 5A). TA-P63 levels did not show significant variations except for HEK293T, where a 50% increase and a 25% decrease was observed upon 5FU or Nutlin treatment, respectively (Figure 5A).
In cell lines that are mutated or null for P53, ∆N-P63 was expressed at much higher levels compared to TA-P63 (Figure 5B). DXR led to a marked decrease of ∆N-P63 in all cell lines, while 5FU treatment decreased ∆N-P63 in HaCat and JHU-011 but not in JHU-029; Nutlin did not impact on ∆N-P63 levels in these cell lines (Figure 5B). TA-P63 level was not particularly affected by the treatments, also considering the low expression of this P63 isoform (Figure 5B).
DXR and 5FU led to the induction of the well-known P53 target P21 in P53 wild type cell lines, while Nutlin was effective only in HepG2 cells that do not contain the P53-inhibiting SV40 T-large antigen. These clear expression changes were not detected in the P53 mutated or null cells, with the exception of an induction of P21 in DXR-treated JHU-011 cells.
We selected JHU-029 cells for further studies as they showed the highest endogenous levels of ∆N-P63 through the comparison of the Cycle Threshold in qPCR assays. Consistent with the qPCR results (Figure 5B), also ∆N-P63 protein levels were reduced by genotoxic stress, particularly after the treatment with DXR (Figure 5D). To provide evidence of a direct contribution of P53 in the regulation of endogenous ∆N-P63, we ectopically expressed the wild type protein in JHU-029 cells, which resulted in a severe down-regulation of ∆N-P63 proteins (Figure 5E).
Figure 5: Relative expression of endogenous P63 isoforms in cells lines treated with DNA damaging or P53 inducing agents. Experiments were conducted in the indicated cell lines with known P53 status (wild type, mutant or null, as indicated). Cells were treated with Doxorubicin (DXR), 5-FluoroUracil (5FU) or Nutlin-3a (Nutlin) for 16 hours, as described in Materials and Methods. B2M and GAPDH served as reference genes. Three biological replicates were performed. Measurements of the endogenous levels of ∆N-P63 or TA-P63 mRNAs in P53 wild type A) or mutant/null cells B). For each cell line the graph indicates the relative expression changes and is normalized over the most abundant isoform transcript, set to 1 for the DMSO (mock solvent condition, broken line). C) The endogenous levels of the P53 target gene P21 were measured as a positive control of the efficacy of the treatments. Values are indicated as fold change relative to DMSO treated cells (set to 1 and indicated by broken line). D) ∆N-P63 protein levels were determined by Western blot in JHU-029 cells after treatment with DXR, 5FU or Nutlin. GAPDH was used as loading control. Immunoreactive bands specific for the ∆N-P63α or ∆N-P63β isoforms are indicated with black arrows. E) Western blot experiment demonstrating the over-expression of P53 after transfection and its effect on endogenous ∆N-P63α or ∆N-P63β isoforms in JHU-029 cells. GAPDH was used as loading control.
Taken collectively, the results indicate that ∆N-P63 can be negatively regulated at the mRNA and protein levels by DNA damage both in a P53-dependent and -independent manner.
DISCUSSION
The P53 family of proteins is a paradigmatic group of sequence-specific transcription factors that are master regulators of many different biological responses through the modulation of a very large number of target genes. While genetic approaches indicate a clear separation of functions acquired after the gene duplication events throughout evolution that led to the three-genes TP53 family, biochemical and molecular biology approaches, including genome level occupancy and transcriptome analyses, indicate a high degree of conservation of the core function, i.e. the sequence-specific DNA binding [10, 27, 46-51]. Another contribution to the complexity of this scenario is represented by the modulation of these transcription factors in the context of acute stress conditions, such as DNA damage, that can impact on transactivation selectivity in vivo. These layers of regulation have been studied more in-depth for P53, for which post-translational modifications affecting protein stability, localization and interactions, along with the contribution of context dependent trans-factors, are recognized elements that modulate transcriptional specificity [52-54]. The contribution of cis-factors, namely specific features of the target DNA sites, here referred as p53 REs, has also been extensively studied [52, 53]. Less is known about the regulation of P63 and P73, although DNA damage is a recognized activation stimulus and specific post-translational modifications have been identified [55-57]. Recently, several studies have uncovered that cooperative DNA binding coupled to RE sequence differences at specific target genes can modulate P53 transactivation selectivity [28, 58-60]. For example, TP53 mutations or post-translational protein acetylation were associated to changes in cooperative DNA binding and to a selective modulation of apoptotic target genes [61]. Also, biophysical measurements led to classify p53 REs in terms of low or high cooperativity; kinetic features, especially off-rates, correlate with transactivation potential more than DNA binding affinity measured with purified P53 DBDs at low protein concentrations [30].
To establish the role of RE sequence in transactivation specificity of P63 promoter- (∆N and TA) and splice-variants (α and β), an experimentally defined assay in yeast was used, where the sequence of the RE under study and the expression of a chosen P63 isoform are the only variable and quantitative luciferase reporter assays are the functional endpoint. Our results revealed that ∆N-P63α and TA-P63α not only differ in transactivation potential, but also in transactivation specificity (Figure 1A-D; Table S1A, S1B). By using a large number of REs, we were able to highlight sequence features that correlated with the changes in transactivation specificity. In particular, ∆N-P63α-preferred REs were associated with CWWG core motifs characterized by 1) a lower torsional flexibility, relative to generic B-DNA sequences, as determined from cyclization kinetics experiments, [30, 38], 2) higher number of mismatches (Table S1A), and 3) in some cases, presence of a short spacer among decameric half-sites or absence of one pentameric quarter site (Table S1B). Hence, ∆N-P63α, compared to TA-P63α, exhibited an intrinsic preference for lower affinity (Figure 1E and 1F), high cooperativity target sequences. A possible explanation for the observed differences is that the preferred quaternary structure would be different for these two P63 isoforms, with ∆N-P63α, but not TA-P63α, being a tetramer. Notably, it was recently reported that P63 isoforms in solution can adopt different quaternary structures and specifically that TA-P63α, unlike ∆N-P63α, would not form tetramers due to an intramolecular inhibiting interaction [8, 62]. Interestingly, DNA damage-induced post-translational changes can lead to TA-P63α tetramer formation [8, 63]. In this regard, ∆N-P63α could be considered as a constitutively active tetramer. Our functional results in yeast are consistent with those data, in that a tetrameric conformation is expected to retain the capacity to interact with REs that, due to mismatches or non-canonical sequence features, would not efficiently enable independent binding of two protein dimers. The lack of the entire TA1 transactivation domain in ∆N-P63α led, however, to a lower extent of reporter genes induction, evident for the REs where both ∆N- and TA-P63α were active (Table S1). Also consistent with this interpretation are our recent results with P63 mutations associated to ectodermal dyplasia syndromes that exhibited a more severe defect when expressed in yeast as TA-P63α compared to ∆N-P63α isoform (Figure S3) [39].
Importantly, ectopic expression of ∆N- and TA-P63α in an otherwise untreated HCT116 cell clone, that is null for P53 and expresses negligible levels of endogenous P63 or P73 (9,46), confirmed the differences in transactivation specificity using gene reporter assays and, in part, also following modulation of endogenous genes. Previous data in mammalian systems had also observed higher activity of ∆N-P63α compared to TA-P63α for selected target genes [11, 64]. Hence, the specific responsiveness of the REs can, to some extent, impact on the relative expression changes even when the RE is embedded in the natural promoter context. However, yeast-based assays and also results with cloned promoter fragments in mammalian cells were not completely predictive of the responsiveness of endogenous target genes, in particular when multiple REs are present in the regulatory regions, such as in the case of BAX.
Our results regarding the ∆N-P63β isoform are strikingly inconsistent between yeast and human cells: in fact in yeast ∆N-P63β showed much lower transactivation capacity than ∆N-P63α (Figure 2A and Table S2) (Figure 3). The low activity of ∆N-P63β in yeast did not appear to be caused by reduced protein expression (Figure S1), but we cannot exclude reduced affinity for components of the transcription machinery, or lack of necessary cofactors, or even a defect in nuclear import. Conversely, in HCT116 P53-/- cells, ∆N-P63β was much more active than ∆N-P63α and in some cases equal to TA-P63β, but without a clear evidence for changes in transactivation specificity (Figure 3). The results with P63β isoforms in HCT116 and with TA-P63β in yeast are also consistent with the reported tetrameric nature of P63β variants in solution [8, 65]. Furthermore, the comparison between TA-P63α and TA-P63β in yeast and in mammalian cells evidenced the higher transactivation potential of the β splice variant lacking the SAM domain and the carboxy-terminal inhibitory domain as well as the capacity of TA-P63β to retain activity towards REs containing spacers or non-canonical three-quarter site REs. This feature was particularly evident in HCT116 cells with the PG13 RE, consisting of a tandem repeat of 13 copies of a low affinity RE of 19 nucleotides.
We also studied TA- and ∆N-, α and β variants of P73. In the yeast system ∆N-P73α was an extremely weak transcription factor and ∆N-P73β was inactive (Figure 2B and Table S3). In HCT116 P53-/-, although results might be biased by the higher protein expression, ∆N-P73β exhibited activity with all reporters tested and was even more active than TA-P73β with the AIP1 and BAX promoter constructs (Figure 4). However, ectopic expression of both ∆N-P73α and ∆N-P73β did not lead to the induction of most of the endogenous target genes tested. Further studies are needed to examine more precisely the possibility that promoter and/or splice variants of P73 would exhibit differences in transactivation specificity and to elucidate the potential for ∆N-P73s to modulate gene expression in a physiological setting. TA-P73α has been reported to be tetrameric by size exclusion chromatography, suggesting a closer similarity with P53 than with TA-P63α in stress-induced regulation of transcriptional activity [65].
All P63 and P73 isoforms tested in yeast were inactive on half-site REs (Tables S2, S3 and S4). When expressed at high levels both in yeast and in human cells, P53 was instead shown to be capable of modulating transcription from half-site REs as well as to bind to half-site REs by ChIP and ChIP-sequencing experiments [66, 67]. However, the oligomeric state of P53 protein bound to half-sites remains to be established.
Several P53 isoforms have also been identified, resulting from alternative splicing of intron 2 or alternative translation initiation (∆N-P53, ∆C-P53), an internal promoter (P53-∆133), or alternative splicing at the carboxy terminus (P53β, P53γ) [68]. Of these isoforms, only ∆N-P53 retains the entire DBD and oligomerization domains [47]. Very recently, a mouse model with homozygous deletion of the basic domain in the P53 C-ter was reported to have severe phenotypes, very reminiscent of Dyskeratosis congenita [69]. In derived MEFs (mouse embryonic fibroblasts), constitutive nuclear levels of this C-ter deleted-P53 were slightly higher compared to wild type controls, but the possibility of changes also at the level of relative transactivation specificity was not conclusively investigated. An independent mouse model of the same P53 C-ter deletion construct reported tissue-specific differences on the transactivation of target genes [70]. Those findings along with data obtained with P63α isoforms and on the oligomeric state of P53 family proteins [65], stimulated us to examine P53 N-ter (∆N-P53) and C-ter deletion (∆C-P53) constructs using the yeast assay in order to investigate possible changes in transactivation specificity. The results obtained with ten selected REs showed that both P53 deletion constructs retained some level of transactivation potential but they were less active than wild type P53 and did not exhibit differences in specificity (Figure S2). However, the possibility of RE-driven allosteric changes that are not impacting on transactivation in yeast due to missing cofactors cannot be entirely excluded.
To explore the potential for evolutionary conservation of transactivation capacity and specificity of P63 proteins, ∆N-P63 from zebrafish (Danio rerio, Dr), an organism where the P2 promoter could be predominantly or exclusively active [71] was cloned and tested in yeast. Moreover, phylogenetic studies have not conclusively demonstrated which TP63 promoter is more ancient between P1 (TA) and P2 (∆N) [71, 72]. We were able to clone a single ∆N-P63 isoform, corresponding to ∆N-P63β, and considered the main or the exclusive isoform expressed in zebrafish [71]. Nevertheless, the protein was an active transcription factor in yeast. A comparison of Dr-∆N-P63 with human TA-P63α indicated an overall conservation of relative transactivation specificity as seen for human ∆N-P63α in the comparison with TA-P63α (Figure S4 and Table S1).This observation may suggest that Dr ∆N-P63, like human ∆N-P63, constitutively adopts a tetrameric conformation.
Alternative TP63 promoter usage in mammals has been clearly established as part of differentiation programs, notably in the context of squamous cell epithelia, where basal layer cells express predominantly ∆N-P63, while TA-P63 expression is prevalent in the differentiated cells above [41]. We were interested in exploring whether changes in relative promoter activity could also occur in the context of DNA damage, also in the light of the established presence of a p53 RE in the TP63 promoter P2 [42-44]. Interestingly, P53 has also been reported to modulate the activity of the TP73 P2 promoter [73] [42]. Experiments in five cell lines expressing endogenously TA- or ∆N-P63 and differing for P53 status indicated that genotoxic stress, or P53 activation with Nutlin in P53 wild type cells, or P53 ectopic expression in a P53 null cell line, can lead to a decrease in ∆N-P63 protein levels, thereby changing the ratio between TA and ∆N isoforms (Figure 5). Given the different transactivation specificities and biological functions of ∆N- and TA-P63 these changes in relative expression might influence the cellular response by modulation of expression of target genes.
Our results indicate that the regulation of P63 promoters’ usage could affect P63-dependent transactivation specificity, exploiting the differences in oligomeric state between ∆N- and TA-P63α [65] and not simply tune the levels of target gene expression.This regulatory mechanism would however require a co-evolution of promoter RE features among different functional groups of P63 target genes to be able to impact on biological outcomes. The apparent enrichment for low affinity/high cooperativity REs among established P53 family target genes involved in apoptosis [30, 58, 59, 61] could reflect this coevolution and be interpreted in the light of the results establishing the ancestral function of P63-like proteins as the quality control of the germline via induction of apoptosis.
MATERIALS AND METHODS
Yeast strains, vectors and functional assay.
For analysis of the transactivation ability of P53 family members (P53, ∆N- and TA-, -α and -β variants of P63 and P73) we used 87 haploid S. cerevisiae yeast strains (yLM-REs) [28, 34-36, 45, 74]. All strains are isogenic except for the different RE located upstream of the luciferase reporter gene [29, 36, 66, 75]; new yLFM-RE strains were constructed, taking advantage of the Delitto perfetto approach [76, 77]. The panel of REs includes 53 canonical REs (20 bp) without spacer (Table S1A), 15 canonical REs with a spacer and 19 non-canonical REs comprising three-quarter sites, half-sites, multimers and altered-structure REs (Table S1B).
Yeast manipulations were performed as previously described [28, 34, 39]. Inducible expression of P53 family members was achieved under a GAL1-10 promoter using a pTSG-based (TRP1) vector. Vectors expressing human ∆N-P63α, TA-P63α and P53 were already available [29, 31, 39]. New pTSG-based vectors expressing human ∆40-P53, ∆C-P53 (∆369-393), ∆N-P63β, TA-P63β, ∆N-P73α, TA-P73α, ∆N-P73β and TA-P73β were constructed as previously described [39]. ∆N-P63β and TA-P63β cDNAs were provided by Prof. Roberto Ravazzolo and Dr. Renata Bocciardi (IRCCS G. Gaslini, Genoa, Italy). pTSG-based vectors expressing TP63 mutants as ∆N-P63α isoform were already available, while the same mutants as TA-P63α variants were constructed as previously described for the pTSG-TA-P63α vector (43). A pTSG vector harboring the ∆N-P63β isoform expressed in zebrafish (Danio rerio) was also constructed. Plasmid pRS314 (TRP1) was used as empty vector. Primer sequences are available upon request.
The transactivation ability of the different P53 family variants was analyzed by transforming yLFM-REs strains with the different expression vectors using the Lithium acetate method, as previously described [34]. The luciferase assay was conducted according to the miniaturized protocol we recently developed [34]. Yeast transformants were grown (8 or 16 hours as indicated) in media containing raffinose (2%) without or with different concentrations of galactose as inducer. The transactivation ability of P53 family members was measured using the Bright Glo Luciferase assay kit (Promega, Milan, Italy) and expressed as relative light unit (RLU) normalized to optical density (600 nm), subtracting the luminescence obtained by the cells transformed with the empty vector in each reporter strain.
Mammalian cell lines and vectors.
Human colon cancer HCT116 P53-/- cell line was a gift of Prof. B. Vogelstein (John Hopkins University, Baltimore, MD, USA) and was used for transfecting constructs expressing P53 family members, followed by gene reporter assays, quantitative PCR and Western blotting experiments. HepG2 (derived from a human hepatocellular carcinoma and wild-type at the TP53 locus) and HEK293T (derived from human embryonic kidney and wild-type at the TP53 locus, but expressing the P53-inhibiting SV40 T-large antigen) were obtained from Dr. Alessandro Provenzani (CIBIO, University of Trento, Italy) and Prof. Juergen Borlak (Hannover Medical School, Germany), respectively. HaCat cells (human immortalized keratinocytes harboring the compound heterozygous TP53 mutations H179Y and R282W) as well as the P53-null JHU-011 and JHU-029 cells (Head and Neck Squamous Cell Carcinoma-derived human cell lines) were obtained from Prof. David Sidransky’s laboratory at John Hopkins University (Baltimore, MD, USA). HepG2, HEK293T, HaCat, JHU-011 and JHU-029 were used to test the effect of P53 activating molecules on P63 isoforms expression levels. Cells were cultured in DMEM or RPMI (GIBCO, Invitrogen, Milan, Italy) supplemented with 10% fetal bovine serum (FBS), 2mM L-Glutamine and antibiotics (100 units/ml penicillin plus 100 µg/ml streptomycin) and maintained in a humidified atmosphere at 37°C with 5% CO2. Cells were routinely checked to exclude the presence of mycoplasm.
pCI-neo plasmids for the expression of ∆N-P63α and TA-P63α were already available [39]. pCI-neo plasmids expressing ∆N-P63β, TA-P63β, ∆N-P73α, TA-P73α, ∆N-P73β and TA-P73β were obtained by XhoI/NotI double digestion of pTSG-vectors containing the desired cDNAs and subsequent ligation of the insert into equally digested pCI-neo plasmid. P53 expression was obtained using a pCI-neo-derived plasmid similarly generated. The empty pCI-neo plasmid was used as control vector.
The pGL3-P21 (2.3 Kb promoter fragment, containing both the 5’ and the 3’ p53 REs of the P21 gene), pGL3-MDM2 (350 bp region with both p53 REs present in intron 1 of the MDM2 gene), PGL3-BAX (400 bp region of intron 1 of the BAX gene), PG13 (13 direct repeats of the sequence 5’-CCAGGCAAGTCCAGGCAGG-3’) and pGL3-AIP1 (containing the p53 RE from yFLM-AIP1 yeast strain) reporter vectors were used for luciferase reporter assay in mammalian cells after transient transfection [28]. pRL-SV40 plasmid, harboring the luciferase gene from Renilla reniformis under the control of a constitutive promoter, was used to normalize for transfection efficiency.
Luciferase assay in the mammalian cell line HCT116 P53-/-.
HCT116 P53-/- cells (2x105) were seeded twenty-four hours prior transfection onto 24-well plates. Cells were co-transfected at 50-70% confluence using TransIT-LT1 (Mirus, Tema Ricerca, Bologna, Italy) with: i) 200 ng of the expression vectors (for P53 expression, 50 ng of vector plus 150 ng of empty vector were used -200 ng total DNA amount- in order to avoid excessive toxicity); ii) 250 ng of the reporter vector and iii) 50 ng of the normalization vector. After additional 24 hours, cells were washed with PBS, lysed with Passive Lysis Buffer (PLB) 1X (Promega). Firefly and Renilla luminescence were measured as described previously [78].
Transfections and treatments with P53 inducing/activating compounds.
HCT116 P53-/- and JHU-029 cells were seeded onto 6 well plates at the concentration of 9x105 cells per well and transfected the following day with 2 μg of expression vectors for P53 family proteins (in pCI-neo) or pCI-neo empty vector, using Fugene HD (Promega). Twenty four hours later, for transfections, or when indicated, for treatments, cells were harvested and processed for RNA or protein extraction.
P53 wild-type (HEK293T and HepG2) as well as P53 mutated (HaCat) or P53 null (JHU-011 and JHU-029) cell lines that endogenously express P63 were treated with compounds able to stabilize and activate P53. Twenty-four hours after seeding, cells were treated with Doxorubicin (DXR -1.5μM), 5-FluoroUracil (5FU -375μM) and Nutlin-3A (Nutlin -10μM) for 16 hours and then processed as indicated above. All compounds were purchased from Sigma Aldrich (Milan, Italy). DMSO was used as control as 5FU and Nutlin were dissolved in such solvent.
Quantitative PCR analyses.
To determine optimal transfection efficiency pEGFP-N1 vector was used (obtained from Prof. Paolo Macchi, CIBIO, University of Trento). Twenty-four hours post-transfection cells were harvested and total RNA was isolated using the RNeasy mini kit, according to the manufacturers’ recommendations (Qiagen, Milan, Italy). Two μg of RNA were converted into cDNA using the M-MuLV reverse transcriptase (Thermo Scientific, Milan, Italy). Quantitative PCR was performed on 25 ng of cDNA as previously described [79] using KAPA Sybr Green Master mix (Kapa Biosystems, Resnova, Ancona, Italy) and specific primers to measure the expression of TP63 (ΔN and TA), P21, MDM2, AIP1, BAX, KILLER and NOXA genes. B2M (β2-microgobulin) and GAPDH (Glyceraldehyde 3-Phosphate DeHydrogenase) were used as reference genes. Primer sequences are available upon request. Fold changes respect to the empty vector were calculated using the ΔΔCT method [80].
Antibodies and Western blotting.
Protein extraction from yeast cells.
Yeast transformants were grown for 8 hours in selective medium containing 0.128% galactose to induce the expression of P63 and P73 specific isoforms. An equivalent amount of cells, based on the culture absorbance measurement (2.5 OD, OD600nm), was collected by centrifugation. Cells were treated with 0.2N NaOH at room temperature following the extraction protocol described in [81] and 25 μl of extracts were loaded on 7.5% poly-acrylamide gels. Transfer onto nitrocellulose membranes was achieved using the i-Blot semi-dry system (InVitrogen, Life Technologies, Milan, Italy). Specific antibodies directed against P63 (clone 4A4: sc-8431, Santa Cruz Biotechnology, Milan, Italy) or P73 isoforms (clone ER-15: OP109, Calbiochem, Millipore, Milan, Italy), were diluted in 1% non-fat skim milk dissolved in PBS-T. PGK1 (Phospho Glycerate Kinase 1) was used as loading control (clone 22C5D8, Life Technologies, Milan, Italy).
Protein extraction from mammalian cells.
Pellets from transfected or treated cells were washed with PBS and used for total protein extraction in RIPA lysis buffer supplemented with Protease Inhibitors cocktail (Roche, Milan, Italy). Besides the antibodies against P63 and P73 described above, antibodies against P53 (clone DO-1: sc-126), P21 (clone C-19: sc-397), MDM2 (clone SMP-14: sc-965), GAPDH (clone 6C5: sc-32233) or α-actinin (clone H-2: sc-17829) (all from Santa Cruz Biotechnology) were used for immunodetection after dilution in 1% non-fat skimmed milk dissolved in PBS-T. Immuno-reactive bands from yeast as well as mammalian extracts were detected by the ChemiDoc XRS+ System (BioRad, Milan, Italy), using the ECL Select detection reagent (Amersham, GE Health Care, Milan, Italy).
Acknowledgements
We wish to thank Drs. Umberto Cardellino, Raffaella Cinquetti, Silvano Paternoster and Francesca Precazzini for excellent technical support.
Funding
This work was supported by the Italian Association for Cancer Research, AIRC (IG grant # 12869 to AI, IG#5506 to G.F.) Partially supported by Compagnia S. Paolo, Turin, Italy (Project 2012.1590 “Le interazioni molecolari della proteina p53 mutata come bersaglio di nuove terapie antitumorali personalizzate”). IR is supported by a Fellowship from the Pezcoller Foundation, Trento, Italy
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http://snailstales.blogspot.com/2011/05/protoconchs-of-assiminea.html
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## 30 May 2011
### Protoconchs of Assiminea
There is a generalization that among closely related, especially congeneric, marine snails, the species with smaller protoconchs have planktonic larvae that go thru a free-swimming stage, while those with larger protoconchs have direct developing larvae that hatch out of their eggs as tiny crawling snails. The idea seems to go back at least to Verduin (1977) with possibly even earlier versions.
To compare the sizes of the protoconchs of related species, Verduin (1977) measured the following 2 dimensions of a protoconch, where Dn is the diameter of the nucleus of the protoconch and D1/2 is the diameter of the 1st half whorl.
Since Dn is within D1/2, the 2 measurements are tightly, in fact, linearly, correlated. Nevertheless, a plot of Dn versus D1/2 is a useful way to separate groups of supposedly planktonic versus supposedly direct-developing species as Verduin (1977) showed to be the case with the species in the genus Alvania.
Recently, Aartsen (2008) noted that the application of Verduin's method to the Atlantic and Mediterranean species of Assiminea revealed the existence of 2 groups. However, he did not present a plot. So I added my own measurements of Assiminea succinea to Aartsen's measurements and did a Verduin plot.
To illustrate the intrinsic variability in the dimensions of such traits, I show here the measurements of 4 specimens rather than the mean value.
As far as I know, among these species, life history information is available only for A. grayana, which has planktonic larvae and for A. succinea, which has direct developing larvae. In the plot, the protoconchs of the former are smaller than those of the larger. So at least with those 2 species, we have agreement with the generalization that direct developing larvae are larger than planktonic larvae.
Aartsen. 2008. Basteria 72:165.
Verduin. 1977. Basteria 41:91.
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https://icsecbsemath.com/2017/08/07/icse-board-problems-solved-class-10-ratio-and-proportion/
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$\displaystyle \text{Question 1: If } x, y \text{ and } z \text{ are in continued proportion, prove that: } \\ \\ \frac{(x+y)^2}{(y+z)^2} = \frac{x}{y} \text{. [2010]}$
If $\displaystyle x, y \text{ and } z$ are in continued proportion, then
$\displaystyle \frac{x}{y} = \frac{y}{z} \Rightarrow x = \frac{y^2}{z}$
Applying componendo and dividendo
$\displaystyle \frac{x+y}{x-y} = \frac{y+z}{y-z}$
$\displaystyle \Rightarrow \frac{x+y}{y+z} = \frac{x-y}{y-z}$
Squaring both sides
$\displaystyle \Rightarrow \frac{(x+y)^2}{(y+z)^2} = (\frac{x-y}{y-z})^2$
$\displaystyle \Rightarrow \frac{(x+y)^2}{(y+z)^2} = (\frac{x-y}{y-z})^2$
Substituting
$\displaystyle \Rightarrow \frac{(x+y)^2}{(y+z)^2} = (\frac{\frac{y^2}{z}-y}{y-z})^2$
$\displaystyle \Rightarrow \frac{(x+y)^2}{(y+z)^2} = (\frac{y^2-yz}{z(y-z)})^2 = \frac{y^2}{z^2} = \frac{zx}{z^2} = \frac{x}{z}$
$\displaystyle \\$
$\displaystyle \text{Question 2: Given } x= \frac{\sqrt{a^2+b^2}+\sqrt{a^2-b^2}}{\sqrt{a^2+b^2 }-\sqrt{a^2-b^2 }} \\ \\ \text{ Use componendo and dividendo to prove that: } x^2= \frac{2a^2 x}{x^2+1} \text{ [2010 ] }$
$\displaystyle \text{Given } x= \frac{\sqrt{a^2+b^2}+\sqrt{a^2-b^2}}{\sqrt{a^2+b^2 }-\sqrt{a^2-b^2 }}$
Applying componendo and dividendo
$\displaystyle \frac{x+1}{x-1} = \frac{(\sqrt{a^2+b^2}+\sqrt{a^2-b^2})+(\sqrt{a^2+b^2 }-\sqrt{a^2-b^2 })}{(\sqrt{a^2+b^2}+\sqrt{a^2-b^2})-(\sqrt{a^2+b^2 }-\sqrt{a^2-b^2 })}$
Simplifying
$\displaystyle \frac{x+1}{x-1} = \frac{\sqrt{a^2+b^2}}{\sqrt{a^2-b^2 }}$
Square both sides
$\displaystyle \frac{x^2+1+2x}{x^2-2x+1} = \frac{a^2+b^2}{a^2-b^2}$
Applying componendo and dividendo
$\displaystyle \frac{x^2+1+2x+x^2-2x+1}{x^2+1+2x-x^2+2x-1} = \frac{a^2+b^2+a^2-b^2}{a^2+b^2-a^2+b^2}$
$\displaystyle \frac{2(x^2+1)}{4x} = \frac{2a^2}{2b^2}$
$\displaystyle \frac{x^2+1}{2x} = \frac{a^2}{b^2}$
Simplifying
$\displaystyle b^2 = \frac{2a^2x}{x^2+1}$
$\displaystyle \\$
$\displaystyle \text{Question 3: If } \frac{x^2+y^2}{x^2-y^2} =2 \frac{1}{8} \text{ , find: }$
$\displaystyle \text{i) } \frac{x}{y}$ $\displaystyle \text{ii) } \frac{x^3+y^3}{x^3-y^3 } \text{ [2010] }$
$\displaystyle \text{i) } \text{Given } \frac{x^2+y^2}{x^2-y^2} =2 \frac{1}{8} = \frac{17}{8}$
Applying componendo and dividendo
$\displaystyle \frac{x^2+y^2+x^2-y^2}{x^2+y^2-x^2+y^2} = \frac{17+8}{17-8}$
$\displaystyle \frac{2x^2}{2y^2} = \frac{25}{9}$
Simplifying, we get
$\displaystyle \frac{x}{y} = \frac{5}{3}$
$\displaystyle \text{ii) } \frac{x^3+y^3}{x^3-y^3 }$
Applying componendo and dividendo
$\displaystyle \frac{x^3+y^3+x^3-y^3 }{x^3+y^3-x^3+y^3 }$
$\displaystyle \frac{2x^3}{2y^3} = \Big( \frac{x}{y} \Big)^3 = \Big( \frac{5}{3} \Big)^3 = \frac{125}{9}$
$\displaystyle \\$
Question 4: Using componendo and dividendo, find the value of $\displaystyle x$ : $\displaystyle \frac{\sqrt{3x+4}+\sqrt{3x-5}}{\sqrt{3x+4}-\sqrt{3x-5}} =9 \text{ [2010] }$
$\displaystyle \text{Given } \frac{\sqrt{3x+4}+\sqrt{3x-5}}{\sqrt{3x+4}-\sqrt{3x-5}} =9$
Applying componendo and dividendo
$\displaystyle \frac{(\sqrt{3x+4}+\sqrt{3x-5})+(\sqrt{3x+4}-\sqrt{3x-5})}{(\sqrt{3x+4}+\sqrt{3x-5})-(\sqrt{3x+4}-\sqrt{3x-5})} = \frac{9+1}{9-1}$
$\displaystyle \frac{2\sqrt{3x+4}}{2\sqrt{3x-5}} = \frac{10}{8}$
Simplifying
$\displaystyle \frac{\sqrt{3x+4}}{\sqrt{3x-5}} = \frac{5}{4}$
Square both sides
$\displaystyle \frac{3x+4}{3x-5} = \frac{25}{14}$
$\displaystyle 42x+56 = 75x-125$
Simplifying we get $\displaystyle x = 7$
$\displaystyle \\$
$\displaystyle \text{Question 5: If } x= \frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a+1}-\sqrt{a-1}} \\ \\ \text{using properties of proportion show that: } x^2-2ax+1 \text{ [2010] }$
$\displaystyle \text{Given } x= \frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a+1}-\sqrt{a-1}}$
Applying componendo and dividendo
$\displaystyle \frac{x+1}{x-1} = \frac{(\sqrt{a+1}+\sqrt{a-1})+(\sqrt{a+1}-\sqrt{a-1})}{(\sqrt{a+1}+\sqrt{a-1})-(\sqrt{a+1}-\sqrt{a-1})}$
Simplify
$\displaystyle \frac{x+1}{x-1} = \frac{\sqrt{a+1}}{\sqrt{a-1}}$
Now square both sides
$\displaystyle \frac{x^2+1+2x}{x^2-2x+1} = \frac{a+1}{a-1}$
Simplifying
$\displaystyle x^2+1 = 2ax$
$\displaystyle \text{or } x^2-2ax+1 = 0$
$\displaystyle \\$
$\displaystyle \text{Question 6: Given, } \frac{a}{b} = \frac{c}{d} \text{, prove that: } \frac{3a-5b}{3a+5b} = \frac{3c-5d}{3c+5d} \text{ [2000] }$
$\displaystyle \text{Given } \frac{a}{b} = \frac{c}{d}$
$\displaystyle \Rightarrow \frac{3a}{5b} = \frac{3c}{5d}$
By componendo and dividendo
$\displaystyle \frac{3a+5b}{3a-5b} = \frac{3c+5d}{3c-5d}$
By Alternendo
$\displaystyle \frac{3a-5b}{3a+5b} = \frac{3c-5d}{3c+5d}$
$\displaystyle \\$
$\displaystyle \text{Question 7: If } x=\frac{\sqrt{a+3b}+\sqrt{a-3b}}{\sqrt{a+3b}-\sqrt{a-3b}} \text{, prove that: } 3bx^2-2ax+3b=0 \text{ [2007] }$
$\displaystyle \text{Given } x= \frac{\sqrt{a+3b}+\sqrt{a-3b}}{\sqrt{a+3b}-\sqrt{a-3b}}$
Applying componendo and dividendo
$\displaystyle \frac{x+1}{x-1} = \frac{(\sqrt{a+3b}+\sqrt{a-3b})-(\sqrt{a+3b}-\sqrt{a-3b})}{(\sqrt{a+3b}+\sqrt{a-3b})-(\sqrt{a+3b}-\sqrt{a-3b})}$
$\displaystyle \frac{x+1}{x-1} = \frac{2\sqrt{a+3b}}{-2\sqrt{a-3b}}$
Squaring both sides
$\displaystyle \frac{x^2+2x+1}{x^2-2x+1} = \frac{a+3b}{a-3b}$
Applying componendo and dividendo once again
$\displaystyle \frac{(x^2+2x+1)-(x^2-2x+1)}{(x^2+2x+1)-(x^2-2x+1)} = \frac{(a+3b)-(a-3b)}{(a+3b)-(a-3b)}$
simplifying
$\displaystyle \frac{x^2+1}{2x} = \frac{a}{3b}$
$\displaystyle 3b(x^2+1) = 2ax$
$\displaystyle 3bx^2-2ax+3b=0$ Hence proved.
$\displaystyle \\$
$\displaystyle \text{Question 8: Using the properties of proportion, solve for x Given: } \\ \\ \frac{(x^4+1)}{2x^2} = \frac{17}{8} \text{ [2013] }$
$\displaystyle \text{Given } \frac{(x^4+1)}{2x^2} = \frac{17}{8}$
Applying componendo and dividendo
$\displaystyle \frac{(x^4+1)+2x^2}{(x^4+1)-2x^2} = \frac{17+8}{17-8}$
$\displaystyle \frac{(x^2+1)^2}{(x^2-1)^2} = \frac{25}{9}$
Taking the square root of both sides
$\displaystyle \frac{x^2+1}{x^2-1} = \frac{5}{3}$
$\displaystyle 3x^2+3=5x^2-5$
$\displaystyle x^2 = 4 or x = \pm 2$
$\displaystyle \\$
Question 9: What least number must be added to each of the numbers $\displaystyle 6, 15, 20, \text{ and } 43$ to make them proportional. [2005, 2013]
Let the number added be $\displaystyle x$
$\displaystyle \text{Therefore } (6+x): (15+x) = (20+x): (43+x)$
$\displaystyle \Rightarrow (6+x) \times (43+x) = (20+x) \times (15+x)$
$\displaystyle \Rightarrow x^2+49x+258 = x^2+ 35x +300$
$\displaystyle \Rightarrow x = 3$
$\displaystyle \\$
Question 10: The monthly pocket money of Ravi and Sanjeev are in the ratio of 5:7 Their expenditures are in the ratio of 3:5. If each saves Rs. 80 per month, find their monthly pocket money. [2012]
Let monthly pocket of Rave and Sanjeev by $\displaystyle x \text{ and } y$ respectively.
$\displaystyle \frac{x}{y} = \frac{5}{7} \Rightarrow x = \frac{5}{7} y$
$\displaystyle \frac{x-80}{y-80} = \frac{3}{5}$
Substituting
$\displaystyle \frac{ \frac{5}{7} y-80}{y-80} = \frac{3}{5}$
$\displaystyle \frac{25}{7} x-400=3x-240 \Rightarrow x=280$
Substituting
$\displaystyle y = \frac{5}{7} \times 280 = 200$
$\displaystyle \\$
Question 11: If $\displaystyle (x-9):(3x+6)$ is the triplicate ratio of $\displaystyle 4:9$ , find $\displaystyle x$ . [2014]
$\displaystyle \frac{x-9}{3x+6} = \frac{4^2}{9^2} = \frac{16}{81}$
$\displaystyle 81x-729=48x+96$
$\displaystyle x=25$
$\displaystyle \\$
Question 12: If $\displaystyle a:b=5:3$ , find $\displaystyle (5a+8b):(6a-7b)$ . [2002]
$\displaystyle \text{Given } a:b=5:3$
$\displaystyle \text{or } \frac{a}{b} = \frac{5}{3} \Rightarrow a = b \frac{5}{3}$
Now substituting
$\displaystyle \frac{5a+8b}{6a-7b} = \frac{5 \times b\frac{5}{3}+8b}{6 \times b\frac{5}{3}-7b} = \frac{25+24}{30-21} = \frac{49}{9}$
Hence $\displaystyle (5a+8b):(6a-7b) = \frac{49}{9}$
$\displaystyle \\$
Question 13: The work done by $\displaystyle (x-3)$ men in $\displaystyle (2x+1)$ days and the work done by $\displaystyle (2x+1)$ men in $\displaystyle (x+4)$ days are in the ratio $\displaystyle 3:10$ . Find the value of $\displaystyle x$ . [2003]
Amount of work done by $\displaystyle (x-3)$ men in $\displaystyle (2x+1)$ days $\displaystyle = (x-3)(2x+1)$
Similarly, amount of work done by $\displaystyle (2x+1)$ men in $\displaystyle (x+4)$ days $\displaystyle = (2x+1)(x+4)$
$\displaystyle \text{Given } \frac{(x-3)(2x+1)}{(2x+1)(x+4)} = \frac{3}{10}$
$\displaystyle 10(2x^2+x-6x-3)=3(2x^2+8x+x+4)$
Simplifying
$\displaystyle 2x^2-11x-6=0$
$\displaystyle (x-6)(2x+1) = 0 \Rightarrow x = 6 or x=- \frac{1}{2} (not possible)$
$\displaystyle \text{Therefore } x = 6$
$\displaystyle \\$
Question 14: What number should be subtracted from each of the numbers $\displaystyle 23, 30, 57 \text{ and } 78$ ; so that the ratios are in proportion. [2004]
Let the number subtracted $\displaystyle = x$
$\displaystyle \text{Therefore } (23-x):(30-x)=(57-x):(78-x)$
$\displaystyle \frac{23-x}{30-x} = \frac{57-x}{78-x}$
Simplifying
$\displaystyle x^2-101x+1794 = x^2-87x+1710 \Rightarrow x =6$
$\displaystyle \\$
Question 15: $\displaystyle 6$ is the mean proportion between two numbers $\displaystyle x \text{ and } y \text{ and } 48$ is the third proportion to $\displaystyle x \text{ and } y$ . Find the numbers. [2011]
$\displaystyle \text{Given } 6$ is the mean proportion between two numbers $\displaystyle x \text{ and } y$
$\displaystyle \text{Therefore } \frac{x}{6} ={6}{y} \Rightarrow xy=36 \Rightarrow x = \frac{36}{y}$ … … … … … … i)
Also $\displaystyle \text{Given } 48$ is the third proportion to $\displaystyle x \text{ and } y$
$\displaystyle \text{Therefore } \frac{x}{y} = \frac{y}{48} \Rightarrow y^2=48x$ … … … … … … ii)
Solving i) and ii)
$\displaystyle y^2 = 48 \frac{36}{y}$
$\displaystyle y^3 = 2^3 \times 6^3 \Rightarrow y = 12$
$\displaystyle \text{Hence } x = \frac{36}{12} = 3$
Hence the numbers are $\displaystyle 3 \text{ and } 12$ .
$\displaystyle \\$
$\displaystyle \text{Question 16: If } \frac{8a-5b}{8c-5d} = \frac{8a+5b}{8c+5d} \text{, prove that } \frac{a}{b} = \frac{c}{d} \text{ [2008] }$
$\displaystyle \text{Given } \frac{8a-5b}{8c-5d} = \frac{8a+5b}{8c+5d}$
$\displaystyle \text{or } \frac{8c+5d}{8c-5d} = \frac{8a+5b}{8a-5b}$
Applying Componendo and Dividendo
$\displaystyle \frac{8c+5d+8c-5d}{8c+5d-8c+5d} = \frac{8a+5b+8a-5b}{8a+5b-8a+5b}$
$\displaystyle \frac{16c}{10d} = \frac{16a}{10b}$
$\displaystyle \frac{c}{d} = \frac{a}{b}$
$\displaystyle \text{or } \frac{a}{b} = \frac{c}{d} \text{ Hence proved.}$
|
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|
https://codegolf.stackexchange.com/questions/50791/sourcecode-selfie
|
# Sourcecode selfie
Objective:
A guru once said a perfect code selfie is best shot diagonally from upper left corner. A code selfie is almost like a Quine - but rotated 45 degree clockwise. Your mission is to code a program that outputs a code selfie.
Rules:
1. You can use any programming language.
2. Your programs should not take any input from file, filename, network or anything else.
Mandatory criterias:
Selfies is about the motive and background, so blankspaces (and other not visible content like linefeeds and such) does not count as being part of the character count. All visible characters is restricted to be outputted on the correct 45 degree rotated position while all non-visible characters is not restricted to the correct 45 degree rotated position. Just like a color palette on a normal selfie, mandatory to a code selfie is that it contains atleast 16 of these characters: {a-zA-Z0-9}
Example:
If this example is valid sourcecode:
Output abcd
Output efgh
Output ijkl
Output mnop
The example code should output this:
O
O u
O u t
O u t p
u t p u
t p u t
p u t
u t a
t e b
i f c
m j g d
n k h
o l
p
This is code-golf, shortest sourcecode in bytes wins!
• 16 of unique [a-zA-Z0-9] ? May 26, 2015 at 9:04
• How would we score a submission in Whitespace? May 26, 2015 at 9:20
• Whitespace is not possible, since blankspaces does not count. Yes, 16 of unique [a-zA-Z0-9], not a total least 16 chars. May 26, 2015 at 9:50
• This challenge is biased against languages which require linebreaks in their code. It's much harder to support multiline for this challenge. May 26, 2015 at 16:20
• @nderscore Perhaps you are correct about that. Well, we are all here for fun, right? Do the best out of the situation! ;) May 26, 2015 at 20:12
# Javascript (ES6), 72 bytes
16 unique alphanumeric character pallete: fjalert0plcgmixn
(f=j=>alert((f=${f})(0).replace(/./gmi,x=>' '.repeat(j++)+x+'\n')))(0) m and i flags are added to the regexp to meet minimum palette requirements. # CJam, 30 28 25 bytes {95c103ic]seeSf.*N*Xmr}_g This is kind of long due to the 16 character from A-Za-z0-9 limit. This is a bit non-trivial variant of a standard quine in CJam. Will add explanations soon. UPDATE - 2 bytes saved thanks to Martin, 3 bytes saved thanks to Dennis Try it online here # Java, 312 class Z{public static void main(String[]a){String s="class Z{public static void main(String[]a){String s=%c%s%1$c,t;for(int i=0,j;i<326;System.out.println(t+s.format(s,34,s).charAt(i++)))for(j=i,t=%1$c%1$c;j-->0;)t+=' ';}}",t;for(int i=0,j;i<326;System.out.println(t+s.format(s,34,s).charAt(i++)))for(j=i,t="";j-->0;)t+=' ';}}
There are actually 326 bytes, but if I understand the rules correctly, I don't have to count the 14 spaces.
The program is basically a standard Java quine, plus a lot of whitespace.
# Python 3, 139 characters - 10 spaces = 129 characters
sjxd='sjxd=%r;[print(" "*i+(sjxd%%sjxd)[i]) for i in range(len(sjxd%%sjxd))]';[print(" "*i+(sjxd%sjxd)[i]) for i in range(len(sjxd%sjxd))]
Since my code was one line, all I had to do was print the program diagonally. My string has the weird name 'sjxd' so that my code could have the 16 unique alphanumeric characters.
# CSS, 69 bytes
<style>:before,*{transform:rotate(45deg;display:block;content:'<style>
Put in a blank html page to avoid conflict with other tags.
Palette: stylebfortanm45dgiplck (22 chars)
# MATLAB, 40 bytes
Bit difficult with the whole recursion thing - how do you print your own source code when adding the code to a string to be printed increases the size of the source code itself. But, never the less, the following will do it:
123456;disp(diag('123456;disp(diag())'))
The 123456; bit is there to meet the required 16 unique characters. The following are used:
'()123456;adgips
The above code doesn't work on Octave for some reason, but does work in MATLAB. Below is the output:
1
2
3
4
5
6
;
d
i
s
p
(
d
i
a
g
(
)
)
Now if you don't mind the ans= bit that MATLAB enjoys putting, the following would work for 32 bytes:
12345678;diag('12345678;diag()')
|
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|
https://solvedlib.com/how-do-you-solve-for-y-in-ax-by-c,320356
|
1 answer
# How do you solve for y in Ax - By = C?
###### Question:
How do you solve for y in Ax - By = C?
## Answers
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-- 0.023845--
|
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|
https://tex.stackexchange.com/questions/342736/how-to-typeset-mathematical-set-with-conditions-in-multiple-rows
|
# How to typeset mathematical set with conditions in multiple rows [duplicate]
With the TeX code
$$\max\left\{ \sum_{i=1}^{n}|\langle x^{(i)},b\rangle|^2\;\mid\vbox{ \hbox{Condition 1} \hbox{Condition 2} \hbox{Condition 3} }\right\}$$
I get the output
How can I achieve the following two things?
1. The separator produced by \mid shall have the same height as the curly brackets.
2. The three conditions shall be centered vertically.
## marked as duplicate by egreg math-mode StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 6 '16 at 20:58
You can use a tabular instead of the \vbox:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
$$\max\left\{ \sum_{i=1}^{n}|\langle x^{(i)},b\rangle|^2\; \begin{tabular}{|l} Condition 1 \\ Condition 2 \\ Condition 3 \end{tabular} \right\}$$
\end{document}
Use \middle\vert to draw a full-height vertical bar. Use an array environment to hold the conditions -- I assume the conditions will be mostly math-mode material.
\documentclass{article}
\usepackage{amsmath} % for "\text" macro
\begin{document}
$$\max \left\{ \sum_{i=1}^{n}|\langle x^{(i)},b\rangle|^2 \;\middle\vert\; \begin{array}{@{}l@{}} \text{Condition 1}\\ \text{Condition 2}\\ \text{Condition 3} \end{array} \right\}$$
\end{document}
• This is also a perfect answer. Can \middle be also applied to different things than \vert to stretch them? – phinz Dec 6 '16 at 19:41
• @phinz: \middle can be applied to any stretchable "fence" symbol, not just \vert. Give \Vert a try, say. – Mico Dec 6 '16 at 19:44
|
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|
http://pillars.che.pitt.edu/student/slide.cgi?course_id=10&slide_id=29.0
|
# LTR: Convective Mass Transfer
Convective mass transfer refers to the transport of mass due to a moving fluid. Like heat convection, this typically refers to transport across phases, however, here solid-fluid transport is on equal footing with liquid-gas transport (rather than being the dominant example of convection). As with heat transport, it is clear that the rate of mass transfer will depend on the character of the fluid flow.
In analogy with Newton's "Law" of Cooling, we can write an expression for the molar flux due to convection as:
##### EQUATION:
$\displaystyle{N_a = \frac{M_a}{A} = k_c\Delta C_a}$
where $k_c$ is the convective mass transfer coefficient, $N_a$ is the molar flux of species $a$, $M_a$ is the molar flow of a, and A is the interphase area of contact.
##### NOTE:
We could write essentially the same expression based on mass concentrations, but will try to denote mass fluxes/flows with lower case letters. Also, for transport int he gas phase, we will often use partial pressures instead of molar concentrations.
As with heat transfer $k_c$ may also sometimes be referred to as a "film coefficient".
$k_c$ will depend on:
• the geometry of the phase boundaries (unlike heat transport, if we have gas-liquid transport this is a very difficult thing to calculate/measure!)
• the nature of the fluid (here the diffusivity)
• the nature of the flow (fluid mechanics!)
##### NOTE:
Again, determining the parameter, $k_c$, will often be the bulk of the work (or at least the only hard part) in a given convection problem.
##### OUTCOME:
Perform convective mass transfer calculations
##### EXAMPLE:
An aspirin sitting in your stomach has a solubility of 0.15 mol/L (so this is the concentration at the solid-liquid surface). Assuming that the concentration in the bulk of the stomach is zero and that the pill does not shrink, but stays a sphere with a 0.5cm diameter, calculate the molar flow into the stomach when the mass transfer coefficient is 0.1 m/s
|
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|
http://tex.stackexchange.com/questions/26843/cite-bibliography-in-beamer
|
# Cite bibliography in beamer
I am preparing a beamer presentation. In some slides (4 of 30) I want to cite one or two bibliographic references and put them at the bottom of each slide. They lie there just as a bulleted list, not being cited in the text of the slide.
I wonder what is the best approach to do so.
PS: I forgot to mention that bibliographic entries are stored in a .bib file
-
From your previous question, you know how to create references as "bulleted lists" using biblatex. Have you tried the beamer/biblatex combo? – lockstep Aug 29 '11 at 13:47
yes, I have tried this, sorry. The problem now is that I do not get the same format for bibliography in both latex document and beamer presentation; in beamer it gets a bit weird; I do not like ; authors, ... IN Journal ... – flow Aug 30 '11 at 11:04
Have a look at tex.stackexchange.com/q/10682/510. – lockstep Aug 30 '11 at 11:06
ok, but now the problem continues; I get "pp" for the page numbers, like if I was to cite a proceeding instead of a journal, and the volume number appers weird like 13.4 instead of 13(4) ... can not tell beamer to put them in some concrete style? – flow Aug 30 '11 at 11:38
beamer has nothing to do with bibliography styles. For customizing biblatex styles, have a look at this question (e.g. it tells you how to remove "pp."). Anyhow, you should ask a new question if you need additional advice. – lockstep Aug 30 '11 at 11:43
## 1 Answer
EDIT
Here a minimal example, which you should provide.:
\RequirePackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@book{test,
author={John Smith},
title={A book},
publisher={Puplisher},
year={1742},
}
\end{filecontents*}
\documentclass{beamer}
\begin{document}
\begin{frame}
asd\cite{test}
\end{frame}
\begin{frame}
\bibliographystyle{alpha}
\bibliography{\jobname}
\end{frame}
\end{document}
ALSO NO PROBLEM -- but the same problem you have to provide a minimal example.
-
thanks a lot, I forgot to mention that bibliographic entries are stored in a .bib file – flow Aug 29 '11 at 13:37
@flow: Same as usual. – Marco Daniel Aug 29 '11 at 13:42
thanks, now it works. The problem now is that I do not get the same format for bibliography in both latex document and beamer presentation; in beamer it gets a bit weird; I do not like ; authors, ... IN Journal ... – – flow Aug 30 '11 at 11:04
so I wonder how can I give it the exact format for the biliography, I just would like; authors, title, journal, etc (similar so American Medical Association style) – flow Aug 30 '11 at 11:05
Ok, I reformulated my question and put an example here: tex.stackexchange.com/questions/26959/… – flow Aug 30 '11 at 12:42
|
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|
https://brilliant.org/problems/an-easy-way-and-a-hard-way/
|
# An Easy Way And A Hard Way I
Algebra Level 4
$\Large \left | \sum_{j=0}^{100} x^{2^j} + \dfrac1{x^{2^j}}\right |$
Given that $$x$$ is a complex number satisfying the constraint $$x + \dfrac1x = 1$$, find the value of the expression above.
×
|
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|
http://mathhelpforum.com/trigonometry/52818-vector-problem.html
|
# Math Help - Vector Problem
1. ## Vector Problem
So I have this small problem. Two vectors A and B added together give the vector S. Show that S (which I'm assuming is the scalar of S) is equal to (A^2 + B^2 + 2AB Cos (theta) ) / 2 remembering that S . S = S^2 and S= A+B. I'm really not sure how to make a start on it. I'm thinking that it has something to do with the rule a.b= [a][b]cos theta, but I can't seem to make the first leap. Any suggestions
2. Hi,
I'm not sure the question quite makes sense - vectors can't usually be "squared" - you can multiply them with the dot or cross product, but the notation in your question suggests otherwise.
Does the answer refer to the magnitude of the two vectors?
3. I think your right about the squaring of the vector, not being possible. However I don't theink the squraed parts are refering to the vectors. They weren't in bold, which I believe is the usual convention for vectors. So I'm assuming that the are the values or the scalar. ie Ai + Aj or possible the scalar product. Sorry I'm really at a loss about this stuff.
4. It follows at once from this.
$S = A + B \Rightarrow \left( {S \cdot S} \right) = \left( {A + B} \right) \cdot \left( {A + B} \right) = A \cdot A + 2A \cdot B + B \cdot B$
5. So the complete derivation would be something like this:
S.S= (A+B).(A+B)
=> A.A + B.B + 2(A.B)
=> A.A = [A][A] cos theta = [A]^2 because cos 0 = 1
=> B.B = [B][B] cos theta = [B]^2 as above
=> 2(A.B) = 2[A][B] cos theta
putting all the parts together
A^2 + B^2 + 2AB cos theta
then taking the square root bcause S.S = S^2
giving S = (A^2 + B^2 + 2AB cos theta)^1/2
Is that how it goes? Thanks for the help
|
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|
https://tex.stackexchange.com/questions/255416/circumflex-over-numeral-with-fontspec
|
# Circumflex over Numeral with fontspec
For a music analysis paper I'm working on, I need to enter an arabic numeral with a circumflex over it to represent a musical scale degree. Normally, I can just use \^3, but as soon as I load the fontspec package, even if I don't change the default font, the circumflex collides with the glyph for the number. (I am using LuaLaTeX.) Here is a minimal example that illustrates the problem:
\documentclass{article}
\usepackage{fontspec}
\begin{document}
\^5 causes a collision!
\end{document}
• Welcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. – Symbol 1 Jul 15 '15 at 5:38
• It seems like LuaLaTeX+fontspec cannot handle \^ well. On the other hand XeLaTeX+fontspec produce good result with Times, Helvetica, and Courier. (bad with Times New Roman and Arial). If you try to copy the result string you will find they are all character 5 following a U+0302, a combining character. The problem might be some PDF/Unicode-wise setting. – Symbol 1 Jul 15 '15 at 5:46
• If possible, I'd really like to use the Linux Libertine O font, and with XeLaTex this still results in the same collision. – musicanalyst Jul 15 '15 at 5:54
• Could you use math mode, i.e., type $\hat{5}$? – Mico Jul 15 '15 at 6:01
• What about $\hat{\text{5}}$ or $\hat{\mathrm{5}}$?. – Bernard Jul 15 '15 at 6:43
It seems a bug somewhere. A temporary workaround:
\documentclass{article}
\usepackage{fontspec}
%\setmainfont{Linux Libertine O}
\newcommand{\hdigit}[1]{%
\accent\string"02C6 #1%
}
\begin{document}
\^5 causes a collision! And 5^^^^0302 too.
But \hdigit{5} doesn't.
\end{document}
|
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|
http://biomechanical.asmedigitalcollection.asme.org/article.aspx?articleid=2678341&resultClick=1
|
0
Research Papers
# Pelvic Construct Prediction of Trabecular and Cortical Bone Structural ArchitectureOPEN ACCESS
[+] Author and Article Information
Dan T. Zaharie
The Royal British Legion Centre for
Blast Injury Studies,
Imperial College London,
London SW7 2AZ, UK;
Structural Biomechanics,
Department of Civil and
Environmental Engineering,
Imperial College London,
Skempton Building, South Kensington Campus,
London SW7 2AZ, UK
e-mail: [email protected]
Andrew T. M. Phillips
The Royal British Legion Centre for
Blast Injury Studies,
Imperial College London,
London SW7 2AZ, UK;
Structural Biomechanics,
Department of Civil and
Environmental Engineering,
Imperial College London,
Skempton Building, South Kensington Campus,
London SW7 2AZ, UK
e-mail: [email protected]
1Corresponding author.
Manuscript received December 5, 2016; final manuscript received November 10, 2017; published online May 24, 2018. Assoc. Editor: Steven D. Abramowitch.
J Biomech Eng 140(9), 091001 (May 24, 2018) (11 pages) Paper No: BIO-16-1497; doi: 10.1115/1.4039894 History: Received December 05, 2016; Revised November 10, 2017
## Abstract
The pelvic construct is an important part of the body as it facilitates the transfer of upper body weight to the lower limbs and protects a number of organs and vessels in the lower abdomen. In addition, the importance of the pelvis is highlighted by the high mortality rates associated with pelvic trauma. This study presents a mesoscale structural model of the pelvic construct and the joints and ligaments associated with it. Shell elements were used to model cortical bone, while truss elements were used to model trabecular bone and the ligaments and joints. The finite element (FE) model was subjected to an iterative optimization process based on a strain-driven bone adaptation algorithm. The bone model was adapted to a number of common daily living activities (walking, stair ascent, stair descent, sit-to-stand, and stand-to-sit) by applying onto it joint and muscle loads derived using a musculoskeletal modeling framework. The cortical thickness distribution and the trabecular architecture of the adapted model were compared qualitatively with computed tomography (CT) scans and models developed in previous studies, showing good agreement. The sensitivity of the model to changes in material properties of the ligaments and joint cartilage and changes in parameters related to the adaptation algorithm was assessed. Changes to the target strain had the largest effect on predicted total bone volumes. The model showed low sensitivity to changes in all other parameters. The minimum and maximum principal strains predicted by the structural model compared to a continuum CT-derived model in response to a common test loading scenario showed good agreement with correlation coefficients of 0.813 and 0.809, respectively. The developed structural model enables a number of applications such as fracture modeling, design, and additive manufacturing of frangible surrogates.
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## Introduction
The pelvic construct is the region of transition between the trunk and the lower limbs and plays multiple roles, from protecting the organs and vessels in the lower abdomen to distributing the upper body weight to the lower limbs. Furthermore, it facilitates the transfer of forces between the lower limbs and the upper body during activities such as walking, running, and stair climbing, withstanding loads at the hip joint up to around six times body weight for running [13].
The structure of bone has been studied extensively since the 19th century, with an emphasis on the femur. It was hypothesized that the trabecular architecture of the proximal femur follows a set of trajectories corresponding to the compressive and tensile stresses [46]. The general consensus is that bone continuously updates its structure in response to the mechanical environment it is exposed to [7] until it reaches a state in which it is able to withstand the forces acting on it with a minimum amount of material [8].
As opposed to long bones, which have thick cortical shafts to optimally resist the forces and moments applied on them [911], the pelvis can be viewed as a sandwich structure consisting mainly of trabecular bone encased within a thin layer of cortical bone [1214].
###### Continuum Modeling Approaches.
Finite element (FE) modeling is a common approach to investigate the behavior of pelvic bone in a wide range of scenarios. Early pelvic models were either two-dimensional [15,16], axisymmetric [17,18], or used simplified geometries [1921]. A common approach over the past three decades has been to develop models based on geometries and material properties extracted from medical imaging data (e.g., computed tomography (CT)) and to build the model using continuum solid elements.
The first computational model developed using subject specific geometry and material properties from a CT scan was presented by Dalstra et al. [22]. The model was validated against experimentally obtained strains from a cadaveric pelvis, although the specimen used for the experiment was different from the one used for generating the FE mesh and material properties. Similarly, Anderson et al. [14] developed an FE model of the pelvis based on CT data with location-dependent cortical thickness and validated it against experimental results obtained from the same specimen. In both studies, specimens were fixed at the superior part of the ilium with the load applied on the acetabulum.
The majority of published models only partially represent the pelvic ring, focusing primarily on either the hip joint [1921,2325] or the hemi-pelvis [14,21,22,2628]. For the purpose of investigating the structural response of the pelvis in a variety of loading conditions, the development of a complete model of the pelvic construct is considered to provide an improved understanding of its behavior in complex loading scenarios, compared to models which represent an isolated part of the construct. An important aspect of the pelvic girdle is the ligaments holding the three components of the pelvis (sacrum and two pelvic innominates) together, the sacroiliac ligaments, and the pubic ligament. A number of other ligaments (sacrospinous, sacrotuberous, iliolumbar, and inguinal) provide additional stability to the pelvic ring under load [29]. Phillips et al. [30] developed a model of the pelvis including both hemipelves, the ligaments mentioned above, a structure representing the sacrum, and the muscles attached to the pelvis in an explicit manner. They observed that the use of ligaments and muscles in the model as opposed to fixed boundary conditions resulted in a more even stress distribution across the bone. A number of complete models have been developed and used to investigate pelvic injury mechanisms [3134].
###### Structural Modeling Approaches.
An alternative approach to continuum FE modeling is to look at bone from a structural perspective and represent its architecture using a combination of idealized elements such as trusses, beams, and shells. An important aspect of this method is the increased computational efficiency [35], while maintaining the ability to capture the overall structural behavior of bone [11]. Furthermore, structural modeling enables the user to visualize and investigate the trajectories formed by trabecular bone.
This study presents the development of a predictive mesoscale structural model of the pelvic girdle in which the pubic and sacroiliac joints, as well as the ligaments forming the pelvic ring, are included. The development of the model consists of adapting its structure to a physiological loading regime. Phillips et al. [11,36] have previously employed this approach to model the femur, showing promising results in terms of bone architecture and structural response to loading of the bone.
## Methods
###### Musculoskeletal Modeling.
To obtain a structural representation of the pelvis, the base model was subjected to physiological loads derived from musculoskeletal simulations of the five most frequent daily activities [40]: walking, stair ascent, stair descent, sit-to-stand, and stand-to-sit. The bilateral musculoskeletal model was based on an ipsilateral model developed by Modenese et al. and validated at the hip joint [39]. The model was based on an anatomical dataset published by Horsman et al. [41] and implemented in OpenSim [42]. It includes thirteen body segments: pelvis, two femurs, two patellas, two tibias, two hindfeet, two midfeet, and two sets of phalanges (Fig. 1). The segments are connected by nine joints: pelvis-ground (6 degrees-of-freedom (DOFs)), hip joints (3 DOF each), knee joints (1 DOF each), patellofemoral joints (1 DOF each), and ankle joints (1 DOF each). The local reference systems of the segments were defined in accordance with the recommendations of the International Society of Biomechanics [43]. The muscle attachment points on the right side of the model were the same as in Ref. [37], while those on the left side were obtained by mirroring the right side points with respect to the sagittal plane. Thus, the total number of muscles and actuators in the bilateral model was 76 and 326, respectively.
Gait data for walking, stair ascent, stair descent, sit-to-stand, and stand-to-sit activities were collected from a male volunteer (age: 23 yr, weight: 93 kg, and height: 188 cm). The activities chosen were identified as the most common daily physical activities [40]. The 59 markers positioned on bony landmarks were tracked using a Vicon system (Oxford Metrics, Oxford, UK) equipped with 10 infrared cameras. Three force plates (Type 9286BA, sampling rate 1000 Hz, and Kistler Instruments Ltd., Hook, UK) were used to measure the ground reaction forces (GRFs). The force plates were positioned to form a walkway which was used to record GRFs during walking (speed: 1.34 m/s, stride length: 0.64 m, and cadence: 115.54 steps/min). To measure stair ascent and descent GRFs, the three force plates were positioned on a staircase (step height 15 cm and step depth 25 cm). A stool with a height of 52.5 cm from the floor instrumented with one force plate along with two force plates on the floor was used to record GRFs during sit-to-stand and stand-to-sit. All data were collected in the Human Biodynamics Lab in the Imperial College Research Labs at Charing Cross Hospital and processed using Vicon Nexus and the Biomechanical ToolKit [44].
The segments of the model were scaled to the anatomical dimensions of the volunteer by calculating the ratios of the lengths between experimental and virtual markers, with the inertial properties of the segments being updated according to the regression equations of Dumas et al. [45]. An inverse kinematics approach [46] was used to calculate the joint angles describing the activities recorded. The muscle forces were estimated using static optimization. The optimization problem was solved by minimizing the sum of squared muscle activations for each frame of the kinematics under the constraints of joint moment equilibrium and physiological limits of the muscle forces [3739]. The hip joint contact forces were calculated using the JointReaction tool available in OpenSim [47]. All simulations were performed in OpenSim (Version 3.3) [42]. For each activity, the loads applied on the pelvis throughout the duration of the gait cycle (hip joint reaction forces and muscle forces) were determined to be applied to the FE model. The muscle insertion points on the pelvis along with the direction and magnitude of each muscle force were extracted using the MuscleForceDirection (v1.0) plugin [11,48].
###### Base Model.
A CT scan (399 × 3 mm thick slices, 512 × 512 pixels, 0.91 mm/pixel) of a cadaveric pelvis and lower limbs (Male, age: 55, weight: 94.3 kg, and height: 188 cm) provided by the Royal British Legion Centre for Blast Injury Studies of Imperial College London was processed in mimics (Materialise, Leuven, Belgium) to generate a volumetric mesh of the pelvic girdle, composed of 377,362 four-noded tetrahedral elements with an average edge length of 3.76 mm. The external surface of the mesh was used to define three-noded linear triangular shell elements, representing cortical bone. These were arbitrarily assigned an initial thickness of 0.1 mm. The internal nodes were used to define two-noded truss elements connecting each node to its nearest 16 neighboring nodes. The trusses were arbitrarily assigned an initial radius of 0.1 mm, with the network representing trabecular bone. The initial minimum connectivity of each node was 16, with a maximum of 62 and a mean value of 21.32. Previous work in the research group has shown that a nodal connectivity of 16 provides a sufficient range of element directionalities to enable trabecular trajectories to develop during adaptation [49]. In addition, a mesh sensitivity study, described in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection, was performed to assess the sensitivity of the adaptation algorithm to mesh refinement.
The base mesh consisted of 33,466 cortical shell elements (12,601 for each innominate and 8264 for the sacrum) and 588,028 trabecular truss elements (247,268 for each innominate and 93,492 for the sacrum) (Fig. 2). The initial structural model was generated from the tetrahedral mesh using matlab (The MathWorks, Inc., Natick, MA). The shell and truss elements were assigned linear isotropic material properties with Young's modulus and Poisson's ratio values of E = 18,000 MPa and ν = 0.3, respectively [14,30,50].
###### Ligaments.
In addition to the pelvic bones, five ligaments were added to the model as bundles of truss elements: sacroiliac, pubic, sacrospinous, sacrotuberous, and inguinal (Fig. 3). The ligaments are required as they facilitate the load transfer between the three bones of the pelvic construct. The anatomical attachment areas of each ligament were visually defined on the model based on descriptions in the literature [30,5153].
The behavior of ligaments is complex and models which aim to capture their biphasic and viscoelastic nature focus on soft tissue behavior or joint mechanics [5456], rather than the behavior of the pelvis as a whole. Ligaments included in the previous pelvic models were either assigned linear elastic material properties [30,31] or modeled as hyperelastic components [33,34,57,58].
The ligaments and cartilage at the joints were assigned linear elastic material properties and were modeled using truss elements by connecting each surface node on an origin area to the closest node on the corresponding insertion area. The material properties of each ligament were derived from previously published data on ligament stiffnesses [30]. Young's moduli were derived from the stiffnesses, with E being Young's modulus, k the stiffness of the ligament, l the length, and A the origination area of the ligament Display Formula
(1)$E=klA$
Display Formula
(2)$r=AπNl$
The radii of each truss element, r, were calculated based on the origination area of the ligaments, A, and the number of truss elements, Nl, forming each ligament (Eq. (2)). As ligaments do not have compressive stiffness, they were assigned zero stiffness in compression and were overlaid with identical sets of elements given a low Young's modulus value of 0.1 MPa to ensure numerical stability. Additional sets of trusses were added at the sacroiliac and pubic joints and assigned compressive stiffness to account for the presence of cartilage. The tensile and compressive material properties assigned to the ligaments and cartilage are shown in Tables 1 and 2, respectively.
Similar to the load applicators, four layers of wedge elements were constructed on the top of the sacrum, representing the L5S1 interface between the lumbar spine and the sacrum. The two layers closer to bone were assigned Young's modulus of 10 MPa (ν = 0.49) to represent cartilage, while the outermost two were assigned stiffnesses of 10 MPa (ν = 0.3) and E = 500 MPa (ν = 0.3). The boundary conditions of the model were applied at the top of the sacrum by constraining the nodes on the outermost layer of the construct in the three translational DOF. This was done to avoid fixing nodes on the bone, which would potentially lead to stress concentrations developing over the surface of the pelvis.
The load cases obtained from the musculoskeletal model were subsampled to maintain computational efficiency. The subset of frames was selected such that the difference between the integrated hip JCF for the full set of frames and the selected set of frames was lower than 5% (Fig. 5). Data reported by Bergmann et al. [3] for the same activities are included for comparison. The derived JCFs show a reasonable match for stair ascent and descent. During walking, the second peak was higher compared to the range reported by Bergmann et al. [3]. The selected load cases were applied in consecutive steps to the FE model.
Based on the Mechanostat principle [8], the base structural pelvis model was subjected iteratively to the load cases obtained from the musculoskeletal model for the activities mentioned in Sec. 2.1 (methodology adapted from Ref. [11]). At each iteration, the thickness of each shell element and the cross-sectional area of each truss element were adjusted according to a strain criterion. The iterative process was implemented using matlab (The MathWorks, Inc.) and python (Python Software Foundation, Beaverton, OR) scripts, and the successive FE models were run using the abaqus/standard solver (Dassault Systèmes Simulia, Johnston, RI) until convergence was achieved (Fig. 6).
For each iteration i, the absolute maximum normal strain value for the jth truss element and the jth shell element over all load cases (λ = 1,…, n) was defined using Eqs. (3) and (4), respectively, Display Formula
(3)$|εi,j|max=max(|ε11,j,λ|)$
with ε11,j,λ being the axial strain in the jth truss element for load case λDisplay Formula
(4)$|εi,j|max=max(|εmax,j,λt|,|εmin,j,λt|,|εmax,j,λb|,|εmin,j,λb|)$
with $εmax,j,λt,εmin,j,λt,εmax,j,λb,εmin,j,λb$ being the maximum and the minimum principal strains in the top and bottom surfaces of the jth shell element for load case λ.
The strain ranges associated with, bone resorption, the lazy zone and bone apposition [8,36,59] are given in the following equation: Display Formula
(5)$ϕi,j={1,if 0≤|εi,j|max<1000με (Bone resorption)0,if 1000≤|εi,j|max≤1500με (Lazy zone)1,if |εi,j|max>1500με (Bone apposition)$
For the subsequent iteration (i + 1), the cross-sectional area of the jth truss element (Ai+1,j) was adjusted based on the value of ϕi,j (Eq. (6)). The thickness of the jth shell element (Ti+1,j) was modified using Eq. (7). The target strain, εt, was given a value of 1250 με, corresponding to the middle of the lazy zone Display Formula
(6)$Ai+1,j={Ai,j|εi,j|maxεt,if ϕi,j=1Ai,j,if ϕi,j≠1$
Display Formula
(7)$Ti+1,j={Ti,j2(1+|εi,j|maxεt),if ϕi,j=1Ti,j,if ϕi,j≠1$
The adaptation equations for the two types of elements were defined such that trabecular bone adapts in preference to cortical bone. This is important in particular during the first few iterations as it prevents the oscillation of the shell elements thicknesses. In addition, truss elements with a strain lower than 250 με were considered to be in the dead zone.
To increase computational efficiency, the thicknesses of the shell elements were linearly discretized into 256 categories. Based on CT scans, cortical thickness throughout the pelvis has been reported to vary between 0.5 and 4 mm [14,22,60]. The thickness range for the shell elements in this model was set between 0.1 mm and 5 mm as for a thickness lower than 0.5 mm the accuracy of clinical CT scanners decreases, meaning that there could be areas with a cortical thickness lower than 0.5 mm which would not be captured due to the image resolution [61].
Similarly, the cross-sectional areas of the truss elements were linearly discretized in 255 categories. The range for the truss elements was set between π(0.1)2 mm2 and π(2)2 mm2, which is considered reasonable to characterize trabecular bone at mesoscale level [11,62]. In addition, a 256th category with a cross-sectional area of π(0.001)2 mm2 was added to allow for effective removal of the elements in the dead zone while maintaining numerical stability. The elements in the dead zone were assigned a radius of 1 μm, making their stiffness contribution to the model negligible. This enabled regeneration of the elements in the dead zone as they could be reassigned in one of the other 255 categories at a later iteration.
###### Sensitivity Study.
A sensitivity study was performed to assess the effects of varying model parameters on its structure. The parameters which were varied were Young's modulus of each ligament and joint cartilage, the initial assigned thickness of the cortical shell elements, the target strain, and the number of discrete categories used for section definitions. Each case model was iteratively adapted until convergence was achieved. The final structure and trabecular and cortical bone volumes were compared to the baseline model. The stiffness of each pair of ligaments and joints in turn was increased and decreased by 50%. The initial assigned thickness of the cortical shell elements was increased to 1 mm. The target strain was given values of 1000 με and 1500 με, the two ends of the lazy zone. Finally, 128 and 512 discrete categories for section definitions were assessed.
To ensure that the mesh resolution of the pelvis model was appropriate, a mesh sensitivity study was performed to assess the effect of mesh resolution on the outcome of the adaptation process. Given that a mesh of the full pelvis geometry at a smaller scale than the current one would have a very high computational cost, an analogous structural model of a cantilever beam was developed. Three different meshes were tested. The baseline mesh was built using the same specifications as for the pelvis model, while fine and coarse meshes were built by varying the maximum tetrahedral edge length by −50% and +50%, respectively. The three models were then subjected to the adaptation process with for a bending load case. The bone volumes of the converged structures were compared to the baseline model. Further details of the sensitivity study are given in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.
###### Comparison With Computed Tomography Scan Derived Model.
The tetrahedral mesh of the right hemipelvis was assigned varying material properties derived from the CT data. The response of the resulting FE model and the corresponding structural hemipelvis to a vertical load applied at the acetabulum were compared. Details regarding the development of the CT scan derived model and the loading scenario applied to both models are made available in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.
## Results
###### Converged Model Structure.
Cortical thickness was highest at the superior sacrum, along the gluteal surface, around the sacroiliac joint, superior pubic ramus, posterior iliac crest, and greater sciatic notch. Cortical bone had a thickness between 0.1 mm and 0.5 mm at the ischium, acetabulum, pubic tubercle, iliac fossa, and posterior superior iliac spine (Fig. 7). The results for the cortical thickness distribution across the hip bone compare well with the study published by Anderson et al. [14].
Clusters of elements with large radii were found at the superior sacrum, supra acetabular region, pubic tubercle, and greater sciatic notch. High trabecular density in these regions was also reported in a previous study [63]. Regions with a small number of active elements were found at the iliac fossa, ischial tuberosity, and inferior sacrum (Fig. 7). Distinct trajectories were formed in the supra acetabular region (Figs. 8(a) and 8(c)), around the sacroiliac joint extending from the ilium to the sacrum (Figs. 8(d)8(f)), and along the direction between the two locations mentioned earlier (Figs. 8(b) and 8(c)). Similar trajectories were found in a study published by Machiarelli et al. [64], where the authors observed a distinct walking-related trabecular architecture formed at the ilium. The trabecular trajectories reported in the paper were identified in the model as the areas where clusters of truss elements developed the most (Fig. 9).
To obtain a more comprehensive view of the optimized bone architecture, a number of transverse slices of the model were compared to the corresponding CT scans used to define the initial surface geometry (Fig. 10). Across the acetabulum, the model predicted a thickening of the cortex at the ascending pubis ramus and a thin cortex along the acetabular notch, which is in agreement with the CT scan (Figs. 10(a) and 10(b)). Another similarity can be found along the inferior gluteal line, where the model showed a distinct thickening of the cortex, which is also present in the scan (Figs. 10(g) and 10(h)). The region around the sacroiliac joint compares well with the scan for both the sacrum and the hip bones (Figs. 10(e) and 10(f)). The main discrepancies in terms of cortical thickness between the model and scan can be seen between the anterior superior iliac spine and the anterior inferior iliac spine (Figs. 10(c)10(h)), toward the greater sciatic notch (Figs. 10(g) and 10(h)) and ischial body (Figs. 10(a) and 10(b)). On the other hand, the areas mentioned have a higher trabecular density compared to the CT scan, meaning that the stiffness of the region could still be considered similar to the scan.
In terms of trabecular architecture, a number of trajectories were clearly formed. In the region surrounding the sacroiliac joint, thick trabecular elements were formed following trajectories which intersect both hip bones and the sacrum (Figs. 8(e) and 8(f)). As a consequence, the highest trabecular density in the sacrum was found on the load-bearing S1 segment (Figs. 8(d) and 8(e)), which is in agreement with previous findings [65,66]. Another region with high trabecular density was identified above the acetabulum (Figs. 8(a)8(c)). Given that the loads are transmitted from the hip joint towards the lumbar spine through the sacroiliac joint and sacrum, the presence of these trabecular trajectories seems justified.
###### Influence of Activities.
Walking was primarily responsible for the thickening of the cortex along the ischial spine, the inferior ramus of the pubis and the posterior part of the iliac fossa. The cortex development between the anterior inferior iliac spine and the iliac crest and along the superior pubic ramus was primarily influenced by stair ascent, while stair descent was the main factor in the thickening of the cortex across the sacrum, between the anterior and the posterior gluteal lines, at the ischium, and along the posterior superior iliac spine. The increased trabecular structure present at the upper sacrum was influenced by both stair ascent and descent (Fig. 11). Walking was mainly responsible for the appearance of trabecular structures along the superior pubic ramus, around the acetabulum, and at the anterior inferior iliac spine and anterior gluteal line. The asymmetric distribution of activity influence can be attributed to the subject specific geometry and gait used in the musculoskeletal model. Additional figures illustrating the level of influence each activity had on the regions of the pelvis can be found in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.
###### Sensitivity Study.
To assess the sensitivity of the ligaments and cartilage, the trabecular and cortical volumes obtained adapting each case model were compared to the baseline model. For all cases, the model showed low sensitivity, with the largest difference in trabecular bone volume being 0.86% for the case when the stiffnesses of all ligaments and cartilage were increased by 50%. The largest difference in cortical bone volume was found when the stiffnesses of all ligaments and cartilage were reduced by 50%, with a value of 3.59%. Data for all cases can be found in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection. Similarly, for the sensitivity cases regarding adaptation related parameters, the total trabecular and cortical volumes were compared to the baseline model. The models with a higher initial thickness and different numbers of categories for section definitions predicted a maximum difference in trabecular bone volume of 2.18% and a maximum difference in cortical bone volume of 2.27%. The model was most sensitive to the target strain value. For a target strain of 1000 με, the model predicted a gain of trabecular and cortical bone of 32.70% and 18.51%, respectively. For a target strain of 1500 με, the model predicted a loss of trabecular and cortical bone of 19.28% and 13.88%, respectively. The complete data are made available in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.
The sensitivity of the adaptation algorithm to mesh refinement was assessed by comparing the total trabecular and cortical volumes obtained from the adaptation process of the two sensitivity cases with the baseline cantilever beam model. The finer mesh predicted an increase of 0.91% in trabecular bone volume and 2.93% in cortical bone volume, while the coarse mesh predicted a decrease of 6.36% in trabecular bone volume and 1.82% in cortical bone volume. The distribution of the cortical thicknesses and trabecular radii, and overall structure were consistent across all three models and are illustrated in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.
###### Comparison With Computed Tomography Derived Model.
The maximum and minimum principal strains across the surface of the structural model and continuum model derived from the CT data were compared (Fig. 12). The coefficients of correlation for both minimum (0.813) and maximum (0.809) principal strains show a fairly good agreement between the strains predicted by the structural model and CT-derived model. A detailed diagram showing the surface strains across both models and Bland–Altman plots [67] showing the 95% confidence intervals can be found in the Supplemental Material which is available under “Supplemental Data” tab for this paper on the ASME digital collection.
## Discussion
Given that a direct validation of the model was not possible, sensitivity studies were performed to assess the effects of material properties and adaptation parameters on the model. The model was not found to be particularly sensitive to changes in material properties of the joints and ligaments or to changes in initial element thickness and number of categories used for section definitions. The most sensitive parameter was found to be the target strain, as changing its value led to considerable changes in trabecular and cortical bone volume, respectively. This result was expected as the main driver of the adaptation algorithm is strain. Overall, the structural model and the adaptation algorithm were found to be robust in terms of predicting cortical and trabecular architecture caused by a loading environment associated with common daily activities.
In addition to the sensitivity study, the response of the converged structural model to a loading scenario was compared to an analogous continuum model with varying material properties derived from the CT data. Although the comparison showed that the two models were in fairly good agreement, the correlation was not found to be as strong as reported in the previous studies which have compared continuum models with experimental data [14,58]. However, the sample size used for comparison in the studies mentioned was limited to the number of strain gauges used, meaning that a comparison between strains across the whole bone surface was not possible. Differences between the structural model and the CT-derived model could be attributed to trabecular bone in the CT-derived model having a minimum stiffness of 50 MPa based on the conversion from the Hounsfield units values, while corresponding regions in the structural model had very sparse trabecular architecture.
The limitations of the musculoskeletal and finite element modeling methodologies presented in this study must be acknowledged. The muscle actuators contained in the musculoskeletal model do not take into account contraction dynamics and force–length–velocity relationships [37]. Although it was shown that these assumptions do not influence the predicted outcome for walking [69], they might have an impact on other activities. A limitation of the combined modeling approach is that the loads applied on the FE model are derived from a rigid multibody system, meaning that the equilibrium condition satisfied in the musculoskeletal model is compromised by the deformation and displacement occurring in the FE model. Furthermore, the muscle forces are not spread over attachment areas and the muscle compression forces onto bone are not accounted for. The musculoskeletal model partially overcomes this limitation by having a high number of actuators for muscles. Finally, the model was developed to contain only the pelvic bones and ligaments. Although the main focus was on bone adaptation, the lack of internal organs and soft tissue connected to the pelvis can have an impact on bone adaptation as loads in certain regions are not spread and compressive loads caused by the presence of muscles and organs are not taken into account. Future work will aim to assess the behavior of bone-soft tissue complex in the pelvic region.
To the authors' knowledge, this is the first application of a bone adaptation algorithm to the pelvis and the first structural model of the pelvis developed using an adaptive approach. The resulting structure shows good agreement with previous findings on pelvic bone architecture and the structural model was found to be computationally efficient. A future application of the model will be to examine fractures occurring in scenarios such as falls, vehicular collisions, or solid blast. Furthermore, the structural model enables the use of additive manufacturing to design and produce frangible surrogates. Finally, massive endoprostheses and scaffolds may be designed to allow bone regeneration in specific areas of the pelvis affected by trauma or disease while maintaining mechanical properties of the region.
## Acknowledgements
The authors thank the Human Performance and Musculoskeletal Biomechanics groups at Imperial College London for assistance with the gait analysis, and Royal British Legion Centre for Blast Injury Studies at Imperial College London for providing the CT data. The authors acknowledge and thank Dr. Luca Modenese for providing assistance with the musculoskeletal modeling and Professor Jon Clasper for providing clinical input throughout the modeling process.
## Funding Data
The Royal British Legion Centre for Blast Injury Studies at Imperial College London.
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Almeida, E. S. , and Spilker, R. L. , 1998, “ Finite Element Formulations for Hyperelastic Transversely Isotropic Biphasic Soft Tissues,” Comput. Methods Appl. Mech. Eng., 151(3–4), pp. 513–538.
Ehlers, W. , and Markert, B. , 2001, “ A Linear Viscoelastic Biphasic Model for Soft Tissues Based on the Theory of Porous Media,” ASME J. Biomech. Eng., 123(5), pp. 418–424.
Li, Z. , Alonso, J. E. , Kim, J.-E. , Davidson, J. S. , Etheridge, B. S. , and Eberhardt, A. W. , 2006, “ Three-Dimensional Finite Element Models of the Human Pubic Symphysis With Viscohyperelastic Soft Tissues,” Ann. Biomed. Eng., 34(9), pp. 1452–1462. [PubMed]
Leung, A. S. O. , Gordon, L. M. , Skrinskas, T. , Szwedowski, T. , and Whyne, C. M. , 2009, “ Effects of Bone Density Alterations on Strain Patterns in the Pelvis: Application of a Finite Element Model,” Proc. Inst. Mech. Eng. H, 223(8), pp. 965–979. [PubMed]
Aamodt, A. , Lund-Larsen, J. , Eine, J. , Andersen, E. , Benum, P. , and Husby, O. S. , 1997, “ In Vivo Measurements Show Tensile Axial Strain in the Proximal Lateral Aspect of the Human Femur,” J. Orthop. Res., 15(6), pp. 927–931. [PubMed]
Zhang, Q. H. , Wang, J. Y. , Lupton, C. , Heaton-Adegbile, P. , Guo, Z. X. , Liu, Q. , and Tong, J. , 2010, “ A Subject-Specific Pelvic Bone Model and Its Application to Cemented Acetabular Replacements,” J. Biomech., 43(14), pp. 2722–2727. [PubMed]
Prevrhal, S. , Engelke, K. , and Kalender, W. A. , 1999, “ Accuracy Limits for the Determination of Cortical Width and Density: The Influence of Object Size and CT Imaging Parameters,” Phys. Med. Biol., 44(3), pp. 751–764. [PubMed]
Nägele, E. , Kuhn, V. , Vogt, H. , Link, T. M. , Müller, R. , Lochmüller, E.-M. , and Eckstein, F. , 2004, “ Technical Considerations for Microstructural Analysis of Human Trabecular Bone From Specimens Excised From Various Skeletal Sites,” Calcif. Tissue Int., 75(1), pp. 15–22. [PubMed]
Dalstra, M. , Huiskes, R. , and Section, B. , 1995, “ Load Transfer Across the Pelvic,” J. Biomech., 28(6), pp. 715–724. [PubMed]
Macchiarelli, R. , Bondioli, L. , Galichon, V. , and Tobias, P. V. , 1999, “ Hip Bone Trabecular Architecture Shows Uniquely Distinctive Locomotor Behaviour in South African Australopithecines,” J. Hum. Evol., 36(2), pp. 211–232. [PubMed]
Peretz, A. M. , Hipp, J. A. , and Heggeness, M. H. , 1998, “ The Internal Bony Architecture of the Sacrum,” Spine, 23(9), pp. 971–974. [PubMed]
Mahato, N. K. , 2010, “ Trabecular Architecture in Human Sacra: Patterns Observed in Complete Sacralisation and Accessory Articulation With the Fifth Lumbar Vertebrae,” J. Morphol. Sci., 27(1), pp. 19–22.
Bland, J. M. , and Altman, D. G. , 1999, “ Measuring Agreement in Method Comparison Studies,” Stat. Methods Med. Res., 8(2), pp. 135–160. [PubMed]
Boyle, C. , and Kim, I. Y. , 2011, “ Three-Dimensional Micro-Level Computational Study of Wolff's Law Via Trabecular Bone Remodeling in the Human Proximal Femur Using Design Space Topology Optimization,” J. Biomech., 44(5), pp. 935–942. [PubMed]
Anderson, F. C. , and Pandy, M. G. , 2001, “ Static and Dynamic Optimization Solutions for Gait are Practically Equivalent,” J. Biomech., 34(2), pp. 153–161. [PubMed]
View article in PDF format.
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Almeida, E. S. , and Spilker, R. L. , 1998, “ Finite Element Formulations for Hyperelastic Transversely Isotropic Biphasic Soft Tissues,” Comput. Methods Appl. Mech. Eng., 151(3–4), pp. 513–538.
Ehlers, W. , and Markert, B. , 2001, “ A Linear Viscoelastic Biphasic Model for Soft Tissues Based on the Theory of Porous Media,” ASME J. Biomech. Eng., 123(5), pp. 418–424.
Li, Z. , Alonso, J. E. , Kim, J.-E. , Davidson, J. S. , Etheridge, B. S. , and Eberhardt, A. W. , 2006, “ Three-Dimensional Finite Element Models of the Human Pubic Symphysis With Viscohyperelastic Soft Tissues,” Ann. Biomed. Eng., 34(9), pp. 1452–1462. [PubMed]
Leung, A. S. O. , Gordon, L. M. , Skrinskas, T. , Szwedowski, T. , and Whyne, C. M. , 2009, “ Effects of Bone Density Alterations on Strain Patterns in the Pelvis: Application of a Finite Element Model,” Proc. Inst. Mech. Eng. H, 223(8), pp. 965–979. [PubMed]
Aamodt, A. , Lund-Larsen, J. , Eine, J. , Andersen, E. , Benum, P. , and Husby, O. S. , 1997, “ In Vivo Measurements Show Tensile Axial Strain in the Proximal Lateral Aspect of the Human Femur,” J. Orthop. Res., 15(6), pp. 927–931. [PubMed]
Zhang, Q. H. , Wang, J. Y. , Lupton, C. , Heaton-Adegbile, P. , Guo, Z. X. , Liu, Q. , and Tong, J. , 2010, “ A Subject-Specific Pelvic Bone Model and Its Application to Cemented Acetabular Replacements,” J. Biomech., 43(14), pp. 2722–2727. [PubMed]
Prevrhal, S. , Engelke, K. , and Kalender, W. A. , 1999, “ Accuracy Limits for the Determination of Cortical Width and Density: The Influence of Object Size and CT Imaging Parameters,” Phys. Med. Biol., 44(3), pp. 751–764. [PubMed]
Nägele, E. , Kuhn, V. , Vogt, H. , Link, T. M. , Müller, R. , Lochmüller, E.-M. , and Eckstein, F. , 2004, “ Technical Considerations for Microstructural Analysis of Human Trabecular Bone From Specimens Excised From Various Skeletal Sites,” Calcif. Tissue Int., 75(1), pp. 15–22. [PubMed]
Dalstra, M. , Huiskes, R. , and Section, B. , 1995, “ Load Transfer Across the Pelvic,” J. Biomech., 28(6), pp. 715–724. [PubMed]
Macchiarelli, R. , Bondioli, L. , Galichon, V. , and Tobias, P. V. , 1999, “ Hip Bone Trabecular Architecture Shows Uniquely Distinctive Locomotor Behaviour in South African Australopithecines,” J. Hum. Evol., 36(2), pp. 211–232. [PubMed]
Peretz, A. M. , Hipp, J. A. , and Heggeness, M. H. , 1998, “ The Internal Bony Architecture of the Sacrum,” Spine, 23(9), pp. 971–974. [PubMed]
Mahato, N. K. , 2010, “ Trabecular Architecture in Human Sacra: Patterns Observed in Complete Sacralisation and Accessory Articulation With the Fifth Lumbar Vertebrae,” J. Morphol. Sci., 27(1), pp. 19–22.
Bland, J. M. , and Altman, D. G. , 1999, “ Measuring Agreement in Method Comparison Studies,” Stat. Methods Med. Res., 8(2), pp. 135–160. [PubMed]
Boyle, C. , and Kim, I. Y. , 2011, “ Three-Dimensional Micro-Level Computational Study of Wolff's Law Via Trabecular Bone Remodeling in the Human Proximal Femur Using Design Space Topology Optimization,” J. Biomech., 44(5), pp. 935–942. [PubMed]
Anderson, F. C. , and Pandy, M. G. , 2001, “ Static and Dynamic Optimization Solutions for Gait are Practically Equivalent,” J. Biomech., 34(2), pp. 153–161. [PubMed]
## Figures
Fig. 1
Lower limb musculoskeletal model in (a) standing and (b) seated positions
Fig. 2
Transverse 2.5 mm slice of the pelvic for the base FE model. Shell elements representing cortical bone are shown in gray in the border areas; truss elements representing trabecular bone are shown in the interior in red (see color figure online).
Fig. 3
Pelvis model with ligaments included: Sacroiliac ligaments in blue (center top), pubic ligament in red (center bottom), sacrospinous ligaments in yellow (middle), sacrotuberous ligaments in purple (outer edge of sacrospinous ligaments) and inguinal ligaments in green (diagonal front) (see color figure online)
Fig. 4
(a) Load applicator at the hip and (b) fixator at the top of the sacrum. The elements representing the bone are shown in light gray, with the two layers of soft elements (from left to right in (a) and top to bottom in (b)) in red and the two layers of stiffer elements in blue (dark blue represents the stiffest layer of elements in the hip applicator). In addition, the black arrow illustrates the peak load applied on the hip joint during walking (see color figure online)
Fig. 5
Hip JCFs derived from musculoskeletal model for cycles of (a) walking, (b) stair ascent, (c) stair descent, and (d) sit-to-stand and stand-to-sit are shown in green lines for the right leg and blue lines for the left leg. The frames selected for each activity to be used in the FE simulations are highlighted using solid circles. Hip JCFs recorded and reported by Bergmann et al. [3] for the right leg are shown in dashed lines for each activity (see color figure online)
Fig. 6
Modeling framework
Fig. 7
Contours of position-dependent cortical thickness ranging from 0.1 mm to 5 mm are shown in the top row: (a) frontal view of pelvis, (b) anterior and (c) medial views of the right innominate. Trabecular elements' radii distribution ranging from 0.1 mm to 2 mm is shown in the bottom row: (d) frontal view, (e) anterior, and (f) medial views of the right innominate. Trabecular elements with a radius < 0.1 mm were excluded for clarity.
Fig. 8
Selected 5 mm slices of the converged pelvic model illustrating the trabecular structure and cortical thickness. Truss elements with a radius >0.1 mm are shown in red (largest clusters), with elements with a radius of 0.1 mm shown in blue. Elements with a radius of 1 μm are not shown for clarity.
Fig. 9
Trabecular trajectories formed in the ilium adapted from (a) walking and (b) all activities. The trajectories highlighted correspond to regions in the ilium with a higher trabecular density caused by walking observed by Machiarelli et al. [64]: superior bundle (sb), anterior bundle (ab), sacropubic bundle (spb), iliocotyloid bundle (icb), ilioischial bundle (iib), and trabecular chiasma (tc). Truss elements with a radius >0.1 mm are shown in red (clusters with thick elements). Truss elements with a radius of 0.1 mm appear in blue (see color figure online).
Fig. 10
Figure illustrating transverse slices from the CT scans (a, c, e, g) shown along with corresponding slices of the pelvic model (b, d, f, h) generated using an in-house code developed in matlab. Shell elements and truss elements with a radius > 0.1 mm are shown in light gray. Truss elements with a radius of 0.1 mm appear in dark gray. Elements with a radius of 1 μm are not shown for clarity.
Fig. 11
Influence of activities on cortical (a, b, c) and trabecular (d, e, f) structure of converged pelvic model. The elements are color mapped based on the activity which has been most influential in determining their thickness. Further information on the influence of activities on the cortical and trabecular architecture can be found in the supplemental Fig. S1 which is available under the “Supplemental Data” tab for this paper on the ASME digital collection.
Fig. 12
Comparison between the minimum (a) and the maximum (b) principal strains across the surface of the structural model (x axis) and CT-derived model (y axis). Lines of best fit are shown in solid lines for each case and do not differ greatly from the y = x line shown in dash line.
## Tables
Table 1 Material properties of the ligaments in tension (taken from Ref. [30]), with k being the ligament stiffness, l the length of the ligament, A the origination area of the ligament, Nl the number of elements forming each ligament, E Young's modulus, and r the radius of each truss element
Table 2 Material properties of the cartilage in compression (taken from Ref. [30]), with k being the ligament stiffness, l the length of the ligament, A the origination area of the ligament, Nl the number of elements forming each ligament, E Young's modulus, and r the radius of each truss element
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# Combinatorics
## Authors and titles for math.CO in Jun 2021, skipping first 50
[ total of 422 entries: 1-50 | 51-100 | 101-150 | 151-200 | 201-250 | ... | 401-422 ]
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[51]
Title: The maximum number of 10- and 12-cycles in a planar graph
Subjects: Combinatorics (math.CO)
[52]
Title: On minimal doubly resolving sets in graphs
Authors: Mohsen Jannesari
Subjects: Combinatorics (math.CO)
[53]
Title: Discrete-to-Continuous Extensions: Lovász extension, optimizations and eigenvalue problems
Subjects: Combinatorics (math.CO); Optimization and Control (math.OC); Spectral Theory (math.SP)
[54]
Title: Normalized Sombor indices as complexity measures of random graphs
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[55]
Title: Closed Ziv-Lempel factorization of the $m$-bonacci words
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Formal Languages and Automata Theory (cs.FL)
[56]
Title: Which graphs can be counted in $C_4$-free graphs?
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[57]
Title: Understanding lettericity I: a structural hierarchy
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[58]
Title: The Szemerédi-Trotter type theorem in matrix rings and its application
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[59]
Title: On near-MDS codes and caps
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[60]
Title: Improving Lower Bounds for Equitable Chromatic Number
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[61]
Title: Intertwining connectivities for vertex-minors and pivot-minors
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[62]
Title: Decomposing graphs into interval colorable subgraphs and no-wait multi-stage schedules
Subjects: Combinatorics (math.CO)
[63]
Title: 2-distance 4-coloring of planar subcubic graphs with girth at least 21
Comments: 21 pages, 14 figures. arXiv admin note: text overlap with arXiv:2103.11687
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
[64]
Title: The Amazing Chromatic Polynomial
Authors: Bruce E Sagan (Michigan State University)
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[65]
Title: Families of convex tilings
Authors: Richard Kenyon
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[66]
Title: Partial Sums of the Fibonacci Sequence
Authors: Hung Viet Chu
Journal-ref: Fib. Quart., 59:2 (May 2021), 132-135
Subjects: Combinatorics (math.CO); Number Theory (math.NT)
[67]
Title: Spectrum of Strongly Regular Graphs under Graph Operators
Subjects: Combinatorics (math.CO)
[68]
Title: Multivariate blowup-polynomials of graphs
Subjects: Combinatorics (math.CO); Classical Analysis and ODEs (math.CA)
[69]
Title: A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula
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[70]
Title: Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. V
Comments: 46 pages; v.2,3 - notation explained, misprints corrected, and references added; v.4-6 - more misprints corrected, main notation changed for better readability, minor improvements
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[71]
Title: Ratio sets of random sets
Journal-ref: The Ramanujan Journal, 43(2), 2017
Subjects: Combinatorics (math.CO); Number Theory (math.NT)
[72]
Title: On the Average (Edge-)Connectivity of Minimally $k$-(Edge-)Connected Graphs
Comments: 16 pages, 3 figures. This version includes revisions based on referee comments
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[73]
Title: A Note on Sumsets and Restricted Sumsets
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Journal-ref: Journal of Integer Sequences, 24 (2021), Article 21.4.2
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[74]
Title: Discrete-to-Continuous Extensions: piecewise multilinear extension, min-max theory and spectral theory
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[75]
Title: Expansion, long cycles, and complete minors in supercritical random subgraphs of the hypercube
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Title: Quasi-Stirling Polynomials on Multisets
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[77]
Title: Quasi-Stirling Permutations on Multisets
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[78]
Title: A Note on Distinguishing Trees with the Chromatic Symmetric Function
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[79]
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[80]
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Comments: Proceedings of Blockchain in Kyoto 2021, JPS Conference Proceedings, To appear (2021)
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[81]
Title: Geometric and o-minimal Littlewood-Offord problems
Comments: 22 pages, minor edits. To appear in the Annals of Probability
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[82]
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[83]
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[84]
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Subjects: Combinatorics (math.CO)
[85]
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[86]
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Subjects: Combinatorics (math.CO)
[87]
Title: Uniform intersecting families with large covering number
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
[88]
Title: A note on explicit constructions of designs
Subjects: Combinatorics (math.CO)
[89]
Title: Best possible bounds on the number of distinct differences in intersecting families
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
[90]
Title: Coxeter Pop-Tsack Torsing
Subjects: Combinatorics (math.CO)
[91]
Title: Hermitean matrices of roots of unity and their characteristic polynomials
Subjects: Combinatorics (math.CO)
[92]
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Subjects: Combinatorics (math.CO); Number Theory (math.NT)
[93]
Title: Cameron-Liebler k-sets in subspaces and non-existence conditions
Journal-ref: Designs, Codes and Cryptography, 2022
Subjects: Combinatorics (math.CO)
[94]
Title: On some graph-cordial Abelian groups
Authors: Sylwia Cichacz
Journal-ref: Discrete Mathematics 345 (2022) 112815
Subjects: Combinatorics (math.CO)
[95]
Title: Edge Domination Number and the Number of Minimum Edge Dominating Sets in Pseudofractal Scale-Free Web and Sierpiński Gasket
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[96]
Title: Minimal Regular graphs with every edge in a triangle
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Subjects: Combinatorics (math.CO)
[97]
Title: Update: Some new results on lower bounds on $(n,r)$-arcs in $PG(2,q)$ for $q\le 31$
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[98]
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[99]
Title: Graphs that are minor minimal with respect to dimension
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https://911weknow.com/12-islanders-puzzle-from-brooklyn-99
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# 12 Islanders Puzzle from Brooklyn 99
A Puzzle is presented where there are twelve identical looking islanders and a seesaw. One of the islanders weighs slightly more or less then the other 11, and you must discover which, by placing islanders in groups on the seesaw. However only three measurements are allowed.
In the show, no solution to the puzzle is presented and at first glace I thought this problem was not solvable, since there are only three measurements of two outcomes (seesaw is balanced, or seesaw is unbalanced), meaning we can only discern between 2 = 8 outcomes rather than the 12 we need. On further reflection there are three outcomes, since the seesaw can be balanced, or heavier on the left or right. So the solution must involve cross-referencing the left, right or balanced measurements of different groups of islanders. Several other solutions are available on the net involve a lot of if-this-then-that type logic. Presented below is a simpler solution.
Each islander is given a position to sit on the seesaw for each round. L: sit on the left, R: sit on the right, -: don?t sit on the seesaw.
The pattern for all islanders is below:
person: A B C D E F G H I J K Lround 1: L L L L R R R R ? ? ? -round 2: L L R R R ? ? ? L R L -round 3: L R R ? ? L R ? L L ? R
For example person F will sit on the right, then stand out, then on the left.
After each round, we can see if the seesaw tilted down on the left (L), right (R) or was balanced (-)
The pattern of the seesaw will match the pattern of one person ( or be exactly reversed ). That person is the heavier (or lighter)
The nice thing about this approach is that the logic is very simple (you just need to know the pattern) and you will always find out whether the person is lighter or heavier.
The tricky bit for me was figuring out that the left/right/balance needed to be cross-referenced, then discovering a pattern for each islander, where;
• the same number of islanders are always on each side of the seesaw
• no pattern is repeated
• the opposite of each used pattern is not used
Note: I?ve since found out this is a variation on a puzzle involving 12 balls and a set of scales, but the same solution works here too.
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|
http://physics.stackexchange.com/questions/36027/maxwell-equations-invariant-under-lorentz-transformation-but-not-galilean-transf/36040
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# Maxwell equations invariant under Lorentz transformation but not Galilean transformations
Why Maxwell equations are not invariant under Galilean transformations, but invariant under Lorentz transformations? What is the deep physical meaning behind it?
-
Are you looking for an explicit demonstration of these properties, or what....I mean, that set of equation simple has those mathematical properties. It's sort of like asking why a square ninety degree angles and not sixty degree one. The deep physical meaning is that physics is Einsteinian and not Galilean. – dmckee Sep 11 '12 at 0:52
You can make a quasistatic approximation when the timescale of the sources $T$ and the size of the system $L$ are such that $L/T \ll c$, or $L/c \ll T$. This means that the field propagates through the entire system much faster than the sources vary. This is a non-relativistic approximation since we get action at a distance. Of course, we can't capture all EM phenomena in this approximation. In particular, radiation is usually studied in the opposite regime $r/c \gg T$ where $r$ is the distance to the source. – Robin Ekman Jul 6 '14 at 18:05
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https://learn.careers360.com/engineering/question-set-a-has-m-element-and-set-b-has-n-elements-if-the-total-no-of-the-subset-of-a-is-112-more-than-the-total-no-of-subsets-of-b-then-m-n-option-1-2option-2-3option-3-6option-4-none-of-the-above-111837/
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# Set A has m element and set B has n elements If the total no of the subset of A is 112 more than the total no of subsets of B then m-n =? Option: 1 2 Option: 2 3 Option: 3 6 Option: 4 None of the above
$2^m-2^n=112$
m= 7 and n = 4
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|
https://www.physicsforums.com/threads/proof-of-limit-by-definition.267096/
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# Proof of limit by definition
1. Oct 26, 2008
1. The problem statement, all variables and given/known data
find the limit as x -> 0 of (sin^2(x))/(x^2)
2. Relevant equations
limit as x -> xo (fx) = L iff for every epsilon (>0) there exists a delta (>0) st if
| x - xo | < delta then |f(x) - L | < epsilon
3. The attempt at a solution
Let epsilon be positive. I believe the limit equals one, so I will proceed there.
then
| (sin^2(x))/x^2 - 1| < epsion if | x | < delta
But | f(x) - 1| <= |1/x^2 - 1| . And this is where I get stuck. If I pick delta to be small, and |x| < delta,
| (f(x)) - 1 | becomes very large, and thus is not being bound by any epsilon.
2. Oct 26, 2008
### Dick
|f(x)-1| does NOT become very large as x->0. Did you experiment with some numbers, like x=0.01,x=0.0001, etc? Do you know anything about the limit of sin(x)/x as x->0?
3. Oct 26, 2008
You are right, I meant to say that
|1/x^2 - 1| becomes large as x -> 0, which confused me, because
| f(x) - 1| <= |1/x^2 - 1|, which i get because | sin^2(x) | <= 1, so then | sin^2(x)/x^2 - 1 | <= |1/x^2 - 1 |
The limit of sin(x)/x equals 1 when x approaches 0. Are you suggesting that I use the fact that
sin(x)/x has limit 1 at x = 0 and the theorem that if lim f exists and equals L1 at some point xo and lim g exists at the same xo and equals L2, then lim f*g = L1*L2, which then I could use to lim sin^2(x)/x^2 =
lim (sin(x)/x)*(sin(x)/x) = 1 * 1 = 1. Yes, but then I need to prove that lim sin(x)/x equals 1. How would I do this by epsilon-delta definition of limits ( I know how to with squeeze theorem)? I would basically be stuck in the same mess I have above.
4. Oct 26, 2008
### Dick
|sin(x)^2/x^2-1|<=|1/x^2-1| is just not a very good estimate of |f(x)-1|. How do you prove sin(x)/x=1 using the squeeze theorem? You should be able to express that logic in epsilon-delta form. You might also notice |f(x)-1|=|sin(x)/x-1|*|sin(x)/x+1|.
5. Oct 26, 2008
| cos(x) | <= | (sin x)/x | <= | 1 |
Is how I would use the squeeze theorem to solve this one. But then I have no idea how to get from there to a delta, or reduction of | (sin x)/x | to | x | from which I can pull a delta. I have been searching the web for a few hours and have found 0 proofs that (sin x)/x has limit equal to 1. There was a place that had it listed as an exercise of finding deltas, but gave no explanation of how to find one. So I remain stuck.
One other idea I had is
| sin(x)/x | <= | sin(x) | <= | x |
Then you get stuck by the triangle inequality trying to bring back (sin x)/x - 1, the closest I got is:
| sin(x)/x - 1 + 1 | <= | sin(x)/x - 1 + 1 | <= | x - 1 + 1 |
6. Oct 26, 2008
### Dick
I think I'm better at searching the web than you are. I found this:
Multiply 1-cos(x) by (1+cos(x))/(1+cos(x)) and get (1-cos(x)^2)/(1+cos(x))=sin(x)^2/(1+cos(x))<=sin(x)^2 (for small x). You also have sin(x)<=x. So put it all together and get 1-cos(x)<=x^2. So:
1-x^2<=sin(x)/x<=1. Does that look like a form you can use?
7. Oct 26, 2008
Oh, I never said I was good at searching the web, only that I had done it for a while...
Anyways, I like where you went with this. Basically:
|sin x| < |x| . This implies that |(sin x)/x| < |x/x| = 1.
On the other hand, if x is in the open interval (0,pi/2) then | x | < | tan x |, so we get that
| x/sin(x) | < | (tan x) / (sin x) | = | 1/(cos x) |. Then you reciprocate the rationals over the inequality to get that:
| cos x | < | (sin x) / x|. Then you put the two of these together to get:
| cox x | < | (sin x) / x| < 1. (1)
Now for the next bit, we will need again that (sin anynumber) < anynumber, and the famous identity sin^2(x) + cos^2(x) = 1, and then we can use your trick:
(1- cos x) * (1 + cos x)/(1 + cos x) = (1 - (cos x)^2)/(1+ cos x) = ((sin x)^2)/(1+ cos x) <= (sin x)^2
But because (sin x) < x, we get that (1 - cos x) < (x)^2. Then:
- cos x < x^2 -1 (2) , and further
cos x > 1 - x^2. Now because we have limited x to the open interval (0,pi/2), and from the inequalities (1) and (2)
1 - x^2 < (sin x)/x < 1 , and now the big one
- x^2 < (sin x)/x - 1 <0 (3)and zero is obviously less than epsilon (which is chosen positive). However, this is not complete, the above will not hold on all intervals, particularly the above holds when x is in (0,pi/2) in other words | x | < pi/2. Notice that,
(sin x) / x - 1 < x^2
And so if delta equals sqrt(epsilon) we get that if | x | < delta
then (sin x) / x - 1 < x^2 < epsilon
We need to pick delta to be the min of pi/2 and sqrt(epsilon).
8. Oct 26, 2008
### Dick
Yes, I think that does it. You shouldn't have any troubles extending that to get the delta for sin(x)^2/x^2.
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|
http://math.stackexchange.com/questions/45642/operators-from-c-0
|
# Operators from $c_{0}$
My question seems to be easy but I cannot spot the answer. I am interested in ranges of operators defined on $c_0$. The celebrated "operator version" of Sobczyk's theorem says that if we are given a separable Banach space $X$ and its subspace $Y$, then every bounded operator $T\colon Y\to c_0$ can be extended to a bounded operator $\overline{T}$ with $\|\overline{T}\|\leq 2\|T\|$ (categorically speaking, $c_0$ is "separably injective"). I am wondering if I could use this theorem (or anything else) to (dis)prove the following conjecture:
If $X$ is a $c_0$-saturated separable Banach, then the range of every operator $T\colon c_0\to X$ embeds into $c_0$. We know that (consult Lindenstrauss and Tzafriri's book) every quotient of $c_0$ embeds into $c_0$ but how about ranges of this sort of operators?
-
There is a quite recent survey article on Sobczyk's theorem by Cabello Sánchez, Castillo and Yost. While I haven't read it in detail, it might contain some pointers to the literature. – t.b. Jun 16 '11 at 0:21
I'm really not sure if I understand your question correctly, but as it is stated I don't really see how the information that $X$ contains a complemented copy of $c_0$ should help. If $T: c_0 \to Y$ is any operator to a separable Banach space, simply consider $X = Y \oplus c_0$ and compose $T$ with the inclusion $Y \to X$. The range of $T$ will remain the same and certainly $X$ is separable. Also, it would be nice if you could make your question a bit more precise (e.g. what exactly does it mean for the range of $T$ to embed into $c_0$)? – t.b. Jun 16 '11 at 0:40
$T(X)$ isomorphic to a subspace of $c_0$. Right, I was thinking about $c_0$-saturation, since if $X$ contains a copy $c_0$ then in the separable setting it is automatically complemented. Btw, I know this paper quite well. – dziobak Jun 16 '11 at 6:45
Like Theo, I don't really understand what exactly it is that is being asked. I am quite sure that the OP knows that regardless of whether or not $X$ is $c_0$-saturated, an operator $T:c_0 \longrightarrow X$ with closed range has the property that $T(c_0)$ embeds into $c_0$ by the Johnson-Zippin result cited above in the comments to the question.
I am somewhat guessing that the question is: for a $c_0$-saturated $X$ and arbitrary $T:c_0 \longrightarrow X$, is the Banach space $\overline{T(c_0)}$ isomorphic to a subspace of $c_0$?
The answer to this question is no. For a counterexample, take $X$ to be $C(\omega^\omega)$ (or replace $\omega^\omega$ by any larger countable ordinal) and let $(\alpha_n)_{n=1}^\infty$ be an enumeration of $\omega^\omega+1$. For each $n\in \mathbb{N}$ let $\alpha_n^- = \min \{\beta \mid \exists \nu \mbox{ such that } \beta + \omega^\nu = \alpha_n\}$, so that the (continuous) characteristic functions $\chi_{(\alpha_n^-, \alpha_n]}$, $n\in\mathbb{N}$, span a dense linear subspace of $C(\omega^\omega)$. The map $e_n \mapsto 2^{-n}\chi_{(\alpha_n^-, \alpha_n]}$ extends to a (compact) continuous linear operator $T: c_0 \longrightarrow C(\omega^\omega)$ - with dense range. Since $C(\omega^\omega)$ does not embed in $c_0$, the claimed counterexample is achieved.
P.S. If you write $\alpha_n$ as a sum of powers of $\omega$ - i.e., in Cantor normal form - then $\alpha_n^-$ is just the ordinal attained by leaving off the last summand.
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https://www.physicsforums.com/threads/schwarzschild-metric.794669/
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# Schwarzschild metric
1. Jan 28, 2015
### TimeRip496
How do you obtain this equation M=Gm/c^2. What does M stand for? Is is newton law at infinity? Again what is this newton law at infinity?
2. Jan 28, 2015
### Staff: Mentor
M is just the mass of the black hole using units in which G and c are both equal to one.
We don't have to make this substitution but if we don't we'll be schlepping factors of G and c around everywhere in our equations, and they're complicated enough already.
3. Jan 28, 2015
### TimeRip496
Do you mind telling me a source for such derivation? Cause all the Internet gives is just the derivation of the schwarzschild radius.
4. Jan 28, 2015
### Staff: Mentor
It's just a conversion factor from mass units to length units; $Gm / c^2$ converts the mass $m$ to an equivalent length. The Schwarzschild radius corresponding to $m$ is just twice that equivalent length.
Similar Discussions: Schwarzschild metric
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https://docs.mosek.com/latest/opt-server/supported-file-formats.html
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# 10 Supported File Formats¶
MOSEK supports a range of problem and solution formats listed in Table 10.1 and Table 10.2. The Task format is MOSEK’s native binary format and it supports all features that MOSEK supports. The OPF format is MOSEK’s human-readable alternative that supports nearly all features (everything except semidefinite problems). In general, text formats are significantly slower to read, but can be examined and edited directly in any text editor.
Problem formats
Table 10.1 List of supported file formats for optimization problems. The column Conic refers to conic problems involving the quadratic, rotated quadratic, power or exponential cone. The last two columns indicate if the format supports solutions and optimizer parameters.
Format Type Ext. Binary/Text LP QO Conic SDP Sol Param
LP lp plain text X X
MPS mps plain text X X X
OPF opf plain text X X X X X
PTF ptf plain text X X X X X
CBF cbf plain text X X X
Solution formats
Table 10.2 List of supported solution formats.
Format Type Ext. Binary/Text Description
SOL sol plain text Interior Solution
bas plain text Basic Solution
int plain text Integer
Jsol format jsol text Solution
Compression
MOSEK supports GZIP and Zstandard compression. Problem files with extension .gz (for GZIP) and .zst (for Zstandard) are assumed to be compressed when read, and are automatically compressed when written. For example, a file called
problem.mps.gz
will be considered as a GZIP compressed MPS file.
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## tanjung find the value of n that satisfy n^2+25n+19 be a perfect square with n is an even number ? one year ago one year ago Edit Question Delete Cancel Submit
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@mukushla @sirm3d
• one year ago
2. mukushla
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unfortunately i dont have enough time for thinkin on this :( try this : $n^2+25n+19=m^2$discriminant of quadratic must be a perfect square too$625-4(19-m^2)=k^2$rearranging gives$(k-2m)(k+2m)=549=3^2\times 61$
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k−2m = 9 k+2m = 61 i got k =35 and m = 13 therefore n^2+25n+19-169=0 or n^2+25n-150=0 (n-15)(n-10)=0 n=15 or n=10 because n even number, so n=10 right ?????
• one year ago
4. mukushla
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right :)
• one year ago
5. mukushla
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how about ?? k−2m = 3 .... k+2m = 183
• one year ago
6. tanjung
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opsss. i think i was mistake factor our n^2+25n-150=0. :)
• one year ago
7. tanjung
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n^2+25n-150=0 (n-5)(n+30)=0 n=5 (is not an even number) hmmm :(
• one year ago
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maybe there is no answer but check other possibilities for m and k
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9. tanjung
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k−2m = 3 k+2m = 183 satisfyied for k=93 and m=45 n^2+25n+19=45^2 n^2+25n+19-2025=0 n^2+25n-2006=0 (n-34)(n-56)=0 n=34 yohohohohohohohoooooo.... i got it hehe.. :))))))
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thank u very much, mukushla...
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welcome :)
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http://nethack.wikia.com/wiki/Power
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## FANDOM
2,035 Pages
Your power, or "Pw", is the amount of magical energy you have:
Dlvl:3 \$:43 HP:68(68) Pw:13(18) AC:-3 Xp:8/1295 T:2358
The above status line shows a character whose current power is 13 and maximum power is 18.
Power is consumed by casting spells, at a rate of 5 times the spell's level. When carrying the Amulet of Yendor, you will expend a random amount from 1 to 3 times the norm. If you fail to cast a spell due to failure rate or confusion, you expend only half the energy required for a success. When you have teleportitis, self-teleporting with ^T always consumes 20 power points, but random teleports consume no power.
Power can be regained slowly over time, quickly by quaffing a non-cursed potion of gain energy or with The Mitre of Holiness, and fully by reading a scroll of charging while confused. The potion also increases your maximum power; the scroll does so only if your power is already at maximum when you read it. Eating a newt corpse occasionally raises your current Power by 1-3 points, with the message "You feel a mild buzz"[1]
The rate at which power regenerates over time increases with your Wisdom and Intelligence. If your encumbrance level is stressed, it will not regenerate. The Eye of the Aethiopica causes your power to regenerate at a much greater rate than usual.
## Energy regeneration Edit
Your energy regenerates every $\lfloor (38 - \mathit{level}) \times 3 / 6 \rfloor$ turns if you are a Wizard, and every $\lfloor (38 - \mathit{level}) \times 4 / 6 \rfloor$ turns for other roles.[2] This regeneration is blocked if your encumbrance is at least Stressed.
If you have energy regeneration (only granted by the carried Eye of the Aethiopica), you will instead regenerate energy every turn.
The amount of energy regenerated each time is between 1 and 1 + (Int+Wis)/15, which for most spellcasters means 1d2 or 1d3.
Level Non-wizard Wizard
1 24 18
2 24 18
3 23 17
4 22 17
5 22 16
6 21 16
7 20 15
8 20 15
9 19 14
10 18 14
11 18 13
12 17 13
13 16 12
14 16 12
15 15 11
16 14 11
17 14 10
18 13 10
19 12 9
20 12 9
21 11 8
22 10 8
23 10 7
24 9 7
25 8 6
26 8 6
27 7 5
28 6 5
29 6 4
30 5 4
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https://git.rockbox.org/cgit/rockbox.git/diff/manual/rockbox_interface/wps.tex?id=2e45ca37ce287b05db12a41e5f81467e7f9e3d2a
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summaryrefslogtreecommitdiffstats log msg author committer range
diff options
context: 12345678910152025303540 space: includeignore mode: unifiedssdiffstat only
Diffstat (limited to 'manual/rockbox_interface/wps.tex')
-rw-r--r--manual/rockbox_interface/wps.tex8
1 files changed, 5 insertions, 3 deletions
diff --git a/manual/rockbox_interface/wps.tex b/manual/rockbox_interface/wps.texindex 03ec1562e5..6ff3642a6f 100644--- a/manual/rockbox_interface/wps.tex+++ b/manual/rockbox_interface/wps.tex@@ -133,9 +133,11 @@ your WPS (While Playing Screen). \item [The clip indicator:] This is a little black block that is displayed at the very right of the scale when an overflow occurs. It usually does not show up during normal- playback unless you play an audio file that is distorted heavily. If you- encounter clipping while recording, your recording will sound distorted. You- should lower the gain. + playback unless you play an audio file that is distorted heavily.+ \opt{recording}{+ If you encounter clipping while recording, your recording will sound distorted.+ You should lower the gain.+ } \note{Note that the clip detection is not very precise. Clipping might occur without being indicated.} \item [The scale:]
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https://brilliant.org/discussions/thread/fibonacci-help-me-2/
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×
# Fibonacci ??? Help me
Have anyone noticed that when we take 4 consecutive numbers in the Fibonacci list: f(x), f(x+1), f(x+2), f(x+3). We will have:
f(x) * f(x+3) - f(x+1) * f(x+2) =1 or -1
For example: 2 * 8 - 3 * 5 =1
p/s: It is just my view, I do not sure if it is always true.
Note by Khoi Nguyen Ho
1 year, 12 months ago
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That is a great observation.
Let me suggest a way of continuing:
Can you list out the values of $$x$$ where the expression is 1, and the values of $$x$$ where the expression is -1?
Do you know Binet's formula which gives you the value of $$f(x)$$? Staff · 1 year, 11 months ago
oh yeah, just adding x, x+1, x+2, x+3 into the Binet's and then minus two products, I found that f(x) * f(x+3) - f(x+1) * f(x+2)= -1 * (-1)^x. Great. Thank you. · 1 year, 11 months ago
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## anonymous 3 years ago Why is this true? $\vec E \cdot d \vec l=E \hat i \cdot (dx \hat i +dy \hat j +dz \hat k)=E dx$ I understand the first part $\vec E \cdot d \vec l=E \hat i \cdot (dx \hat i +dy \hat j +dz \hat k)$ but how does that equal Edx? Delete Cancel Submit
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• 3 years ago
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@zepdrix @hartnn
2. hartnn
• 3 years ago
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i and j are perpendicular i and k are perpendicular
3. hartnn
• 3 years ago
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i.i = i^2 = 1 i.j =0 i.k=0
4. anonymous
• 3 years ago
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I'm sorry...but how did you determine that they were perpendicular?
5. hartnn
• 3 years ago
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i , j and k are standard unit vectors. i is along x direction j is along y direction k is along z direction and axes are perpendicular to each other, right ?
6. anonymous
• 3 years ago
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yes
7. hartnn
• 3 years ago
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|dw:1361080799293:dw| any more doubts ?
8. anonymous
• 3 years ago
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ahhhh.....nope...no more doubts =D
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http://www.bibsonomy.org/bibtex/253c8ebd9fe28d8518072afa3cf4f09fb/clange
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BibSonomy :: publication :: Measurement of the shape of the boson rapidity distribution for $p p \to Z/gamma^* \to e^+ e^-$ + $X$ events produced at $s$ of 1.96-TeV
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# Measurement of the shape of the boson rapidity distribution for $p p \to Z/gamma^* \to e^+ e^-$ + $X$ events produced at $s$ of 1.96-TeV
V. M. Abazov, and others. Phys. Rev. (2007)
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### Measurement of the shape of the boson rapidity distribution for $p p \to Z/gamma^* \to e^+ e^-$ + $X$ events produced at $s$ of 1.96-TeV
V. M. Abazov, and others. Phys. Rev. (2007)
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https://singingdrjosh.com/anomalisa-stream-wvzg/bellman-equation-paper-1431a9
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# bellman equation paper
The main difference between optimal control of linear systems and nonlinear systems lies in that the latter often requires solving the nonlinear Hamilton–Jacobi–Bellman (HJB) equation instead of the Riccati equation (Abu-Khalaf and Lewis, 2005, Al-Tamimi et … Path Dependent PDEs. ). Initially, the system is assumed to have no faulty components, i.e. (2006) Linear forward-backward stochastic differential equations with random coefficients. A neces- Example 1. Stochastic Linear-Quadratic Control. (2019) Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations. Keyword: Bellman Equation Papers related to keyword: G. Barles - A. Briani - E. Chasseigne (SIAM Journal on Control and Optimization ) A Bellman approach for regional optimal control problems in R^N (2014) G. Barles - A. Briani - E. Chasseigne (ESAIM: Control Optimisation and Calculus of Variation) (2014) Mean field games with partially observed major player and stochastic mean field. (2008) Differentiability of Backward Stochastic Differential Equations in Hilbert Spaces with Monotone Generators. (2012) ε-Nash Mean Field Game theory for nonlinear stochastic dynamical systems with mixed agents. According to the strategy proposed in Theorem Estimation and Control of Dynamical Systems, 395-407. (2014) General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients. (2015) Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. 2013. (1996) Existence, uniqueness and space regularity of the adapted solutions of a backward spde. It is also important to note that and , , are independent Bernoulli random variables with success probability and , respectively. (2004) Quadratic Hedging and Mean-Variance Portfolio Selection with Random Parameters in an Incomplete Market. nothing at zero cost; b) detect the number of faulty components at the cost of Nonlinear Analysis: Theory, Methods & Applications 70:4, 1776-1796. (2019) Multi-dimensional optimal trade execution under stochastic resilience. Two numerical examples are presented to demonstrate the results in the cases of fixed and variable rates. . In the Bellman equation, the value function Φ(t) depends on the value function Φ(t+1). (2015) ε-Nash equilibria for a partially observed mean field game with major player. Stochastic H (2008) Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients. 2015. (1991) Adapted solution of a backward semilinear stochastic evolution equation. In the design of control systems for industrial applications, it is important to achieve a certain level of fault tolerance. The classical Hamilton–Jacobi–Bellman (HJB) equation can be regarded as a special case of the above problem. A Mean Field Games and Mean Field Type Control Theory, 67-87. (2005) Strong, mild and weak solutions of backward stochastic evolution equations. Control, 343-360. shows the optimal course of action for the above setting, in different scenarios in terms of the number of faulty processors (based on the most recent observation). Stochastic Control for Non-Markov Processes. For all s ∈ S: s \in \mathcal{S}: s ∈ S: { + \sum_{i,j} {\sigma_{ij}(x,v,t)\partial _{x_i } \Psi _{j,t} (x)} } \right\}dt - \Psi _t (x)dW_t ,\quad \Phi _T (x) = h(x), \hfill \\ \end{gathered}\] where the coefficients $\sigma _{ij}$, $b_i$, L, and the final datum h may be random. Given any realization , , and , , the following equality holds irrespective of strategy , The proof follows from equations ( (2019) A Weak Martingale Approach to Linear-Quadratic McKean–Vlasov Stochastic Control Problems. ∎. Proceedings of IEEE International Midwest Symposium on Circuits and Systems, 2018. Three courses of action are defined to troubleshoot the faulty system: (i) let the system operate with faulty components; (ii) inspect the system, and (iii) repair the system. Classical Solutions to the Master Equation. Their drawback, however, is that the fixed points may not be reachable. Encyclopedia of Systems and Control, 1-6. (2013) Stochastic H 2/H ∞ control with random coefficients. Inspection and repair with fixed price. The number 51 represents the use of 51 discrete values to parameterize the value distribution ZZZ. Each course of action has an implementation cost. (2015) Stochastic minimum-energy control. In this paper, we presented a fault-tolerant scheme for a system consisting of a number of homogeneous components, where each component can fail at any time with a prescribed probability. Introduction. [■] where is the cost of operating with faulty processors. [■] (2011) Backward linear-quadratic stochastic optimal control and nonzero-sum differential game problem with random jumps. Stochastic Control Theory, 209-244. The Fascination of Probability, Statistics and their Applications, 435-446. The first option is to do nothing and let the system continue operating without disruption at no implementation cost. ∎. Consider a stochastic dynamic system consisting of internal components. Optimization in a Random Environment. 16. ), ( [■] (2006) Weak Dirichlet processes with a stochastic control perspective. (2020) A Stochastic Approximation Approach for Foresighted Task Scheduling in Cloud Computing. As a future work, one can investigate the case where there are a sufficiently large number of components using the law of large numbers (2009) A class of backward doubly stochastic differential equations with non-Lipschitz coefficients. (2020) The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations. Bernoulli random variables with success probability . Stochastic Control Theory, 31-78. Stochastic Hamilton–Jacobi–Bellman Equations, Copyright © 1991 Society for Industrial and Applied Mathematics. (2015) A new comparison theorem of multidimensional BSDEs. The Hamilton–Jacobi–Bellman equation (HJB) is a partial differential equation which is central to optimal control theory. co-state = shadow value Bellman can be written as ˆV(x) = max u2U H(x;u;V′(x)) Hence the \Hamilton" in Hamilton-Jacobi-Bellman Can show: playing around with FOC and envelope condition The optimal solution of the approximate model is obtained from the Bellman equation ( I. C51 works like this. (2012) Maximum principle for quasi-linear backward stochastic partial differential equations. We consider the following numerical parameters: Figure Probabilistic Theory of Mean Field Games with Applications II, 447-539. Probability, Uncertainty and Quantitative Risk, Journal of Network and Computer Applications, Journal of Optimization Theory and Applications, Stochastic Processes and their Applications, Journal of Mathematical Analysis and Applications, Journal de Mathématiques Pures et Appliquées, Discrete and Continuous Dynamical Systems, Acta Mathematicae Applicatae Sinica, English Series, Applied Mathematics-A Journal of Chinese Universities, Journal of Systems Science and Complexity, International Journal of Theoretical and Applied Finance, Nonlinear Analysis: Theory, Methods & Applications, Communications on Pure and Applied Mathematics, Journal of Applied Mathematics and Stochastic Analysis, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Random Operators and Stochastic Equations, SIAM J. on Matrix Analysis and Applications, SIAM/ASA J. on Uncertainty Quantification, Journal / E-book / Proceedings TOC Alerts, backward stochastic differential equation, Society for Industrial and Applied Mathematics. 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With Monotone generators 2011 ) backward stochastic differential equations and Infinite time.! Repairing the faulty components, i.e Burgers PDEs with random coefficients -optimal solution the... For controlled backward stochastic partial differential equations and Infinite time horizons on title above here... Control and viscosity solution of a degenerate backward spde the standard Q-function used in Learning... A fault-tolerant control include power systems and aircraft flight control systems for industrial Applications, 435-446: classical viscosity! And refer, respectively, to improve the search results or fix bugs with a Large number of processors. Number of points in the operating mode or faulty s }: s ∈:. 37 Reinforcement Learning course at the School of AI solution to POMDP variance of importance sampling,., \Psi ) ( x, t ) depends on the Existence of optimal Feedback stochastic! First option is to repair the faulty processors at time that are.! 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https://support.10xgenomics.com/single-cell-gene-expression/software/pipelines/4.0/algorithms/targeted
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HOME › pipelines
# Targeted Gene Expression Algorithms Overview
## Cell Calling for Targeted Gene Expression
Cell Ranger 4.0 includes a new dedicated cell calling algorithm that is applied specifically to identify cells from Targeted Gene Expression datasets. The Targeted Gene Expression cell calling method relies on identification of a transition point based on the shape of the barcode rank plot to separate cell-associated barcodes from background partitions, and is designed to successfully classify cells for a wide variety of potential target gene panels.
Note: This method is distinct from the cell calling algorithm used for non-targeted Whole Transcriptome Analysis (WTA) Chromium single cell datasets, and it does not employ the EmptyDrops approach (Lun et al., 2018) due to the distinct features of Targeted Gene Expression datasets.
The algorithm has three key steps:
1. It identifies an initial set of cells based on the total UMI counts (from both targeted and non-targeted genes) per barcode. This step identifies the primary mode of high RNA content cells, which is directly analogous to the first step in the cell calling process for WTA (non-targeted) gene expression datasets.
2. Then, the algorithm identifies the transition point associated with the minimum first derivative value of the log-transformed barcode rank plot, and calls any additional barcodes with total UMI counts exceeding the transition point as cells. This second step captures additional cells whose barcode ranks precede the steepest ‘cliff’ separating cell-associated barcodes from background partitions.
3. Finally, any barcodes called as cells after the first two steps whose total targeted UMI counts are zero (such that all of their total UMI counts derive from non-targeted genes) are excluded from being classified as cells. This last step ensures that all identified cells contain some positive targeted UMI counts that will be useful for subsequent analysis steps.
In the first step, an approach analogous to the original Cell Ranger cell calling algorithm is used to identify the primary mode of high RNA content cells, using a cutoff based on the total UMI count (from both targeted and non-targeted genes, but excluding any filtered UMIs) for each barcode. Cell Ranger takes as input the expected number of recovered cells, N (see --expect-cells). Let m be the 99th percentile of the top N barcodes by total UMI counts. All barcodes with total UMI counts m/10 or greater are called as cells in the first pass.
In the second step, the algorithm fits a smoothed cubic spline curve to the log-transformed barcode rank plot (defined as the curve whose y-axis is the log of the UMI counts per barcode, and whose x-axis is the log of the number of barcodes with UMI counts greater than or equal to the associated y-value). Next, the algorithm looks for a potential transition point where the gradient (or first derivative value) of this curve has its minimum value, within a defined range of allowed barcode ranks considered for additional cell calling (beyond the number of cells called in the first step). Then, any additional barcodes whose total UMI counts exceed the cutoff value associated with the transition point are added to the set of positive cell calls. This second step retains additional cells from the region of the barcode rank plot preceding the steep ‘cliff’ at the transition point separating cell-associated barcodes from background partitions.
In the third step, any barcodes called as cells whose total targeted UMI counts are zero (such that all of their total UMI counts derive from non-targeted genes) are removed from the final set of positive cell calls. This last step ensures that all identified cells contain positive targeted UMI counts that will be useful for subsequent analysis steps.
## Targeted UMI Filtering
For Targeted Gene Expression data, targeted UMI filtering is performed after the usual UMI Counting step. This additional filtering is only active for sequencing libraries with very high depth, in which spurious molecules can be observed in a very small fraction of reads. Targeted UMI filtering can be disabled if desired using --no-target-umi-filter, although this is not recommended.
The targeted UMI filtering threshold is computed based on the distribution of number of supporting reads per UMI (RPU) over UMIs counted towards targeted genes. The threshold is equal to the 90th percentile RPU value, multiplied by 0.01 and rounded up. For practical reasons, this threshold is computed using a random subset of valid barcodes comprising at least 10% of all barcodes or 1 million reads (whichever is greater). Then, after aligning and annotating the full set of reads, each targeted UMI with read count strictly lower than the targeted UMI threshold is removed. The UMIs removed by this filter will not appear in any Cell Ranger outputs except in the BAM file, where the associated reads are indicated using the xf tag.
Note that when the 90th percentile RPU is lower than 100 reads, the filtering threshold is computed as 1, meaning that no UMIs are filtered. In this case, the targeted UMI filtering threshold appears as N/A in the Web Summary.
## Design of Bait Sequences for Targeted Gene Expression
### Bait Design Overview
In order to specifically recover molecules from targeted genes of interest in gene expression libraries, a set of bait oligonucleotides are designed with sequence complementarity for each gene in a targeted panel. Baits designed for the Targeted Gene Expression assay are 120 base pair (bp) biotinylated oligonucleotides that are used within a hybridization-capture workflow.
Targeted Gene Expression baits were designed with the following broad design principles:
• Compatibility: Targeted baits can be applied to a variety of gene expression technologies, including both 3’ and 5’ Chromium single cell RNA-seq libraries as well as Visium Spatial Gene Expression libraries (coming soon).
• Comprehensiveness: Baits are designed to cover the full set of annotated transcripts for each gene in the target panel, without designing redundant baits across overlapping transcript sequences. This strategy also confers added robustness by enabling capture of molecules derived from unexpected or poorly-annotated isoforms of target genes.
• Specificity: Baits are chosen to avoid unwanted complementarity to repetitive elements or other problematic genomic sequences that may compromise assay performance.
### Which Transcripts are Targeted?
Baits are tiled across the full length of all transcripts annotated in the GRCh38-2020-A reference. The transcript annotations in this reference are based on the GENCODE comprehensive set. See release notes for documentation on this new reference.
### Tiling Approach
For each transcript within a gene, bait sequences are designed starting at the 5’ and 3’ ends and progresses inwards until the center of the transcript is reached. Baits are typically 120bp in length (see the Optimizing Specificity section for more details on exceptions) and are aimed to tile at 1x coverage. If the transcript length is not a multiple of 120bp, a small coverage gap appears in the center of the transcript (away from the annotated 3’ or 5’ ends). In order to prevent the unnecessary selection of redundant baits from regions of identical sequence shared across transcripts from the same gene, a De Bruijn graph data structure is used. By querying the De Bruijn graph constructed using all 120bp subsequences derived from each transcript, the algorithm ensures that any potential new bait sequence is rejected if it would be redundant with an existing previously-designed bait from the same target gene.
A subtle consequence of this procedure is that the ordering of transcripts can, to some extent, impact the exact set of baits designed for a given gene. We utilize the rankings provided by the APPRIS database (Annotating principal splice isoforms) to order transcripts such that higher confidence transcript annotations are tiled first. Other measures are taken to make sure ordering is deterministic when ties occur in categorizations from the APPRIS.
### Introns
Introns are not included in our tiling of annotated transcripts, only UTRs and coding regions. As a consequence, intronic reads from targeted genes will be mostly depleted in Targeted Gene Expression libraries.
### Optimizing Specificity
There are many repetitive elements in the transcriptome that, if overlapped by baits, could result in reduced enrichment in Targeted Gene Expression libraries, particularly within large target gene panels. Before designing baits, K-mers with many near-identical occurrences in the genome are identified so that they can be avoided during bait design.
If a bait intended to be designed at a given position contains one of the K-mers above, a combination of shifting the bait position up or downstream and removing portions of the bait sequence is used to attempt to retain a functional bait that will not overlap the problematic sequence. If this is not possible, the algorithm will not design a bait at this location.
Approximately 2-3% of baits within the pre-designed panels are shorter than 120bp because of the procedure above.
### Bait Designs for Custom Sequences
Via the 10x Genomics Custom Panel Designer users can also design baits for entirely custom exogenous sequences. These sequences follow the same design procedure as endogenous genes, including the procedure described in the Optimizing Specificity section. Each custom sequence is treated independently. No check for redundancy is performed between submitted sequences, as would normally occur for transcripts within the same gene. Similarly, no check for further homology between submitted sequences and endogenous sequences is performed.
### Identifying Genes with Low Mappability or Low Bait Coverage
In order to assign a UMI to a particular gene, Cell Ranger has to be able to confidently identify which gene a given read came from. A small number of genes share a substantial proportion of their sequence content with other genes in the genome, which in extreme cases makes it difficult to assign the corresponding UMI count to a single gene. These genes may have reduced counts in the resulting expression matrices.
The bait design algorithm intentionally avoids designing baits that overlap repetitive sequences as described in the Optimizing Specificity section above. A very small number of genes have low coverage of baits around annotated transcript ends due to the presence of repetitive sequences. These genes are flagged with poor coverage within the last 360bp of the 3’ or 5’ ends of annotated transcripts, where the majority of signal for most genes is expected to be localized in 3'/5' gene expression libraries. While the baits that are provided for these genes may work well, enrichment of these genes may be less robust depending on the particular sample and location of library molecules within annotated transcripts.
Genes with very low mappability or low bait coverage at annotated transcript ends are noted using the mappability_flag column in the gene metadata file provided for predesigned and customized panels. These genes will also trigger a warning if added to customized panels via the 10x Genomics Custom Panel Designer.
### Reviewing Genes with Mappability Warnings
If a gene you want to add to your panel has triggered a mappability warning (or equivalently has the value TRUE in the mappability_flag column of the gene metadata file), the following steps are recommended:
1. Check if this gene has near-zero counts in existing whole transcriptome datasets. If so, this warning may indicate the gene is very poorly mappable and Cell Ranger is not able to assign many reads to this gene. Targeting this gene may not yield the desired results.
2. If you have existing whole transcriptome data, you can use IGV or another genome browser to view coverage within this gene alongside the bait locations (as defined in the panel's BED12 file). If the baits appear to cover the regions where reads have aligned, then enrichment is likely to work well. If baits do not overlap the main regions of coverage, enrichment may not work well for this gene and you may want to consider whether to include this gene on your panel. See our documentation on BED12 files for more information.
## Modeling Targeted Gene Enrichments (count)
In a Targeted Gene Expression sample, it is assumed that there are two classes of genes: genes from the target panel that are enriched by targeting, and non-targeted genes that constitute the background. Since the number of reads per gene in the sample before targeting is not known, standard approaches to directly calculate read enrichments are not available. Instead, the mean number of reads per UMI (or Mean Reads per UMI) for each gene is calculated, which serves as a proxy for read enrichment. The mean number of reads per UMI for a gene is closely related to the sequencing saturation for that gene (mean reads per UMI = 1/(1 - sequencing saturation), also see GEX Metrics). Given there are a finite number of UMIs per gene in a given sample, enrichment will tend to recover more UMIs from those genes as well as more PCR duplicates arising from those UMIs. Therefore enriched genes are likely to have greater sequencing saturation and greater mean reads per UMI.
In order to assess whether targeted genes were enriched in a given sample, the mean reads per UMI values are first calculated for all genes (both targeted and non-targeted genes). Only UMIs in cell-associated barcodes are used and only genes with at least 10 UMIs in cell barcodes are considered for this analysis. Cell Ranger then fits a two-component Gaussian Mixture Model to the mean reads per UMI distribution, grouping genes into one of two classes: enriched genes (those with higher mean reads per UMI) and non-enriched genes (those with lower mean reads per UMI). The numbers of targeted and non-targeted genes that are considered enriched by this method are shown in the Targeted Gene Expression Run Summary Targeted Enrichment dashboard.
Cell Ranger may sometimes be unable to confidently assign genes as enriched or non-enriched using this method, particularly when there are too few targeted and/or non-targeted genes meeting the criteria above, or if sequencing saturation is too low. In these rare cases, enrichment calculations may be disabled, and the metrics reporting the number of genes enriched will be N/A.
## Modeling Targeted Gene Enrichments (Targeted-Compare)
The computation of gene enrichments in targeted-compare is analogous to that described above, with one modification. When running targeted-compare, the parent sample read counts are known, allowing for direct calculation of read enrichments. Cell Ranger therefore uses the same Gaussian Mixture Model described above, substituting Mean Reads per UMI with Read Enrichments. Read Enrichments per Gene are calculated as (Number of reads in the Targeted Sample)/(Number of reads in the Parent Sample), after rescaling both samples to the same number of read pairs from Gene Expression libraries.
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https://www.hpmuseum.org/forum/printthread.php?tid=9887
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Triangular number AND sum of first m factorials - Printable Version +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: Not HP Calculators (/forum-7.html) +--- Forum: Not remotely HP Calculators (/forum-9.html) +--- Thread: Triangular number AND sum of first m factorials (/thread-9887.html) Triangular number AND sum of first m factorials - Joe Horn - 01-09-2018 04:31 PM 153 (my favorite number) is both a triangular number (the sum of the integers 1 through $$n$$; in this case $$n=17$$) as well as the sum of the factorials $$1!$$ through $$m!$$ (in this case $$m=5$$). The first three natural numbers which have both of those properties are 1, 3 (both trivial) and 153. Find the next number in this sequence. For extra credit, find the mathematical relationship between $$n$$ and $$m$$ for all members of this sequence (which apparently is not yet in OEIS). RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-09-2018 08:53 PM 001 LBL A 002 DEC X 003 ENTER 004 INC X 005 LBL 00 006 INC X 007 RCL* Y 008 DSE Y 009 GTO 00 010 PSE 10 011 STO+ X 012 +/- 013 1 014 ENTER 015 ENTER 016 R/\ 017 SLVQ 018 X<>Y 019 FP 020 PSE 10 021 RCL L 022 RTN 2 A -> 3 -> 0 -> 2 5 A -> 153 -> 0 -> 17 The next m should be greater than 31, but I can’t find it using the wp34s, at least not with help of this simple program… Edited to fix a typo pointed out by Dieter below. RE: Triangular number AND sum of first m factorials - Valentin Albillo - 01-09-2018 10:16 PM (01-09-2018 04:31 PM)Joe Horn Wrote: 153 (my favorite number) is both a triangular number (the sum of the integers 1 through $$n$$; in this case $$n=17$$) as well as the sum of the factorials $$1!$$ through $$m!$$ (in this case $$m=5$$) It's also a narcissistic number: 153 = 1^3 + 5^3 + 3^3 V. . RE: Triangular number AND sum of first m factorials - Dieter - 01-09-2018 10:19 PM (01-09-2018 08:53 PM)Gerson W. Barbosa Wrote: 001 LBL A 002 DEC X 003 ENTER 004 INC C ?!? – increment register C ? Is this possibly supposed to mean INC X ? (01-09-2018 08:53 PM)Gerson W. Barbosa Wrote: 005 LBL 00 006 INX X Ah, here is the missing X from line 004, and the "C" from there goes here. ;-) Dieter RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-09-2018 11:00 PM (01-09-2018 10:19 PM)Dieter Wrote: Ah, here is the missing X from line 004, and the "C" from there goes here. ;-) :-) Duly swapped. Thanks! Gerson. RE: Triangular number AND sum of first m factorials - John Keith - 01-09-2018 11:10 PM A quick test of sums of factorials up to 69! finds no triangle numbers larger than 153. If one does exist, it has to have well over 100 digits. I may try later on the emulator. John RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-10-2018 04:03 AM Tn&Sf: « { } 3 ROT FOR n n Sfac 8 * 1 + ISPF? { √ 1 - 2 / + n I→R + } { DROP } IFTE NEXT » Sfac: « DUP 1 - 2 FOR m 1 + m * -1 STEP 1 + » ISPF? « 1 » 7 Tn&Sf -> { '(√73-1)/2' 3. '(√265-1)/2' 4. 17 5. '(√6985-1)/2' 6. '(√47305-1)/2' 7. } 49G or 50g in exact mode ISPF? (IsPerfectSquare?) has yet to be implemented. It should return 1 when the argument is a perfect square and 0 otherwise. Then the output would be a list of n and m pairs, separated by dots. Just in case someone wants to try. RE: Triangular number AND sum of first m factorials - Joe Horn - 01-10-2018 04:58 AM (01-10-2018 04:03 AM)Gerson W. Barbosa Wrote: ISPF? (IsPerfectSquare?) has yet to be implemented. It should return 1 when the argument is a perfect square and 0 otherwise. The HP 50g LongFloat library contains a function like the one you're looking for. It's called ZSqrt, and it returns IP(sqrt(x)) to level 2, and a 0 or 1 to level 1, with 1 meaning that x was a perfect square. It works on integers of any length. Gerald H also posted a program HERE which seems to do essentially the same thing. It returns IP(sqrt(x)) to level 2 of the stack, and on level 1 it leaves a SysRPL TRUE if x was a perfect square, otherwise a FALSE. That's the same idea as HP's FPTR2 ^ZSQRT, but Gerald's program is more accurate; see Gerald's posting for evidence of HP's function's inaccuracy. BTW, LongFloat's ZSqrt function gets the same result as Gerald's program when given the example in Gerald's posting, so I surmise that ZSqrt is trustworthy. RE: Triangular number AND sum of first m factorials - Paul Dale - 01-10-2018 06:35 AM I've got a proof that there are only three such numbers. Consider the last pair of digits in $$\sum_1^n i!$$, from n=9 onwards these never change because subsequent factorial terms will always have a factor of 100 present. These digits are '13'. Note that n is triangular iff 8n+1 is a perfect square. For the sum of factorials to be triangular, the last two digits must therefore be '05'. Checking all possibilities shows that there are no square numbers that end '05'. Thus, numbers of the desired form must have n < 9. Checking all cases reveals that only 1, 3 and 153 have the desired properties. Pauli RE: Triangular number AND sum of first m factorials - Joe Horn - 01-11-2018 03:01 AM Paul: Now THAT is beautiful! Thank you! I can stop my futile hunt now. RE: Triangular number AND sum of first m factorials - Paul Dale - 01-11-2018 10:21 AM Thanks Don't let my proof stop your hunt, you'll be able to wile away many hours looking... I hadn't realised that all square numbers that end in '5' actually end in '25'. I must have seen this before but never noticed or remembered it. Pauli RE: Triangular number AND sum of first m factorials - John Cadick - 01-11-2018 02:22 PM A really great thread Joe et. al. Thank you all. John RE: Triangular number AND sum of first m factorials - Gerson W. Barbosa - 01-11-2018 06:29 PM (01-11-2018 10:21 AM)Paul Dale Wrote: Don't let my proof stop your hunt, you'll be able to wile away many hours looking... At least I can do it a little more efficiently now :-) 100 « { } SWAP 0 1 ROT 1 SWAP FOR m m * SWAP OVER + ROT OVER 8 * 1 + ZSqrt { 1 - 2 / + m I→R + } { DROP } IFTE SWAP ROT NEXT DROP2 » EVAL --> { 1 1. 2 2. 17 5. } (about 17 seconds on the real 50g) ZSqrt from the LongFloat Library Yes, that's a consequence of the ever growing number of trailing zeros in factorials and the properties of perfect squares. Gerson. RE: Triangular number AND sum of first m factorials - John Keith - 01-11-2018 10:30 PM (01-11-2018 10:21 AM)Paul Dale Wrote: Thanks Don't let my proof stop your hunt, you'll be able to wile away many hours looking... I hadn't realised that all square numbers that end in '5' actually end in '25'. I must have seen this before but never noticed or remembered it. Pauli I figured it was hopeless since I tested all sums of factorials up to 1000 (over 2500 digits) with no triangle numbers found. Took almost 20 min. on the emulator. An interesting and educational thread indeed! John RE: Triangular number AND sum of first m factorials - John Keith - 01-11-2018 10:43 PM (01-11-2018 06:29 PM)Gerson W. Barbosa Wrote: Yes, that's a consequence of the ever growing number of trailing zeros in factorials and the properties of perfect squares. Gerson. Note, however, that due to the first four numbers, all of the sums of factorials end in 3.
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https://www.physicsforums.com/threads/emf-induced-in-rotating-rod-inside-uniform-magnetic-field.964615/
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# EMF induced in rotating rod inside uniform magnetic field
• Start date
#### songoku
1,177
23
1. Homework Statement
A 40 cm rod is rotated about its centre inside a region of uniform magnetic field of 6.4 T. Given that the speed of rotation is 15 rad/s, find potential difference between the centre and either end of the rod
2. Homework Equations
emf = - ΔΦ / Δt
ω = 2π / T
3. The Attempt at a Solution
emf = - B cos θ . ΔA / Δt = - B . πr2 / T
I just need to plug the numbers with r = 20 cm (because from center to either end of rod)?
Thanks
Related Introductory Physics Homework Help News on Phys.org
#### Delta2
Homework Helper
Gold Member
2,384
668
In one full period T, the radius of the rod which is $r=20cm$ (since $d=40cm$ is the diameter) covers a surface of a full circle which is $\pi r^2$. You can use the diameter but then you ll have to take the formula $\pi\frac{d^2}{4}$ for the surface of the circle.
Last edited:
#### songoku
1,177
23
In one full period T, the radius of the rod which is $r=20cm$ (since $d=40cm$ is the diameter) covers a surface of a full circle which is $\pi r^2$. You can use the diameter but then you ll have to take the formula $\pi\frac{d^2}{4}$ for the surface of the circle.
For the time, do I use the period because half of the rod travels full circle in one full period or I use half of period because one whole rod covers one full circle in half period?
Thanks
#### Delta2
Homework Helper
Gold Member
2,384
668
For the time, do I use the period because half of the rod travels full circle in one full period or I use half of period because one whole rod covers one full circle in half period?
Thanks
You use the full period for half rod, otherwise if you follow the 2nd approach you find the EMF between the two ends of the rod. But the problem asks for the EMF between one end and the center, that's why we have to take the area that the half rod covers in one full period.
#### songoku
1,177
23
You use the full period for half rod, otherwise if you follow the 2nd approach you find the EMF between the two ends of the rod. But the problem asks for the EMF between one end and the center, that's why we have to take the area that the half rod covers in one full period.
If the question asks the emf between two ends of the rod, will the answer be zero because they have the same value and the difference = 0?
Thanks
#### Delta2
Homework Helper
Gold Member
2,384
668
If the question asks the emf between two ends of the rod, will the answer be zero because they have the same value and the difference = 0?
Thanks
Yes, the emf between the two ends is zero.
#### songoku
1,177
23
Thank you very much
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https://repository.uantwerpen.be/link/irua/110132
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Title Measurement of the $t\bar{t}$ production cross section in the dilepton channel in pp collisions $\sqrt{s}=7TeV$Measurement of the $t\bar{t}$ production cross section in the dilepton channel in pp collisions $\sqrt{s}=7TeV$ Author Chatrchyan, S. Khachatryan, V. Sirunyan, A. M. Bansal, S. Cornelis, T. de Wolf, E.A. Janssen, X. Luyckx, S. Mucibello, L. Roland, B. Rougny, R. Selvaggi, M. van Haevermaet, H. Van Mechelen, P. Van Remortel, N. Van Spilbeeck, A. et al. Faculty/Department Faculty of Sciences. Physics Research group Elementary Particle Physics Publication type article Publication 2012Bristol, 2012 Subject Physics Source (journal) Journal of high energy physics. - Bristol Volume/pages (2012):11, p. 1-40 ISSN 1126-6708 1029-8479 Article Reference 067 Carrier E-only publicatie Target language English (eng) Full text (Publishers DOI) Affiliation University of Antwerp Abstract The t (t) over bar production cross section (sigma(t (t) over bar)) is measured in proton-proton collisions at root s = 7 TeV in data collected by the CMS experiment, corresponding to an integrated luminosity of 2.3 fb(-1). The measurement is performed in events with two leptons (electrons or muons) in the final state, at least two jets identified as jets originating from b quarks, and the presence of an imbalance in transverse momentum. The measured value of sigma(t (t) over bar) for a top-quark mass of 172.5 GeV is 161.9 +/- 2.5 (stat.)(-5.0)(+5.1) (syst.) +/- 3.6 (lumi.) pb, consistent with the prediction of the standard model. Full text (open access) https://repository.uantwerpen.be/docman/irua/ff8dcf/5075.pdf E-info http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000312198500002&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000312198500002&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000312198500002&DestLinkType=CitingArticles&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 Handle
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http://physics.stackexchange.com/users/12412/aclassicalcaseofconfusion?tab=activity
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# AClassicalCaseOfConfusion
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bio website location age member for 1 year, 11 months seen Jul 22 '13 at 7:51 profile views 8
# 14 Actions
Oct23 answered Find total energy and momentum of an moving electron in a rest frame Oct23 revised Can I find a potential function in the usual way if the central field contains $t$ in its magnitude? According to comment, the equation should've been for the force F not U. Oct23 suggested suggested edit on Can I find a potential function in the usual way if the central field contains $t$ in its magnitude? Oct22 comment How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$ Look, I'm not sure this is going to be helpful for anyone to read. Neither of us disagree about the definitions of partial derivatives, nor about the solution to the problem. I only posted as I felt that your first post may have said something unclear. The edit you made to your answer is sufficient to point out the assumption in your first proof, and to avoid any misinterpretation. I'd like to leave it at that. Oct22 comment How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$ It's clear that you know how to do this problem; I wasn't trying to put that in doubt. However, I disagree with the statement I quoted, and again with "treating it as ordinary is fine as there is no explicit reason not to". The explicit reason is that it is not the same operator. Making incorrect statements only leaves the possibility of confusion for those less familiar with this kind of material. Oct18 awarded Teacher Oct18 answered How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$ Oct15 comment Does air resistance ever slow a particle down to zero velocity? No problem. It's not just $\alpha<0$ though! For example, try $\alpha=\frac{1}{2}$. Oct14 awarded Editor Oct14 revised Does air resistance ever slow a particle down to zero velocity? Comment on units. Oct14 answered Does air resistance ever slow a particle down to zero velocity? Sep21 awarded Student Sep21 comment Classical scalar field correlation function Maybe I'm being foolish, but surely your answer only works in a quantum field theory. I was under the impression that Strassler was talking about an analogous quantity in classical field theory. Sep21 asked Classical scalar field correlation function
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https://runestone.academy/ns/books/published/javajavajava/finding-things-withina-string.html
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## Section7.3Finding Things Within a String
Programmers often have to find the location of a particular character or substring in a string. For example, user names and passwords are sometimes stored in a single string in which the name and password are separated from each other by a special character, such as a colon (username:password). In order to get the name or password from such a string, it is convenient to have methods that will search the string and report the index of the colon character.
### Subsection7.3.1indexOf()
The indexOf() and lastIndexOf() methods are instance methods that can be used to find the index position of a character or a substring within a String. There are several versions of each:
public int indexOf(int character);
public int indexOf(int character, int startingIndex);
public int indexOf(String string);
public int indexOf(String string, int startingIndex);
public int lastIndexOf(int character);
public int lastIndexOf(int character, int startingIndex);
public int lastIndexOf(String string);
public int lastIndexOf(String string, int startingIndex);
The indexOf() method searches from left to right within a String for either a character or a substring. The lastIndexOf() method searches from right to left for a character or substring. To illustrate, suppose we have declared the following strings:
String string1 = "";
String string2 = "Hello";
String string3 = "World";
String string4 = string2 + " " + string3;
Recalling that strings are indexed starting at 0, searching for o in the various strings gives the following results:
string1.indexOf('o') ==> -1 string1.lastIndexOf('o') ==> -1
string2.indexOf('o') ==> 4 string2.lastIndexOf('o') ==> 4
string3.indexOf('o') ==> 1 string3.lastIndexOf('o') ==> 1
string4.indexOf('o') ==> 4 string4.lastIndexOf('o') ==> 7
Because string1 is the empty string, “”, it does not contain the letter o. Therefore, indexOf() returns $$-1\text{,}$$ a value that cannot be a valid index for a String. This convention is followed in indexOf() and lastIndexOf().
Because string2 and string3 each contain only one occurrence of the letter o, both indexOf() and lastIndexOf() return the same value when used on these strings. Because string4 contains two occurrences of o, indexOf() and lastIndexOf() return different values in this case. As Figure 7.3.1 shows, the first o in “Hello World” occurs at index 4, the value returned by indexOf(). The second o occurs at index 7, which is the value returned by lastIndexOf().
By default, the single-parameter versions of indexOf() and lastIndexOf() start their searches at their respective (left or right) ends of the string. The two-parameter versions of these methods allow you to specify both the direction and starting point of the search. The second parameter specifies the starting index. Consider these examples:
string4.indexOf('o', 5) ==> 7
string4.lastIndexOf('o', 5) ==> 4
If we start searching in both cases at index 5, then indexOf() will miss the o that occurs at index 4. The first o it finds will be the one at index 7. Similarly, lastIndexOf() will miss the o that occurs at index 7 and will find the o that occurs at index 4.
The indexOf() and lastIndexOf() methods can also be used to find substrings:
string1.indexOf("or") ==> -1 string1.lastIndexOf("or") ==> -1
string2.indexOf("or") ==> -1 string2.lastIndexOf("or") ==> -1
string3.indexOf("or") ==> 1 string3.lastIndexOf("or") ==> 1
string4.indexOf("or") ==> 7 string4.lastIndexOf("or") ==> 7
The substring “or” does not occur in either string1 or string2. It does occur beginning at location 1 in string3 and beginning at location 7 in string4. For this collection of examples, it doesn't matter whether we search from left to right or right to left.
#### ExercisesSelf-Study Exercises
##### 1.String indexOf() expressions.
Suppose the String variable s has been initialized to “mom.” Evaluate each of the following expressions:
1. s.indexOf("m");
2. s.indexOf("o");
3. s.indexOf("M");
Hint.
The indecOf() method returns the index of its argument's location in the string.
##### 2.String indexOf() expressions.
Suppose the String variable s has been initialized to: String s1 = "Java, Java, Java"; Evaluate each of the following expressions:
1. s1.length()
2. String.valueOf(s1.length())
3. s1.indexOf('a')
4. s1.lastIndexOf('a')
##### 3.String indexOf() expressions.
Suppose the String variable s has been initialized to: String s1 = "Java, Java, Java"; Evaluate each of the following expressions:
1. s1.indexOf("av")
2. s1.lastIndexOf("av")
3. s1.indexOf('a', 5)
4. s1.lastIndexOf('a', 5)
##### 4.String indexOf() expressions.
Suppose the String variable s has been initialized to: String s1 = "Java, Java, Java"; Evaluate each of the following expressions:
1. s1.indexOf("av", s1.length() - 10)
2. s1.lastIndexOf("av", s1.length() - 4)
3. s1.indexOf("a", s1.indexOf("va"))
##### 5.Tricky indexOf() expression.
Evaluate the following expression:
String tricky = "abcdefg01234567";
tricky.indexOf(String.valueOf( tricky.indexOf("c") ));
Hint.
Evaluate the expressions starting with the innermost parentheses.
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http://math.stackexchange.com/questions/213316/shock-waves-characteristics
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shock waves characteristics
I'm trying to solve $u_t + u^2u_x = 0$ with $u(x, 0) = 2 + x$.
I'm thinking to proceed by characteristics where we have above that $\frac{dx}{dt} = 1$ and $dy/dt = u^2$, but not sure if this will help. This is from shock waves idea.
Here's what I have
ut + u^2ux = 0
q +u^2 p =0
u^2 p +q =0
dx/ u^2 = dy/1
and u^-1/-1 + C = y
then u(x,t)^-1 + C = y
is this correct?
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Please learn to $\LaTeX$ format your quesitons. There are tutorials on the web and you can see this. Presumably ut is $u_t=\frac {\partial u}{\partial t}$ but I am not sure how to parse u2ux. Is it $\frac {\partial^2 u}{\partial x^2}?$ – Ross Millikan Oct 14 '12 at 0:19
Given that the method of characteristics gives $\frac{dy}{dt} = u^2$, I think she means $u^2u_x$. – Michael Albanese Oct 14 '12 at 0:24
@Michael, that is correct – mary Oct 14 '12 at 1:02
As timur comments, the characteristic equations are wrong. They should be stated against a parameter, not the involved variables. You are mistaking $t$ with $y$. The construction of the characteristics is based on the supposition that if $x = x(\eta)$ and $t = t(\eta)$, then $$\frac{d}{d\eta}u\big(x(\eta),t(\eta)\big) = u_x x'(\eta) + u_t t'(\eta) = u^2 u_x + u_t = 0$$ and then one says $x'(\eta) = u^2$, $t'(\eta) = 1$, $u'(\eta) = 0$. See my answer for a full analysis. – Pragabhava Oct 16 '12 at 8:31
Also, if I understand your work, the method you are using is for solving fully nonlinear first order PDE's, and you are using it wrong. Your problem is quasilinear, and there is no need to introduce $p$ and $q$. This are only introduced in the case the derivatives of $u$ are involved nonlinearly in the equation. I strongly suggest you to study the first chapter of John's Partial Differential Equations, as I believe you are very confused. Any doubts, we can try to help. – Pragabhava Oct 16 '12 at 8:40
The quasilinear first order PDE $$a\big(x,y,u(x,y)\big) u_x(x,y) + b\big(x,y,u(x,y)\big)u_y(x,y) = c\big(x,y,u(x,y)\big)$$ where $a,\,b,\,c \in C^1$ with data $\mathcal{C}(\xi) = \big(x(\xi), y(\xi), u(\xi)\big) \in C^1$ and with $$\begin{vmatrix} \frac{dx}{d\xi} & a \\ \frac{dy}{d\xi} & b\end{vmatrix} \neq 0$$ has a unique solution near $\mathcal{C}$ given by \begin{align} \frac{d x}{d \eta} &= a & x\big|_{\eta = 0}&= x(\xi)\\ \frac{d y}{d \eta} &= b & y\big|_{\eta = 0}&= y(\xi)\\ \frac{d u}{d \eta} &= c & u\big|_{\eta = 0}&= u(\xi)\\ \end{align}
For proof and geometrical interpretation, see F. John's Partial Differential Equations1.4)
In your case, $\mathcal{C}(\xi) = \big(\xi,0,\xi+2\big)$. Near $\eta \sim 0$ $$\begin{vmatrix} \frac{dx}{d\xi} & a \\ \frac{dy}{d\xi} & b\end{vmatrix} = \begin{vmatrix} 1 & u^2 \\ 0 & 1\end{vmatrix} = 1$$ and the solution is unique.
The system of ODE's is \begin{align} \frac{d x}{d \eta} &= u^2 & x\big|_{\eta = 0}&= \xi\\ \frac{d t}{d \eta} &= 1 & t\big|_{\eta = 0}&= 0\\ \frac{d u}{d \eta} &= 0 & u\big|_{\eta = 0}&= \xi + 2\\ \end{align} with solution $$t = \eta, \quad u = \xi + 2, \quad x = (\xi + 2)^2 \eta + \xi.$$
The characteristics are $t = \frac{x - \xi}{(\xi + 2)^2}$ hence $\xi = -2$ is a special point. As $\xi \rightarrow \infty$, $t \rightarrow 0$. As $\xi \rightarrow -\infty$, $t \rightarrow 0$. As $\xi \rightarrow -2$, $t \rightarrow \infty$.
This of course, means that there is no solution when the characteristics meet.
A simple explanation for this is that the transformation $$(x,t) \rightarrow (\xi,\eta)$$ is invertible iff $$\begin{vmatrix} \partial_\xi x & \partial_\eta x \\ \partial_\xi t & \partial_\eta t \end{vmatrix} = 1 + 4\eta + 2\xi \eta \neq 0$$ meaning there is no solution when $\xi = -\frac{1 + 4 \eta}{2\eta}$ or, inverting the transformation, when $$t = - \frac{1}{4(x+2)}$$
$\hskip.75in$
Lastly, inverting for $\xi$
$$\xi = \frac{-(1 + 4t) \pm \sqrt{1 + 4t(2 + x)}}{2 t}$$
and
$$u(x,t) = \frac{-1 \pm \sqrt{1 + 4t(2 + x)}}{2 t}.$$
In order to determine the correct sign, we must look at the initial condition. For the minus sign $\lim_{t \rightarrow 0} u(x,t) = -\infty$, while the plus sign gives the correct answer. Hence
$$u(x,t) = \frac{-1 + \sqrt{1 + 4t(2 + x)}}{2 t}.$$
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Thank you so much for your explanation. – mary Oct 16 '12 at 9:55
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https://mathoverflow.net/questions/352752/quotient-banach-space-whose-dual-map-sends-the-ball-onto-a-given-convex-subset
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# Quotient Banach space whose dual map sends the ball onto a given convex subset
Let $$X$$ be a Banach space and let $$A$$ be a closed, convex and balanced subset of $$B_{X^{*}}$$ (where $$B_{X^{*}}$$ denotes the closed unit ball of the dual $$X^{*}$$). Is there a closed subspace $$M$$ of $$X$$ such that $$Q^{*}_{M}$$ maps $$B_{(X/M)^{*}}$$ onto $$A$$, where $$Q_{M}:X\rightarrow X/M$$ is the quotient map?
• What happens when $X=\mathbb{R}^n$? Suppose $A$ is any closed convex balanced set with nonempty interior, other than the ball itself. If $M \ne 0$ then $Q_M^*$ has rank less than $n$ and so its image cannot cover $A$, and if $M=0$ then $Q_M^*$ is the identity map and it maps the ball to itself. – Nate Eldredge Feb 15 at 13:33
• Thanks, Nate. What happens if $X$ is infinite-dimensional? – Dongyang Chen Feb 15 at 14:36
• I was just thinking about that. More generally, the image of $Q_M^*$ will always equal the annihilator of $M$, right? If $M \ne 0$ this is a proper closed subspace. So take any $A$ which is not contained in a proper closed subspace (e.g. any $A$ with nonempty interior) and is not the ball, and I think that is a counterexample. – Nate Eldredge Feb 15 at 14:39
• Indeed, $Q^{*}_{M}B_{(X/M)^{*}}$ is equal to the closed unit ball of the annilator of $M$. If we take $A$ with nonempty interior, then $M=0$. – Dongyang Chen Feb 15 at 14:57
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https://link.springer.com/chapter/10.1007/978-3-319-18461-6_3
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The Morphological Equivalents of Relativistic and Alpha-Scale-Spaces
Conference paper
DOI: 10.1007/978-3-319-18461-6_3
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)
Cite this paper as:
Schmidt M., Weickert J. (2015) The Morphological Equivalents of Relativistic and Alpha-Scale-Spaces. In: Aujol JF., Nikolova M., Papadakis N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science, vol 9087. Springer, Cham
Abstract
The relations between linear system theory and mathematical morphology are mainly understood on a pure convolution / dilation level. A formal connection on the level of differential or pseudo-differential equations is still missing. In our paper we close this gap. We establish the sought relation by means of infinitesimal generators, exploring essential properties of the slope and a modified Cramér transform. As an application of our general theory, we derive the morphological counterparts of relativistic scale-spaces and of $$\alpha$$-scale-spaces for $$\alpha \in [\frac{1}{2}, \infty )$$. Our findings are illustrated by experiments.
Keywords
Mathematical morphology Alpha-scale-spaces Relativistic scale-spaces Cramér transform Slope transform
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http://export.arxiv.org/abs/2102.11785
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nucl-th
(what is this?)
# Title: Bjorken flow attractors with transverse dynamics
Abstract: In the context of the longitudinally boost-invariant Bjorken flow with transverse expansion, we use three different numerical methods to analyze the emergence of the attractor solution in an ideal gas of massless particles exhibiting constant shear viscosity to entropy density ({\eta}/s) ratio. The fluid energy density is initialized using a Gaussian profile in the transverse plane, while the ratio \chi = P_L / P_T between the longitudinal and transverse pressures is set at initial time to a constant value \chi_0 throughout the system using the Romatschke-Strickland distribution. We highlight the transition between the regimes where the longitudinal and transverse expansions dominate. We find that the hydrodynamization time required for the attractor solution to be reached increases with the distance from the origin, as expected based on the properties of the 0+1D system defined by the local initial conditions. We argue that hydrodynamization is predominantly the effect of the longitudinal expansion, being significantly influenced by the transverse dynamics only for small systems or for large values of {\eta}/s.
Comments: 6 pages, 3 figures Subjects: Nuclear Theory (nucl-th); High Energy Physics - Phenomenology (hep-ph); Fluid Dynamics (physics.flu-dyn) Cite as: arXiv:2102.11785 [nucl-th] (or arXiv:2102.11785v1 [nucl-th] for this version)
## Submission history
From: Victor Eugen Ambruş [view email]
[v1] Tue, 23 Feb 2021 16:42:11 GMT (729kb,D)
Link back to: arXiv, form interface, contact.
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https://math.stackexchange.com/questions/1125213/definable-real-numbers
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# Definable real numbers
A real number $a$ is first-order definable in the language of set theory, without parameters, if there is a formula $\phi$ in the language of set theory, with one free variable, such that $a$ is the unique real number such that $\phi(a)$ holds in the standard model of set theory.
A few lines later we find the statement:
Assuming they form a set, the definable numbers form a field....
But, since they are a subset of the set of real numbers, why shouldn't they be a set?
Coming back from this question to the definition, I've another doubt: if ZFC is consistent this does not means that every set-theoretic object (and so any real number) is definable in some model?
Reading the whole article does not lessen my confusion .... and the ''talk'' is too difficult for me and it does not help.
More generally, this Wikipedia article is "disputed" ad has many "!" So I doubt that it is not reliable.
A brief surf on the web give me many pages on this subject but I've found nothing that I can understand and give a response to the question: we can well define what is a definable real number?
• See also the answer to this post at MathOverflow. – Mauro ALLEGRANZA Jan 29 '15 at 17:08
There are several problems here:
1. There is not "the standard model of set theory". There are notions of "standard models" (note the plural), but there is no "the standard model". With respect to the real numbers there are several possible scenarios:
• It might be the case that there is a standard model containing all the reals. This model, if so, has to be uncountable.
• It might be that every real is a real number of some standard model, but there is no standard model containing all the reals.
• It might be that there are real numbers which cannot be members of any standard model, and some that can be.
• It might be that there are no standard models at all.
So this is really a delicate issue here. But in any case, one shouldn't qualify "standard model of set theory" with "the". At all.
2. The notion of "definable real number" often means definable over $\Bbb R$ as a real number in a language augmented by all sort of things we are used to have in mathematics, integrals, sines and cosines, etc. In that case, there are generally only countably many definable reals, since there are only countably many formulas to define reals with.
Once you add the rest of the set theoretic universe into play, you can have that every real number is definable. This is a delicate issue, and known to be consistent, see Joel Hamkins, David Linetsky and Jonas Reitz's paper "Pointwise Definable Models of Set Theory" (and Joel Hamkins' blog post on the paper which has a nice discussion on the topic).
3. And this brings us to the problem at hand. It might be the case that the collection of all definable reals is not itself definable internally. Namely, we can recognize whether a real number is definable or not; but there is no formula whose content is "$x$ is a definable real number". This can be the case because we cannot match a real number to its definition, and we cannot really quantify over formulas to say "There exists a definition".
But sometimes we are in a case where we can in fact identify the definable real numbers, either we know that they form a set (which was defined using some other formula) or that we managed to circumvent the inability to match a real to its definition by adding further assumptions that make things like that possible. And in those cases the set of definable reals, the Wikipedia article states, is a subfield of $\Bbb R$ of that model of set theory.
• When the Wikipedia article says "the standard model" I don't think the set theorist's notion of "standard model" is even close to what it intends. I think it's trying to refer to an "intended interpretation" of the language of set theory, that is, an assumed Platonically existing universe of actual sets. There are numerous problems with that concept, but I don't think it illuminates those problems to point to the (more or less unrelated) technical use of the words "standard model". – Henning Makholm Jan 29 '15 at 17:24
• Henning, that might be; doesn't mean that one should say things like "the standard model of set theory", since unlike the natural numbers or the real numbers, it's nearly impossible for set theorists to decide what is "the intended interpretation" (and most mathematicians simply don't care). – Asaf Karagila Jan 29 '15 at 17:27
• Sure sure, hence the "numerous problems" I alluded to. – Henning Makholm Jan 29 '15 at 17:30
• Asaf, do you know an example of a model of ZFC where the collection of definable reals in the model does not belong to the model? – user203787 Jan 29 '15 at 17:46
• Never mind, the pointwise definable reals in any model always form a definable set in that model. – user203787 Jan 29 '15 at 18:17
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https://www.physicsforums.com/threads/gm-m-mass-of-earth-g-gravitational-constant.125978/
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GM - M mass of earth , G gravitational constant
1. Jul 13, 2006
dopey9
im trying to find the maximum distance of a spacecraft from the earth, whre GM is used ( M is the mass of the earth and G is the gravitational constant) ....
i was just wondering if there is a general formual for this?
2. Jul 13, 2006
Staff: Mentor
Not clear what it is you are looking for. Please state the exact problem you are trying to solve.
3. Jul 13, 2006
HallsofIvy
Staff Emeritus
Unless there is some additional information, there is NO "maximum distance of a space craft from earth". Are you think of the case where the spacecraft is in orbit around the earth and you are given the position and speed of the spacecraft? In that case, you could solve for the apogee (point in the orbit farthest from earth).
4. Jul 13, 2006
dopey9
bascailly im meant to get to this formula which they have given
1/R[a] = 8GM/R^2*(V[a] + V)^2 - 1/R
where V[a] and V are speeds of two spacecrafts
G is gravitational constant
M is mass of earth
R is the distance from the centre of the earth
R[a] is the max distance of wreckage from the earth...because the two spacecrafts collided and where stuck together as one lump of wreckage...this part is continued from another of a question i posted earlier on spacecrafts of which i have solved...but this one iv come close to getting the answer but i dont know how they got the 8... also i got a hint that a mass M[2] orbits around a fixed mass M[1] according to the formula 1/r=Acos(theta) + G*M[1]*M[2]^2 / L^2
so basically iv been given the formula to derive but iv tried but cant get to it
Similar Discussions: GM - M mass of earth , G gravitational constant
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https://www.scienceforums.net/profile/140766-ragordon2010/
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# RAGORDON2010
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1. ## An Invitation to Visit My Blogs
The Forum members are invited to visit two blogs I have created that expand on my earlier postings under the Special Relativity and Quantum Mechanics categories: “Special Relativity from the Inside Out” LINK DELETED “Introduction to Schrodinger Ensemble Theory” LINK DELETED
2. ## Schrodinger Ensemble Theory
Introduction to Schrodinger Ensemble Theory Some years ago, I chose to pursue a different approach to the study of the time-independent Schrodinger equation, particularly as it is commonly applied to the following situations: a particle in an infinite potential well, a particle in a finite potential well, the harmonic oscillator, the hydrogen atom. The first group of examples I will discuss are all one-dimensional. The work will generalize when I deal with the hydrogen atom. My concept is simple. For a given potential V(x), suppose $$\psi(x)$$ is the solution to the Schrodinger equation in the form: $$(E)(\psi) = ((h/2(\pi))^2/2m)(d^2(\psi)/dx^2) + (V)(\psi)$$. Suppose further that an ensemble of identical, non-interacting particles is distributed in real space at time t=0 such that the fraction of particles in the region (x, x + dx) is given by $$\psi\psi*dx$$. Suppose, in addition, that these particles exhibit an initial momentum distribution such that the fraction of particles with momentum in the range (p, p + dp) is given by $$\phi\phi*dp$$, where $$\phi(p)$$ and $$\psi(x)$$ are Fourier transforms of each other according to the usual rules. I then require that the fractional density functions be consistent across the two spaces - real space and momentum space. That is, I insist that the fraction of ensemble particles initially positioned in the region (x, x + dx) equals the fraction of ensemble particles with initial values of momentum in the region (p, p + dp). That is, I require that my consistency relationship $$\psi(x)\psi(x)*dx = \phi(p)\phi(p)*dp$$ is satisfied. Finally, I use this consistency relationship to seek a momentum function p(x). On the one hand, it may be possible to find p(x) by inspection or via trial and error. Otherwise, it might be possible to integrate each side of the relationship separately and isolate p(x) from the result. Even then, there will still be some freedom left to decide on the direction of the momentum vectors. Please note that for these Schrodinger ensembles, total particle energy is not a "sharp" variable. The expectation energy averaged across the entire ensemble remains the eigenvalue E, but the energy of any individual particle is always computed from $$p(x)^2/2m + V(x)$$ in the usual manner. Also note that since psi(x) and phi(p) only represent initial conditions placed on the ensemble, the subsequent development of the ensemble over time is determined by applying Liouville's theorem to the ensemble. I have not found a way to develop Schrodinger Ensemble Theory for the time-dependent Schrodinger Equation.
3. ## Another way of looking at Special Relativity
Must be my browser. Regarding the comments, I ask the Forum to be patient. I think most concerns being raised will self-resolve eventually. A little more background on Related Experiments, and then I will focus on the question of invariance. (Most of us are home-bound anyway because of this damn virus, so it's probably healthy to have some anonymous person on the outside to argue with.) Note to all who view this thread - The count of views to this thread surpasses 60,000. I take this view count very seriously. It is my intention that every one of my posts be an accurate and clear reflection of my thinking. To this end, if I draft a post on, say, Monday, the draft is read/edited and read/edited until, say, Thursday or Friday when I finally submit it to the Forum Unfortunately, this discipline does not hold for my responses to individual comments from Forum members. Those responses tend to be “off the cuff” and ill thought out. In particular, my disrespectful comment on a frame-driven physics in the context of the expression: $$(dS/2)^2 + (vdt/2)^2 = (cdt/2)^2$$. I interprete this expression as pointing to the formation of “Minkowski ellipsoids” that mark off the progress of a particle as it moves along its path of motion under the influence of applied fields. Instead of commenting the way I did, what I should have said, upon reflection, is that neither these ellipsoids nor their defining expressions are intended to be viewed as transformation invariants across a pair of related experiments, or in the conventional sense, across the associated “rest” and “moving” frames of reference. I hope all of this will become clearer to the Forum in my future posts. Special Relativity - A Fresh Look, Part 5 This post begins with the Related Experiments treatment of the “In-Line” Relativistic Doppler Effect and follows with the Related Experiments treatment of the “Transverse” Relativistic Doppler Effect. In his 1905 paper, Einstein* begins his analysis by imagining a monochromatic source placed at rest at a point some distance from the origin of his “rest” frame, Frame K. If we only wish to focus on the in-line Doppler effect, we may limit the positioning of the source to somewhere along the Frame K negative x-axis. *(ref. “Einstein’s Miraculous Year - Five Papers That Changed the Face of Physics", Edited by John Stachel and Published by Princeton University Press, 1998, pgs. 146-149.) Following Einstein’s approach, we write the wave function argument for a light wave emanating from the source and traversing in the positive x direction with frequency f, period T = 1/f, and wavelength w = c/f = cT as it would be recorded by a stationary detector positioned at the origin: $$2(\pi)(f)(t - x/c)$$. For our Related Experiments analysis, we assign the above set-up to our image experiment. We place a monochromatic source with frequency f’, period T’ = 1/f’, and wavelength w’ = c/f’ = cT’ at rest at a distant point somewhere along the negative x’-axis and we place a stationary detector at the origin. We expect that the detector will record a wave function argument equal to $$2(\pi)(f’)(t’ - x’/c)$$. Moving over to our object experiment, we use the a similar set-up, but here we place the source in motion with velocity v in the direction of the stationary detector. We now determine what the detector would record in the object experiment as follows: We substitute for t’ and x’ in the argument $$2(\pi)(f’)(t’ - x’/c)$$ using the Lorentz transformations in the form: $$t’ = (\gamma)(t - vx/c^2)$$, and $$x’ = (\gamma)(x - vt)$$, with $$\gamma$$ defined in the usual way. After some simplification, we will find that the stationary detector records a wave with argument: $$2(\pi)(\gamma)(f’)(1 + v/c)(t - x/c)$$, giving a frequency of $$(\gamma)(f)’(1 + v/c)$$. This represents the relativistic Doppler shift for a source moving toward a fixed observer (or equivalently, for an observer moving toward a fixed source.) To determine the frequency transformation for the case where the source moves away from a fixed observer (or equivalently, where the observer moves away from the fixed source), we need only replace v in the above with -v. For the case of the Transverse Relativistic Doppler Effect (TDE), we follow Einstein’s general analysis, but, for Frame K, we position the source at the origin and place the detector at rest at an arbitrary point, point P, on the positive z-axis some distance from the origin. For source frequency f, period T = 1/f, and wavelength w = c/f = cT, we would expect that this detector will record a plane wave emanating from the source with argument $$2(\pi)(f)(t - z/c)$$. For our Related Experiments analysis, we assign Einstein’s Frame K set-up to our image experiment. We place a monochromatic source with frequency f’, period T’ = 1/f’, and wavelength w’ = c/f’ = cT’ at rest at the origin, and we place the detector at rest on the positive z’-axis at a point P some distance from the origin. As in the Einstein model, we expect that this detector will record a plane wave emanating from the source with argument $$2(\pi)(f’)(t’ - z’/c)$$. Moving over to our object experiment, we use the a similar set-up, but here we locate the source somewhere along the negative x-axis and set it in motion with velocity v in the positive x direction. We now ask how the wave emitted by the moving source as it passes the origin would appear to the detector at point P. We substitute for t’ and z’ in the argument $$2(\pi)(f’)(t’ - z’/c)$$ using the Lorentz transformations in the form: $$t’ = (\gamma)(t - vx/c^2)$$ and z’ = z, with $$\gamma$$ defined in the usual way. After some simplification, we find that the detector records a plane wave with argument $$2(\pi)(f’)(\gamma)(t - (vx/c + z/(\gamma))/c)$$. This represents a plane wave with frequency $$f = f’(\gamma)$$ and direction cosines, (l, m, n), with l = v/c, m = 0, and n = $$1/(\gamma)$$. We see that the light detected by the receiver is blue-shifted by a factor of gamma. Also, we see that the light beam will appear to be emanating from a displaced source, an example of “aberration”. Let $$\theta$$ = angle between the light beam and the x-axis. Let $$\phi$$ = angle between the light beam and the z-axes. Then $$cos (\theta)$$ = l = v/c, and $$cos \(phi) = n = 1/(\gamma)$$. Since $$\theta$$ and $$\phi$$ are complementary angles, $$cos (\phi) = sin (\theta)$$, and we would expect $$(l)^2 + (n)^2 = 1$$, which is true here.
10. ## Another way of looking at Special Relativity
I wish to continue presenting my insights into a different view of Special Relativity. I took a cue from an invitation from Swansont to open a new thread in Speculations where I posted the following. I found today that Strange has stopped that thread and it seems that I am being directed back to this one. So, for the sake of consistency, I am repeating this post here and will follow up shortly with another one. Special Relativity - a Fresh Look: Overview A fresh look at the underpinnings of Special Relativity is merited for the following reasons - 1. In earlier posts, I’ve shown how to view SR applications as Related Experiments - a pair of matched experiments in which charged particles are subjected to external electromagnetic fields. In the object experiment, the particle is given an initial velocity v and subjected to fields E and H. In the image experiment, fields E’ and H’ are applied to the particle at rest, where E’ and H’ are the transformed images of E and H under the SR field transformations. The 4-space motion of the particle in the image experiment (t’,x’,y’,z’) will then match up with the transformed 4-space motion (t, x, y, z) of the particle in the object experiment under a Lorentz time and space transformation with parameter v. In approaching SR this way, we avoid any discussions or dependencies on clocks that run slow or fast, and meter sticks that shrink or grow, as we move from one experiment to the other. 2. The Relativistic form of Newton’s Second Law of Motion is a Classical Physics formulation. We are given a set of initial conditions, a set of prescribed forces and a differential equation from which we can compute the position, velocity and energy of the particle for any time in the future to any degree of accuracy, and, if we insert negative values of time, we can compute the position, velocity and energy of the particle for any time in the past to any degree of accuracy. This is classical Classical Physics - given knowledge of the initial conditions and applied forces, the entire past and future of the particle is completely determinable. Contrast this with the stochastic behavior of Modern Physics, where SR plays a major role in nuclear physics, the physics of high energy particle collisions, and quantum field theory (QFT). 3. The mention of QFT brings me to my final point - QFT speaks of relationships between particles and fields characterized by a series of minute, discrete interactions in which the particles are accelerated slightly or decelerated slightly and/or deflected slightly and/or rotated, twisted or spun slightly. In contrast, conventional SR theory is marked by functions that are everywhere smooth and continuous. I intend to develop a model of SR which addresses all of the above, stays well within conventional bounds of discussion on the subject, and, here and there, introduces key, defensible ideas. Finally, I ask that the Forum members allow me to retain control over my terminology. For example, I shall refer to Minkowski’s S function as a “Minkowski interval”, and I shall refer to his dS function as a “Minkowski differential interval”.
11. ## Special Relativity - A Fresh Look
A fresh look at the underpinnings of Special Relativity is merited for the following reasons - 1. In earlier posts, I’ve shown how to view SR applications as Related Experiments - a pair of matched experiments in which charged particles are subjected to external electromagnetic fields. In the object experiment, the particle is given an initial velocity v and subjected to fields E and H. In the image experiment, fields E’ and H’ are applied to the particle at rest, where E’ and H’ are the transformed images of E and H under the SR field transformations. The 4-space motion of the particle in the image experiment (t’,x’,y’,z’) will then match up with the transformed 4-space motion (t, x, y, z) of the particle in the object experiment under a Lorentz time and space transformation with parameter v. In approaching SR this way, we avoid any discussions or dependencies on clocks that run slow or fast, and meter sticks that shrink or grow, as we move from one experiment to the other. 2. The Relativistic form of Newton’s Second Law of Motion is a Classical Physics formulation. We are given a set of initial conditions, a set of prescribed forces and a differential equation from which we can compute the position, velocity and energy of the particle for any time in the future to any degree of accuracy, and, if we insert negative values of time, we can compute the position, velocity and energy of the particle for any time in the past to any degree of accuracy. This is classical Classical Physics - given knowledge of the initial conditions and applied forces, the entire past and future of the particle is completely determinable. Contrast this with the stochastic behavior of Modern Physics, where SR plays a major role in nuclear physics, the physics of high energy particle collisions, and quantum field theory (QFT). 3. The mention of QFT brings me to my final point - QFT speaks of relationships between particles and fields characterized by a series of minute, discrete interactions in which the particles are accelerated slightly or decelerated slightly and/or deflected slightly and/or rotated, twisted or spun slightly. In contrast, conventional SR theory is marked by functions that are everywhere smooth and continuous. I intend to develop a model of SR which addresses all of the above, stays well within conventional bounds of discussion on the subject, and, here and there, introduces key, defensible ideas. Finally, I ask that the Forum members allow me to retain control over my terminology. For example, I shall refer to Minkowski’s S function as a “Minkowski interval”, and I shall refer to his dS function as a “Minkowski differential interval”.
12. ## Influence of the Universe on Physical Laws
The article that has had the greatest effect on my thinking about physics over the years is “Extended Mach Principle” by Professor Joe Rosen, then at Tel-Aviv University, Israel (AJP, Volume 49, March 1981, pp. 258-264). Of all the fundamental principles Professor Rosen addresses, these three stand out for me - The origin of all laws of physics lies with the universe as a whole. Every single physical property and behavior aspect of isolated systems is determined by the whole universe. If the rest of the universe is taken away leaving only an isolated system, all laws of physics will cease to hold for it, and even space and time will lose their meaning for it. What I would like to do here is pursue these principles with regard to the following two questions: What role does the surrounding universe play in the decay of a single unstable particle at rest in an inertial frame of reference? What role does the surrounding universe play in the retarded rate of decay of these unstable particles as they move rapidly within this inertial frame of reference? I’ve always thought that the decay of an unstable particle is the strongest illustration of Eddington’s “Arrow of Time” - There is a BEFORE, there is an instant of NOW, and there is an AFTER. With respect to my Question 1, it is not hard to point to numerous examples of interactions where particle decay is in some way connected to the surrounding environment - an atomic pile comes to mind, so do particles struck by random photons, neutrinos, or miscellaneous other particles, real or virtual, that “exist” in the wilds of the universe. I intend instead to focus on Question 2. Retarded rate of decay as a function of pure motion is defined by Einstein’s time dilation formula appearing in his Theory of Special Relativity and also by an identical formula appearing in his Theory of General Relativity. Interestingly enough, the time dilation formula applies to any type of unstable particle regardless of mass, charge, spin or any of the other parameters generally applied to unstable particles by particle physicists and depends only on a relative velocity v and light speed c. If we exclude the class of retarded decay rates associated with General Relativity on the basis that the Universe is interacting with these unstable particle through gravity, we are left with the class of retarded decays associated with Special Relativity. I’ve made the point in earlier posts that I believe that Special Relativity Theory belongs firmly in the house of Electromagnetic Theory, including phenomena related to light such as the constancy of light speed in any inertial frame and the Relativistic Doppler Effect. Given this, I would think that retarded decay of speeding unstable particles, a hallmark of SR time dilation, would be in some way connected to the charge and/or magnetic moment, i.e., spin, of the unstable particle. The theoretical physics community has a large storehouse of weaponry with which to attack this phenomenon - QED, QFT, the Standard Model with its quark/gluon interactions, interactions with the universal background radiation, interactions with fields of passing neutrinos, interactions with the Higgs Boson, the influence of Dark Matter and/or Dark Energy. Just for starters, there are the unstable particles detected down here on the Earth’s surface that are created in collisions between atoms in the upper atmosphere and high-energy particles and gamma rays coming in from outer space. They should decay long before reaching the Earth’s surface. If we approach this problem from the point of view of interactions with external electromagnetic fields, then we might look at interactions with the Earth's magnetic field as well as with miscellaneous electric and magnetic fields in the upper atmosphere. Another aspect of the problem is that particle decay is a stochastic process. Any single unstable particle can exist over a range of time intervals - all that can be determined in the laboratory is the mean time to decay from observations of many instances. Accordingly, the application of the SR time dilation factor has to be applied to the mean time observation, which becomes even more tenuous when we account for the fact that there will be some statistical distribution in the velocities of the observed particles relative to the laboratory frame. Be that as it may, as I’ve pointed out in earlier posts, I am still hoping for an explanation for retarded particle decay times that goes beyond simply stating that Special Relativity requires it.
13. ## Another way of looking at Special Relativity
Studiot , thank you for bringing the Wangsness material to my attention. Oddly, I think his spherical light shell approach to deriving the Lorentz transformations was the vehicle I was first introduced to as a freshman undergrad. I never was happy with it, and I think this dissatisfaction was a prime motivator for me to seek out Einstein’s original 1905 paper to read what the master actually wrote. Currently, I am working on a post targeted for the General Philosophy category. Perhaps we’ll meet up again over there.
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https://xinitrc.de/2014/02/23/Of-rabbits-and-hats.html
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Of rabbits and hats
In the last blog post I showed you what you can do with just implementing the interface for a $*$-semiring and than using the matrix over that semiring. Up until now, we can compute the transitive connection relation, the length of the shortest path, the maximum throughput between two nodes and the reliability on the the most reliable path. But what we can’t do right now is producing the path(s) that instatiate these properties. So this is exactly what I will show you in this post.
Regular Expressions
As you might have to expected, especially since I wrote it in the last blog post, we will use our new best friend the $*$-semiring to do so. We acutally come to the premier example of a semiring, the regular expression. You migth know regular expressions from other programming languages, which is as helpful as it is unfortunate. Why is that you might ask and the answer is simple: There is a singificant difference between regular expressions used in theoretical computer science and regular expressions as implemented in programming languages. The later are strictly more powerful. The nice part is they are also a syntactical super set of the first and by simply dropping this extra syntax we get exactly the former. So for you, this might result in some unlearning, I can’t help that, sorry.
So for everybody not knowing regular expressions from computer science curriculum, let’s give a briefe explanation.
First we need a finite set of “letters” called alphabet usually for illustrative purposes one uses the regular letters of a usual alphabet, like $a, b, c, d, \ldots$. But any alphabet will do, we could for example use the natural numbers up to some number, the edges of a finite graph or all subsets of $\{1,2,3,4\}$ as letters, it doesn’t matter.
Let $l$ be any letter of the alphabet $A$, then we can give the following recursive definition of the regular expressions $re$ as follows:
$re,re_1, re_2 ::= \emptyset | \epsilon | l | (re_1 + re_2) | (re1 \cdot re2) | (re^{*})$
Let’s go through this and decipher what it means.
1. $\emptyset$ this is simple. If we write nothing, it is a valid regular expression. And since we can’t leave blank space in a definition and think that everybody picks up that this is a significant syntactical element we write this as $\emptyset$.
2. $\epsilon$ we need to construct a word that has no letters, to do that we use the symbol $\epsilon$. You might ask what is the difference between nothing and a a word of no letters. It is the same difference as between ${0}$ and $\emptyset$ one is a set containing something of no value, the other is simply empty, same goes here.
3. If we have a letter $l$, writing just that letter is a valid regular expression.
4. $re_1 + re_2$ If we have two regular expressions already, we can either use the left or the right, appropriatly this is usually called alternative or choice. The plus-sign is somewhat of a convention for regular expressions, but you might have already guessed what will make it’s way into our semiring in the end.
5. $re_1 \cdot re_2$ this is called concatenation or sequential compositon it simply states that we can look for something that matches the first regular expression followed by something that matches the second regular expression. If there is no danger of misreading it we usually ommit the $\cdot$ just like with multiplication.
6. $re^*$ this finally is arbitrary iteration, if we have a regular expression we can match it an arbitrary amount of times.
Usually you define some form of rules that say we can ommit some of the parens by saying $*$ binds stronger than $\cdot$ which in turn binds stronger than $+$. Since this just looks like basic arithmetic I trust you can follow.
Example
Let’s give an example, suppose we have this regular expression:
$re_{example}=a (bd + ce)$
Then let’s first check that it is a valid regular expression for the alphabet $\{a,b,c,d,e\}$. After stating this, we know that the individual letters $a,b,c,d$ are valid regular expressions. Next we combine $b$ and $d$ with $\cdot$ and get $(b\cdot d)$ same for $c$ and $e$ now we can apply $+$ and get $(b\cdot d+c\cdot e)$. As a last step we combine $a$ and $(b\cdot d+c\cdot e)$ and get $a\cdot (bd+ce)$. Ommiting the $\cdot$’s we get that the regular expression is valid.
Now what does it “mean”
Let’s assume we have a graph and the edges are labeled with letters so there is an edge between two nodes labeld with $a$, one labeled $b$ and so on, like in this graph.
Then the regular expression above $a(bd+ce)$ describes all pathes from $N1$ to $N5$. We first have to take the edge labeled $N1\rightarrow N2$ which is labeld $a$ then either the edges labeled $b$ followed by the edge labeled $d$ or the edges labeled $c$ and $e$.
I think you see how this is useful for our case. So let’s go on and define a semiring.
The Regular Expression $*$-Semiring
Acutally regular expressions form a family of $*$-semirings, one for each underlying alphabet. So let’s call our alphabet $\Sigma$, which is the usual name for the alphabet in computer science. Then
$(\Sigma, +, \cdot, \emptyset, \epsilon)$ is a $*$-semiring with the rules above and two corner cases which give rise to the following two (standard) definitions:
1. $\emptyset + x = x + \emptyset = x$
2. $\emptyset \cdot x = x \cdot \emptyset = \emptyset$
Let’s do a quick check if our properties hold.
1. $a\oplus b = b\oplus a$ since $+$ is the same as or this holds.
2. $(a\oplus b)\oplus c = a\oplus (b\oplus c)$, by almost the same argument, if we first decide between $a$ or $b$ and then between the result of that decission or $c$ it’s the same as deciding the other way around.
3. $a \oplus \mathbf{0}=\mathbf{0}\oplus a=a$, this is by the definition given above.
4. $(a \otimes b)\otimes c = a \otimes (b \otimes c)$ the concatenation of $a$ and $b$ yields $ab$ that concatenated with $c$ yields $abc$ which is the same as first concatenating $b$ and $c$ to $bc$ and then prepending $a$.
5. $a \otimes \mathbf{1} = \mathbf{1} \otimes a = a$ concatenating the empty word to anything will not change anything, so this is ok too.
6. $a \otimes \mathbf{0} = \mathbf{0} \otimes a = \mathbf{0}$ that’s by the definition above.
7. $a \otimes (b \oplus c) = (a\otimes b) \oplus (a\otimes b)$, this is simply either first doing $a$ and then deciding between $b$ or $c$ or first deciding to go $ab$ or $ac$ so this holds true to.
8. $(a \oplus b) \otimes c = (a\otimes c) \oplus (b\otimes c)$, almost the same as the one before, either decided between $a$ and $b$ and then do $c$ or deciding to do $ac$ or $bc$ should be the same.
Ok, we are satisfied, this is a semiring. Now for the $*$ part, which this time is a little more interesting than before.
If we want to describe all ways from one node to another, there might be loops on the way. For example let’s modify the graph from above slightly that it looks like this.
Now on a way from $N1$ to $N5$ we would be allowed to take the loop labeled $f$ arbitrarily often, or writing it in the syntax of regular expression $a(bd+cf*e)$. We have to take this into account for our $*$-semiring definition of regular expressions. So let’s do it.
Defining $*$s
I have a small problem since now we have two different $*$ operations, one from the semiring and one from the regular expressions. To keep those apart I will use $*_{sr}$ for the semiring and $*_{re}$ for the regular expression star. I hope this isn’t to confusing for you.
Let us define a $*$.
$x^*_{sr}=\left\{\begin{array}{ll}\epsilon & \text{falls } x = \emptyset\\\epsilon & \text{falls } x = \epsilon\\y^{*_{sr}} & \text{falls } x = y^{*_{re}}\\x^{*_{re}} & \text{sonst}\end{array}\right.$
Even though this is a bit tricky I think you are by now fully capable of checking that this will actually give us a valid $*$-semiring.
Now for a Haskell implementation. Since I don’t want to conflict with other operators I use Or, Concat and Star instead of $+, \cdot$ and $*$.
data StarSemiringExpression a =
Var a
| Or (StarSemiringExpression a) (StarSemiringExpression a)
| Concat (StarSemiringExpression a) (StarSemiringExpression a)
| Star (StarSemiringExpression a)
| None
| Empty
newtype RE a = RE (StarSemiringExpression a)
re :: a -> RE a
re = RE . Var
instance Semiring (RE a) where
zero = RE None
one = RE Empty
RE None <+> x = x
x <+> RE None = x
RE Empty <+> RE Empty = RE Empty
RE Empty <+> RE (Star a) = RE (Star a)
RE (Star a) <+> RE Empty = RE (Star a)
RE x <+> RE y = RE (x Or y)
RE Empty <.> x = x
x <.> RE Empty = x
RE None <.> _ = RE None
_ <.> RE None = RE None
RE x <.> RE y = RE (x Concat y)
instance StarSemiring (RE a) where
star (RE None) = RE Empty
star (RE Empty) = RE Empty
star (RE (Star x)) = star (RE x)
star (RE x) = RE (Star x)
The helper function re in combination with leaving out the actual alphabet for our regular expression allows us to use any type, as I’ve shown above, and implicitly creating the alphabet from the “letters” that are used in the regular expression. (For mathmatically inclined readers: Yes, this is only ok, as long as we only use finite regular expression, but I would doubt that you can write an infinite one ;-) )
There is one thing left to do to put this to use, we need another small helper function:
reGraph :: (Ix i) => Matrix i (Maye a) -> Matrix (Re a)
reGraph = fmap (maybe zero re)
This simply takes a matrix, where there might be an entry at any index and transforms that into a regular expression of one letter and takes the entry itself as letter, if there is no entry present at that point it just uses $\epsilon$.
What remains is to clear up is what our $*$ operation matrixes does now. What we saw in the post from last week is that the algorithm minimized or maximized some property we calculated. This time it’s a little different. Operations like $max$, $min$ and $||$ “discard” one of their operatands but the $+$ operation from regular expressions doesn’t it creates an alternative. So for any node we can use as an intermediate what we get is “Either use the path we already know OR the path we can construct by using the intermediate node we are currently testing.” Now there might not be a path leading from our start to our target node using the specific intermediate node, then we get back a $\emptyset$ as an alternative path, which is the only value that is acutally discarded by $+$. So what we get in the end is a regular expression specifying all pathes we can take to get from a start to a target node.
Oh by the way with this we implemented another well known algorithm with the same 7 lines of code I presented in the last blog post. It’s called the McNaughton-Yamada algorithm or the Kleene-Construction and is taught to probably every computer science major on the planet. (And forgotten about 5 minutes later ;-) )
Wrapping up
In the last blog post we started to put our $*$-semiring knowledge to use on graphs and found that the same 7 lines of code got us various properties, depending on which semiring we plugged in. This blog post showed how we can describe path between nodes and how we can get all path between two given nodes. Again with the very same 7 lines of code. What remains is to put both of those together to get all path exibiting a given property and that is what we’ll do in the next installment.
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https://codebycase.com/algorithm/a12-design-and-scalability.html
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# Algorithm 12 - Design and Scalability
In an interview, when someone asks a system design question. You should have a conversation in which you demonstrate an ability to think creatively, understand design trade-offs, and attack unfamiliar problems. You should sketch key data structures and algorithms, as well as the technology stack (programming language, libraries, OS, hardware, and services) that you would use to solve the problem.
## System Design Patterns
Design Principle Key Points
Algorithms and Data Structures Identify the basic algorithms and data structures
Decomposition and Design Patterns Split the architecture, functionality and code into manageable, reusable components. The subject of design patterns is concerned with finding good ways to achieve code-reuse. Broadly speaking, design patterns are grouped into creational, structural, and behavioral patterns.
Scalability and Parallelism Break the problem into subproblems that can be solved relatively independently on different machines. Shard data across machines to make it fit. Decouple the servers that handle writes from those that handle reads. Use replication across the read servers to gain more performance. Consider caching computation and later look it up to save work.
### Design a Spell Checker
Design a good spelling correction system can be challenging.
Solution: The basic idea behind most spelling correction system is that the misspelled word’s Levenshtein (edit) distance from the intended word tends to be very small (one or two edits). Hence, if we keep a hash table for all the words in the dictionary and look for all the words that have a Levenshtein distance of 2 from the text, it is likely that the intended word will be found in this set.
The total number of ways of selecting any two characters is n(n-1)/2, and each character can be changed to one of (m-1) other chars. Therefore, the number of lookups is n(n-1)(m-1)^2/2.
It is important to provide a ranked list of suggestions to the users, with the most likely candidates are at the beginning of the list.
• Context based: track what words used in the text. (Viterbi Algorithm)
• Language words differences, or culture differences.
• Typing errors model: can be modeled based on keyboard layouts.
• Phonetic modeling: when knows how the words sounds but does not know the exact spelling.
• History of refinements: by first entering a misspelled word and then correcting it.
• Stemming: by keeping only the stemmed version of each word.
### Design Gmail System
Based on Reactive Design Pattern (Responsive, Resilient, Elastic, Message-Driven).
• Coping with load by splitting a system into distribution parts. Sharding datasets or computational resources solves the problem of providing sufficient resources for the nominal case. (Sharding Pattern): Combine Multiple IDs
• Coping with failure by using active-passive replication. Replicas agree on which one of them can accept updates. Fail-over to a different replica requires consensus among the remaining ones when the active replica no longer responds. Especially Gmail services should provide consistency to the user.
• Making the system responsive by employing circuit breakers and flow control between front end and web servers, also between web servers and back-end services. We can even switch the entire application to offline mode when only a small part of the back-end are unavailable.
• Avoiding the ball of mud by decomposing the multitude of services. Such as the principle of message-flow design would be that the service that handles email composition probably should not talk directly to the contact pop-up service. Also consider Micro services pattern.
• Integrating nonreactive components by measuring the latency of external API. Dedicate a process/thread to this encapsulated form of external API call. If latency exceeds the acceptable threshold, will either response to requests with a rejection or a temporary failure code.
Reacting to users and failures, losing strong consistency
Systems with an inherently distributed design are built on a set of principles called BASE: Basically available; Soft state (state needs to be actively maintained instead of persisting by default); Eventually consistent.
The last point means that modifications to the data need time to travel between distributed replicas, and during this time it is possible for external observers to see data that are inconsistent. The qualification “eventually” means the time window during which inconsistency can be observed after a change is bounded; when the system does not receive modifications any longer and enters a quiescent state, it will eventually become fully consistent again.
In the example of editing a shared document, this means although you see your own changes immediately, you might see the other’s changes with some delay; and if conflicting changes are made, then the intermediate states seen by both users may be different. But once the incoming streams of changes end, both views will eventually settle into the same state for both users.
The inconsistency observed in eventually consistent systems is also short-lived; the delay between changes being made by one user and being visible to others is on the order of tens or maybe hundreds of milliseconds, which is good enough for collaborative document editing.
Apply the Actor Model
The Actor model is a model of concurrent computation in which all communication occurs between entities called Actors, via message passing on the sending side and mailbox queues on the receiving side. The Erlang programming language, one of the earliest to support Reactive application development, uses Actors as its primary architectural construct. With the success of the Akka toolkit on the JVM, Actors have had a surge in popularity of late.
### Handle the Stemming Problem
Stemming is the process of reducing all variants of a given word to one common root, like {computers, computer, compute, computation} to compute. How to design a stemming algorithm that is fast and effective.
Solution: Most stemming systems are based on simple rewrite rules, e.g., remove suffixes of the form “es”, “s”, and “ation”. sometimes need to replaces suffix. Other approaches include the use of stochastic methods to learn rewrite rules and n-gram based approaches where we look at the surrounding words.
One way of efficiently performing the transformation rules is to build a finite state machine based on all the rules.
### Design a Scalable Priority System
Design a system for maintaining a set of prioritized jobs that implements Insert, Delete and Fetch the highest priority job. Assume the set cannot fit into a single machine’s memory.
Solution: If we have enough RAM on a single machine, the most simple solution would be to maintain a min-heap where entries are ordered by their priority. An additional hash table can be used to map jobs to their corresponding entry in the min-heap to make deletions fast.
A more scalable solution entails partitioning the problem across multiple machines. One approach is to apply a hash function to the job ids and partition the resulting hash codes into ranges, one per machine. Insert as well as delete require communication with just one server. To do extract-min, we send a lookup minimum message to all the machines, infer the min from their responses, and the delete it.
If many clients are trying to do this operation at the same time, we may run into a situation where most clients will find that the min event they are trying to extract has already been deleted. If the throughput of this service can be handled by a single machine, we can make one server solely responsible for responding to all the requests. This server can prefetch the top hundred or so events from each of the machines and keep them in a heap.
In many applications, we do not need strong consistency guarantees. A client could pick one of the machines at random (or round-robin), and request the highest priority jobs. This would work well for the distributed crawler application, but not suited to event-driven simulation because of dependencies.
Consider resilience: if a node fails, all list of work on that node fails as well. It is better to have nodes to contain overlapped lists (replicate a copy to it’s predecessor) and the dispatching node in this case will handle duplicates. The lost of a node shouldn’t result in full re-hashing – the replacement node should handle only new jobs. Consistent hashing can be used to achieve this.
A front-end caching server can become a bottleneck. This can be avoided by using replication, i.e, multiple servers which duplicate each other. There could be possible several ways to coordinate them: use non-overlapping lists, keep a blocked job list, return a random job from the jobs with highest priority.
### Design a Recommendation System
Design a system that automatically generates a sidebar of related articles.
Solution:
1. Some simple solutions: Add articles that have proved to be popular recently; Link to recent news articles; Tag articles with related keywords (i.e., finance, sports, and politics); These tags could also come from the HTML meta-tags or page titles.
2. We could also provide randomly selected articles to a random subset of readers and see how popular these articles proved to be. The popular articles could then be shown more frequently. (Leads to No.3)
3. Build a scoring mechanism that takes various features as signals and computes a final score for each article (link/like/comment numbers).
4. A more sophisticated level, use automatic textual analysis, where a similarity is defined between pairs of articles: This similarity is a real number and measures how many words are common to the two. Several issues come up, such as the fact that frequently occurring words such as “for” and “the” should be ignored and that having rare words such as “arbitrage” and “diesel” in common is more significant that having say, “sale” and “international”.
5. Textual analysis has problems, such as the fact that two words may have the same spelling but completely different meanings (anti-virus means different things in the context of articles on medical and computer). One way to augment textual analysis is to use collaborative filtering–using information gleaned from many users. For example, by examining cookies and timestamps in the web server’s log files, we can tell what articles individual users have read. If we see many users have read both A and B in a single session, we might want to recommend B to anyone reading A.
### Design an Online Advertising System
Solution: (skipped some other details)
• Consider the stakeholders separately (Users, Advertisers, Search Engine Companies).
• The ad-serving system would build a specialized data structure, such as a decision tree from the ads database. It chooses ads from the database of ads based on their “relevance” to the search. In addition to keywords, the ad-serving systems can use knowledge of the user’s search history, bidding amount, the time scheduler, user location, device type, gender, etc. Many strategies can be envisioned here for estimating relevance.
• The ads could be added to the search results by embedding JavaScript in the results page. Which pulls in the ads from the ad-serving system directly. This helps isolate the latency of serving search results from the latency of serving ad results.
### Optimize Large Files Distribution
Design an efficient way of copying one thousand files each 100 kilobytes in size from a single lab server to each of 1000 servers in a distant data center.
Solution:
• Assume that the bandwidth from the lab machine is a limiting factor. We can do some trivial optimizations, such as combining the articles into a single file and compressing this file.
• We can copy the file from the lab machine to a single machine in the data center first. And have each machine that has received the file initiate copies to the machines that have not yet received the file. (In theory, this leads to an exponential reduction.)
• How should the knowledge of machines which do not yet have copies of the file be shared? There can be a central repository or servers can simply check others by random selection.
• If the bandwidth between machines in a data center is not a constant, Servers close to each other, e.g. in the same rack, should prefer communicating with each other.)
• Finally, please note there are open source solutions to this problem, such as Unison and BitTorrent.
### Encode and Decode TinyURL
TinyURL is a URL shortening service where you enter a URL such as https://leetcode.com/problems/design-tinyurl and it returns a short URL such as http://tinyurl.com/4e9iAk.
Solution:
1. Random fixed-length encoding, can be encoded up to 62^6! And also can increase the number of encodings possible as well by increasing the length. It’s hard to predict since the random numbers are used.
2. To build the index, instead of using Map, we could build index with a 64-way B-tree or Trie-tree, and each leave has the address pointer which points to the long url in disk. So we can handle large volume of long urls and distribute to different partitions or nodes.
3. We can even build a LFU (Least Frequently Used Cache) to cache with popular urls and also periodically clean up the legacy URLs.
public class TinyUrl { String alphabet = “0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ”; Map<String, String> map = new HashMap<>(); Random rand = new SecureRandom(); String key = getRand();
public String getRand() {
StringBuilder sb = new StringBuilder();
for (int i = 0; i < 6; i++) {
sb.append(alphabet.charAt(rand.nextInt(62)));
}
return sb.toString();
}
public String encode(String longUrl) {
while (map.containsKey(key)) {
key = getRand();
}
map.put(key, longUrl);
return "http://tinyurl.com/" + key;
}
public String decode(String shortUrl) {
return map.get(shortUrl.replace("http://tinyurl.com/", ""));
} }
### MapReduce Design Pattern
MapReduce is a computing paradigm for processing data that resides on hundreds of computers, which has been popularized recently by Google, Hadoop, and many others. MapReduce is more of a framework than a tool. You have to design and fit your solution into the framework of map and reduce. Several other open source projects have been built with Hadoop at their core, like Pig, Hive, HBase, Mahout, and ZooKeeper.
The input to a MapReduce job is a set of files in the data store that are spread out over the Hadoop Distributed File System (HDFS). In Hadoop, these files are split with an input format, which defines how to separate a file into input splits. An input split is a byte-oriented view of a chunk of the file to be loaded by a map task.
Each map task in Hadoop is broken into the following phases: record reader, mapper, combiner, and partitioner. The output of the map tasks, called the intermediate keys and values, are sent to the reducers. The reduce tasks are broken into the following phases: shuffle, sort, reducer, and output format. The nodes in which the map tasks run are optimally on the nodes in which the data rests. This way, the data typically does not have to move over the network and can be computed on the local machine.
Pig and Hive are higher-level abstractions of MapReduce. They provide an interface that has nothing to do with “map” or “reduce,” but the systems interpret the higher-level language into a series of MapReduce jobs. Much like how a query planner in an RDBMS translates SQL into actual operations on data, Hive and Pig translate their respective languages into MapReduce operations.
public class CommentWordCount {
public static class WordCountMapper extends Mapper<Object, Text, Text, IntWritable> {
private final static IntWritable one = new IntWritable(1);
private Text word = new Text();
public void map(Object key, Text value, Context context) throws IOException, InterruptedException {
// Parse the input string into a nice map
Map<String, String> parsed = MRDPUtils.transformXmlToMap(value.toString());
// Grab the "Text" field, since that is what we are counting over
String txt = parsed.get("Text");
// .get will return null if the key is not there
if (txt == null) {
// skip this record
return;
}
// Unescape the HTML because the data is escaped.
txt = StringEscapeUtils.unescapeHtml(txt.toLowerCase());
// Remove some annoying punctuation
txt = txt.replaceAll("'", ""); // remove single quotes (e.g., can't)
txt = txt.replaceAll("[^a-zA-Z]", " "); // replace the rest with a space
// Tokenize the string by splitting it up on whitespace into
// something we can iterate over,
// then send the tokens away
StringTokenizer itr = new StringTokenizer(txt);
while (itr.hasMoreTokens()) {
word.set(itr.nextToken());
context.write(word, one);
}
}
}
public static class IntSumReducer extends Reducer<Text, IntWritable, Text, IntWritable> {
private IntWritable result = new IntWritable();
public void reduce(Text key, Iterable<IntWritable> values, Context context) throws IOException, InterruptedException {
int sum = 0;
for (IntWritable val : values) {
sum += val.get();
}
result.set(sum);
context.write(key, result);
}
}
public static void main(String[] args) throws Exception {
Configuration conf = new Configuration();
String[] otherArgs = new GenericOptionsParser(conf, args).getRemainingArgs();
if (otherArgs.length != 2) {
System.err.println("Usage: CommentWordCount <in> <out>");
System.exit(2);
}
Job job = new Job(conf, "StackOverflow Comment Word Count");
job.setJarByClass(CommentWordCount.class);
job.setMapperClass(WordCountMapper.class);
job.setCombinerClass(IntSumReducer.class); // use combiner to reduce in local first!
job.setReducerClass(IntSumReducer.class);
job.setOutputKeyClass(Text.class);
job.setOutputValueClass(IntWritable.class);
FileOutputFormat.setOutputPath(job, new Path(otherArgs[1]));
System.exit(job.waitForCompletion(true) ? 0 : 1);
}
}
### Bloom Filter Pattern
An empty Bloom filter is a bit array of m bits, all set to 0. There must also be k different hash functions defined, each of which maps or hashes some set element to one of the m array positions, generating a uniform random distribution.
To add an element, feed it to each of the k hash functions to get k array positions. Set the bits at all these positions to 1.
To query for an element (test whether it is in the set), feed it to each of the k hash functions to get k array positions. If any of the bits at these positions is 0, the element is definitely not in the set – if it were, then all the bits would have been set to 1 when it was inserted. If all are 1, then either the element is in the set, or the bits have by chance been set to 1 during the insertion of other elements, resulting in a false positive. In a simple Bloom filter, there is no way to distinguish between the two cases, but more advanced techniques can address this problem.
This classic example is using bloom filters to reduce expensive disks (or network) lookups for non-existent keys.
If the element is not in the bloom filter, then we know for sure we don’t need to perform the expensive lookup. On the other hand, if it is in the bloom filter, we perform the lookup, and we can expect it to fail some proportion of the time (the false positive rate).
The false positive rate is a function of the bloom filter’s size and the number and independence of the hash functions used. As more elements are added to the set, the probability of false positives increases.
Problem: Given a list of user’s comments, filter out a majority of the comments that do not contain a particular keyword. We can train a bloom filter with a hot list of keywords.
https://www.jasondavies.com/bloomfilter/ https://en.wikipedia.org/wiki/Bloom_filter
### Top Ten Pattern
Problem: Given a list of user information, output the information of the top ten users based on reputation.
The top ten pattern utilizes both the mapper and the reducer. The mappers will find their local top K, then all of the individual top K sets will compete for the final top K in the reducer. Since the number of records coming out of the mappers is at most K and K is relatively small, we’ll only need one reducer.
### Microservices Design
Microservices is a form of service-oriented architecture style wherein applications are built as a collection of different smaller services rather than one whole app. These independent applications can run and scale on their own and may be created using different coding and languages.
Advantages of microservice architecture: Focus on a specific requirement; developed by a small team with productivity; Loosely coupled, can be developed, deployed and scaled on their own; Easy to integrate with 3rd parties.
The popular microservice framework:
• Spring Boot. Which has all the infrastructures that you applications need: Framework, Cloud, Data, Batch, Security, Social, Mobile, and REST Docs.
• Play Framework: Play Framework gives you an easier way to build, create and deploy Web applications using Scala and Java. Play Framework is ideal for RESTful application that requires you to handle remote calls in parallel. It is also very modular and supports async. Play Framework also has one of the biggest communities out of all microservices frameworks.
• Swagger. Helps you in documenting API as well as gives you a development portal, which allows users to test your APIs.
## Game Design Questions
### Zobrist Hashing for Chess
Design a hash function for chess game states. Your function should take a state and the hash code for that state, and a move, and efficiently compute the hash code for the updated state.
Solution:
The state of a game of chess is determined by what piece is present on each square, each square may be empty, or have one of six classes of pieces; each piece may be black or white. Thus log(1+6x2) = 4 bits suffice per square, which means a total of 64 x 4 = 256 bits can represent the state of the chess board. But this way uses more store space and can not be efficiently computed based on incremental changes to the board.
A straightforward hash function is to treat the board as a sequence of 64 base 13 digits. There is one digit per square, with the squares numbered from 0 to 63. Each digit encodes the state of a square: blank, white pawn, white rook,…,white king, blank pawn,…,blank king. We use the hash function as $\sum_{i=0}^{63}c_ip^i$, where $c_i$ is the digit in location i, and p is a prime number.
Note that this hash function has some ability to be updated incrementally. If, for example, a black knight taken by a white bishop the new hash code can be computed by subtracting the terms corresponding to the initial location of the knight and bishop, and adding a term for a blank at the initial location of the bishop and a term for the bishop at the knight’s original position.
Now let’s describe the Zobrist Hashing function which is much faster to update. The main purpose of Zobrist hash codes in chess programming is to get an almost unique index number for any chess position/state (chess fingerprint), with a very important requirement that two similar positions generate entirely different indices. These index numbers are used for xor-operation to allow a fast incremental update of the hash key during moves.
At program initialization, we generate an array of random numbers:
• One number for each piece at each square.
• One number to indicate the side to move is black.
• Four numbers to indicate the castling rights.
• Eight numbers to indicate the file of a valid En passant square, if any.
This leaves us with an array with 781 (12 * 64 + 1 + 4 + 8) random numbers. Since pawns don’t happen on first and eighth rank, 12 * 64 should be fine. Usually 64bit are used as a standard size in modern chess programs.
To get a Zobrish hash code of the starting position, we can do the xor-operation for all pieces:
[Hash for White Rook on a1] xor [White Knight on b1] xor [White Bishop on c1] xor ... ( all pieces )
... xor [White castling short] xor [White castling long] xor ... ( all castling rights )
E.g., for a White Knight that jumps from b1 to c3 capturing a Black Bishop, these operations are performed:
[Original Hash of position] xor
[Hash for White Knight on b1] (removing the knight from b1) xor
[Hash for Black Bishop on c3] (removing the captured bishop from c3) xor
[Hash for White Knight on c3] (placing the knight on the new square) xor
[Hash for Black to move] (change sides)
An important issue is the question of what size the hash keys should have. Smaller hash keys are faster and more space efficient, while larger ones reduce the risk of a hash collision. The chance of collisions approaches certainty at around the square root of the number of possible keys. With a 64 bit hash, you can expect a collision after about 2^32 or 4 billion positions.
Zobrist Hashing Implementation
// A program to illustrate Zobrist Hashing Algorithm
#include <bits/stdc++.h>
using namespace std;
unsigned long long int ZobristTable[8][8][12];
// Generates a Randome number from 0 to 2^64-1
unsigned long long int randomInt()
{
uniform_int_distribution<unsigned long long int>
dist(0, UINT64_MAX);
return dist(mt);
}
// This function associates each piece with
// a number
int indexOf(char piece)
{
if (piece=='P')
return 0;
if (piece=='N')
return 1;
if (piece=='B')
return 2;
if (piece=='R')
return 3;
if (piece=='Q')
return 4;
if (piece=='K')
return 5;
if (piece=='p')
return 6;
if (piece=='n')
return 7;
if (piece=='b')
return 8;
if (piece=='r')
return 9;
if (piece=='q')
return 10;
if (piece=='k')
return 11;
else
return -1;
}
// Initializes the table
void initTable()
{
for (int i = 0; i<8; i++)
for (int j = 0; j<8; j++)
for (int k = 0; k<12; k++)
ZobristTable[i][j][k] = randomInt();
}
// Computes the hash value of a given board
unsigned long long int computeHash(char board[8][9])
{
unsigned long long int h = 0;
for (int i = 0; i<8; i++)
{
for (int j = 0; j<8; j++)
{
if (board[i][j]!='-')
{
int piece = indexOf(board[i][j]);
h ^= ZobristTable[i][j][piece];
}
}
}
return h;
}
// Main Function
int main()
{
// Uppercase letters are white pieces
// Lowercase letters are black pieces
char board[8][8] =
{
"---K----",
"-R----Q-",
"--------",
"-P----p-",
"-----p--",
"--------",
"p---b--q",
"----n--k"
};
initTable();
unsigned long long int hashValue = computeHash(board);
printf("The hash value is : %llu\n", hashValue);
//Move the white king to the left
char piece = board[0][3];
board[0][3] = '-';
hashValue ^= ZobristTable[0][3][indexOf(piece)];
board[0][2] = piece;
hashValue ^= ZobristTable[0][2][indexOf(piece)];
printf("The new hash vlaue is : %llu\n", hashValue);
// Undo the white king move
piece = board[0][2];
board[0][2] = '-';
hashValue ^= ZobristTable[0][2][indexOf(piece)];
board[0][3] = piece;
hashValue ^= ZobristTable[0][3][indexOf(piece)];
printf("The old hash vlaue is : %llu\n", hashValue);
return 0;
}
Output
The hash value is : 14226429382419125366
The new hash vlaue is : 15124945578233295113
The old hash vlaue is : 14226429382419125366
### Design the Chess Game
Basic Object Design
• Game:
• Contains the Board and 2 Players
• Commands List (for history tracking)
• isOver(), isStaleMate(), isCheckMate()
• Board (No Singleton):
• Hold spots with 8*8
• Initialize the piece when game start
• Move Piece
• Remove Piece
• Replace Piece
• Can move to location(int startX, int startY, int endX, int endY).
• Spot:
• Hold Pieces
• Piece (Abstract):
• Hold the color to represent the affiliation.
• Extended by concreted classes with 8 Pawns, 2 Rooks, 2 Bishops, 2 Knights, 1 Queen, 1 King.
• Concreted classes define the detail step approach.
• Defined move rules: isValidMove(), checkPromote()
• Player (Abstract):
• Has a list of piece reference it owns.
• Concreted classes for Human and Computer players
• isChecked()
• Command
• Piece
• Destination x, y
• Game Engine
• Transposition Table
• Zobrist Hashing
• Minimax Evaluation
• Game Tree
Web Version: Consider Session (Authentication), Ajax, WebSocket, or Comet (Streaming or Long Polling), Controller (RESTful API), Scalability, Data Model. Design Pattern: MVC, Observer, Listener, Singleton.
Make a Move
What a full “move” entails in chess:
• Player chooses piece to move.
• Piece makes legal move according to its own move rules.
• In addition to purely move-based rules, there’s also capture logic, so a bishop cannot move from a1-h8 if there’s a piece sitting on c3.
• If the player was previous under check and the move does not remove the check, it must be undone.
• If the move exposes check, it must be undone / disallowed.
• If player captures a piece, remove the piece (including en passant!)
• If the piece is a pawn reaching the back rank, promote it.
• If the move is a castling, set the new position of the rook accordingly. But a king and rook can only castle if they haven’t moved, so you need to keep track of that. And if the king moves through a check to castle, that’s disallowed, too.
• If the move results in a stalemate or checkmate, the game is over.
And the move method would contain all the code to validate the steps above:
• Check Piece.isValidMove(currentSpot, newSpot); - probably need castling logic here since king moves more than 1 space and rook jumps the king)
• Check Player.isChecked() (which is just sugar for Player.Pieces[“King”].CanBeCaptured() - more fun logic here!)
• Check if newSpot contains a piece and if so, newSpot.Piece.Remove();
• Build some logic to call Piece.checkEnPassant() (Piece is pawn, first move, 2 steps, past an enemy pawn who moved into capturing position on previous move - have fun with that!)
• Piece.checkPromote() (Piece is pawn, move ends on opposing player’s back rank)
• Check if Game.isOver(), which checks Game.isStaleMate() and Game.isCheckMate().
Transposition Table
The chess game will often have to consider the same position several times. So most chess engines implement a Transposition Table that stores previously searched positions and evaluations.
The transposition table should store 15%-20% positions/entries ahead of time (NO NEED ALL), and it has to be efficient because we will be storing and searching more useless entries.
To calculate the unique identify for the game state. We can use Zobrist Hashing as described above.
Transposition Table Contents:
• Hash: This is a Zobrist Hash representing the chess position (game state)
• Depth: The depth remaining in the alpha beta search. So depth 5 would mean the score is recorded for a 5 ply search. This can also be referred to as the Depth of the Search Tree.
• Score: The evaluation score for the position.
• Ancient: Boolean flag, if false the node will not be replaced with a newer entry.
• Node Type: There are 3 node types, Exact, Alpha and Beta. Exact means this is an exact score for the tree. An Alpha Node Type means the value of the node was at most equal to Score. The Beta Node Type means the value is at least equal to score.
Minimax Evaluation
Implement a function that calculates the value of the board depending on the placement of pieces on the board. This function is often known as Evaluation Function. We can use Minimax Algorithm.
A Minimax algorithm is a recursive algorithm for choosing the next move in an n-player game, usually a two-player game. A value is associated with each position or state of the game. This value is computed by means of a position evaluation function and it indicates how good it would be for a player to reach that position. The player then makes the move that maximizes the minimum value of the position resulting from the opponent’s possible following moves. If it’s A’s turn to move, A gives a value to each of this legal movies.
The algorithm can be thought of as exploring the nodes of a game tree. The effective branching factor of the tree is the average number of children of each node (i.e., the average number of legal moves in a position). The number of nodes to be explored usually increases exponentially with the number of plies (it is less than exponential if evaluating forced moves or repeated positions). The number of nodes to be explored for the analysis of a game is therefore approximately the branching factor raised to the power of the number of plies. It is therefore impractical to completely analyze games such as chess using the minimax algorithm.
### Design Battleship Game
Basic Object Design
• Game:
• Contains Boards (4 Grids) and 2 Players
• loadGame(), startGame(), saveGame(), isGameOver()
• Commands list (for history tracking)
• Use a variable shift to count each round and decide whose turn, i.e. shift = 3 -> shift % 2 = 1 -> player 1
• Board/Grid:
• ShipGrid (primary) and ShotGrid
• Hold spots and status int[10][10]
• Set<Coordinate> taken
• placeShip(Ship s)
• placePeg(Peg p)
• setShipHitAt(int x, int y)
• Ship (Abstract):
• numHitPoints, numHitsTaken
• x, y location and Orientation
• Concreted classes: Carrier(5), Battleship(4), Cruiser(3), Submarine(3) and Destroyer(2)
• isHit(), isSunk()
• Peg
• Color (White, Red)
• With x, y location
• Build a picture of the opponent’s fleet
• Player (Abstract):
• setHits(), setMisses(), setShotsFired()
• Concreted classes for Human and Computer players
• The Computer player can be configured to different level
• Game Engine
• Probability Calculator
• Transition Matrix
• Game Tree
Web Based Application
1. Design UI with Board/Grid, Buttons
2. For every placement, make an Ajax call to update player board object.
3. Once user clicks the button, submit the request to Servlet.
4. Maintain an available players and occupied players list in an object. (Singleton)
5. If the user is first to click the button, generate a key/sessionId and assign it to user, put to the available list.
6. When opponent clicks button, check for available user having a key/sessionId. and use the same id to opponent and move both players to occupied list and redirect theme to same url.
7. Now every click in the board, send Coordinate to servlet by making an Ajax call, Blacken the Coordinate for both the boards. Also update the turn status based on Ajax response.
8. have session timeout set incase one or both users close their browser.
9. Design pattern: MVC, Listener, Observer
Probability Calculator
On a 10 x 10 grid, players hide ships of lengths: 5(Carrier), 4(Battleship), 3(Submarine), 3(Cruiser), 2(Destroyer), which results in 17 possible targets out of the total 100 squares.
The first possible strategy is to make shots totally at random. The game will take almost 100 shots to complete as the majority of squares have to be hit in order to ensure that all the ships are sunk. Mathematically, the changes of playing a perfect game will be 17!/(100!/83!).
17!/(100!/83!) = (17x16x15...3x2x1)/(100x99x98...86x85x84)
Initially, shots can be fired at random (Hunt Mode), but once part of a ship has been hit, it’s possible to search up, down, left and right looking for more of the same ship (Target Mode). After a hit, the four surrounding squares are added to a stack of ‘potential’ targets (or less than four if the cell was on an edge/corner or already visited).
Once in Target mode the computer pops off the next potential target off the stack, fires at this location, actions on this (either adding more potential targets to the stack, or popping the next target location off the stack), until either all ships have been sunk or there are no more potential targets in the stack, at which point it returns to Hunt mode and starts firing at random again looking for another ship.
Hunt Mode Algorithm
If it’s early in the game and there are too many configurations to check all of them, shots can be fire at random.
• Starts with Center: With no information about the board, the algorithm selects one of the center squares because of the edge effect , the middle of the board will score higher than an edge or a corner.
• Parity: Because the min length of a ship is two units long, no matter how the two unit destroyer is placed on the 1-100 squares, it will cover one odd and one even square. So we can only randomly fire into location with even parity.
Target Mode Algorithm
• Density:
How to calculate the Density score?
We know which ships (and even more importantly what the lengths of the ships) are still active. The algorithm will calculate the most probably location to fire at next based on a superposition of all possible locations the enemy ships could be in.
We start in the top left corner, and try placing a target ship horizontally. If it fits, we increment a value for each cell it lays over as a ‘possible location’. Then we try sliding it over one square and repeating … and so on util we reach the end of the row. Then we move down a line and repeat. Next we repeat the exercise with the ship oriented vertically.
When in hunt mode, there are only three states to worry about: unvisited space, misses and sunk ships. Misses and Sunk ships are treated the same (obstructions that potential ships needed to be placed around). In target mode (where there is at least one hit ship that has not been sunk), the ships can by definition, pass through this location, and so hit squares are treated as unvisited square for deciding if a ship ‘could’ pass through this square, and then a heavy score weighting is granted to possible locations that pass through this hit.
The Linear Theory of A.I.
$P_{i,\alpha}$ denotes the probability of there being the given ship on the given square. $i$ ranges from 0 to 99 and $\alpha$ is one of the ship.
If we had such a matrix, we could figure out the probability of there being at hit on every square by summing over all the ships we have left, i.e.
$P_i = \sum_{ships left} P_{i,\alpha}$
How do we get $P_{i,\alpha}$? We can predict the probability of a particular ship being in a particular square by (1) noting the background probability of that being true, and (2) adding up all of the information I have, weighting it by the appropriate factor. The equation looks as below:
$P_{i,\alpha} = B_{i,\alpha} + \sum_{j,\beta} W_{i,\alpha,j,\beta}I_{j,\beta}$
$B_{i,\alpha}$ denotes the background probability of a particular ship being on a particular ship being on a particular spot on the board.
Below battleship board reflects the sum of all the ship background probabilities.
$B_i = \sum_{all ships} B_{i,\alpha}$
As a game unfolds, we learn a good deal of information about the board, so we need to incorporate this information into our theory of battleship. We call this info matrix $I_{j,\beta}$.
$\beta$ marks the kind of information we have about a square: M means a miss, H means a hit, but we don’t know which ship, and CBSDP mark a particular ship hit, which we would know once we sink a ship.
Let’s say, after a few turns, we were told the spot 34 was a hit and also sunk the ship submarine. we would set:
$I_{34,H} = 1, I_{34,S} = 1, I_{34,M} = I_{34,C} = I_{34,B} = I_{34,D} = I_{34,P} = 0$
Also, we need to take weight into consideration, “the extra probability of there being ship alpha at location i, given the fact that we have the situation beta going on at location j”.
This is a picture of $W_{i,C,33,M}$, the extra probabilities for each square (i is all of them), of there being a carrier (alpha=C), given that we got a miss (beta=M) on square 33, (j=33).
This is a picture of $W_{i,S,65,H}$, showing the extra probability of there being a submarine (alpha=S), at each square (i is all of them, since its a picture with 100 squares), given that we registered a hit (beta=H) on square 65 (j=65).
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https://crypto.stackexchange.com/questions/20974/proving-the-existence-of-a-pseudorandom-function
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# Proving the existence of a pseudorandom function
I've been reading the Introduction to Modern Cryptography book by Katz and Lindell as part of my own learning and have come across this exercise which I am not sure how to approach. The exercise is: (exercise 3.8)
Prove unconditionally the existence of an efficient pseudorandom function $F:\{0,1\}^* \times \{0,1\}^* \mapsto \{0,1\}^*$ where the input-length is logarithmic in the key-length (i.e., $F(k,x)$ is defined only when $|x| = log |k|$, in which case $|F(k,x)| = |k|$).
There is also a hint which states that you should use the fact that any random function is also pseudorandom.
This is my initial train of thought:
We require the pseudorandom function to be indistinguishable from a function chosen uniformly at random from the set of functions that map $log|k|$ bit strings to $|k|$ bit strings (let's say this set is called $Func_{log\,|k|\, \mapsto |k|}$). I'm guessing that we need to work how many functions in this set in order to work out the probability of picking a random function, $f$, from this set.
I know that the set of functions $Func_{n \mapsto n}$ mapping $n$ bit strings to $n$ bit strings contains $2^{n*2^n}$ functions. However my first obstacle is calculating how many functions are in $Func_{log\,|k|\, \mapsto |k|}$ since the functions in this set are not bijective as they are in $Func_{n \mapsto n}$.
If I could calculate this value then I would approach the rest of the problem by calculating the amount of possible pseudorandom functions (clearly given by $|k|$ since $k$ is chosen uniformly at random). I was then hoping, if there was a similar number of functions in $Func_{log\,|k|\, \mapsto |k|}$ (although I speculate there is way more than $|k|$ functions in this set), then eventually try to show that it would be hard for any ppt distinguisher to tell between the pseudorandom function and the randomly chosen one.
I have no idea if this is along the right line and I also don't really know how to bring the hint in to play. All I can think is that it may turn out to be easier to prove that $F$ is indistinguishable from another pseudorandom function which also happens to have been chosen at random.
If anyone could provide a hint as to how to calculate the amount of functions in $Func_{log\,|k|\, \mapsto |k|}$ or pointers for how to approach this then that would be great. As I said, I am doing the exercises for my own good so I'm not massively keen on being given a full solution straight away.
• Is this the exact wording of the exercise? – Guut Boy Dec 24 '14 at 0:49
• Btw. let $n = |k|$ then there are $2^{n^2}$ functions from $log(n)$ to $n$ bits (where log is taken to be base 2). To see this note that all elements in $\{0,1\}^{log(n)}$ can be mapped to $2^n$ different values (all the strings in $\{0,1\}^n$). There are $n$ distinct elements in $\{0,1\}^{log(n)}$ so you have $\Pi^{n}_{i = 1}2^n = (2^n)^n = 2^{n^2}$ possible functions. – Guut Boy Dec 24 '14 at 1:09
• @Guut Boy - Thanks that makes sense! Yes it is the exact wording. – Alex Dec 24 '14 at 9:47
Though this is an 4-year old topic, it seems the following should work:
We can construct a function $F_k(x)$ with output length $l_{out}(n)=l_{key}(n)/2^{l_{in}(n)}=n/2^{O(\log n)}$.
Didivde the key $k$ into $2^{O(\log n)}$ blocks with equal length, denoted by $k_i$ with $i=1,2,\dots, 2^{O(\log n)}$. Because $k$ is uniformly distributed in $\{0,1\}^n$, so is $k_i$.
$F_k(x)= k_x$ is the pseudorandom function.
Preface: of course the following tests alone are not unconditional tests. See below for clarification.
Interesting post Alex. Dan Boneh from Stanford and the Coursera classes Cryptography I and II discusses the statistical test of PRG's which are related to PRF's in terms of an algorithm as the following:
$\{0,1\}^n$ bit strings such that $A(x)$ where $x$ is the input string--outputs $0$ or $1$. Where $0$ = not random and $1$ = random. As Katz discusses too PRF's are indistinguishable from a truly random function.
Examples include $A(x) = 1$ iff (if and only if) the # of $0$'s in the given string $x$ and the number of $1$'s in the string $x$ is not very different. #$0(x)$ - #$1<= 10 * \sqrt n$.
Here is a second example from Boneh: $A(x) = 1$ iff the number of consecutive 0's is the difference between $00x$ and $n/4<= 10*\sqrt n$. $n/4$ just represents the 25% chance from the uniform distribution.
In Boneh's third example we now see logs:
$A(x) = 1$ iff max-run-of-0$(x) <= 10* log_2 (n)$
In order to begin to actually test for a secure PRF/PRG we must look at the concept of advantage:
Where $G:K--> [0,1]^n$ is a PRG and A a statistical test on $[0,1]^n$ then we can define the following:
adv PRG [A,G]= Pr [A(G(k)) = 1 K<--R K - Pr [A(r) = 1] r<-- R {0,1}^n where we are calculating a specific probability of summing to 1 within the seed space and how likely an output from a ST outputed from a generator and 1 from a truly random string.
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https://www.sierrachart.com/index.php?page=doc/StudiesReference.php&ID=475&Name=Moving_Average_-_Arnaud_Legoux
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# Technical Studies Reference
### Moving Average - Arnaud Legoux
This study calculates and displays an Arnaud Legoux Moving Average (ALMA) of the data specified by the Input Data Input.
Let $$X$$ be a random variable denoting the Input Data, and let $$X_j$$ be the value of the Input Data at Index $$j$$. Let the Inputs Length, Sigma, and Offset be denoted as $$n$$, $$\sigma$$, and $$k$$, respectively. Then we denote the Moving Average - Arnaud Legoux at Index $$t$$ for the given Inputs as $$ALMA_t(X, n,\sigma, k)$$, and we compute it for $$t \geq n - 1$$ as follows.
$$\displaystyle{ALMA_t(X, n, \sigma, k) = \frac{\sum_{j = 0}^{n - 1}\exp\left(-\frac{(j - \lfloor k(n - 1) \rfloor)^2}{2n^2/\sigma^2}\right) \cdot X_{t - n + 1 + j}}{\sum_{j = 0}^{n - 1}\exp\left(-\frac{(j - \lfloor k(n - 1) \rfloor))^2}{2n^2/\sigma^2}\right)}}$$
For an explanation of the Sigma ($$\Sigma$$) notation for summation, refer to our description here.
For an explanation of the Floor Function ($$\lfloor \cdot \rfloor$$), refer to our description here
ALMA is a weighted moving average with Gaussian weights. It is advertised as a Gaussian filter, however caution should be exercised in this interpretation. The Input $$\sigma$$ does not play the role of the standard deviation of the Gaussians. Rather, the standard deviation is determined by $$\frac{n}{\sigma}$$. The mean, or center, of each Gaussian is determined by $$\lfloor k(n - 1) \rfloor$$
#### Inputs
• Input Data
• Length
• Sigma: This Input controls the width of the Gaussian distribution of the weights.
• Offset: This Input controls the center of the Gaussian distribution of the weights.
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https://socratic.org/questions/if-x-3y-0-and-5x-y-14-then-what-is-6x-4y
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Algebra
Topics
# If x - 3y = 0 and 5x - y = 14, then what is 6x - 4y?
$6 x - 4 y = \left(x - 3 y\right) + \left(5 x - y\right) = 0 + 14 = 14$.
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https://www.studyadda.com/solved-papers/jee-main-paper-held-on-09-4-2019-morning_q48/819/374832
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• question_answer The standard Gibbs energy for the given cell reaction in $kJ\,mo{{l}^{-1}}$ at 298 K is : $Zn(s)+C{{u}^{2+}}(aq)\to Z{{n}^{2+}}(aq)+Cu(s),$ ${{E}^{o}}=2\,V\,at\,298K$ (Faraday's constant, $F=96000\,C\,mo{{l}^{-1}}$) [JEE Main 9-4-2019 Morning] A) -384B) -192C) 192 D) 384
$\Delta {{G}^{o}}=-nF{{E}^{o}}_{cell}$ $=2\times 96000\times 2$ $=384000\text{ }J$ $=384\text{ }kJ$ $\therefore$ Ans. is [a]
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http://math.stackexchange.com/questions/11384/what-is-the-order-of-the-set-of-distinct-up-to-similarity-nxn-matrices-over-r
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# What is the order of the set of distinct (up to similarity) nxn matrices over R?
What is the order of the set of distinct (up to similarity) nxn matrices over $\mathbb{R}$ with determinant equal to some non-zero scalar... say 6? (eg. countable, uncountable etc.)
The set of matrices over $\mathbb{R}$ is uncountable. So is the order of the set consisting of classes of matrices with the same determinant. In each of those classes we have further subsets, the equivalence classes formed by grouping similar matrices. Each equivalence class of similar matrices represents one linear transformation expressed in terms of all possible bases of $\mathbb{R}^n$, so the size of each equivalence class is also uncountable.
What I'm not certian about is the size of the set of "different" transformations that have the same determinant. Is it uncountable too?
What can I put it in a correspondence with to show this?
Apologies if this is poorly worded-- let me know if there is a better way to ask this question.
-
No apologies necessary. I think the question is very clearly worded. Minor nitpick: the similarity class of a scalar multiple of the identity matrix is not uncountable. – Jonas Meyer Nov 22 '10 at 21:01
It has the same cardinality as $\mathbb{R}$ if $n\gt 1$. Consider diagonal matrices with entries $(x,6/x,1,1,\ldots,1)$, $x\neq0$. Two such are similar only in a situation where $\{x,6/x\}=\{y,6/y\}$, hence the conclusion on cardinality. (Of course $6$ is not special.)
(The only additional detail worth mentioning is that ${}|{\mathbb R}|$ is an upper bound, and so this argument gives equality. Asaf's answer addresses this detail.) – Andres Caicedo Nov 23 '10 at 2:34
Asaf's answer was deleted. Anyway, just in case: The set of $n\times n$ matrices with real entries is obviously in bijection with ${\mathbb R}^{n^2}$, which has the same size as ${\mathbb R}$. This gives the upper bound. – Andres Caicedo Nov 23 '10 at 6:50
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https://ehp.niehs.nih.gov/ehp959/
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## Research September 2017 | Volume 125 | Issue 9
Environ Health Perspect; DOI:10.1289/EHP959
# Changes in Transportation-Related Air Pollution Exposures by Race-Ethnicity and Socioeconomic Status: Outdoor Nitrogen Dioxide in the United States in 2000 and 2010
Lara P. Clark,1,2 Dylan B. Millet,1,3 and Julian D. Marshall2
Author Affiliations open
1Department of Civil, Environmental, and Geo-Engineering, University of Minnesota, Minneapolis, Minnesota, USA
2Department of Civil and Environmental Engineering, University of Washington, Seattle, Washington, USA
3Department of Soil, Water, and Climate, University of Minnesota, St. Paul, Minnesota, USA
• Background:
Disparities in exposure to air pollution by race-ethnicity and by socioeconomic status have been documented in the United States, but the impacts of declining transportation-related air pollutant emissions on disparities in exposure have not been studied in detail.
Objective:
This study was designed to estimate changes over time (2000 to 2010) in disparities in exposure to outdoor concentrations of a transportation-related air pollutant, nitrogen dioxide (NO2), in the United States.
Methods:
We combined annual average NO2 concentration estimates from a temporal land use regression model with Census demographic data to estimate outdoor exposures by race-ethnicity, socioeconomic characteristics (income, age, education), and by location (region, state, county, urban area) for the contiguous United States in 2000 and 2010.
Results:
Estimated annual average NO2 concentrations decreased from 2000 to 2010 for all of the race-ethnicity and socioeconomic status groups, including a decrease from 17.6 ppb to 10.7 ppb (−6.9 ppb) in nonwhite [non-(white alone, non-Hispanic)] populations, and 12.6 ppb to 7.8 ppb (−4.7 ppb) in white (white alone, non-Hispanic) populations. In 2000 and 2010, disparities in NO2 concentrations were larger by race-ethnicity than by income. Although the national nonwhite–white mean NO2 concentration disparity decreased from a difference of 5.0 ppb in 2000 to 2.9 ppb in 2010, estimated mean NO2 concentrations remained 37% higher for nonwhites than whites in 2010 (40% higher in 2000), and nonwhites were 2.5 times more likely than whites to live in a block group with an average NO2 concentration above the WHO annual guideline in 2010 (3.0 times more likely in 2000).
Conclusions:
Findings suggest that absolute NO2 exposure disparities by race-ethnicity decreased from 2000 to 2010, but relative NO2 exposure disparities persisted, with higher NO2 concentrations for nonwhites than whites in 2010. https://doi.org/10.1289/EHP959
Revised: 07 June 2017
Accepted: 09 June 2017
Published: 14 September 2017
Please address correspondence to J.D. Marshall, Dept. of Civil, Environmental, and Geo-Engineering, University of Washington, 201 More Hall, Seattle, WA 98195 USA. Telephone: (206) 685-2591. Email: [email protected]
Supplemental Material is available online (https://doi.org/10.1289/EHP959).
The authors declare they have no actual or potential competing financial interests.
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### Introduction
Environmental injustice describes conditions in which more vulnerable communities experience disproportionate burdens of environmental health risks, such as exposure to air pollution. Environmental injustice in air pollution has been widely documented in the United States: many (>140) studies, covering a range of pollutants and U.S. locations, found higher air pollution exposures for lower-income groups and/or for race-ethnicity minority groups (Marshall et al. 2014). A key knowledge gap is whether environmental injustice has changed over time in the United States (Mohai and Saha 2015; Hajat et al. 2015). Longitudinal studies are needed to evaluate impacts of environmental policies on equity (Bento et al. 2015; Post et al. 2011), to explore the underlying causes of environmental injustice (Pastor et al. 2001), to enable tracking of environmental justice outcomes over time (Payne-Sturges and Gee 2006), and to test relationships between health disparities and exposure disparities over time (Mohai et al. 2009).
The goal of the present study was to estimate changes over time in environmental injustice in exposure to outdoor concentrations of a transportation-related air pollutant (TRAP) for the contiguous United States. Previous studies explored environmental injustice aspects of distributions of benefits (e.g., accessibility) and costs (e.g., noise) of transportation (Schweitzer and Valenzuela 2004). We focused on exposure to air pollution as a cost of transportation emissions that often differs by race-ethnicity and/or socioeconomic status in the United States. Racial minorities and low-income households are disproportionately likely to live near a major road [e.g., 27% of racial minorities vs. 19% of the total population lived near high traffic volume roads in the United States in 2010 (based on an analysis of national census and traffic data; Rowangould 2013)], where TRAP concentrations are typically highest (e.g., nitrogen dioxide concentrations were on average 2.9 times higher near major roads than urban background levels [based on a synthesis of monitoring studies in multiple cities; Karner et al. 2010]).
Previous U.S.-based longitudinal air pollution environmental justice studies have focused on exposure to industrial air pollution or proximity to polluting industrial facilities. Ard (2015) studied annual average concentrations of industrial air pollution nationwide during 1995–2004 and found that exposures decreased for all race-ethnicity groups over time, but African Americans remained more exposed than whites and Hispanics (by a factor of ∼50%). Longitudinal case studies on residential proximity to polluting industrial facilities [e.g., Seattle, 1990–2007 (Abel and White 2011); southern California, 1990–2000 (Hipp and Lakon 2010); in a national cohort, 1990–2007 (Pais et al. 2014)] found that race-ethnicity minority groups and/or lower socioeconomic status groups experienced closer average proximity to industrial facilities compared with other groups, and this pattern persisted over time. Few U.S.-based studies have explored temporal trends in environmental injustice for ambient air pollution or for transportation-related air pollution. Brajer and Hall (2005), studying ozone and coarse particulate matter in southern California during 1990–1999, found that on average, as air pollution decreased over time, Asians and Hispanics experienced larger reductions in ozone concentrations but smaller reductions in coarse particulate matter concentrations, compared with other groups. Kravitz-Wirtz et al. (2016), studying nitrogen dioxide and particulate matter exposures in the United States for a cohort of ∼9,000 families during 1990–2009, found that as exposures decreased over time, exposures remained higher for blacks and Hispanics than for whites.
We focused on nitrogen dioxide (NO2) as a TRAP. Transportation sources accounted for an estimated 60% of anthropogenic NOx emissions in the United States in 2010 (U.S. EPA 2016), and NO2 is an indicator of local transportation-related emissions (Brook et al. 2007; Burnett et al. 2004; Levy et al. 2014) with high within-urban spatial variability (Hewitt 1991; Apte et al. 2017). The U.S. EPA regulates outdoor annual NO2 as one of six criteria pollutants, in part because exposure to NO2 (together with other co-emitted TRAPs) is associated with health impacts, including low birth weight (Brauer et al. 2008), asthma in children (Takenoue et al. 2012), and cardiovascular mortality (Jerrett et al. 2013).
Air quality improved substantially in the United States after the 1990 Clean Air Act Amendments (Clean Air Act Amendments of 1990). From 2000 to 2010, estimated annual anthropogenic NOx emissions in the United States decreased by ∼50% (U.S. EPA 2016). It is unknown to what extent these estimated emission-reductions impacted NO2 exposure disparities by race-ethnicity and by socioeconomic status. To investigate, we combined NO2 air pollution data from a spatially precise (Census block scale) temporal land use regression model (Bechle et al. 2015) with Census demographic data (MPC 2011) and then estimated changes in TRAP environmental injustice over a decade (2000 to 2010) for the contiguous United States.
### Methods
#### Study Area and Time Points
Analyses covered the contiguous United States (48 states plus the District of Columbia; selected based on availability of air pollution data) for two time points (selected based on availability of decennial Census demographic data): year 2000 (population: 280 million) and year 2010 (population: 306 million).
#### Datasets
##### Air pollution data.
Air pollution estimates were annual average NO2 concentrations for 2000 and 2010. These values were from a monthly land use regression (LUR) model incorporating satellite-based and ground-based observations (Bechle et al. 2015) for Census blocks [in 2010, n=8.2 million; mean area=0.97 km2 (total), 0.048 km2 (urban), 1.8 km2 (rural)].
##### Demographic data.
Demographic data were population estimates from the Census by race-ethnicity, socioeconomic status, language, and age. Demographic data included race (seven categories: white alone, black or African American alone, Asian alone, Native Hawaiian or other Pacific Islander alone, American Indian or Alaska Native alone, other race alone, two or more races), ethnicity (two categories: Hispanic, non-Hispanic), per capita income (continuous variable), household income [five categories (approximate annual household income quintiles): <$20,000,$20,000–$35,000,$35,000–$50,000,$50,000–$75,000, >$75,000], poverty (two categories: below poverty level, at or above poverty level), highest level of education for population >25 y (five categories: less than high school degree, high school degree, some college, college degree, graduate degree), employment for population >16 y (two categories: employed, unemployed), household language (five categories: English only, Spanish, other Indo-European language, Asian language, other language), household linguistic isolation [two categories: linguistically isolated (no one >14 y speaks English “well” or “very well”), not linguistically isolated], and age [four categories: younger children (<5 y), older children (5–18 y), younger adults (18–65 y), older adults (>65 y)]. Demographic data for 2000 were from the Decennial Census (2000 estimated populations for all demographic characteristics) and for 2010 were from the decennial Census (2010 estimated populations by race-ethnicity and by age) and the American Community Survey (2008–2012 five-year estimated populations for all other demographic characteristics not reported in the 2010 decennial Census) at the Census block group level [in 2010, n=210,000 (total); area mean (interquartile range)=36 km2 (0.49 km2–9.1 km2) (total); 1.1 km2 (0.34 km2–1.3 km2) (urban); 200 km2 (32 km2–150 km2) (rural)], the finest spatial scale for which detailed Census data are publicly available.
#### Spatial and Temporal Matching of Air Pollution and Demographic Data
To match the air pollution data (block level) with demographic data (block group level), we calculated population-weighted mean annual NO2 concentrations for all block centroids within each block group boundary, for 2000 and 2010. Boundaries for Census urban areas (defined based on population, population density, land cover, and other criteria; U.S. Census Bureau 2011) and boundaries for smaller Census geographies (blocks and block groups) changed during 2000 to 2010. For analyses comparing consistent block group boundaries over time, we applied the National Historic Geographic Information System time-series data: estimates of 2000 population counts and race-ethnicity within 2010 block group boundaries (MPC 2011). To match urban area data over time, we applied the 2010 urban area definitions to both 2000 and 2010 block groups, including block groups for which all blocks were inside the urban area boundary.
#### Urban and Rural Block Group Definitions
For urban versus rural comparisons, we applied the following definitions based on 2010 Census urban definitions: urban block groups contain only urban blocks (65%; n=140,000 in 2010), rural block groups contain only rural blocks (13%; n=28,000 in 2010), and mixed block groups contain both urban and rural blocks (22%; n=47,000 in 2010).
#### Exposure Assessment
Exposure assessment was based on residential block group LUR estimates of outdoor annual average NO2 concentrations.
#### Analyses Estimating Changes in NO2 Environmental Injustice over Time
We applied three related approaches to estimating changes in NO2 environmental injustice over time: a) we estimated and compared NO2 concentrations for populations defined by demographic characteristics (e.g., race-ethnicity groups); b) we estimated and compared NO2 concentrations for block groups (as proxies for “neighborhoods” or “local areas”) by demographic characteristics (e.g., per capita income); and c) we estimated and compared NO2 environmental injustice metrics on a national basis and for regions, states, counties, and urban areas.
##### Estimated changes in NO2 concentrations by demographic groups.
Our analyses by demographic groups focused on categories of race-ethnicity (14 groups), age (4 groups), household income (5 groups), and educational attainment (5 groups). We also performed analyses with race-ethnicity dichotomized as “white” or “nonwhite,” where the white population was defined as the race-ethnicity majority group (i.e., the “white alone, non-Hispanic” population; 69% of population in 2000, 64% in 2010), and the nonwhite population included all other race-ethnicity minority groups combined (i.e., the non-“white alone, non-Hispanic” population). In addition, we performed supplemental analyses of populations by household primary language and linguistic isolation (combined, 13 groups), employment status (unemployed, employed), and poverty (below or above poverty level). For all analyses by demographic groups, we conducted analyses for the total population, and separately for the urban and rural populations.
We estimated the population-weighted mean annual NO2 concentration C for each demographic group j in each year t (2000 or 2010) as
where ci is the annual mean NO2 concentration for block group i, pji is the population of demographic group j in block group i, and n is the total number of block groups.
To compare population-weighted mean NO2 concentrations between demographic groups j1 and j2 in year t (cross-sectional comparisons), we estimated absolute differences as Cj1Cj2, and relative percent differences as {100(Cj1Cj2)/[(Cj1+Cj2)/2]}. To compare population-weighted mean NO2 concentrations between 2000 (t1) and 2010 (t2) for group j (temporal comparisons), we estimated the absolute change as Ct2Ct1, and the relative percent change as {100(Ct2>−Ct1)/[(Ct2+Ct1)/2]}. Changes were calculated such that negative values indicate a decrease in NO2 concentration over time.
##### Estimated changes in NO2 concentrations by block group demographic characteristics.
To quantify differences by local (i.e., block group) demographic characteristics, we compared estimated mean NO2 concentrations in each year between block groups with proportions of nonwhite residents in the highest and lowest 5% of the distribution for all block groups in the United States in each year. We analyzed data for all block groups combined and separately for urban and rural block groups.
To explore block group differences in NO2 concentrations by race-ethnicity, income, and size of urban area, we categorized urban block groups by percent nonwhite in each year (quintiles), average per capita income in 2010 (20 equal groups), and total urban area population in 2010 (tertiles; large: 4.2 million to 18 million residents, n=8, total population=61 million; medium: 830,000 to 3.8 million residents, n=35, total population=63 million; and small: 14,000 to 800,000 residents, n=438, total population=61 million). We then compared estimated mean NO2 concentrations according to average per capita income (by approximate interquartile range in 2010 per capita income: $18,000 to$33,000) between urban block groups with percent nonwhite populations in the highest and lowest quintile of the national distribution for each year, after stratifying by small, medium, or large urban area population size.
##### Estimated changes in NO2 environmental injustice metrics.
To quantify how environmental injustice has changed over time on a national basis and for different U.S. geographies, we calculated and compared environmental injustice metrics in 2000 and 2010 on a national basis and by region, state, county, and urban area. Our core environmental injustice metric is the difference in estimated population-weighted mean NO2 concentration (Equation 1) for nonwhites versus whites [i.e., (population-weighted mean NO2 concentration for nonwhites) − (population-weighted mean NO2 concentration for whites)], hereafter referred to as the “nonwhite–white NO2 disparity.” As supplements to the nonwhite–white NO2 disparity, we calculated alternate environmental injustice metrics by race-ethnicity {for the three largest minority race-ethnicity groups: black–white NO2 disparity [difference in estimated population-weighted mean NO2 concentration for non-Hispanic blacks and non-Hispanic whites], Hispanic–white NO2 disparity [difference in estimated population-weighted mean NO2 concentration for Hispanics of any race(s) and non-Hispanic whites], and Asian–white NO2 disparity [difference in estimated population-weighted mean NO2 concentration for non-Hispanic Asians and non-Hispanic whites]} and by income (difference in estimated population-weighted mean NO2 concentration for the population with income below the poverty level and the population with income two times the poverty level). We calculated correlations (Pearson’s correlation coefficient; Spearman’s rank coefficient) among the changes in the alternate environmental injustice metrics for states, counties, and urban areas.
#### Potential Influence of Changes in NO2 Emissions and Changes in Demographic Patterns to Changes in Environmental Injustice over Time
As a preliminary step in understanding underlying mechanisms for changes over time in TRAP environmental injustice, we explored potential contributions of two factors: emission-reductions and residential demographic patterns. To estimate the potential extent to which each factor separately contributed to changes in NO2 environmental injustice, we considered two counterfactual scenarios with the following assumptions: a) NO2 concentrations changed as observed (from 2000 to 2010), but residential demographic patterns remained constant (at year-2000 values); and b) residential demographic patterns changed as observed (from 2000 to 2010), but NO2 concentrations remained constant (at year-2000 values). We then calculated the core national environmental injustice metric (nonwhite–white NO2 disparity) for each scenario. To estimate the contribution of changes in NO2 concentrations alone, we divided the predicted change in the national nonwhite–white NO2 disparity calculated under counterfactual scenario a by the observed change in the national nonwhite–white NO2 disparity. To estimate the contribution of changes in residential demographic patterns alone, we divided the change in national nonwhite–white NO2 disparity calculated under counterfactual scenario b by the observed change in the nonwhite–white NO2 disparity.
#### Potential Relevance of Changes in Environmental Injustice for Public Health
As a preliminary step to explore the potential health relevance of the observed gaps in NO2 exposures, we a) compared estimated exposures to health-based air quality guidelines and b) conducted an illustrative (“back-of-the-envelope”) health impact calculation. We compared the proportion of nonwhites versus whites living in block groups with NO2 concentrations above the WHO annual guideline [>40 μg/m3 (corresponds approximately to >21 ppb) NO2; WHO 2005] and below 50% of the WHO guideline (<11 ppb). [All block groups were below the U.S. EPA annual standard for NO2 (53 ppb) in 2000 and 2010.] We estimated potential health impacts for one outcome [ischemic heart disease (IHD) mortality, the most common cause of death in the United States (CDC 2015)] attributable to the difference in national mean NO2 concentration for nonwhites and whites in 2000 and 2010. We assumed the relative risk (RR) of IHD mortality associated with outdoor annual average NO2 concentration was 1.066 [95% confidence interval (CI): 1.015, 1.119] per 4.1 ppb NO2 (based on a cohort of 74,000 adults in California during 1982–2000; Jerrett et al. 2013). Relative risks (RR) for NO2 concentrations experienced by nonwhites and whites were calculated using: RR=exp (βC), where C is the population-weighted mean NO2 concentration (Equation 1), and β=ln(1.066)/(4.1 ppb)=0.0156 ppb−1. To obtain a simplified estimate that reflects only the estimated potential impact of changes in NO2 exposure over time experienced on average by nonwhites and whites (all else equal), our health risk calculations assumed that the underlying IHD mortality rate was constant over time [using the year-2011 estimate: 109 deaths per 100,000 (CDC 2012), although IHD mortality rates decreased during this time period in the United States (Finegold et al. 2013; WHO 2016)], and that the underlying mortality rate was the same for nonwhites and whites and the same by U.S. location [although IHD mortality rates differed by race-ethnicity and by U.S. location during this time period (CDC 2016)].
#### Sensitivity Analyses on Uncertainty in NO2 LUR Model Estimates
To assess the potential impact of exposure misclassification on our findings, we tested whether NO2 LUR model residuals showed systematic bias with respect to demographic characteristics. We compared annual average NO2 concentrations based on measurements from 366 U.S. EPA monitors in 2006 (the base year for the temporal LUR model; Bechle et al. 2015) with the LUR-based estimates for each block group in which a monitor was located. We then compared the distributions of the LUR model residuals (i.e., the measured – predicted values) among block groups categorized by tertiles of percent nonwhite residents and tertiles of average per capita income in 2010. In addition, we compared the nonwhite–white NO2 disparity (core environmental injustice metric) based on U.S. EPA monitor data versus LUR model estimates for the 366 block groups with U.S. EPA monitors.
### Results
#### Estimated Changes in NO2 Concentrations by Demographic Groups
Consistent with national trends, outdoor annual average NO2 concentrations decreased substantially across all race-ethnicity, income, education, and age groups during 2000 to 2010. Overall, on a national basis, the estimated population-weighted mean NO2 concentration decreased from 14.1 ppb in 2000 to 8.9 ppb in 2010, an absolute change of −5.2 ppb and a relative change of −37% (Table 1). Estimated changes among groups defined by race-ethnicity, income, age, and education ranged from −3.5 ppb to −8.6 ppb (−33% to −42%).
Table 1. Estimated NO2 population-weighted mean concentration (ppb) for year 2000, year 2010, and estimated change over time (year 2010–year 2000), by race-ethnicity, household income quintile, educational attainment, and age.
Demographic characteristic Population (%) Mean NO2 concentration (ppb) Change in mean NO2 concentration:
Absolute (ppb) Relative (%)
2000 2010 2000 2010 2010–2000 2010–2000
Total 100 100 14.1 8.9 −5.2 −37
Race-ethnicity
Non-Hispanic 87 84 13.4 8.4 −5.0 −37
White alone 69 64 12.6 7.8 −4.7 −38
Black or African American alone 12 12 16.2 10.0 −6.1 −38
American Indian or Native American alone 0.7 0.7 10.1 6.6 −3.5 −35
Asian alone 3.4 4.5 20.2 12.1 −8.1 −40
Native Hawaiian or other Pacific Islander alone 0.1 0.1 17.7 10.6 −7.1 −40
Other race alone 0.2 0.2 17.9 10.8 −7.1 −40
Two or more races 1.6 1.8 16.1 9.3 −6.8 −42
Hispanic 13 16 18.9 11.2 −7.7 −41
White alone 6.0 8.7 17.6 10.6 −7.0 −40
Black or African American alone 0.3 0.4 20.8 12.2 −8.6 −41
American Indian or Native American alone 0.1 0.2 18.8 11.2 −7.6 −41
Asian alone 0.04 0.1 19.3 11.8 −7.5 −39
Native Hawaiian or other Pacific Islander alone 0.01 0.02 18.4 10.8 −7.6 −41
Other race alone 5.3 6.0 20.2 12.0 −8.2 −41
Two or more races 0.8 1.0 19.3 11.3 −8.0 −41
Household income quintilea
<$20,000 8.3 6.7 14.2 9.0 −5.2 −36$20,000–$35,000 7.3 5.9 13.7 8.7 −5.0 −37$35,000–$50,000 6.2 5.1 13.7 8.6 −5.0 −37$50,000–$75,000 7.3 6.8 13.8 8.6 −5.2 −38 >$75,000 8.4 13 14.6 9.0 −5.7 −39
Educational attainmentb
<High school degree 13 19 14.9 9.3 −5.6 −37
High school degree 19 10 13.2 8.8 −4.4 −33
Some college 18 12 13.7 8.9 −4.9 −35
College degree 10 5.5 14.6 9.3 −5.3 −36
Graduate degree 5.7 6.2 14.9 9.3 −5.6 −38
Age (y)
<5 6.8 6.5 14.4 9.0 −5.4 −38
5–17 19 17 14.0 8.8 −5.2 −37
18–65 62 63 14.2 9.0 −5.2 −37
>65 12 13 13.7 8.4 −5.3 −38
aHousehold income quintiles are based on year-2000 population and income data. Income is reported for householders (38% of the total population in year 2000).
bEducational attainment data is reported for population >25 y (65% of the total population in year 2000).
In general, the groups with the highest estimated NO2 exposures in 2000 experienced the largest reductions in NO2 concentrations from year 2000 to year 2010 (see Figures S1 and S2). As an example, the Hispanic black group, the group with the highest estimated mean NO2 exposure in 2000 [20.8 ppb; 6.6 ppb (38%) higher than the national mean] experienced the largest estimated reduction in NO2 exposure from 2000 to 2010 [−8.6 ppb, a 3.3 ppb (48%) greater concentration reduction than the national mean reduction].
In 2000 and 2010, disparities in estimated mean NO2 concentrations were larger by race-ethnicity group than by income, education, or age group (Table 1). For example, in 2000, mean NO2 concentrations for race-ethnicity groups ranged from 10.1 ppb (non-Hispanic American Indian group) to 20.8 ppb (black Hispanic group), a maximum difference of 10.7 ppb, compared with maximum differences of 1.7 ppb, 0.9 ppb, and 0.7 ppb between the education, income, and age groups with the highest and lowest mean exposures, respectively. In 2010, mean NO2 concentrations for race-ethnicity groups ranged from 6.6 ppb to 12.2 ppb (a maximum difference of 6.5 ppb), whereas mean values for all individual education, income, and age subgroups were within 1.0 ppb of the national average.
On a national basis, rankings (most to least exposed groups) remained fairly consistent over time (Figure 1). For the six largest race-ethnicity groups, rank-order by estimated population-weighted mean NO2 concentration remained constant with time: the non-Hispanic Asian group was most exposed and the non-Hispanic American Indian group was least exposed over time. Differences by age, income, and education were small compared with differences by race-ethnicity in both time periods.
After controlling for urban versus rural location (see Figures S3 and S4, Table S1), disparities in NO2 concentrations by race-ethnicity persisted (with higher concentrations and higher disparities in urban than in rural locations), with some differences in exposure patterns for demographic groups by urban versus rural location in each year. For example, estimated population-weighted mean NO2 concentrations were lower for non-Hispanic American Indians than non-Hispanic whites in rural locations (−1.3 ppb in 2000; −0.5 in 2010) but higher in urban locations (+0.2 ppb 2000; +0.1 ppb in 2010).
Results for supplemental measures of socioeconomic status (poverty, employment) and language (see Table S2) were generally consistent with the core demographic characteristics (race-ethnicity, income, education, and age). NO2 concentrations were higher for people below the poverty level than above the poverty level, for households with a language other than English than households with only English, and for linguistically isolated than nonlinguistically isolated households. NO2 concentrations were higher for employed than for unemployed populations.
#### Estimated Changes in NO2 Concentrations by Block Group Demographic Characteristics
Consistent with population-based results, block groups with a higher proportion of race-ethnicity minority residents tended to have higher concentrations of NO2, and this pattern was consistent over time (Figure 2). In 2000, the 5% of block groups with the highest proportion of nonwhite residents had 2.5 times higher [+13.2 ppb (22.1 ppb vs. 8.9 ppb)] estimated mean NO2 concentrations than the 5% of block groups with the lowest proportion of nonwhite residents; in 2010, the 2.5-fold gap had increased slightly, to 2.7-fold [+8.9 ppb (14.1 ppb vs. 5.2 ppb)]. Considering urban versus rural block groups separately (see Figure S5), urban results were consistent with national results [the 5% of urban block groups with the highest versus lowest proportion of nonwhite residents had 1.8 times higher [+10.3 ppb (23.6 ppb vs. 13.3 ppb)] mean NO2 concentration in 2000 and 1.8 times higher [+6.9 ppb (15.0 ppb vs. 8.1 ppb)] mean NO2 concentration in 2010), whereas rural results had the reverse pattern to a minor extent: NO2 concentrations were lower in block groups with a higher proportion of nonwhite residents (the 5% of rural block groups with the highest vs. lowest proportion of nonwhite residents had 0.7 times lower [−1.9 ppb (5.4 ppb vs. 7.3 ppb)] mean NO2 concentrations 2000] and 0.8 times lower [−0.9 (3.8 ppb vs. 4.6 ppb)] mean NO2 concentration in 2010).
In urban areas, disparities in block group estimated mean NO2 concentrations by race-ethnicity (for nonwhites vs. whites) persisted over time, regardless of average block group per capita income or the size of the urban area (large, medium, or small), and were generally larger than disparities by income (see Figure S6). For example, in large urban areas in 2010, estimated mean NO2 concentrations were 3.0 ppb higher (16.8 ppb vs. 13.8 ppb) for block groups with the highest versus lowest quintile percent nonwhite residents at the 25th percentile income ($18,000) and 4.2 ppb higher (16.4 ppb vs. 12.2 ppb) for block groups with the highest versus lowest quintile nonwhite residents at the 75th percentile income ($33,000). Estimated mean NO2 concentrations were 1.6 ppb higher (13.8 ppb vs. 12.2 ppb) for the block groups at the 25th percentile income than at the 75th percentile income among lowest quintile percent nonwhite block groups, and 0.4 ppb (16.8 ppb vs. 16.4 ppb) higher among the highest quintile percent nonwhite block groups. In large urban areas, in 2000, the estimated mean NO2 concentration was 2.9 ppb higher for highest income category block groups with the highest quintile nonwhite residents (mean per capita income: $74,000; mean percent nonwhite residents: 88%; mean NO2: 25.4 ppb; population: 56,000) than the lowest income block groups with the lowest quintile nonwhite residents (mean per capita income:$6,400; mean percent nonwhite residents: 2.9%; mean NO2: 22.4 ppb; population: 14,000), and in 2010, 1.2 ppb higher (16.7 ppb vs. 15.5 ppb).
#### Estimated Changes in NO2 Environmental Injustice Metrics
Nationally, on an absolute basis, environmental injustice declined from 2000 to 2010. The nonwhite–white NO2 disparity decreased from 5.0 ppb in 2000 to 2.9 ppb in 2010 (−2.1 ppb [−42%]; Table 2). However, nationally, on a relative basis, environmental injustice persisted. Nonwhites remained more exposed to outdoor NO2 air pollution than whites on average in 2010, and there was little change in the relative NO2 difference between nonwhites and whites between 2000 and 2010: The nonwhite–white NO2 difference was 33% in 2000 (nonwhites were 40% more exposed than whites) and 31% in 2010 (nonwhites were 37% more exposed than whites).
Table 2. Estimated population-weighted mean NO2 concentrations (ppb) for nonwhites and whites: year 2000, year 2010, and change over time (year 2010–year 2000).
Race-ethnicity 2000 2010 Change: 2010–2000
Nonwhitesa 17.6 10.7 −6.9 (−39%)
Whitesb 12.6 7.8 −4.7 (−38%)
Difference: nonwhites–whites 5.0 (33%) 2.9 (31%) −2.1 (−42%)
aNonwhites includes all race-ethnicity minority groups (i.e., people who reported any race-ethnicity other than white alone, non-Hispanic).
bWhites includes people who reported white alone, non-Hispanic race-ethnicity.
Environmental injustice declined in most, but not all, locations. In all regions and in most (>75%) states, counties, and urban areas, the nonwhite–white NO2 disparity decreased over time (Figure 3). The nonwhite–white NO2 disparity decreased by >1 ppb in 16 urban areas (accounting for 32% of the urban area population; 49 million in year 2000), including Detroit (Michigan), Los Angeles (California), New Orleans (Louisiana), and Chicago (Illinois). The nonwhite–white NO2 disparity increased by >1 ppb in two urban areas (accounting for <1% of the urban population): Watertown (New York) and Delano (California): both are urban areas for which mean NO2 concentrations were higher for whites than nonwhites in 2000, and for which concentrations decreased to a greater extent for whites than for nonwhites during 2000 to 2010. Similar patterns hold among counties: the nonwhite–white NO2 disparity decreased by >1 ppb in 75 counties (accounting for 16% of the population in 2000), and increased by >1 ppb in 6 counties (accounting for <0.1% of the population in 2000), for all of which NO2 concentrations were higher for whites than for nonwhites in 2000.
The alternate environmental injustice metrics considered (see Figures S7–S10) were moderately correlated (see Tables S3–S5). For example, for urban areas, changes in alternate environmental injustice metrics were moderately correlated (Pearson’s correlation coefficient, r, range: 0.3–0.8; Spearman’s rank coefficient, s, range: 0.2–0.9). New York and California had large reductions (high decile reductions) in all five environmental injustice metrics, and North Dakota had increases (low decile reductions) in all five environmental injustice metrics. Similar to the patterns for the nonwhite–white NO2 disparity, the black–white, Hispanic–white, and Asian–white NO2 disparity decreased in most (>75%) regions, states, counties, and urban areas from 2000 to 2010. In contrast, the poverty-based NO2 disparity increased in nearly half of states and counties, although in general, the poverty-based NO2 disparities were smaller than the race-based NO2 disparity metrics (e.g., among states the mean change in the poverty-based NO2 disparity was −0.2 ppb vs. −1.0 ppb for the Asian–white NO2 disparity). Estimated population-weighted mean NO2 concentrations and environmental injustice metrics for each region, state, county, and urban area included in our analyses are available in Supplemental Material (Excel Tables A-D).
#### Potential Influence of Changes in NO2 Emissions and Changes in Demographic Patterns to Changes in Environmental Injustice over Time
When we estimated what population-weighted mean NO2 concentrations in 2010 would have been if residential demographic patterns changed as observed but NO2 concentrations were fixed as in 2000, we predicted a decrease in mean NO2 exposure for nonwhites from 17.6 ppb to 16.6 ppb (−1.0 ppb) and for whites from 12.6 ppb to 12.1 ppb in whites (−0.5 ppb), for a change of −0.6 ppb in the nonwhite–white NO2 disparity over time (5.0 ppb in 2000, 4.5 ppb in 2010), in contrast with the estimated change of −2.1 ppb in the nonwhite–white NO2 disparity (Table 2). When we estimated what population-weighted mean NO2 concentrations in 2010 would have been if residential demographic patterns were fixed as in 2000 but NO2 concentrations decreased as observed, we predicted a decrease in mean NO2 exposure for nonwhites to 11.4 ppb (−6.3 ppb) and for whites to 8.1 ppb (−4.5 ppb), for a change of −1.8 ppb in the nonwhite-white NO2 disparity over time (5.0 ppb in 2000, 3.3 ppb in 2010). This analysis of counterfactual scenarios suggests that both changes in NO2 and changes in residential demographic patterns contributed to the observed reductions in the national nonwhite-white NO2 disparity, with changes in NO2 contributing to a larger extent (83%, i.e., −1.8 ppb of the −2.1 ppb observed change in environmental injustice metric) than changes in residential demographic patterns (26%, i.e., −0.6 ppb of the −2.1 ppb observed change in environmental injustice metric). [The individual contributions of these two factors sum to greater than 100%, indicating interaction effects (9%) due to air pollution and population changing together.]
#### Potential Relevance of Changes in Environmental Injustice for Public Health
In 2000 and in 2010, nonwhites were more likely than whites to live in block groups with NO2 concentrations above international health-based guidelines. In 2000, 30% of nonwhites and 10% of whites lived in block groups with NO2 concentrations above the WHO annual guideline (>21 ppb), compared with 5% of nonwhites and 2% of whites in 2010 (see Figures S11–S12, Table S6). Thus, nonwhites were three times as likely as whites to live in a block group above the WHO guideline in 2000, and 2.5 times as likely in 2010. Conversely, 23% of nonwhites and 44% of whites lived in block groups with NO2 concentrations below 50% of the WHO guideline (<11 ppb) in 2000, compared with 56% of nonwhites and 80% of whites in 2010. Thus, nonwhites were 0.5 and 0.7 times as likely as whites to live in a block group with population-weighted mean NO2 concentrations <50% of the WHO guideline in 2000 and 2010, respectively. For the urban population in 2000 and 2010, nonwhites were 2.1 times and 3.5 times as likely, respectively, to live in a block group with mean NO2 concentrations above the WHO guideline. Most of the rural population (95% of whites and 97% of nonwhites in 2000; 99% of whites and nonwhites in 2010) lived in blocks groups with NO2 concentrations below 50% of the WHO guidelines.
Based on the simplified health impact calculation, the estimated mean NO2 concentration burden for nonwhites relative to whites (5.0 ppb in 2000, 2.9 ppb in 2010) was associated with an estimated ∼7,000 (95% CI: 2000, 10,000) additional premature IHD deaths for nonwhites in the United States in 2000 and an estimated ∼5,000 (95% CI: 1,000, 9,000) in 2010 (calculations presented in Table S7). Thus, the reduction in the mean nonwhite–white NO2 disparity (−2.1 ppb between 2000 and 2010) was associated with preventing an estimated ∼2,000 (95% CI: 400, 3,000) premature IHD deaths per year among nonwhites. The purpose of this simplified (back-of-the-envelope) calculation was to provide background and context for concentration disparities reported here. This health impact calculation was limited by several important simplifying assumptions and considerations [i.e., this calculation assumed that the U.S. population breathed the national mean NO2 concentration, considered only one health impact (IHD mortality), assumed that the IHD mortality rate is constant over time and by race-ethnicity and U.S. location, and did not adjust for differences in age by race-ethnicity or over time]. This simplified health impact calculation suggests that the estimated nonwhite–white NO2 disparity may have been associated with potentially large health impacts (i.e., thousands of IHD deaths per year in the United States); more detailed analyses are needed to fully investigate the implications of NO2 disparities for public health.
#### Sensitivity Analyses on Uncertainty in NO2 LUR Model Estimates
When we compared LUR model-based NO2 estimates for the 366 block groups with U.S. EPA monitors to the monitor-based NO2 observations, median model-based NO2 concentrations were lower for block groups in the middle and highest tertiles of percent nonwhite residents, and higher for block groups in the lowest tertile of percent nonwhite residents (see Figure S13). Median model-based estimates were also higher than monitor-based estimates for block groups in the highest tertile of average per capita income. When we estimated the nonwhite–white NO2 disparity for these block groups in 2006 (the year for which monitor data were available; 670,000 people, 48% nonwhite) the disparity was larger when based on monitor data (3.3 ppb; 13.4 ppb vs. 10.1 ppb for nonwhites and whites, respectively) than LUR model predictions (2.3 ppb; 12.2 ppb vs. 9.8 ppb for nonwhites and whites, respectively). These findings suggest that our model-based results may under-estimate disparities in exposures.
### Discussion
Estimated average NO2 concentrations decreased for almost all U.S. populations and locations from 2000 to 2010. Disparities in average NO2 concentrations by race-ethnicity decreased on an absolute basis (e.g., the nonwhite–white difference decreased from 5.0 ppb in 2000 to 2.9 ppb in 2010). However, despite these improvements, estimated average annual concentrations continued to be higher for nonwhite populations than for white populations in 2010 (nonwhite–white difference: 31% in 2010, 33% in 2000). In 2010, the estimated average concentration in the 5% of block groups with the highest proportion of nonwhite residents was 2.7 times higher than in the 5% of block groups with the lowest proportion of nonwhite residents (2.5 times higher in 2000). Therefore, our findings suggest that over time, NO2 concentrations decreased; disparities by race-ethnicity decreased on an absolute basis but on a relative basis have persisted.
Our finding that, on a relative basis, NO2 air pollution disparities by race-ethnicity persisted in the United States over time is consistent with a recent U.S. cohort study that reported that estimated NO2 concentrations during 1990 to 2009 were ∼10% higher for blacks and Hispanics than whites, even after controlling for individual socioeconomic characteristics (income, employment, home ownership) and metropolitan area characteristics (residential segregation, industry) (Kravitz-Wirtz et al. 2016). Our findings are also consistent with a national study of industry-related air pollution that reported that, although estimated exposures to industrial hazardous air pollutants (HAPs) decreased in the United States during 1994–2005, HAPs exposures remained ∼1.5 times higher for African Americans than whites (Ard 2015); in our study, NO2 exposures remained ∼1.3 times higher for African Americans than whites.
Our findings suggested that most of the reduction in nonwhite–white NO2 disparities between 2000 and 2010 was attributable to overall reductions in outdoor NO2 concentrations. Emissions-reductions were achieved in part via emission-control technology in motor vehicles (particularly in gasoline vehicles during this time period; McDonald et al. 2012) and stationary sources (e.g., power plants) (U.S. EPA 2016). In addition, U.S. metropolitan regions became more suburban, and suburban areas became more racially diverse during 2000 to 2010 (Howell and Timberlake 2014). Shifts in demographic residential patterns leading to larger proportions of race-ethnicity minorities in suburban locations (where TRAP concentrations are typically lower compared with central cities or downtown locations) also may have contributed to reductions in NO2 disparities by race-ethnicity during this time period.
Our evidence of larger NO2 disparities by race-ethnicity than by income is consistent with previous studies of environmental injustice in TRAP (e.g., Clark et al. 2014) and with persistent patterns of residential segregation in U.S. metropolitan regions, which remain more segregated by race than by income (Reardon et al. 2015). Additional work is needed to further investigate potential underlying causes (e.g., changes in patterns of residential segregation) of changes in environmental injustice in exposure to TRAP over time.
Although absolute NO2 exposure disparities reduced substantially during this period, there remain potentially large public health benefits from eliminating these disparities: nonwhites remained 2.5 times more likely than whites to live in block groups above WHO guidelines for NO2 in 2010, and based on the back-of-the-envelope calculation described above, the estimated nonwhite–white NO2 disparity may have been associated with thousands of premature IHD deaths among nonwhites in 2010.
Our analyses have several important limitations. Due to limitations in the spatial resolution of the Census data, we were unable to explore spatial patterns in air pollution and demographics at spatial scales finer than Census block groups. We focused on outdoor air pollution exposures, and we were unable to explore the potential influence of time-activity patterns for which air pollution exposure gradients by race-ethnicity and socioeconomic status may exist, including exposures during commuting, at work, or indoors (O’Neill et al. 2003). In addition, we evaluated only one pollutant at only two time points. Spatial patterns may differ for other TRAPs or for cumulative exposures to multiple pollutants. We also did not account for joint effects (interactions) of race-ethnicity and socioeconomic characteristics. Finally, our estimates were limited by uncertainties in the NO2 LUR model estimates and Census data. The impact of uncertainties in the Census data, particularly for national race-ethnicity data that represent an almost complete sample of ∼300 million people, is likely to be small relative to the potential impact of uncertainties in NO2 LUR model estimates. Findings from a sensitivity analysis comparing results when NO2 exposure estimates were based on U.S. EPA monitor data instead of our LUR model suggested that exposure misclassification may have varied in a way that would have caused us to underestimate true disparities by race-ethnicity in outdoor NO2 concentrations in the United States. However, we were unable to directly test the potential consequences of exposure misclassification on our national-scale estimates of environmental injustice.
### Conclusion
During 2000 to 2010, estimated annual average exposures to outdoor NO2 air pollution declined across all race-ethnicity and socioeconomic groups [range of mean change: −33% to −42% (−3.5 ppb to −8.6 ppb)]. The most exposed groups in 2000 experienced, on average, the largest reductions in NO2 during 2000 to 2010. Disparities in NO2 exposure were larger by race-ethnicity than by other demographic characteristics (income, education, age) in 2000 and 2010, with higher exposures for race-ethnicity minorities. The estimated national mean nonwhite–white NO2 disparity decreased from 5.0 ppb in 2000 to 2.9 ppb in 2010. Most of this reduction in the national mean nonwhite–white NO2 disparity over time is attributable to reductions in outdoor NO2 concentrations, suggesting that existing efforts to reduce TRAP are also reducing TRAP exposure disparities by race-ethnicity over time. Despite these improvements in absolute exposures, relative exposure disparities persisted, with nonwhites remaining exposed to 37% more NO2 than whites on average in 2010, and 2.5 times more likely than whites to live in a block group with NO2 concentration above WHO guidelines in 2010. Overall, these findings suggest that continued improvements to air quality may further reduce TRAP exposure disparities by race-ethnicity. However, eliminating disparities may require additional policies and interventions that target the underlying causes of environmental injustice.
### Acknowledgments
The authors thank M. Bechle for providing the calculated Census block descriptive statistics and for his contributions to method design and data interpretation.
The authors are grateful for support from the National Science Foundation (NSF; Sustainability Research Network award 1444745 and grant 0853467) and the U.S. Environmental Protection Agency (EPA; Assistance Agreement RD83587301). This article has not been formally reviewed by the NSF or the U.S. EPA; views expressed herein are solely those of authors and do not necessarily reflect those of either agency.
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https://www.davideaversa.it/blog/turn-based-battle-systems-chapter-2-analytical-analysis/
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# Turn-based Battle Systems - Chapter 2 - Analytical Analysis
Hello everybody! It is time to continue with our series on Pokemon-like battle systems. In this chapter I will be more general: now we have our beautiful damage formula and a draft of our characters sheet, it is time to quickly check if the game the formula is balanced. There are a lot of questions that we want to answer as quick as possible. How much HP the enemy must have? What is a good value for the attack power? Is the damage formula fair enough? The critical strike is too much? The randomness is too much? Is it fun?
Except for the last question, we can try to answer them just looking at the damage formula. Obviously we can not answer all the questions now, but if there is some evident mistake, we will be able to catch it as soon as possible. There are several ways to analyse the tuning and balancing problem for a damage formula (and thus for a large part of the combat system): analytically, with Google Sheet/Excel, and with a quick software prototype.
In this article we will start from the first step: the analytical analysis.
## Analytical Analysis: The Basics
The simplest thing we can do is to look at the formula and understand how it changes with respect to the various parameters. How can we do that? Borrowing tools from the study of a function we used to do in high school (or equivalent). In this way we can quickly identify some critical conditions we want to avoid in your game but, most important, we can understand it. We can understand the effect of each attribute to the damage output and balance the game accordingly.
### Identify the Singularities
The first things is to identify singularities. Let’s look at our damage function. Just looking at it we can clearly see that there is one point in which the damage can be infinite (and infinite damage is not good). If EnemyDefense is 0 we can have infinite damage. So, for instance, this is a point you have to keep in mind when you will implement the Tail Whip move (reduce defense) because you have to guarantee that it is impossible to bring the enemy defense to zero. This is a trivial example but I hope you got the point. Other formulas can be much more complex and identifying singularities could be much more trickier.
Other singularities you should identify could be: negative damage, inconsistent damage (when the increase of a “positive” stats such attack power can reduce the damage) or “imaginary” damage (yep… this happened to me once…).
### Understanding the flow
Second, we need to understand how the damage output changes with the parameters. You have to ask yourself questions such as: if I have an enemy with 10 defense, how much the damage changes when the player attack increases (or decreases)? If attack and defense are fixed, how the damage change when the base damage of the Pokemon’s attack change? You have to find out this by yourselves! If you don’t, your players will and they will start exploiting every loophole they can find.
Moreover the “shape” of the damage function with respect to the variation of some character’s attributes can tell you a lot on how much important that parameters is and, for instance, guide you when you will start designing equipment, weapons, buffs and so on. There are a limited amount of shapes you can usually find out in this step.
But before we move on, there are two important concept that you have to grasp: the absolute increment or absolute effect (AE) and the marginal effect (ME). The absolute effect is how much the damage increase (or decrease) if I increase (or decrease) the target attribute by one. For instance, if I am looking at the attribute p the AE value is just:
$$AE = Damage(p+1) - Damage(p)$$
The marginal effect, instead, represents how “effective” is the damage increase (or reduction) respect to the previous value. For instance, a ME value of 0.3 means that increasing the attribute by one point you obtain a 30% increased damage. Mathematically:
$$ME = \frac{Damage(p+1) - Damage(p)}{Damage(p)} = \frac{AE}{Damage(p)}$$
For the law of diminishing returns the absolute value of ME must decrease when the attribute increase. If not, there is a problem in your formula! Diminishing returns is important in this kind of games because, in short, is the reason why your sword with + 10 is so good when you are Level 1 and so shitty when you are level 90. Diminishing returns is your only weapon to control players preogression. We will talk a lot of the diminishing returns law in the next chapters.
#### Linear Attributes
When you increase the attribute, the damage changes in a straight line. This means that the effect of the attribute you are exploring is proportional to the attribute itself. Mathematically you can rewrite your function in this form
$$Damage(x) = k x + q$$
In this formula k means: how much more damage I get if I increase the attribute (e.g., the strength) by one point? At the same time q is the residual, that is the damage when the attribute is zero. Usually this value is small and we can forgot about that.
In the Pokemon formula, the PlayerAttack attribute is a linear attribute. In fact we can rewrite the formula in this way:
$$Damage = PlayerAttack \left( \frac{\alpha + Base}{EnemyDefense} Multiplier \right) + (2 \times Multiplier)$$
Where the constant alpha is that strange value who depends on the player’s level.
Assuming q small enough, the ME value of a linear attribute is always:
$$ME = \frac{1}{x}$$
Note that the marginal effect does not depends on the k! So, for example, moving from 2 attack to 3 attack will always increase your damage by 50% (because 1/2 = 0.5). Moving from 10 to 11 will always increase the output damage by 10% (because 1/10 = 0.1) and so on.
Linear attributes are very standard stuff in this kind of games. If your positive attributes are linear, you are in a good spot.
#### Hyperbolic
This is the analogous of a linear attribute for negative attributes, the one that reduce the total damage such as EnemyDefense. Mathematically you can explicitly highlight an hyperbolic attributes rewriting the damage formula in this way:
$$Damage(x) = \frac{k}{x} + c$$
In this case, you need k points in the attribute in order to neutralize the damage and get c + 1 point of damage. c is the amount of unavoidable damage: you can not do less than c even if your enemy has infinite defense.
In the Pokemon damage formula EnemyDefense is an hyperbolic attribute. In fact:
$$Damage = \frac{ (\alpha + Base) \times PlayerAttack \times Multiplier}{EnemyDefense} + (2 \times Multiplier)$$
For hyperbolic attributes diminish returns is different:
$$ME = \frac{k}{x(x+1)}$$
Note that marginal effect for an hyperbolic negative attribute depends on k and, moreover, it decreases much faster than the marginal effect for a linear attribute (ME decrease quadratically). This means that, assuming that PlayerAttack and EnemyDefense grow at the same rate, the amount of damage will increase. This is a common and good effect! You don’t want your player to do 10 damage until the end of the game, you want that your player feels more and more powerful during the game, you want that at higher level your player does ten million damage! So powerful! However, how to balance this? Usually you absorb the extra damage increasing the HP of the enemies and the player. Plus: if you are a math geek you can compute how much more HP is needed in order to compensate the increased diminishing return. That’s why analytic analysis is so interesting!
These are just the two basic shapes we can usually find in damage formulas and, in particular, in the Pokemon formula. This is because we chose to 1) use only the four basic operations 2) we use every character attribute only once in the formula. This is the simplest and safest way to build a damage formula, you can not screw up too much in this way. However, there could be more complex cases who needs a more funny use of the attributes. Sometime we would like to use more “extreme” shapes for our damage-attributes functions. Let’s look a some fancy case.
#### Exponential Attributes
This is an ugly beast. This could be useful in some cases but you do not want exponential attributes in your damage formula. Exponential attributes can easily go out of control. When you have an exponential attribute you can rewrite the damage formula in this way:
$$Damage(x) = k(base)^x + q$$
This is dangerous. To understand how much this is dangerous, let’s look at the marginal effect.
$$ME = base - 1$$
There is no diminishing returns, at all! Every time you increase your attribute by 1 you increase the damage by a fixed percentage. An example will make this clear. Suppose the output damage depends exponentially from the attack attribute and that base = 1.5. If we increase the attack value from 1 to 2, the damage will increase by 50%. Ok, cool. However, if we increase the attack from 100 to 101, we still get a 50% damage increase! We could use a level 1 equip item in order to boost our damage by 50%!
Constant marginal effect is usually dangerous but, for instance, IDLE games use this kind of shape A LOT to handle the game progression and guarantee that you can play the game indefinitely.
#### Nth-Root or Logarithmic Attributes
In both cases this is the shape of the attributes that make the damage increase fast at the beginning and slow as the they increase without reaching a fixed value. Damage will just go slower and slower up to infinity. This is a good way to control and “debuff” attributes that could too easily becomes a problem. A logarithmic attribute can be find rewriting the damage formula in this form:
$$Damage(x) = k Log(x)$$
Marginal effect go really really quickly to zero so at higher level you will need tons of that attribute bonus in order to change the damage in a meaningful way.
## Conclusion
There a lot more functions and shapes we can use in our damage formula. They are just like colors in a painting, they can spice up your game or completely destroy it. However, I spent already too many words on this topic. It is time to start computing something and look at how a spreadsheet application can really help us in understanding and balancing our damage function.
##### How to design a Pokémon-like Combat System – Chapter 1
The Combat System is one of the main gameplay element in a game. Of course there are a lot of games without “combat”: puzzles, simulation games, driving games and so on. However (and you …
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https://blog.pollithy.com/general/LOTUS
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# LOTUS
by Daniel Pollithy
If you want to calculate the expected value of a continuous random variable which was transformed by a monotonic function, then the law of the unconscious statistician provides a convenient shortcut.
## Transforming a random variable
We have a continuous random variable $X$ with a probability density function $f_{X}(x)$. This could for example be our last knowledge about the position of a robot.
Now, we apply a continuous, monotonic function $g(\cdot)$ to the random variable. This could be a simple motion model for the robot.
If we want to calculate $E[g(X)]$ then LOTUS tells us that we don’t have to solve $g(X)$ but instead we can write:
$E[g(X)] = \int_{+\infty}^{+\infty}{g(x) \cdot f_{X}(x) dx}$
## Proof
For the proof let’s assume $g$ to be strictly increasing (decreasing would also be possible). And due to its continuity, $g(x)$ therefore has a positive derivative for every x. This results in $g(\cdot)$ being bijective therefore g is invertible. We call its inverse $g^{-1}$. With $g^{-1}: Y \rightarrow X$
### Prepare change of variable
The derivative of a function and its inverse are related. They are reciprocal:
$\frac{dx}{dy} \cdot \frac{dy}{dx} = 1$ $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$
First we replace $y$ with $g(x)$ on the right side:
$\frac{dx}{dy}= \frac{1}{\frac{d g(x)}{dx}}$
Second we replace $x$ with $g^{-1}(y)$ on the right side:
$\frac{dx}{dy}= \frac{1}{\frac{d g(g^{-1}(y))}{dx}}$
Multiplying with dy on both sides:
$dx = \frac{1}{\frac{d g(g^{-1}(y))}{dx}} dy$
### Expected value with exchanged variable
$\int_{+\infty}^{+\infty}{g(x) \cdot f_{X}(x) dx}$
And now we can switch from x to y. First, replace g(x) with y. Second, replace $f_{X}(x)$ with $f_{X}(g^{-1}(y))$. And third, replace dx with the right hand side from “dx = …” above:
$\int_{+\infty}^{+\infty}{g(x) \cdot f_{X}(x) dx} = \int_{+\infty}^{+\infty}{y \cdot f_{X}(g^{-1}(y)) \frac{1}{\frac{d g(g^{-1}(y))}{dx}} dy}$
We have now switched to integrating over y.
### Cumulative density function
$F_{Y}(y) = Pr(Y \le y)$
Apply g:
$F_{Y}(y) = Pr(g(X) \le y)$
Apply $g^{-1}$ on both sides
$F_{Y}(y) = Pr(X \le g^{-1}(y))$ $F_{Y}(y) = F_{X}(g^{-1}(y))$
### Derivative of CDF
Now we can get the derivative of $F_{Y}(y)$ for y in order to get $f_{Y}(y)$. The chain rule is used. Note that this is the place where we need the derivative of $g^{-1}(y)$ which is $\frac{1}{\frac{d g(g^{-1}(y))}{dx}}$ !
Apply the CDF solution from above:
$f_{Y}(y) = \frac{d}{dy} F_{Y}(y) = \frac{d}{dy} F_{X}(g^{-1}(y))$
Apply the chain rule of derivation:
$= f_{x}(g^{-1}(y)) \cdot \frac{d}{dy} g^{-1}(y)$ $= f_{X}(g^{-1}(y)) \cdot \frac{1}{\frac{d g(g^{-1}(y))}{dx}} =$
### Plug-in
Two sections before, we got to this point:
$E[g(X)] = \int_{+\infty}^{+\infty}{y \cdot f_{Y}(y) dy} = \int_{+\infty}^{+\infty}{y \cdot f_{X}(g^{-1}(y)) \frac{1}{\frac{d g(g^{-1}(y))}{dx}} dy}$
Looking at the last formula from the section “Prepare change of variable”, we find that $f_{X}(g^{-1}(y)) \cdot \frac{1}{\frac{d g(g^{-1}(y))}{dx}} dy$ to be the same as $f_{X}(g^{-1}(y)) dx$
$E[g(X)] = \int_{+\infty}^{+\infty}{y \cdot f_{X}(g^{-1}(y)) dx}$
Per definition $g^{-1}(y)$ can be replaced by x.
$E[g(X)] = \int_{+\infty}^{+\infty}{y \cdot f_{X}(x) dx}$
And also $y$ can be replaced by g(x).
$E[g(X)] = \int_{+\infty}^{+\infty}{g(x) \cdot f_{X}(x) dx}$
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https://link.springer.com/article/10.1134%2FS0032946013040078
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Problems of Information Transmission
, Volume 49, Issue 4, pp 375–381
# Group testing problem with two defectives
• C. Deppe
• V. S. Lebedev
Large Systems
## Abstract
We consider the classical (2,N) group testing problem, i.e., the problem of finding two defectives among N elements. We propose a new adaptive algorithm such that for $$N = \left\lfloor {2\tfrac{{t + 1}} {2} - t \cdot 2\tfrac{t} {4}} \right\rfloor$$ the problem can be solved in t tests.
## Keywords
Search Algorithm Group Test Information Transmission Adaptive Algorithm Transmission Strategy
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
## References
1. 1.
Du, D.-Z. and Hwang, F.K., Combinatorial Group Testing and Its Applications, Singapore: World Sci., 2000, 2nd ed.
2. 2.
Du, D.-Z. and Hwang, F.K., Pooling Designs and Nonadaptive Group Testing. Important Tools for DNA Sequencing, Hackensack, NJ: World Sci., 2006.
3. 3.
Ahlswede, R. and Wegener, I., Suchprobleme, Stuttgart: Teubner, 1979. Translated under the titles Zadachi poiska, Moscow: Mir, 1982; Search Problems, Chichester: Wiley, 1987.
4. 4.
Chang, G.J., Hwang, F.K., and Lin, S., Group Testing with Two Defectives, Discrete Appl. Math., 1982, vol. 4, no. 2, pp. 97–102.
5. 5.
Chang, X.M., Hwang, F.K., and Weng, J.F., Group Testing with Two and Three Defectives, Graph Theory and Its Applications: East and West (Proc. 1st China-USA Int. Graph Theory Conf., Jinan, China, 1986), Capobianco, M.F., Guan, M., Hsu, D.F., and Tian, T., Eds., Ann. New York Acad. Sci., vol. 576, New York: New York Acad. Sci., 1989, pp. 86–96.Google Scholar
6. 6.
Tošić, R., An Optimal Search Procedure, J. Statist. Plann. Inference, 1980, vol. 4, no. 2, pp. 169–171.
7. 7.
Weng, S.Y., The Research of Group Testing Questions in (2, n), Master Thesis, Taiwan: National Central Univ., 1999.Google Scholar
8. 8.
Sobel, M., Binomial and Hypergeometric Group-Testing, Studia Sci. Math. Hungar., 1968, vol. 3, pp. 19–42.
9. 9.
Chang, G.J. and Hwang, F.K., A Group Testing Problem on Two Disjoint Sets, SIAM J. Algebr. Discrete Methods, 1981, vol. 2, no. 1, pp. 35–38.
10. 10.
Lebedev, V.S., Separating Codes and a New Combinatorial Search Model, Probl. Peredachi Inf., 2010, vol. 46, no. 1, pp. 3–8 [Probl. Inf. Trans. (Engl. Transl.), 2010, vol. 46, no. 1, pp. 1–6].Google Scholar
11. 11.
Ahlswede, R., Deppe, C., and Lebedev, V.S., Finding One of D Defective Elements in Some Group Testing Models, Probl. Peredachi Inf., 2012, vol. 48, no. 2, pp. 100–109 [Probl. Inf. Trans. (Engl. Transl.), 2012, vol. 48, no. 2, pp. 173–181].Google Scholar
12. 12.
Garey, M.R. and Hwang, F.K., Isolating a Single Defective Using Group Testing, J. Amer. Statist. Assoc., 1974, vol. 69, no. 345, pp. 151–153.
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https://brilliant.org/problems/another-controversial-question-2/
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# Three Equations But Two Unknowns
Algebra Level 4
$\begin{cases} x^4- y^4 = 24 \\ x^2 + y^2 = 6 \\ x + y = 4 \end{cases}$
If $x$ and $y$ satisfy the above system of equations, find $x-y$.
×
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https://cracku.in/blog/fill-in-the-blanks-questions-for-ssc-chsl-mts-2022/
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0
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# Fill in the Blanks Questions for SSC CHSL & MTS 2022 – Download PDF
Here you can download SSC CHSL & MTS 2022 – important for SSC CHSL & MTS Fill in the Blanks Questions PDF by Cracku. Very Important SSC 2022 exams and These questions will help your SSC preparation. So kindly download the PDF for reference and do more practice.
InstructionsDirections: Sentences are given with blanks to be filled in with an appropriate word(s). Four alternatives are suggested for each question. Choose the correct alternative out of the four.
Question 1: The hotel was not too expensive
a) was it ?
b) wasn’t it?
c) is it ?
d) isn’t it ?
Question 2: Like humans, zoo animals must have a dentist to __ their teeth
a) fill
b) filled
c) filling
d) to be filled
Question 3: It was very kind of you to do the washing-up, but you _ it.
a) didn’t have to do
c) mightn’t have done
d) mustn’t have done
Question 4: He went __ sea alone.
a) in
b) to
c) into
d) on
Question 5: The __ of our civilization from an agricultural society to today’s complex industrial world was accompanied by war.
b) migration
c) route
d) metamorphosis
InstructionsDirections:In thefollowing questions sentences are given with blanks to be filled in with an appropriate word(s). Four alternatives are suggested for each question. Choose the correct alternative out of the four.
Question 6: The court _____ cognisance of the criminal’s words.
a) took
c) gave
d) allowed
Question 7: _____ wins this civil war there will be little rejoicing at the victory.
a) Whichever
b) Whoever
c) Whatever
d) Wherever
Question 8: As he got older his belief in these principles did not_____.
a) wither
b) shake
c) waver
d) dither
Question 9: Everyone in this world is accountable to God___ his actions.
b) for
c) to
d) over
Question 10: Your father used to be the principal of this college,
a) did he ?
b) does he ?
c) didn’t he?
d) doesn’t he ?
Question 11: Many premier educational institutions come forward to have a _______ with flourshing industries.
a) tie-down
b) tie-up
c) tie-in
d) tie-on
Question 12: All is not well _______ the automobile sector.
a) of
b) down
c) in
d) to
Question 13: He slipped _______ his old ways and started drinking again.
a) into
b) off
c) by
d) in
Question 14: They reached the railway station before the train __________ .
c) left
d) was leaving
Question 15: The Information and Communication Technology has _____ age and employees very highly paid technocrats.
a) come of
b) come upon
c) come out of
d) come through
Question 16: In the following question, the sentence given with blank to be filled in with an appropriate word. Select the correct alternative out of the four and indicate it by selecting the appropriate option.
He has a……………..interest in studying human psychology.
a) deep
b) wide
c) vast
d) heavy
Question 17: In the following question, the sentence given with blank to be filled in with an appropriate word. Select the correct alternative out of the four and indicate it by selecting the appropriate option.
After being given _________ warnings for disrupting class, Thomas was finally sent to the principal’s office.
a) singular
b) lone
c) numerous
d) unique
Question 18: In the following question, the sentence given with blank to be filled in with an appropriate word. Select the correct alternative out of the four and indicate it by selecting the appropriate option.
The __________ “pretty ugly” implies that a person can be both attractive and unattractive at the same time.
a) simile
b) metaphor
c) alliteration
d) oxymoron
Question 19: In the following question, the sentence given with blank to be filled in with an appropriate word. Select the correct alternative out of the four and indicate it by selecting the appropriate option.
Finding the comedy routine extremely funny, the family laughed _____________ along with the rest of the crowd.
a) lot
b) hysterically
c) crazy
d) guffaw
Question 20: In the following question, the sentence given with blank to be filled in with an appropriate word. Select the correct alternative out of the four and indicate it by selecting the appropriate option.
The detective’s ability to ________ makes it easy for him to scare suspects into confessing.
a) bluff
b) stuff
c) enough
d) cough
The correct answer is option A.
Question tag -It is used when one wants to reconfirm ones statement.It is a statement followed by a question.
If the statement is positive, question tag will be negative and vice versa.
For ex- Shyam is working in a bank, isn’t he?
Question tags are formed with the auxiliary or modal verb from the statement and the appropriate subject.
A subject pronoun comes after an auxiliary or after to be’ form of the verb & is used to replace the noun.
Here in the given question-
Hotel was not too expensive, wasn’t it?
Since the statement /main part of the sentence is in positive form, question tag will be negative.
Here was’ is used in main part of the sentence,so in question tag we use wasn’t’.
(Same verb/(auxiliary,modal,main)as that of the sentence)
To indicate Hotel’,here we used pronounit’.
..
The correct answer is option number 1st.
To +V1 is called as the infinitive form if the verb.
After to’ we use first form if the verb.
Infinitive form if the verb is used when we want to express the purpose,to answer why.
Ex-He studied hard to make his parents proud.
Why he do study -to make his parents proud.
Infinitive form is used as a subject of verb,as a object of verb,as the complement of the verb,as a object of preposition,to qualify noun,to qualify a sentence.
Here it is used as a Adverb’.It qualifies verb Studied’.
In the given example, options -to be filled,filling,filled’are wrong.Because after to’ we can’t use 3rd form of the verb,rest options are grammatically incorrect.
The correct answer is option number -D
Mustn’t have done is the correct answer.
Mustn’t have -Must is a modal verb,it adds meaning to another verb.
Mustn’t we use to show possibility or necessity.
Must not -We use to talk about things we need to avoid doing or while telling someone to not do it.
Rest options are grammatically & contextual incorrect.
Might -We use might to show prediction, possibility, probability.
Ex- the Prime Minister might visit the town.
Here indicating possibility.
Might & hadn’t is inappropriate because sentence is in the present tense.
Usage of didn’t is not contextually correct.
Didn’t have to do -We say didn’t have to represent specific point of time that has already passed.
Only option Mustn’t have do’ is grammatically & contextually correct.
The correct answer is option B.
Here it’s prepositional error.
Preposition TO‘ is used to show destination.
Also go to sea ‘ is an idiom which means to become a seller,to embark on a voyage.
We see uses of other preposition
Into– we use this to show moment,any action which show moment.
Ex- He jumped into the well.
In -It is use to show place,time.
Ex- There is a pen in the cupboard.
On -it is use to indicate time,place.(When things are in direct contact,at surface area,in rest position)
Ex- there is a book on the table.
The correct answer is option number D.
The sentence implies here the transformation /changes in civilization (human society) from agriculture to today’s industrial world was accompanied by war.
Meaning of remaining options.
a small change, corrections, modification.
Ex -The stock market exchange made an adjustment in their policy to get more investors.
Route (Noun)
Ex- The route to Theni from Madurai is 70kms.
Migration (Noun)
An instance of moving to live in another place for a while.Movement, journey.
Ex- Migration of outsiders to Delhi make the city over- populated.
Option A is the correct answer.
Cognisance or cognizance Means Notice, awareness or knowledge.
Here in this sentence ,the court notices the words of the criminal. That they are considering the words from the Criminal. So gave, allowed and made doesn’t fit in the blank of the sentence.
The correct answer is option -B
Whoever- It is a subject pronoun & work like the pronouns I,We He,She & They.
Ex- Whoever wins this debut will win a prize.
Whoever means -Someone/anyone who does a particular thing or is in a particular situation.
Meaning of the remaining options
1) Whatever- Unexceptional /unimportant,at all, absolutely, whatsoever.No matter which,for any, anything that.
Synonyms- So what.
Ex- Whatever may happen,I will go to Mahabaleshwar today.
2) Wherever-(Adverb) An emphatic form of where.
It also acts as a conjunction -whatever place, anywhere,in all places, everywhere.
Ex- I will come with you wherever you will take me.
3) Whichever- (Adjective / pronoun) – Any one/ a no of a group.
Ex- Whichever pizza you ordered for her, it must have had some very delicious ingredients.
The correct answer is option number – C.
Waver means – To sway back & forth, to fluctuate /vary,to tremble,to flatter,to be indecisive.
Dither- To tremble,the state of being undecided.
Wither-To go against,resist,oppose.
Shake -to cause something to move rapidly in opposite directions,to disturb emotionally,to lose,evade,get rid of
Here sentence implies that -As he got older, beliefs in principles became more strong,he didn’t fluctuate.
Remaining options are contextually incorrect.
The correct preposition is for’.
The given sentence seems to imply that everyone is responsible to God for his actions.
Accountable for -means responsible for.
Meaning of other options
1)about-Near,not far from,on the point of verge of,concerned with,regard to,on account of,on the subject of.We can use About’ in above context.It can refer to movement /position in various directions/places
2) to – It is used to indicate directions.To denote any thing which is at surface area, in rest position.
3) over- it is used to indicate place.
So these are contextually incorrect.
Also the expression Accountable for’ means to responsible for.So,usage of For’ will be correct here.
The correct answer is option -C
The given sentence is of question tag.
Question tag is used to reconfirm the question or the statement.
If the main sentence is positive, question tag will be negative and vice -versa.
Here the given sentence is positive,so question tag will be negative.
The sentence is in the past form.Used to’ we use as a simple present tense but to indicate the things happened in the past.
Ex-I used to go by bus.
As the sentence is in the past form,usage of did’ will be correct.
Did’ is the past form of the Do’
Option A- Did he ? Will be wrong.Because given sentence is in the positive form,so question tag will be negative.
Option B- Does he?- Will be wrong.Because the given sentence is in the past tense & Does’ is use for 3rd person singular & when sentence is in present form.
Option D- Doesn’t he? – Will be wrong.Because the given sentence is in present tense.Though the question tag is negative,it is wrong.Because `Does’ is used when the sentence is in the present tense & subject is 3rd person singular.
The correct answer is option number -B
Tie up -means to connect with,associate with.
Tie in – means something that is related or connected to another.
Tie down – to constrain,to confine within set limits.
Tie on – capable of being fastened or added to something else.
The correct answer is option number -C
In -is the correct presposition.
In is used to show time, certain period of time.In means within, pertaining to.
Here what sentence implies that condition is not good in automobile sector.
Rest options doesn’t fits here.
The correct answer is option number -A
Slipped into -is the correct choice.
Slipped into -is a phrase which means to gradually start to be in a bad phase,state.
Remaining options doesn’t fits here.
The correct answer is option number -C
The correct answer is option number -A
Come of is the correct phrase which means to be the result of something.
Come of age -means it reaches an advanced stage of development.
Remaining options are contextually & grammatically incorrect.
The sentence expresses an interest in ‘human psychology’. Since it is a single subject, ‘wide’ does not fit the blank. Only the word ‘deep’ fits the blank correctly.
‘Singular’ means single. ‘Lone’ means solitary or single. ‘Numerous’ means many. ‘Unique’ means being the only one of its kind. Hence, option C is correct.
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http://mathhelpforum.com/calculus/60434-please-check-my-critical-points-print.html
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# Please check my critical points
• November 19th 2008, 02:52 AM
koalamath
Find the critical points of the equation $xy-7x^2y-9xy^2$
The critical points I found in increasing lexicographic order are
(0.0) (0,1/9),(1/21,1/27) and (1/7, 0)
at (0,0) is it undefined
at (0/1/9) and (1/7,0) you have saddle points
at (1/21,1/2) you have a local max.
Is this correct?
Thank you
• November 19th 2008, 03:00 AM
mr fantastic
Quote:
Originally Posted by koalamath
Find the critical points of the equation $xy-7x^2y-9xy^2$
The critical points I found in increasing lexicographic order are
(0.0) (0,1/9),(1/21,1/27) and (1/7, 0)
at (0,0) is it undefined Mr F says: ?? Do you mean the test fails? ${\color{red}xy-7x^2y-9xy^2}$ is certainly not undefined at (0, 0) ....
at (0/1/9) and (1/7,0) you have saddle points
at (1/21,1/2) you have a local max.
Is this correct?
Thank you
..
• November 19th 2008, 03:20 AM
koalamath
fxx=-14y
fxx(0,0)=0
doesn't that make it undefined?
• November 19th 2008, 03:38 AM
mr fantastic
Quote:
Originally Posted by koalamath
fxx=-14y
fxx(0,0)=0
doesn't that make it undefined?
Why? What part of the second partial derivative test says this? How can $z = xy-7x^2y-9xy^2$ be undefined when x = 0 and y = 0? z = 0, a perfectly well defined value.
In fact, there's a saddle point at (0, 0, 0).
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https://math.stackexchange.com/questions/2437696/non-sparse-solution-for-a-linear-programming-problem
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# Non-sparse solution for a linear programming problem
I have formulated a linear program with equality, inequality and non-negativity constraints. My objective function is the minimization of a linear combination of decision variables (with different coefficients for different variables). I have 7803 decision variables, 51 inequality constraints and 111 equality constraints. I used the simplex method and found the solution. In the solution the values for most decision variables is 0 (like 98%), which is not surprising, based on the model formulation. Because of the nature of the problem, I need to have a way larger ratio of the decision variables with non-zero values in the solution. Is there any formulation technique that can do this for me? In other words, I need to modify my formulation (I guess probably in the OF) to find a less-sparse solution, at the cost of getting a little far from the optimum point. I am okay if it ends up in a non-linear optimization problem.
It is not surprising that your current solution has so many variables with value zero especially if your solver uses the simplex method instead of an interior point method. When you say "Because of the nature of the problem, ..." it means that your current model does not represent your actual problem (it is not the right model) and you should formulate the problem differently. Anyway, here is what you should consider. Given that your current problem is a linear program we can consider two cases:
1. Your linear program has a unique optimal solution.
2. Your linear program has alternative optimal solutions.
In the first case, it is impossible to increase the Number of Positive Variables (NPV) without making the optimal value worse. So you need to change your problem if you really want to increase the NPV. You need to solve a two-objective optimization problem to maximize NPV while optimizing your original objective function. There are different ways to formulate and solve multi-objective problems. For example, you may consider a convex combination (or weighted average) of the two objective function as your new objective function and solve the problem by typical methods used for single-objective problems.
In the second case, you may find another optimal solution for the problem with a better value of the NPV. To find such a solution, you may define a secondary objective for the problem. The new problem is considered as a lexicographic multi-objective problem which can be solved taking two approaches. The first approach is to define an objective function that maximizes the NPV without influencing the primary objective and then use typical single-objective methods to solve the problem (finding such an objective function is sometimes difficult but may be possible). The second approach is to solve a sequence of single-objective problems (once you solve the original problem, you transmit some information to a second problem whose objective is maximizing the NPV).
As you see, in both cases you need two objective functions. Given the number of variables and constraints of your problem it should be easy to solve even if you add some integer variables to your model. Note that your problem becomes a mixed integer program that may be solved by MILP solvers. I suggest to define binary variable $y_i\in\{0,1\}$ corresponding to each $x_i$ and add the constraints $l_i\, y_i \le x_i \le u_i\,y_i$ for each $i$, where $u_i$ is an upper bound on the variable $x_i$ (or a big-M parameter) and $l_i$ is any positive number as the lower bound of $x_i$ (e.g., $l_i = \epsilon$). Note that such a lower bound is effective only if $y_i=1$ and this is a decision that is made by the optimization problem itself. You can now consider $\max (\sum_{i} y_i)$ or $\min (-\sum_{i} y_i)$ as another objective of the problem. You can formulate the problem as a MILP (not a nonlinear program).
Note that you should still find the right single-objective model of your problem and what I mentioned here may give you some insight to do so.
• Thank you for your comprehensive answer. I think doing a two-objective optimization problem to maximize NPV while optimizing my original objective function is the exact thing I need. I found this Matlab function (GAMULTIOBJ) that solves these problems using GA but it seems it doesn't work with integer constraints (that I will have for yi's). Do you know how I can solve multi-objective mixed integer programs? Or are you familiar with a simple method for calculating weights for the two objective functions, so I can use the weights to make it a single-objective mixed integer program and solve it?
– Fred
Sep 21 '17 at 21:33
• I suggest using FPBH for solving multiobjective mixed integer linear programs. It is the most recent and trusted package that I know about. You can find the package and the instructions from github.
– user477602
Sep 21 '17 at 23:53
• I used Johan's trick and it is working so far. I will be able to verify my results later. If I find out the current solution is not good, I will come back and try your suggested method. Thank you for the time you spent on it.
– Fred
Sep 22 '17 at 20:00
A simple trick is to solve the LP resulting in the solution $x^{\star}$. As you don't like this solution, you now solve a new LP with the old constraints and additional constraint that $c^Tx = c^Tx^{\star}$ (i.e., just as good) but minimize the distance to what you think is a good solution (such as $\left\| x-1 \right\|_2^2$).
• isn't the new constraint exactly what old equality constraints were? This is what I understand from cTx=cTx⋆. Also for the new OF, can you make an example please? Like, if the original OF is A1x1 + A2x2 + ... AnXn, what will be the new OF?
– Fred
Sep 21 '17 at 20:46
• You keep all old constraints, but add an equality on the objective, to get a solution with same objective but hopefully more non-zeros (old objective is $c^Tx$, new objective is quadratic here, but you could use linear norm such as sum of absolute values) Sep 21 '17 at 20:51
• Okay, now I understand that I'll need to add the old OF as a equality constraint to the new problem. The new OF will be then (x1-1)^2+(x2-1)^2+...(x7803-1)^2? Is this what you mean? Can you explain how this OF will make the number of non-zero variables in the solution fewer? Thanks.
– Fred
Sep 21 '17 at 21:22
• Because the only thing we are trying to do is to force all variables towards 1, which is not zero. It is a heuristic. There is absolutely no guarantee that it will work Sep 22 '17 at 5:55
• What you probably should do is to tweak solver options. A significant effort is placed in the solver to achieve precisely what you don't want. A final step in advanced solvers is the cross-over, when one tries to move to a vertex of the feasible set, in case the final solution is on a plane or line or something, or in the interior. My advice would be to turn on an interior-point algorithm in the solver, turn off cross-over, and reduce the optimality tolerances to force the solver to terminate prematurely when it is still far into the interior, and thus should have more non-zeros. Sep 22 '17 at 5:59
You could just add constraints of the form $x_i \ge \epsilon$ for some of the variables, where $\epsilon$ is the smallest positive value you would allow as "non-zero". You might choose the variables with smallest shadow prices in your optimal solution of the original problem.
• That enters subjectivity in the problem and I don't know how to justify that. Also, then won't all the decision variables that were originally zero and are forced to be greater than epsilon, equal epsilon in the solution? That would not be helpful. I need an alternate formulation that is more flexible and splits the quantities among more variables, rather than assigning them all to a limited number of variables.
– Fred
Sep 20 '17 at 22:06
• From your description, it's really hard to guess what the actual problem is. Why do you need more nonzero values? Why are equal values not helpful? Sep 21 '17 at 0:34
• Because the problem's solution is not values used for decision making, but are supposed to be approximates of actual real world values. I can simply observe that in the real world case, the number of non-zero variables are way higher than in the solution of my LP. In other words, what is actually taking place is not the optimal solution.
– Fred
Sep 21 '17 at 21:38
• So perhaps this is something that should not be modeled as an optimization problem at all. Without knowing anything about whatever it is you're trying to model, there's not much else we can say. Sep 24 '17 at 7:58
• There are many examples of usage of optimization for modeling behavior of users of a system, assuming that the users try to minimize their costs, such as the traffic assignment problem, based on Wardrop's equilibrium.
– Fred
Sep 24 '17 at 15:17
There is an non convex optimisation based approach used in signal processing : you can use NMF with sparse constraints. Use a multiplicative gradient descent with l1 regularisation step, and project your inequality constraint at each step.
EDIT: Roughly the idea of Lee, DD & Seung, HS (1999) is to use a "multiplicative" gradient update to maintain nonegativity during the factorisation : split the gradient into two positive components $g=g_+−g_−g$ and update using $g_+/g_−$ direction.
Then you can include constraint inside the gradient descent, that would be the fuzzy part of my proposal. See & Seung algorithm as very little mathematical guarantees but somehow performs very well. There are a lot of contrained variants of NMF, you can check J. Le Roux, J. R. Hershey, and F. Weninger. "Sparse NMF – half-baked or well done ?" for sparse constraints.
• Can you provide a reference? I googled the terms you mentioned, but it is hard to fully understand. Also, is there a Matlab function that do this (or any other language)?
– Fred
Sep 20 '17 at 17:02
• sklearn and matlab have a NMF function, but i guess you should implement the iterations yourself. Roughly the idea of Lee, DD & Seung, HS (1999) is to use a "multiplicative" gradient update : split the gradient into two positive components $g = g_+ - g_-$ and update by $g_+/g_-$. Then you can include constraint inside the gradient descent, that would be the fuzzy part of my proposal. See&Seung algorithm as very little mathematical guarantees but somehow performs very well. There are a lot of contrained variants of NMF, you can check J. Le Roux, J. R. Hershey, and F. Weninger. Sep 21 '17 at 12:02
• Thank you for your help. It sounds a little too complicated for me now. I will try other suggestions first, and if they did not work, I will come back and try to understand this.
– Fred
Sep 21 '17 at 21:35
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https://nubtrek.com/maths/integers/integers-multiplication-division/integer-division-simplified-procedure
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mathsIntegersInteger Multiplication & Division
### Integer Division : Simplified Procedure
This page extends the division in first principles into a simplified procedure for division of integers, which is called sign property of integer division.
click on the content to continue..
The division 14-:3is understood as
text(received:)14 is split into 3 equal parts and one part is put-in (positive divisor).
The result of the division is 14-:3= quotient 4 and remainder 2
The division (-14)-:3is understood as
text(given:)14 is split into 3 equal parts and one part is put-in (positive divisor).
The result of the division is (-14)-:3= quotient -4 and remainder -2
The division 14-:(-3)is understood as
text(received:)14 is split into 3 equal parts and one part is taken-away (negative divisor).
The result of the division is 14-:(-3)= quotient -4 and remainder 2
The division (-14)-:(-3) is understood as
text(given:)14 is split into 3 equal parts and one part is taken-away (negative divisor).
The result of the division is (-14)-:(-3)= quotient 4 and remainder -2
Summary of integer division illustrative examples:
• 14-:3 = 4text((Q) & ) 2text((R)) : text(received:)14 is split into 3 parts is quotient text(received:)4 and remainder text(received:)2.
• (-14)-:3 = -4text((Q) & ) -2text((R)) : text(given:)14 is split into 3 parts is quotient text(given:)4 and remainder text(given:)2.
• 14-:(-3) = -4text((Q) & ) 2text((R)) : text(received:)14 is split into -3 parts is quotient text(given:)4 and remainder text(received:)2.
• (-14)-:(-3) = 4text((Q) & ) -2text((R)) : text(given:)14 is split into -3 parts is quotient text(received:)4 and remainder text(given:)2.
Based on this, the division is simplified as
• +ve -: +ve = +ve with +ve remainder
• +ve -: -ve = -ve with +ve remainder
• -ve -: +ve = -ve with -ve remainder
• -ve -: -ve = +ve with -ve remainder
Integer Division -- Simplified Procedure : The sign of the quotient and remainder are decided by the signs of dividend and divisor as:
Sign-property of Integer Division
• positive -: positive = positive with positive remainder
• positive -: negative = negative with positive remainder
• negative -: positive = negative with negative remainder
• negative -: negative = positive with negative remainder
Sign of the remainder is that of the dividend.
The absolute values of the quotient and remainder are calculated by whole number division of absolute values of dividend and divisor.
Solved Exercise Problem:
Find the result of the division 22 -: (-1)
• -11
• -22
• -22
The answer is "-22"
Solved Exercise Problem:
Find the result of the division 0-:(-32)
• 0
• -0
• both the above
• both the above
The answer is "both the above"
Solved Exercise Problem:
Find the result of the division -180 -: (-88)
• quotient 2 and remainder 4
• quotient -2 and remainder -4
• quotient 2 and remainder -4
• quotient 2 and remainder -4
The answer is "quotient 2 and remainder -4".
switch to interactive version
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https://www.physicsforums.com/threads/work-done-by-an-expanding-gas.735198/
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# Work done by an expanding gas
1. Jan 27, 2014
### moogull
Today in an engineering thermodynamics lecture, the professor gave an example of a gas doing work. We had a cylinder full of helium at a pressure of something like 200kPa absolute and the valve was opened so that the gas would flow out against the atmospheric pressure until the pressures were equal. Also the cylinder was assumed to be in thermal equilibrium with its surroundings so the temperature of the gas was equal to the temperature of the ambient air. However, the way he calculated the work perturbed me. He said that this was an isobaric process because the gas was expanding against a constant atmospheric pressure. I was under the assumption that an isobaric process means that the working fluid stays at constant pressure throughout the process which is not the case in this expansion. And in this case, the gas pressure is dropping as it leaves the cylinder.
The professor then proceeded to calculate the work as W = Patm*ΔV. But I don't think that is right and that simple.
Am I correct, or is the professor? Can someone please return me to sanity?
2. Jan 27, 2014
### Jano L.
You are right, the helium expands and its pressure decreases from 200kPa to atmospheric 100 kPa. Helium gas does not undergo isobaric process in the common sense of the word (atmosphere does).
Your professor is right this time - the work helium does on the atmosphere is indeed (approximately) W = Patm*ΔV, where ΔV is the volume of the helium gas outside the cylinder just after it escapes from it. After a while, the helium is heated by the atmosphere and expands even more and does further work, but this work is neglected in the above.
3. Jan 27, 2014
### moogull
Thanks for the response Jano,
If the process is not isobaric, then why is the work not calculated using an integral and instead W = Patm*ΔV. I'm fairly certain he took the system as a control mass/closed system.
4. Jan 27, 2014
### Jano L.
You can write the work as integral, but because the pressure of the atmosphere can be assumed constant during the process, the result is just $P_{atm} \Delta V$.
5. Jan 27, 2014
### moogull
Okay, so in this case, why is the pressure of the atmosphere the pressure used to calculate work and not the pressure of the working fluid?
6. Jan 27, 2014
### moogull
What I mean to say is, why, since this is not an isobaric process, the work is calculated using a pressure that is assumed not to change?
edit: Looking at the atmosphere as the working fluid I agree that the work is defined P_atm*deltaV.
Last edited: Jan 27, 2014
7. Jan 27, 2014
### Staff: Mentor
It depends on what you define as your system. If you define your system as just the gas that remains in the cylinder after equilibration, then that gas has done work on expelling the gas from the cylinder, and the pressure at the interface with the gas that it expelled was not at constant pressure.
If you define your system by surrounding all the helium that was originally inside the cylinder with an imaginary moving boundary, then, throughout this process, different parts of the helium were at different pressures. However, at the imaginary boundary with the surrounding atmospheric air, the pressure was constant (atmospheric). In the first law, you calculate the work done on the surroundings by calculating the integral of the pressure at the interface with the surroundings integrated over the volume change. (See my Blog on my PF home page.) So, in the case of this system, your professor was correct.
Chet
8. Oct 22, 2014
### Odd
Hello
Is it possible then to calculate the same work looking only at the work of the expanding gas inside the cylinder ?
I assume one would have to use d(PV) and then a equation of state for the process.
Odd
9. Oct 22, 2014
### Staff: Mentor
You would just solve it as an isothermal reversible expansion. The real irreversibility occurs within the valve, where the pressure drops from that inside the cylinder to 1 atm.
Chet
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https://hsm.stackexchange.com/tags/energy/hot
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I suspect what you really are after is the electron's mass, since we can then "easily" get its energy via an obscure :-) equation developed by some fellow named Einstein. The path thru measuring the ...
• 2,485
### How did Huygens derive the conservation law for of kinetic energy?
There are several themes in Huygens' unpublished paper De motu corporum ex percussione ("On the motion of bodies out of collisions"), but maybe the most significant is that he frequently investigates ...
• 386
### How did Huygens derive the conservation law for of kinetic energy?
("How did Huygens derive the conservation law for of kinetic energy?") Huygens in 'The Motion of Colliding Bodies' (English translation) contributed ingenious reasoning, mathematics and thought-...
• 2,969
### How did Huygens derive the conservation law for of kinetic energy?
The "mathematics" was a combination of experiments with falling bodies, imaginative thought experiments, common sense, and geometric reasoning. Part of it is explained in the book. Galileo ...
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Accepted
### Are there any direct comments by Isaac Newton on Leibniz's living force / vis-viva?
The controversy was not so much about the tension between vis viva and mechanics, as about what is the "true" quantity of motion, vis viva or momentum, and what is the "metaphysical&...
• 66.4k
### What are the earliest inventions to store and release energy (e.g. fly wheels)?
According to this page, charcoal was in use circa 3750 BCE. That's an energy storage medium, although perhaps not the class of energy-release you were thinking of.
• 2,485
Accepted
### Who discovered the Virial Theorem?
According to the OED, Clausius coined the German word "virial" (from vīs force, strength): a. In Clausius' kinetic theorem of gases: (see quots.). virial theorem, the theorem that for a steady-...
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Accepted
### References about the the development of the concept of mechanical work
You can see : Agamenon Oliveira, A History of the Work Concept: From Physics to Economics, Springer (2014). Also interesting : Danilo Capecchi, History of Virtual Work Laws: A History of Mechanics ...
• 13.8k
### What was the vis viva controversy, including its philosophical aspects?
I must put up a clarifying answer. They groped towards different concepts and talked past each other, in the modern perspective. Much of what we call Newtonian mechanics is due to Euler, who called ...
• 1,182
### History of Energy
This is a very interesting and complex topic that is far from closed. Here are some of the sources I have found. Energy the subtle concept, by Jennifer Coopersmith, is probably the book you are ...
• 211
### How did Newton establish the conservation laws in the Principia?
In Principia Newton presents a picture based on forces rather than energy and momentum, and he did not have the concepts of vector and of mechanical energy at his disposal. Moreover, Newton opposed ...
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### Clausius regarding energy
There is a second sentence to the quote:"The energy of the universe is constant. The entropy of the universe is increasing". The conservation of energy law. The second law of thermodynamics. The quote ...
• 66.4k
### What are the earliest inventions to store and release energy (e.g. fly wheels)?
Tree limb. Before humans became ground bound, they used tree limbs to catapult themselves to the next tree. But seriously, probably a stone, then a stone attached to a stick, then a stone attached ...
• 31
### What are the earliest inventions to store and release energy (e.g. fly wheels)?
How about good-old fashioned pit traps? If 'stored energy' is something where you input work ahead of time and then it is expended at a later date for a use, a pit trap is a perfect example. While ...
• 131
Accepted
### Why did energy-momentum relationship have to wait until 1928 to be established?
It is not that it had to wait to be "established", it is obtainable from what was known by trivial algebra, but rather that it had to wait for a reason to write it that way. In the early ...
• 66.4k
Accepted
### When was relativistic mass first observed?
In theory, relativistic mass was preceded by the "electromagnetic mass" introduced by J.J. Thomson in 1881, and further developed by Heaviside (1888), Searle (1897), Poincaré (1900), Abraham (1902), ...
• 316
### Why did energy-momentum relationship have to wait until 1928 to be established?
Actually, it wasn't Dirac who first found that relation. It was already used by Planck as early as in 1906 while deriving the Hamiltonian equations of motion Planck: The Principle of Relativity and ...
• 316
### How did people figure out the formula for mechanical work, and related it to energy?
It was a side effect of the vis viva controversy described in What was the vis viva controversy, including its philosophical aspects? about what to call the "quantity of motion", momentum (...
• 66.4k
### Landau doubting conservation of energy---and what followed
In 1932 Landau speculated that the conservation of energy is not valid in neutron stars and appealed to the authority of Niels Bohr[1]: Following a beautiful idea of Professor Niels Bohr’s we are ...
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Let us consider the following info on developments prior to 1905. Before 1905 (The forthcoming of $E = mc^2$) 1881 J. J. Thompson proposed that a charged conductor in motion increases its mass by \$\...
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https://infoscience.epfl.ch/record/202277
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Infoscience
Journal article
# Modelling Small-Scale Drifting Snow with a Lagrangian Stochastic Model Based on Large-Eddy Simulations
Observations of drifting snow on small scales have shown that, in spite of nearly steady winds, the snow mass flux can strongly fluctuate in time and space. Most drifting snow models, however, are not able to describe drifting snow accurately over short time periods or on small spatial scales as they rely on mean flow fields and assume equilibrium saltation. In an attempt to gain understanding of the temporal and spatial variability of drifting snow on small scales, we propose to use a model combination of flow fields from large-eddy simulations (LES) and a Lagrangian stochastic model to calculate snow particle trajectories and so infer snow mass fluxes. Model results show that, if particle aerodynamic entrainment is driven by the shear stress retrieved from the LES, we can obtain a snow mass flux varying in space and time. The obtained fluctuating snow mass flux is qualitatively compared to field and wind-tunnel measurements. The comparison shows that the model results capture the intermittent behaviour of observed drifting snow mass flux yet differences between modelled turbulent structures and those likely to be found in the field complicate quantitative comparisons. Results of a model experiment show that the surface shear-stress distribution and its influence on aerodynamic entrainment appear to be key factors in explaining the intermittency of drifting snow.
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https://www.physicsforums.com/threads/partial-wave-unitarity.222047/
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# Partial Wave Unitarity ?
1. Mar 14, 2008
### ophase
Partial Wave Unitarity ??
Is there anyone who deals with "Partial Wave Unitarity" in Quantum Mechanics before??
2. Mar 14, 2008
### pam
"Partial wave unitarity" means that the largest possible scattering amplitude squared (|A|^2) for any partial wave is when sin^2(delta)=1, where delta is the phase shift.
This means that |A|^2 is limited by 4 pi(2L+1)^2/k^2.
That is for spinless particles. Spin factors come in otherwise.
Many people deal with this.
Similar Discussions: Partial Wave Unitarity ?
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https://math.libretexts.org/TextMaps/Applied_Mathematics/Book%3A_Introduction_to_the_Modeling_and_Analysis_of_Complex_Systems_(Sayama)/3%3A_Basics_of_Dynamical_Systems
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$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
3: Basics of Dynamical Systems
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$
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• 3.1: What Are Dynamical Systems?
Dynamical systems theory is the very foundation of almost any kind of rule-based models of complex systems. It consider show systems change over time, not just static properties of observations.
• 3.2: Phase Space
A phase space of a dynamical system is a theoretical space where every state of the system is mapped to a unique spatial location. The number of state variables needed to uniquely specify the system’s state is called the degrees of freedom in the system. You can build a phase space of a system by having an axis for each degree of freedom, i.e., by taking each state variable as one of the orthogonal axes.
• 3.3: What Can We Learn?
You can tell from the phase space what will eventually happen to a system’s state in the long run. For a deterministic dynamical system, its future state is uniquely determined by its current state (hence, the name “deterministic”). Trajectories of a deterministic dynamical system will never branch off in its phase space (though they could merge), because if they did, that would mean that multiple future states were possible, which would violate the deterministic nature of the system.
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https://www.physicsforums.com/threads/general-conditions-for-stokes-theorem.716788/
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# General Conditions for Stokes' Theorem
1. Oct 15, 2013
### Mandelbroth
What is the least restrictive set of conditions needed to utilize the formula $\int\limits_{\Omega}\mathrm{d}\alpha=\int\limits_{\partial\Omega} \alpha$?
2. Oct 16, 2013
### Ben Niehoff
I think the only conditions are those needed to define the integrals (i.e. the same kinds of conditions used to define 1-dimensional integrals). I don't think there are any extra geometrical conditions, provided you're on a differentiable manifold.
You can even relax smoothness of the manifold (and the forms!) if you are careful about using Dirac delta functions. Generally, since a form is something you integrate, they should be thought of as distributions.
The boundary operator can be used in a distributional sense as well. For example, the boundary of a sphere is zero. But it might be helpful to think of a sphere with a point removed, whose boundary is therefore a point; then you can use Stokes' theorem to integrate forms over the sphere.
3. Oct 16, 2013
### Mandelbroth
Could you please expand on this? Thinking of forms as distributions feels foreign, and I don't see where that line of thought would go.
4. Oct 16, 2013
### Ben Niehoff
Expand on it how? Surely you can figure out how to integrate something like
$$\delta(x,y,z) \, dx \wedge dy \wedge dz$$
Do you have a specific question?
5. Oct 16, 2013
### Ben Niehoff
This might be a better example of what I'm talking about. Say we want to find the area of a sphere. The form we want to integrate is
$$\omega = \sin \theta \, d \theta \wedge d \phi$$
Now, the sphere $\Omega$ is a closed surface, so $\partial \Omega = 0$. However, the coordinate patch $\tilde \Omega$ covered by the coordinates $\phi \in (0, 2\pi), \; \theta \in (0, \pi)$ is not a closed surface, and is in fact contractible. We have that $\partial \tilde \Omega$ is the union of the north and south poles of the sphere, and a segment of a great circle that runs between them.
Now, it so happens that
$$\omega = d \big( - \cos \theta \, d \phi \big) = d \alpha$$
so we can use Stokes' theorem. So
$$\int_{\tilde \Omega} \omega = \int_{\partial \tilde \Omega} \alpha = - \int_{\partial \tilde \Omega} \cos \theta \, d \phi$$
To integrate around the "cut" between the north and south poles, we draw a loop around it. On either side there is a vertical part where $d \phi = 0$, and so these parts do not contribute. Then around the north and south poles, there are tiny circles, at which $\cos \theta = \pm 1$ and $\phi$ runs from 0 to $2 \pi$. The tiny circles go opposite directions, so each part contributes positively:
$$- \int_{\partial \tilde \Omega} \cos \theta \, d \phi = 2 \pi + 2 \pi = 4 \pi$$
So you see, if you are careful about how you cut up a manifold, you can apply Stokes' theorem in all sorts of situations.
In this case, we took a closed surface and removed a set of measure zero to turn it into a surface with boundary. The reason this worked is because the form $\omega$ is smooth on the set of measure zero that we removed. If that were not the case (say $\omega$ had a delta-function-like contribution on the "cut"), then you would have to include an extra piece to account for that.
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https://www.physicsforums.com/threads/what-rate-does-the-lift-accelerates-in-the-first-5-sec.846382/
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# What rate does the lift accelerates in the first 5 sec?
Tags:
1. Dec 3, 2015
### henrco
Hi,
I'm trying to solve this problem and it's driving me a little crazy. Any help greatly appreciated.
Q) A lift travels to the top of a tower through a vertical displacement of 48 m. The total journey takes 17 s. The lift accelerates from rest at a constant rate for the first 5 seconds. Then it moves at constant speed and then decelerates to rest at a constant rate for the last 5 seconds.
What rate does the lift accelerate during the first 5 seconds?
My Attempt:
I have the initial and final velocity which are both zero (elevator starts and stops), displacement (48m) and time (overall time 17sec and three time intervals of acceleration, constant velocity and deceleration). Clearly the acceleration and deceleration will be same rates as they occur during the same time intervals. However with the information I have I feel I'm unable to use the usual constant acceleration equations.
The average velocity is 48/17 = 2.8 m/s. However I'm not sure how that helps.
To find the acceleration during the first 5 seconds, I have the time and initial velocity but I need the velocity
it reaches at 5 seconds to obtain the acceleration and I just can't work out how to get it.
2. Dec 3, 2015
### PeroK
There are several approaches. Since you've been thinking about average velocity, you might like to try to solve the problem using the average velocity for each stage of the motion. Then you don't need any of the "usual" equations.
3. Dec 3, 2015
### Staff: Mentor
Hi Conal Henry, Welcome to Physics Forums.
Please retain the formatting headers (provided in the edit window) when you post a problem here.
Start with a graphic approach to gain insight. Have you tried making a sketch of velocity versus time? What does the area under a v vs t graph give you?
4. Dec 4, 2015
### henrco
Hi Perok,
1) For stage 1, the first 5 seconds, the acceleration is constant, so I've taken the angle the acceleration makes to be 45 degrees. (This is intuitive and if it's correct,
would you mind explaining why?).
The velocity for this 5 x Sin (45) = 3.5m/s.
With initial velocity is 0, the acceleration a = (3.5-0)/5 = 0.7m/s2
2) Find the average velocity for each stage. Starting with Stage 1, the first 5 seconds. I obviously have the time but not the displacement.
Avg Vel = Displacement / 5.
Work out area of each stage. Stage 1 = ( 5 x v/2 ) Stage 2 = ( 7 x v) Stage 3 = ( 5 x v/2). This all comes to 12 v
We know Area = displacement, therefore: 12v = 48. So v = 4.
Acceleration = (4-0)/5 = .8 m/s2
My second attempt seems to be correct? As I then worked out the displacement for each stage to be (S1 = 10, S2 = 28 and S3 = 10).
Could you please let know if this is correct?
Also you mentioned that there are several approaches, if you have time would you mind briefly outlining another approach.
I'd like to try to understand this problem from different approaches.
5. Dec 4, 2015
### henrco
Hi gneill,
Point noted, will post future problems using the template. Thank you for your advice regarding the problem.
I replied to another helpful suggestion above and I think the answer below was the direction you were sending me in?
Find the average velocity for each stage. Starting with Stage 1, the first 5 seconds. I obviously have the time but not the displacement.
Avg Vel = Displacement / 5.
Work out area of each stage. Stage 1 = ( 5 x v/2 ) Stage 2 = ( 7 x v) Stage 3 = ( 5 x v/2). This all comes to 12 v
We know Area = displacement, therefore: 12v = 48. So v = 4.
Acceleration = (4-0)/5 = .8 m/s2
Conal
6. Dec 4, 2015
### Staff: Mentor
Yes, you get to the same result. A graphical depiction to begin can sometimes help. For example:
You know the area must be your displacement, you have the time periods, the only thing you don't have is the maximum velocity Vm. But Vm is easily found given the other information. Acceleration is just the slope of a line in the figure.
7. Dec 4, 2015
### PeroK
You seem to have understood the graphical approach. I'd say that's one thing never to forget: displacement = area under the velocity vs time graph.
Regarding average velocity. For constant acceleration from or to rest, the average velocity is half the final or initial velocity respectively. More generally, if you accelerate from $u$ to $v$ at a constant acceleration, then the average velocity is $(u+v)/2$. You can check that graphically.
Another approach was to note the symmetry of the motion. Although, in this case, that just makes things a little easier. Always look out for symmetry in a problem. As problems get harder, using the symmetry of a problem can makes things a lot easier. And I mean a lot!
8. Dec 4, 2015
### henrco
Thank you very much, very helpful.
9. Dec 4, 2015
### henrco
Thank you very much, this was very helpful.
Draft saved Draft deleted
Similar Discussions: What rate does the lift accelerates in the first 5 sec?
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http://math.stackexchange.com/questions/44294/can-one-show-that-sum-n-1n-frac1n-log-n-gamma-leqslant-frac12/44323
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# Can one show that $\sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N}$ without using the Euler-Maclaurin formula?
I would like to prove that $$\sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N}$$ without using the Euler-Maclaurin summation formula. The motivation for this is that I have come very close to doing so (see the answer provided below) but annoyingly have not actually proved the above.
Some may ask why I don't just use the formula. I'm writing a set of analytic number theory notes for my own use and it seems an unwieldy result to introduce and prove, given that the above inequality is all I need, and given that I have gotten so close without using Euler-Maclaurin!
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Let $$\gamma_n = \sum_{k=1}^n \frac{1}{k} - \log n.$$ Our goal is to show that $$\gamma_n - \lim_{m \to \infty} \gamma_m \leq \frac{1}{2n}.$$ It is enough to show that, for $n<m$, we have $$\gamma_n - \gamma_m \leq \frac{1}{2n}.$$ This has the advantage of dealing solely with finite quantities.
Now, $$\gamma_n - \gamma_m = \int_{n}^m \frac{dt}{t} - \sum_{k=n+1}^m \frac{1}{k} =\sum_{j=n}^{m-1} \int_{j}^{j+1} \left( \frac{1}{t} - \frac{1}{j+1} \right) \cdot dt .$$
At this point, if I were at a chalkboard rather than a keyboard, I would draw a picture. Draw the hyperbola $y=1/x$ and mark off the interval between $x=n$ and $x=m$. Divide this into $m-n$ vertical bars of width $1$. Each bar stretches up to touch the hyperbola at its right corner. There is a little wedge, bounded by $x=j$, $y=1/(j+1)$ and $y=1/x$. We are adding up the area of each of these wedges.1
Because $y=1/x$ is convex, the area of this wedge is less than that of the right triangle with vertices at $(j,1/(j+1))$, $(j+1, 1/(j+1))$ and $(j,1/j)$. This triangle has base $1$ and height $1/j - 1/(j+1)$, so its area is $(1/2) (1/j - 1/(j+1))$. So the quantity of interest is $$\leq \sum_{j=n}^{m-1} \frac{1}{2} \left( \frac{1}{j} - \frac{1}{j+1} \right) = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{m} \right) \leq \frac{1}{2n}.$$
Of course, this is just a standard proof of Euler-Maclaurin summation, but it is a lot more geometric and easy to follow in this special case.
1 By the way, since this area is positive, we also get the corollary that $\gamma_n - \gamma_m > 0$, so $\gamma_n - \gamma >0$, another useful bound.
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(+1) Just to point out a typo: In "Draw the hyperbola y=1/x and mark off the interval between x=n and x=n.", the second n should be an m. – John Bentin Jun 9 '11 at 13:52
It is really annoying. This is exactly the kind of geometric proof I went for, using areas, and I always failed! Thanks for showing how it's done +1. – Sputnik Jun 18 '11 at 10:20
What follows is a variant of the method suggested by Fahad Sperinck, which almost gave the desired bound. Although we obtain a pretty short proof of the inequality, I think that the "right" proof is the one in the post by David Speyer. (A proof based on geometry is "right," as is a combinatorial proof.)
Let us start as Fahad Sperinck did, from $$\int_n^{n+1} \frac{x-[x]}{x^2}\: dx = \log\Big(\frac{n+1}{n}\Big) - \frac{1}{n+1} < \frac{1}{n} - \frac{1}{n+1} -\frac{1}{2n^2} +\frac{1}{3n^3}.$$
Ultimately, we will summing from $N$ to infinity. If we keep this fact in mind, the chunk $$\frac{1}{n}-\frac{1}{n+1}$$ sums beautifully to $1/N$, and should be left as is. If we could show that the part that is taken away, namely $$\frac{1}{2n^2}-\frac{1}{3n^3}$$ is bigger than $$\frac{1}{2}\left(\frac{1}{n}-\frac{1}{n+1}\right),$$ we would be finished.
Now I will do some unofficial scribbling, don't look. I want to show that $1/2n^2-1/3n^3 \ge 1/2(n)(n+1)$, so I want to show that $(3n-2)/6n^3\ge 1/2n(n+1)$, so I want to show that $(3n-2)/3n^2 \ge 1/(n+1)$, so I want to show that $(3n-2)(n+1) \ge 3n^2$, and this is clearly true if $n \ge 2$, just multiply out the stuff on the left.
Now if I had the energy I would hide my tracks, and have the desired inequality drop out as if by magic.
Comment: Somehow, one acquires the habit of thinking of $n^2$ and $1/n^2$ as "nice" and of $n(n+1)$ and $1/n(n+1)$ as not so nice. In many ways, the opposite is true. Certainly that is the case from the combinatorial point of view.
The calculations in the post were fine, the problem was that of giving away a tiny bit too much. That was, maybe, because the strategy was directed at getting to something that looks like $1/n^2$, which was viewed as tractable and desirable. But $1/n(n+1)$, aka $1/n-1/(n+1)$, arises naturally in the problem, and is much more tractable.
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Nice exposition! I am a big fan of Penn and Teller's videos where they do magic tricks while showing you how they are done, and this has the same feel. – David Speyer Jun 9 '11 at 14:41
The beauty, clarity and simplicity of your posts are such an enrichment of this site. Many thanks for all the time and effort you invest in your contributions. – t.b. Aug 10 '11 at 14:05
One can check that $S(N):=\sum_{n=1}^N\frac{1}{n} -\log N - \gamma = \int_N^\infty \frac{x-[x]}{x^2} \: dx$, where $[x]$ is the integer part of $x$. Moreover $$\int_n^{n+1} \frac{x-[x]}{x^2}\: dx = \log\Big(\frac{n+1}{n}\Big) - \frac{1}{n+1} < \frac{1}{n} - \frac{1}{n+1} -\frac{1}{2n^2} +\frac{1}{3n^3}, \qquad (1)$$ by the Taylor series for $\log(1+x)$. But we have that $$n(n+1)(3n-1) = 3n^3 + 2n^2 -n > 3n^3$$ so $$\frac{1}{n(n+1)} < \frac{3n-1}{3n^3} = \frac{1}{n^2} - \frac{1}{3n^3}.$$ Therefore, from equation $(1)$ we find $$S(N) < \sum_{n=N}^\infty \frac{1}{n(n+1)} -\frac{1}{2n^2} +\frac{1}{3n^3} < \sum_{n=N}^\infty \frac{1}{n^2} - \frac{1}{3n^3} -\frac{1}{2n^2} +\frac{1}{3n^3},$$ and so finally, $$S(N) < \frac{1}{2}\sum_{n=N}^\infty \frac{1}{n^2} < \frac{1}{2(N-1)},$$ for all $N \in \mathbb{N}$, by a standard approximation for $\sum \frac{1}{n^2}$.
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By the way, does the difference between $1/2(N-1)$ and $1/2N$ really matter? Unless you are heading for hard bounds in the end, I would guess that $1/2N + O(1/N^2)$ is good enough for whatever you want, and you already have that. – David Speyer Jun 9 '11 at 12:11
@David: I'll admit that it doesn't really matter, but it is just nicer to be able to write that something is $O(\frac{1}{N})$ with a simple implied constant like $\frac{1}{2}$, which is actually the best possible constant as well. I guess it was more a matter of elegance for me! – Sputnik Jun 18 '11 at 10:18
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https://engineeringlibrary.org/calculators
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## Structural Mechanics
The beam calculator allows for the analysis of stresses and deflections in straight beams.
The 2D Finite Element Analysis (FEA) calculator can be used to analyze any structure that can be modeled with 2D beams.
The bolted joint calculator allows for stress analysis of a bolted joint, accounting for preload, applied axial load, and applied shear load.
The bolt torque calculator can be used to calculate the torque required to achieve the desired preload on a bolted joint.
The bolt pattern calculator allows for applied forces to be distributed over bolts in a pattern.
The lug calculator allows for analysis of lifting lugs under axial, transverse, or oblique loading.
The column buckling calculator allows for buckling analysis of long and intermediate-length columns loaded in compression.
The Mohr's circle calculator provides an intuitive way of visualizing the state of stress at a point in a loaded material.
The stress-strain curve calculator allows for the calculation of the engineering stress-strain curve of a material.
The cross section builder allows for the calculation of properties for a custom cross section.
The stress concentration calculator provides a set of interactive plots for common stress concentration factors.
## Failure Mechanisms
The fracture mechanics calculator allows for fracture analysis of a cracked part.
The fatigue crack growth calculator allows for fatigue crack growth analysis of a cracked part.
## Math
The unit conversion calculator allows for conversion between various units, with a focus on engineering units.
## Systems Engineering
The trade study calculator provides a systematic method for making a decision among competing alternatives.
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http://mathhelpforum.com/calculus/110913-am-i-doing-right.html
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# Math Help - Am I doing this right?
1. ## Am I doing this right?
Let f(x)= xe^(x). Use the limit definition to compute f'(0) and find the equation of the tangent line at x=0
f(x)= xe^(x)
limit definition f'(x)= f(x+h)-f(x) / h as lim h-->0
so
f'(x) = (x+h)* (e^(x+h) - xe^(x) / h
f'(0) = (0+h)* (e^(0+h) - xe^(0) / h
= h*e(h)- 0 / h
= e^(h) = e(0) =1
So, the derivative is 1.
Is this right?
2. Originally Posted by skboss
Let f(x)= xe^(x). Use the limit definition to compute f'(0) and find the equation of the tangent line at x=0
f(x)= xe^(x)
limit definition f'(x)= f(x+h)-f(x) / h as lim h-->0
so
f'(x) = (x+h)* (e^(x+h) - xe^(x) / h
f'(0) = (0+h)* (e^(0+h) - xe^(0) / h
= h*e(h)- 0 / h
= e^(h) = e(0) =1
So, the derivative is 1.
Is this right?
close enough from what I can discern from your syntax ... this might be a bit easier.
$f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}$
$f'(0) = \lim_{x \to 0} \frac{xe^x - 0}{x - 0}$
$f'(0) = \lim_{x \to 0} \frac{xe^x}{x}$
$f'(0) = \lim_{x \to 0} e^x = e^0 = 1$
3. Thanks. BTW, Can I use that a limit definition to find derivative? I wasn't sure about it.
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http://chalkdustmagazine.com/regulars/crossnumber/prize-crossnumber-issue-06/
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# Prize crossnumber, Issue 06
Win £100 of Maths Gear goodies by solving our famously fiendish crossnumber
Our original prize crossnumber is featured on pages 52 and 53 of Issue 06.
Correction: The pdf was incorrect and 5D did not match the clues below. This has now been fixed.
Clarification: Added brackets to 29A and 34D to reduce ambiguity.
### Rules
• Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
• One randomly selected correct answer will win a £100 Maths Gear goody bag. Three randomly selected runners up will win a Chalkdust t-shirt. The prizes have been provided by Maths Gear, a website that sells nerdy things worldwide, with free UK shipping. Find out more at mathsgear.co.uk
• To enter, submit the sum of the across clues via this form by 8 January 2018. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 22 January 2018.
### Crossnumber
Crossnumber #6, set by Humbug:
Click to enlarge
### Clues
#### Across
• 1. This number is divisible by both the sum and the product of its digits. (2)
• 3. All the digits of this number are different. (4)
• 5. The product of 1A and 5D. (3)
• 6. The difference between 22D and 23D. (5)
• 11. A prime number. (2)
• 12. A square number. (2)
• 13. The maximum number of pieces a cube can be cut into with 36 planar cuts. (4)
• 15. A factor of 7D. (4)
• 17. When written in hexadecimal, this number spells a common English word. (4)
• 18. This number and 19D are a pair of triangle numbers whose sum and difference are also triangle numbers. (2)
• 20. A palindrome. (5)
• 23. A multiple of 7 whose digits add to 7. (4)
• 24. A factor of 33D. (2)
• 25. The largest number whose square has 12 digits. (6)
• 29. A factor of (12 more than 22D). (2)
• 30. A multiple of 26D. (3)
• 31. The sum of the digits of 29A. (2)
• 32. The product of 36D and 12A. (3)
• 33. A Fibonacci number. (2)
• 37. A palindrome that is a multiple of 1111. (7)
• 41. An odd number. (2)
• 42. Each digit of this number is one more than the previous digit. (4)
• 43. This number and 29D are a pair of triangle numbers whose sum and difference are also triangle numbers. (3)
• 44. The mean of 45D and 41A. (2)
• 46. This number and 47A are a pair of triangle numbers whose sum and difference are also triangle numbers. (7)
• 47. see 46A. (7)
#### Down
• 1. Take this number’s digits as the first two numbers in a sequence. Form a Fibonacci-like sequence. This number is in this sequence. (2)
• 2. This number is divisible by both the sum and the product of its digits. (2)
• 4. The square root of 3A. (2)
• 5. The product of the first two digits of 35D. (2)
• 7. Not a prime number. (5)
• 8. Ten times the sum of the across clues in this crossnumber. (8)
• 9. A factor of 45D. (2)
• 10. The product of 11A and 12A. (3)
• 11. Three more than a multiple of four. (3)
• 14. A multiple of 45D. (7)
• 16. The product of 11A and 36D. (3)
• 19. see 18A. (2)
• 21. A factor of 20A. (2)
• 22. A multiple of 36D. (6)
• 23. A multiple of 44A and 28D. (6)
• 24. An integer $x$ that satisfies $x^2+1=2y^4$, for some integer $y$. (3)
• 26. Not equal to 44A. (2)
• 27. The price of a £100 coat after discounting 10% then adding 10% tax. (2)
• 28. The price of a £100 coat after adding 10% tax then discounting 10%. (2)
• 29. see 43A. (3)
• 33. A multiple of 17A. (5)
• 34. A multiple of (one more than 15A). (5)
• 35. An even number. (3)
• 36. The mean of 11A, 12A and 36D. (2)
• 37. The sum of the digits of this number’s cube, sixth power and seventh power are all equal. (2)
• 38. In a base other than 10, this number can be written as 110001. (4)
• 39. The product of the digits of 34D. (2)
• 40. This number is equal to the product of its digits plus the sum of its digits. (2)
• 45. A multiple of 5. (3)
• ### Book of the Year 2019
We announce the winner of this coveted prize
• ### 2019 Book of the Year vote
Have your say, and help to choose the 2019 Readers' Choice
• ### The Hidden Half
We review the ninth of this year's nominees for the Book of the Year
• ### Maths on the Back of an Envelope
We review the eighth of this year's nominees for the Book of the Year
• ### The Maths of Life and Death
We review the sevnth of this year's nominees for the Book of the Year
• ### Humble Pi
We review the sixth of this year's nominees for the Book of the Year
|
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http://mathhelpforum.com/algebra/133811-radical-multiplication.html
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# Math Help - radical multiplication.
$\sqrt[3]{2}\sqrt{3}$
I feel extremely stupid for not being able to simplify this.
The answer is $\sqrt[6]{108}$
Any help on how they got this answer?
2. Originally Posted by integral
$\sqrt[3]{2}\sqrt{3}$
I feel extremely stupid for not being able to simplify this.
The answer is $\sqrt[6]{108}$
Any help on how they got this answer?
$2^{\frac{1}{3}} \cdot 3^{\frac{1}{2}}$
$2^{\frac{2}{6}} \cdot 3^{\frac{3}{6}}$
$(2^2 \cdot 3^3)^{\frac{1}{6}}$
finish it
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https://www.hepdata.net/search/?q=title%3A%22photon+collisions%22&page=1&phrases=Single+Differential+Cross+Section
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Showing 25 of 36 results
#### Observation of Charmed Mesons in Photon-photon Collisions
The collaboration Bartel, W. ; Becker, L. ; Felst, R. ; et al.
Phys.Lett.B 184 (1987) 288-292, 1987.
Inspire Record 235696
The inclusive production of D ∗± mesons in single tagged photon-photon collisions is investigated using the JADE detector at PETRA. D ∗± mesons are reconstructed through their decay into D 0 +π ± where the D 0 decays via D 0 →Kππ 0 . The event rate and topology are compared to the expectations of c quark production in the quark-parton model: γγ→c c .
0 data tables match query
#### High $p_T$ Hadron Production in Photon - Photon Collisions
The collaboration Brandelik, R. ; Braunschweig, W. ; Gather, K. ; et al.
Phys.Lett.B 107 (1981) 290-296, 1981.
Inspire Record 167417
We have studied the properties of hadron production in photon-photon scattering with tagged photons at the e + e − storage ring PETRA. A tail in the p T distribution of particles consistent with p T −4 has been observed. We show that this tail cannot be due to the hadronic part of the photon. Selected events with high p T particles are found to be consistent with a two-jet structure as expected from a point-like coupling of the photons to quarks. The lowest-order cross section predicted for γγ → q q , σ = 3 Σ e q 4 · σ γγ → μμ , is approached from above by the data at large transverse momenta.
0 data tables match query
#### rivet Analysis Inclusive jet production in two-photon collisions at LEP
The collaboration Achard, P. ; Adriani, O. ; Aguilar-Benitez, M. ; et al.
Phys.Lett.B 602 (2004) 157-166, 2004.
Inspire Record 661114
Inclusive jet production, e+e- -> e+e- \ee$jet X, is studied using 560/pb of data collected at LEP with the L3 detector at centre-of-mass energies between 189 and 209 GeV. The inclusive differential cross section is measured using a k_t jet algorithm as a function of the jet transverse momentum, pt, in the range 3<pt<50 GeV for a pseudorapidity, eta, in the range -1<eta<1. This cross section is well represented by a power law. For high pt, the measured cross section is significantly higher than the NLO QCD predictions, as already observed for inclusive charged and neutral pion production. 0 data tables match query #### Inclusive lambda production in two photon collisions at LEP The collaboration Achard, P. ; Adriani, O. ; Aguilar-Benitez, M. ; et al. Phys.Lett.B 586 (2004) 140-150, 2004. Inspire Record 637287 The reactions e^+e^- -> e^+e^- Lambda X and e^+e^- -> e^+e^- Lambda X are studied using data collected at LEP with the L3 detector at centre-of-mass energies between 189 and 209 GeV. Inclusive differential cross sections are measured as a function of the lambda transverse momentum, p_t, and pseudo-rapidity, eta, in the ranges 0.4 GeV < p_t < 2.5 GeV and |\eta| < 1.2. The data are compared to Monte Carlo predictions. The differential cross section as a function of p_t is well described by an exponential of the form A exp (- p_t / <p_t>)$.
0 data tables match query
#### Pion and Kaon Pair Production in Photon-Photon Collisions
The collaboration Aihara, H. ; Alston-Garnjost, M. ; Avery, R.E. ; et al.
Phys.Rev.Lett. 57 (1986) 404, 1986.
Inspire Record 228072
We report measurements of the two-photon processes e+e−→e+e−π+π− and e+e−→e+e−K+K−, at an e+e− center-of-mass energy of 29 GeV. In the π+π− data a high-statistics analysis of the f(1270) results in a γγ width Γ(γγ→f)=3.2±0.4 keV. The π+π− continuum below the f mass is well described by a QED Born approximation, whereas above the f mass it is consistent with a QCD-model calculation if a large contribution from the f is assumed. For the K+K− data we find agreement of the high-mass continuum with the QCD prediction; limits on f′(1520) and θ(1720) formation are presented.
0 data tables match query
#### Inclusive production of charged hadrons in photon-photon collisions
The collaboration Abbiendi, G. ; Ainsley, C. ; Akesson, P.F. ; et al.
Phys.Lett.B 651 (2007) 92-101, 2007.
Inspire Record 734955
The inclusive production of charged hadrons in the collisions of quasi-real photons e+e- -> e+e- +X has been measured using the OPAL detector at LEP. The data were taken at e+e- centre-of-mass energies from 183 to 209 GeV. The differential cross-sections as a function of the transverse momentum and the pseudorapidity of the hadrons are compared to theoretical calculations of up to next-to-leading order (NLO) in the strong coupling constant alpha{s}. The data are also compared to a measurement by the L3 Collaboration, in which a large deviation from the NLO predictions is observed.
0 data tables match query
#### Measurement of eta eta production in two-photon collisions
The collaboration Uehara, S. ; Watanabe, Y. ; Nakazawa, H. ; et al.
Phys.Rev.D 82 (2010) 114031, 2010.
Inspire Record 862260
We report the first measurement of the differential cross section for the process gamma gamma --> eta eta in the kinematic range above the eta eta threshold, 1.096 GeV < W < 3.8 GeV over nearly the entire solid angle range, |cos theta*| <= 0.9 or <= 1.0 depending on W, where W and theta* are the energy and eta scattering angle, respectively, in the gamma gamma center-of-mass system. The results are based on a 393 fb^{-1} data sample collected with the Belle detector at the KEKB e^+ e^- collider. In the W range 1.1-2.0 GeV/c^2 we perform an analysis of resonance amplitudes for various partial waves, and at higher energy we compare the energy and the angular dependences of the cross section with predictions of theoretical models and extract contributions of the chi_{cJ} charmonia.
0 data tables match query
#### Double tag events in two photon collisions at LEP
The collaboration Achard, P. ; Adriani, O. ; Aguilar-Benitez, M. ; et al.
Phys.Lett.B 531 (2002) 39-51, 2002.
Inspire Record 565440
Double-tag events in two-photon collisions are studied using the L3 detector at LEP centre-of-mass energies from root(s)=189 GeV to 209 GeV. The cross sections of the e+e- -> e+e- hadrons and gamma*gamma* -> hadrons processes are measured as a function of the photon virtualities, Q1^2 and Q2^2, of the two-photon mass, W_gammagamma, and of the variable Y=ln(W_gammagamma^2/(Q1 Q2)), for an average photon virtuality <Q2> = 16 GeV2. The results are in agreement with next-to-leading order calculations for the process gamma*gamma* -> q qbar in the interval 2 <= Y <= 5. An excess is observed in the interval 5 < Y <= 7, corresponding to W_gammagamma greater than 40 GeV . This may be interpreted as a sign of resolved photon QCD processes or the onset of BFKL phenomena.
0 data tables match query
#### Inclusive $D^{*+-}$ production in two photon collisions at LEP
The collaboration Achard, P. ; Adriani, O. ; Aguilar-Benitez, M. ; et al.
Phys.Lett.B 535 (2002) 59-69, 2002.
Inspire Record 585623
Inclusive D^{*+-} production in two-photon collisions is studied with the L3 detector at LEP, using 683 pb^{-1} of data collected at centre-of-mass energies from 183 to 208 GeV. Differential cross sections are determined as functions of the transverse momentum and pseudorapidity of the D^{*+-} mesons in the kinematic region 1 GeV < P_T < 12 GeV and |eta| < 1.4. The cross sections sigma(e^+e^- -> e^+e^-D^{*+-}X) in this kinematical region is measured and the sigma(e^+e^- -> e^+e^- cc{bar}X) cross section is derived. The measurements are compared with next-to-leading order perturbative QCD calculations.
0 data tables match query
#### Inclusive charged hadron production in two photon collisions at LEP
The collaboration Achard, P. ; Adriani, O. ; Aguilar-Benitez, M. ; et al.
Phys.Lett.B 554 (2003) 105-114, 2003.
Inspire Record 605973
Inclusive charged hadron production, e+e- -> e+e- h+- X, is studied using 414 pb-1 of data collected at LEP with the L3 detector at centre-of-mass energies between 189 and 202 GeV. Single particle inclusive differential cross sections are measured as a function of the particle transverse momentum, pt, and pseudo-rapidity, eta. For p_t < 1.5 GeV, the data are well described by an exponential, typical of soft hadronic processes. For higher pt, the onset of perturbative QCD processes is observed. The pi+- production cross section for pt > 5 GeV is much higher than the NLO QCD predictions.
0 data tables match query
#### Production of the F(0) Meson in Photon-photon Collisions
The collaboration Behrend, H.J. ; Fenner, H. ; Schachter, M.J. ; et al.
Z.Phys.C 23 (1984) 223, 1984.
Inspire Record 199731
The production of thef0 in two photon collisions, with the subsequent decayf0→π+π− has been observed in the CELLO detector at PETRA. Thef0 peak was found to lie on a dipion continuum and to be shifted downwards in mass by ≃50 MeV/c2. The ππ mass spectrum from 0.8 to 1.5 GeV/c2 was well fitted by the model of Mennessier using only a unitarised Born amplitude and helicity 2f0 amplitude. The previously observed mass shift and distortion of thef0 peak are explained by strong interference between the Born andf0 amplitudes. The only free parameter in the fit of the data to the model is the radiative widthΓγγ(f0). It was found that:Γγγ(f0)=2.5±0.1±0.5 keV where the first (second) quoted errors are statistical (systematic).
0 data tables match query
#### Exclusive Production of Proton Anti-proton Pairs in Photon-photon Collisions
The collaboration Bartel, W. ; Becker, L. ; Cords, D. ; et al.
Phys.Lett.B 174 (1986) 350-356, 1986.
Inspire Record 231554
Total and differential cross sections for exclusive production of proton-antiproton pairs in photon-photon collisions have been measured using the JADE detector at PETRA. The total cross section in the CM angular |cos θ ∗ | < 0.6 reaches a maximum value of 3.8 nb for a γγ invariant mass of W γγ = 2.25 GeV, and decreases rapidly for higher values of W γγ . In the range 2.0 GeV < W γγ < 2.6 GeV the angular distribution is not isotopic. The nucleons are preferentially emitted at large angles to the collision axis.
0 data tables match query
#### $p\bar{p}$ pair production in two photon collisions at LEP
The collaboration Achard, P. ; Adriani, O. ; Aguilar-Benitez, M. ; et al.
Phys.Lett.B 571 (2003) 11-20, 2003.
Inspire Record 620433
The reaction e^+e^- -> e^+e^- proton antiproton is studied with the L3 detector at LEP. The analysis is based on data collected at e^+e^- center-of-mass energies from 183 GeV to 209 GeV, corresponding to an integrated luminosity of 667 pb^-1. The gamma gamma -> proton antiproton differential cross section is measured in the range of the two-photon center-of-mass energy from 2.1 GeV to 4.5 GeV. The results are compared to the predictions of the three-quark and quark-diquark models.
0 data tables match query
#### Formation of Delta (980) and A2 (1320) in Photon-photon Collisions
The collaboration Antreasyan, D. ; Aschman, D. ; Besset, D. ; et al.
Phys.Rev.D 33 (1986) 1847, 1986.
Inspire Record 217547
The reaction γγ→π0η has been investigated with the Crystal Ball detector at the DESY storage ring DORIS II. Formation of δ(980) and A2(1320) has been observed with γγ partial widths Γγγ(A2)=1.14±0.20±0.2 6 keV and Γγγ(δ)B(δ→πη)=0.19±0.07 −0.07+0.10 keV.
0 data tables match query
#### Measurement of inclusive $D^{*+-}$ production in two photon collisions at LEP
The collaboration Acciarri, M. ; Achard, P. ; Adriani, O. ; et al.
Phys.Lett.B 467 (1999) 137-146, 1999.
Inspire Record 505281
Inclusive production of $\mathrm{D^{*\pm}}$ mesons in two-photon collisions was measured by the L3 experiment at LEP. The data were collected at a centre-of-mass energy $\sqrt{s} = 189$ GeV with an integrated luminosity of $176.4 \mathrm{pb^{-1}}$. Differential cross sections of the process $\mathrm{e^+e^- \to D^{*\pm} X}$ are determined as functions of the transverse momentum and pseudorapidity of the $\mathrm{D^{*\pm}}$ mesons in the kinematic region 1 GeV $< p_{T}^{\mathrm{D^*}} < 5$ GeV and $\mathrm{|\eta^{D^*}|} < 1.4$. The cross section integrated over this phase space domain is measured to be $132 \pm 22(stat.) \pm 26(syst.)$ pb. The differential cross sections are compared with next-to-leading order perturbative QCD calculations.
0 data tables match query
#### A Measurement of $\pi^0 \pi^0$ Production in Two Photon Collisions
The collaboration Marsiske, H. ; Antreasyan, D. ; Bartels, H.W. ; et al.
Phys.Rev.D 41 (1990) 3324, 1990.
Inspire Record 294492
The reaction e+e−→e+e−π0π0 has been analyzed using 97 pb−1 of data taken with the Crystal Ball detector at the DESY e−e+ storage ring DORIS II at beam energies around 5.3 GeV. For the first time we have measured the cross section for γγ→π0π0 for π0π0 mvariant masses ranging from threshold to about 2 GeV. We measure an approximately flat cross section of about 10 nb for W=mπ0π0<0.8 GeV, which is below 0.6 GeV, in good agreement with a theoretical prediction based on an unitarized Born-term model. At higher invariant masses we observe formation of the f2(1270) resonance and a hint of the f0(975). We deduce the following two-photon widths: Γγγ(f2(1270))=3.19±0.16±0.280.29 keV and Γγγ(f0(975))<0.53 keV at 90% C.L. The decay-angular distributions show the π0π0 system to be dominantly spin 0 for W<0.7 GeV and spin 2, helicity 2 in the f2(1270) region, with helicity 0 contributing at most 22% (90% C.L.).
0 data tables match query
#### High-statistics measurement of neutral pion-pair production in two-photon collisions
The collaboration Uehara, S. ; Watanabe, Y. ; Adachi, I. ; et al.
Phys.Rev.D 78 (2008) 052004, 2008.
Inspire Record 786406
We report a high-statistics measurement of differential cross sections for the process gamma gamma -> pi^0 pi^0 in the kinematic range 0.6 GeV <= W <= 4.0 GeV and |cos theta*| <= 0.8, where W and theta* are the energy and pion scattering angle, respectively, in the gamma gamma center-of-mass system. Differential cross sections are fitted to obtain information on S, D_0, D_2, G_0 and G_2 waves. The G waves are important above W ~= 1.6 GeV. For W <= 1.6 GeV the D_2 wave is dominated by the f_2(1270) resonance while the S wave requires at least one additional resonance besides the f_0(980), which may be the f_0(1370) or f_0(1500). The differential cross sections are fitted with a simple parameterization to determine the parameters (the mass, total width and Gamma_{gamma gamma}B(f_0 -> pi^0 pi^0)) of this scalar meson as well as the f_0(980). The helicity 0 fraction of the f_2(1270) meson, taking into account interference for the first time, is also obtained.
0 data tables match query
#### High-statistics study of neutral-pion pair production in two-photon collisions
The collaboration Uehara, S. ; Watanabe, Y. ; Nakazawa, H. ; et al.
Phys.Rev.D 79 (2009) 052009, 2009.
Inspire Record 815978
The differential cross sections for the process $\gamma \gamma \to \pi^0 \pi^0$ have been measured in the kinematic range 0.6 GeV $< W < 4.1$ GeV, $|\cos \theta^*|<0.8$ in energy and pion scattering angle, respectively, in the $\gamma\gamma$ center-of-mass system. The results are based on a 223 fb$^{-1}$ data sample collected with the Belle detector at the KEKB $e^+ e^-$ collider. The differential cross sections are fitted in the energy region 1.7 GeV $< W <$ 2.5 GeV to confirm the two-photon production of two pions in the G wave. In the higher energy region, we observe production of the $\chi_{c0}$ charmonium state and obtain the product of its two-photon decay width and branching fraction to $\pi^0\pi^0$. We also compare the observed angular dependence and ratios of cross sections for neutral-pion and charged-pion pair production to QCD models. The energy and angular dependence above 3.1 GeV are compatible with those measured in the $\pi^+\pi^-$ channel, and in addition we find that the cross section ratio, $\sigma(\pi^0\pi^0)/\sigma(\pi^+\pi^-)$, is $0.32 \pm 0.03 \pm 0.05$ on average in the 3.1-4.1 GeV region.
0 data tables match query
#### High-statistics study of $K^0_S$ pair production in two-photon collisions
The collaboration Uehara, S. ; Watanabe, Y. ; Nakazawa, H. ; et al.
PTEP 2013 (2013) 123C01, 2013.
Inspire Record 1245023
We report a high-statistics measurement of the differential cross section of the process gamma gamma --> K^0_S K^0_S in the range 1.05 GeV <= W <= 4.00 GeV, where W is the center-of-mass energy of the colliding photons, using 972 fb^{-1} of data collected with the Belle detector at the KEKB asymmetric-energy e^+ e^- collider operated at and near the Upsilon-resonance region. The differential cross section is fitted by parameterized S-, D_0-, D_2-, G_0- and G_2-wave amplitudes. In the D_2 wave, the f_2(1270), a_2(1320) and f_2'(1525) are dominant and a resonance, the f_2(2200), is also present. The f_0(1710) and possibly the f_0(2500) are seen in the S wave. The mass, total width and product of the two-photon partial decay width and decay branching fraction to the K bar{K} state Gamma_{gamma gamma}B(K bar{K}) are extracted for the f_2'(1525), f_0(1710), f_2(2200) and f_0(2500). The destructive interference between the f_2(1270) and a_2(1320) is confirmed by measuring their relative phase. The parameters of the charmonium states chi_{c0} and chi_{c2} are updated. Possible contributions from the chi_{c0}(2P) and chi_{c2}(2P) states are discussed. A new upper limit for the branching fraction of the P- and CP-violating decay channel eta_c --> K^0_S K^0_S is reported. The detailed behavior of the cross section is updated and compared with QCD-based calculations.
0 data tables match query
#### Study of $\pi^0$ pair production in single-tag two-photon collisions
The collaboration Masuda, M. ; Uehara, S. ; Watanabe, Y. ; et al.
Phys.Rev.D 93 (2016) 032003, 2016.
Inspire Record 1390112
We report a measurement of the differential cross section of $\pi^0$ pair production in single-tag two-photon collisions, $\gamma^* \gamma \to \pi^0 \pi^0$, in $e^+ e^-$ scattering. The cross section is measured for $Q^2$ up to 30 GeV$^2$, where $Q^2$ is the negative of the invariant mass squared of the tagged photon, in the kinematic range 0.5 GeV < W < 2.1 GeV and $|\cos \theta^*|$ < 1.0 for the total energy and pion scattering angle, respectively, in the $\gamma^* \gamma$ center-of-mass system. The results are based on a data sample of 759 fb$^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy $e^+ e^-$ collider. The transition form factor of the $f_0(980)$ and that of the $f_2(1270)$ with the helicity-0, -1, and -2 components separately are measured for the first time and are compared with theoretical calculations.
0 data tables match query
#### Exclusive production of $p\bar{p}$ pairs in two photon collisions at PEP
The collaboration Aihara, H. ; Alston-Garnjost, M. ; Avery, R.E. ; et al.
Phys.Rev.D 36 (1987) 3506, 1987.
Inspire Record 246557
We report cross sections for the process γγ→pp¯ at center-of-mass energies W from 2.0 to 2.8 GeV. These results have been extracted from measurements of e+e−→e+e−pp¯ at an overall center-of-mass energy of 29 GeV, using the TPC/Two-Gamma facility at the SLAC storage ring PEP. Cross sections for the untagged mode [both photons nearly real] are shown to lie well above QCD predictions. Results are also presented for the single-tagged mode [one photon in the range 0.16<Q2<1.6 (GeV/c)2].
0 data tables match query
#### Inclusive production of charged hadrons and K0(S) mesons in photon-photon collisions
The collaboration Ackerstaff, K. ; Alexander, G. ; Allison, John ; et al.
Eur.Phys.J.C 6 (1999) 253-264, 1999.
Inspire Record 472639
The production of charged hadrons and K_s mesons in the collisions of quasi-real photons has been measured using the OPAL detector at LEP. The data were taken at e+e- centre-of-mass energies of 161 and 172 GeV. The differential cross-sections as a function of the transverse momentum and the pseudorapidity of the charged hadrons and K_s mesons have been compared to the leading order Monte Carlo simulations of PHOJET and PYTHIA and to perturbative next-to-leading order (NLO) QCD calculations. The distributions have been measured in the range 10-125 GeV of the hadronic invariant mass W. By comparing the transverse momentum distribution of charged hadrons measured in gamma-gamma interactions with gamma-proton and meson-proton data we find evidence for hard photon interactions in addition to the purely hadronic photon interactions.
0 data tables match query
#### Measurement of the proton - anti-proton pair production from two photon collisions at TRISTAN
The collaboration Hamasaki, H. ; Abe, K. ; Amako, K. ; et al.
Phys.Lett.B 407 (1997) 185-192, 1997.
Inspire Record 443677
The cross section of the γγ → p p reaction was measured at two-photon center-of-mass energy ( W γγ ) between 2.2 and 3.3 GeV, using the two-photon process at an e + e − collider, TRISTAN. The W γγ dependence of the cross section integrated over a c.m. angular region of | cos θ ∗ | < 0.6 is in good agreement with the previous measurements and the theoreticalv prediction based on diquark model in the high W γγ region.
0 data tables match query
#### Exclusive production of pion and kaon meson pairs in two photon collisions at LEP
The collaboration Heister, A. ; Schael, S. ; Barate, R. ; et al.
Phys.Lett.B 569 (2003) 140-150, 2003.
Inspire Record 626022
Exclusive production of π and K meson pairs in two photon collisions is measured with ALEPH data collected between 1992 and 2000. Cross-sections are presented as a function of cos θ ∗ and invariant mass, for | cos θ ∗ |<0.6 and invariant masses between 2.0 and 6.0 GeV/ c 2 (2.25 and 4.0 GeV/ c 2 ) for pions (kaons). The shape of the distributions are found to be well described by QCD predictions but the data have a significantly higher normalization.
0 data tables match query
#### Inclusive $\pi^0$ and $K^0_{S}$ production in two photon collisions at LEP
The collaboration Achard, P. ; Adriani, O. ; Aguilar-Benitez, M. ; et al.
Phys.Lett.B 524 (2002) 44-54, 2002.
Inspire Record 563335
The reactions ee->ee+pi0+X and ee->ee+K0s+X are studied using data collected at LEP with the L3 detector at centre-of-mass energies between 189 and 202 GeV. Inclusive differential cross sections are measured as a function of the particle transverse momentum pt and the pseudo-rapidity. For pt < 1.5 GeV, the pi0 and K0s differential cross sections are described by an exponential, typical of soft hadronic processes. For pt > 1.5 GeV, the cross sections show the presence of perturbative QCD processes, described by a power-law. The data are compared to Monte Carlo predictions and to NLO QCD calculations.
0 data tables match query
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http://upem.tripod.com/miktex.html
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Links and documentation related to LaTeX, TeX and MikTeX
AMSTeX/AMSLaTeX
Problems with amstex. The command "\input amstex" in a file seems to throw off miktex. How does one get around this problem? I need to process this command for TeXing any file to be published in the Proceedings of the AMS
Are you trying to use AMSTeX or AMS-Latex. These are two different programs. AMS-Latex is included in miktex, and is accessed using the miktex latex command. The relevant file used to be called amstex.sty. The current version of this is called amsmath.sty. It is invoked in latex with the command \usepackage{amsmath}.
AMSTEX is apparently not included in miktex, which may be why you are having trouble. This is a package that can be used in TeX rather than Latex, and is usually invoked with the command \input amstex. I downloaded AMSTEX, and installed in in my localtexmf directory. It works fine for me (I just checked). I am using miktex 1.11, but have not installed the various updates to miktex. I think you can download AMSTEX from either CTAN or the AMS. Be sure to refresh the filename database after you install it.
BibTeX
If you need interactive editing of BibTeX, *.bib files, look at http://www.tcisoft.com/bibdb.html
Spanish BibTeX documentation sites: http://feynman.faii.etsii.upm.es/seidel/ and ftp://tex.unirioja.es/pub/tex/doc
Babel, spanish version 1.0. Look CTAN at: ftp://ftp.cdrom.com/pub/tex/ctan/macros/latex/packages/babel/contrib/spanishb/
The Excalibur Standard Dictionary ftp://ftp.eg.bucknell.edu//pub/mac/Excalibur_2.5.2.sit.hqx
If you only need the updated version: ftp://ftp.eg.bucknell.edu//pub/mac/Excalibur_2.5.2-nodict.sit.hqx
Spanish Dictionary to be usesd with Ispell http://www.datsi.fi.upm.es/~coes/coesesp.html
Spanish Dictionary: ftp://ftp.dante.de/tex-archive/systems/win32/winedt/dict/es.zip. Look also at ftp://tex.unirioja.es or at CTAN:...\systems\win32\winedt\contributions.
How to include a reference with the help of bibtex.
The Documentation is in your miktex installation under doc/bibtex. You might try also the book "The LaTeX Companion", published by Addison-Wesley. The following steps will help.
1) Decide on a directory where you will keep your bibtex database, which are all your *.bib files. You can have one *.bib file or dozens filed by category or particular authors. The same database files will be used for ALL your LaTeX papers. DO NOT keep separate *.bib files in the directories w/your papers. BiBTeX is smart, and you don't want redundancy.
Now see the MiKTeX local guide or the help files in doc/miktex to configure miktex with a bibtex path.
For instance, my *.bib files are in c:\users\xxx\Bibliographies and here is an excerpt from my miktex.ini file:
[BibTeX]
Input
Dirs=.;c:\users\xxx\Bibliographies//;%R\bibtex//
2) Following is a file with 3 entries, save it as test.bib in the directory you just decided upon. See if you can have library
search results emailed to yourself, then you can convert the results you receive in the mail directly to BiBTeX format. For instance, see r2b and ref2bib they convert from REFER format to BiBTeX format. They reside at BiBNeT, described in (3)
% Here is the start of test.bib
@String{pub-SIAM = "Society for Industrial and Applied Mathematics"}
@InCollection{Berger:1983:DSA,
author = "Marsha Berger",
title = "Data Structures for Adaptive Mesh Refinement",
crossref = "Babuska:1983:ACM",
publisher = pub-SIAM,
pages = "??",
year = "1983",
bibsource = "ftp://ftp.math.utah.edu/pub/bibnet/authors/b/berger-marsha-j.bib",}
@Book{Babuska:1983:ACM,
editor = "Ivo Babuska and Jagdish Chandra and Joseph E. Flaherty",
title = "Adaptive computational methods for partial differential equations",
publisher = pub-SIAM,
pages = "xii + 251",
year = "1983",
ISBN = "0-89871-191-6",
LCCN = "QA377 .A29 1983",
bibdate = "Tue Oct 11 12:35:12 1994",
bibsource = "ftp://ftp.math.utah.edu/pub/bibnet/authors/b/berger-marsha-j.bib",}
@TechReport{Bethe-HA42,
author = {H. A. Bethe},
title = {The theory of shock waves for an arbitrary equation of state},
institution = {Office of Scientific Research and Development},
year = 1942,
number = 545,
abstract = {You can have an abstract, it will not appear in your LaTeX'd paper}}
% Here is the end of test.bib
3) Here is a LaTeX file, save it as it.tex. Then at the command prompt, latex it <- No .extension required.
bibtex it
latex it
latex it
(I'm going on a limb here, I didn't try this, I hope I have no mistakes.)
% Here is the start of it.tex
\documentclass{article}
\begin{document}
While this \cite{Bethe-HA42} is a masterful report, it is unfortunate that Bethe didn't have LaTeX.
You can find bibliography files of many authors as part of the BibNet Project. The master copy is available for public access on ftp.math.utah.edu in the directory tree /pub/bibnet/authors. For instance, the paper \cite{Berger:1983:DSA} is from Marsha Berger's file at that site. Her file is particularly sophisticated, with strings defined and cross-references used.
%\bibliographystyle{you could uncomment this and change the style in these braces}
% Now tell bibtex what *.bib file(s) to readin.
% You could have a comma seperated list of files, no extensions required.
% No spaces before or after filenames, just
% \bibliography{file1,file2,file3}
\bibliography{test}
\end{document}
% This is the end of it.tex
A perl script to convert Refer format to bibtex format. The script is from:
http://www.math.utah.edu/ftp/pub/bibnet/index.html
Dviwin
Dviwin previewer: ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/minten/soft/NTemacs:
For installation tips of dviwin previewer, read Minten's page http://www.esat.kuleuven.ac.be/~minten/NTTeXing/NTTeXing.html
Here is info about xdviwin. You may need to update your commctl32.dll from:
ftp://ftp.microsoft.com/Softlib/MSLFILES/COM32UPD.EXE
or at ftp://free.kaist.ac.kr/TeX/beta/
To use DVIPS,alone with the RedMon "Redirection Port Monitor" (created by Russell Lang, the author of
GSview). What it does: RedMon allows transparent PostScript printing from Windows 95(/98?) and NT. See http://www.cs.wisc.edu/~ghost/redmon/ for details. Once you have set up a redirected port (named "RPT1:" for example), you should be able to print directly to the printer from DVIPS using the command line option "-o!RPT1:" So your users can still print using a single "DOS box" command.
There is a nice documentation in the Web for installation dictionaries. Look for it at:
http://www.gap.baynet.de/members/werdenfels.gym/tex/
Emacs, AucTeX, NTMacs, GNU
Emacs, LaTeX etc, for your PC (Win 95/NT) http://web.math.auc.dk/~dethlef/Tips/likehere.html:
Installation instructions for emacs http://www.esat.kuleuven.ac.be/%7Eminten/NTTeXing/NTTeXing.html and http://www.esat.kuleuven.ac.be/~minten/NTTeXing/NTTeXing.html
Sooner or later, you will find out that Emacs is better than any other editor. To install NTemacs together with AuCTeX(major model for Latex in emacs), ReFTeX(for label, reference and citation) and Ispell can be found at:
http://www.esat.kuleuven.ac.be/~minten/NTTeXing/NTTeXing.html
Place where you can find information on the emacs and auctex: http://www.esat.kuleuven.ac.be/%7Eminten/NTTeXing/NTTeXing.html
AucTeX provides the best TeX environment to be ever found. Check: http://www.cs.washington.edu/homes/voelker/ntemacs.html
Auctex, the standard with xemacs, but not included in emacs. It works fine with NTemacs. See: http://sunsite.auc.dk/auctex/ and also http://www.esat.kuleuven.ac.be/%7Eminten/NTTeXing/NTTeXing.html
http://ourworld.compuserve.com/homepages/hoenicka_markus/sgmlhtfiles.html
For emacs and auctex installation see:
http://ourworld.compuserve.com/homepages/hoenicka_markus/sgmlhtemacs.html
http://ourworld.compuserve.com/homepages/hoenicka_markus/sgmlhtauctex.html
If you have any question in relation to Emacs(NT FAQ) go to:
ftp://ftp.sunet.se/pub/os/Win32/ntEmacs/docs/ntemacs.html#windows-like-cua
Port of GNU Emacs to windows NT at: http://www.cs.washington.edu/homes/voelker/ntemacs.html
An internet page which describes the implementation of miktex and emacs: Native 32-bit TeXing on PC.
Iinformation available about Emacs for win95/NT: GNU Emacs for Windows NT and Windows 95
The Windows 95/NT GNU RCS Component Software: http://www.componentsoftware.com/
For GNU latest release: The GNU-Win32 Project
Site for NTEmacs: ftp://ftp.sunet.se/pub/os/Win32/ntEmacs/docs/ntemacs.html
Ntemacs, a free text editor for LaTeX. To be used in conjunction with AUC-TeX package.
Notes: what’s the advantages of AUC over LaTeX? AucTeX does not replace LaTeX, rather it is an editing-and-compiling mode for LaTeX in emacs. A few nice features: it runs LaTeX, and then parses the log file. So, instead of running LaTeX, finding an error, stopping, and doing the whole thing again, you can run LaTeX all the way through, and then have AucTeX bring you to each error LaTeX finds in succession. Also, some nice input features: command completion or menu-driven entry, matching s, etc, and some formatting of the input (.tex) file. And syntax highlighting, etc. In short, AUCTeX is a TeX-editing environment with many useful features like syntax-shortcuts, and so on. It sits on top of LaTeX. Get it at: http://www.cs.washington.edu/homes/voelker/ntemacs.html#tex A self-extracting version of NTEmacs, that should work on Win95 also. Its version 19.34.6 and weights in as a 7.5mb download – This installation contains the full binaries, but no source. http://www.tardis.ed.ac.uk/~skx/win/ Try Ispell for NTemacs : Ispell 4.0 can be found at http://mohawk.cat.rpi.edu/~tibbetts/ispell_toc.html and ispell 3.1 at ftp://ftp.tue.nl/pub/tex/GB95/ispell-dutch96/ Gnuplot in unix and Windows versions. See http://www.cs.dartmouth.edu/gnuplot_info.html A new version of Emacs can be found at ftp://ftp.cs.washington.edu/pub/ntemacs/i386/README Graphics Software GnuPlo: To plot graphs from equations or data files. It outputs many formats includung Postscript and LaTeX picture environment. See ftp://ftp.dartmouth.edu/pub/gnuplot/gptwin32.zip or http://www.cs.dartmouth.edu/gnuplot_info.html or http://www.comnets.rwth-aachen.de/doc/gnu/gnuplot.html. DPlot Home pagehttp://smd4d.wes.army.mil/ For plotting, look at the various packages at www.gnu.org A good design package running under WindowsNT suitable for drawing and particularly graphs: Mayura Draw,see http://www.mayura.com. See the instructions at http://www.mayura.com/ps2ai.htm Another graphic pack: tkpaint is easier to use with more functionality. The url: http://netanya.ac.il/~samy/tkpaint.html Drawings are saved in a special format, but you can easily export to encapsulated postscript without any problems. It's free, too. If you need to download jpeg2ps go to Thomas Merz web page http://www.ifconnection.de/~tm/. If you REALLY need to include JPEG files on the fly, there is a way. First put jpeg2ps.exe in your path (probably texmf/miktex/bin). Here's an example that just now worked for me (using MiKTeX): \documentclass{article} \usepackage{graphicx} \DeclareGraphicsRule{.JPG}{eps}{*}{jpeg2ps #1} \begin{document} \includegraphics [width=\textwidth , bb= 20 20 575 575]{slope.JPG} \end{document} The bb= stuff is the bounding box information TeX needs. So where did these numbers come from? I ran jpeg2ps in a dos command line and created an eps file, then opened the file to get the numbers. Even though tex calls jpeg2ps on the fly and must generate these numbers, apparantly there's no way to get tex to use these numbers; they must be in the \includegraphics command. Thus, this method is worthwhile only if you have a large number of jpeg image files all of the same size. If you scanned an image, convert/save the scanned image to eps format. For instructions go to ftp://ctan.tug.org/tex-archive/info/epslatex.ps. They will tell you everything else you need to know. I think it is often still best to convert things to eps and generate Postscript files with dvips. To do this I like to convert bitmaps to jpeg format and then use jpeg2ps to convert this to eps. This makes small eps files since it keeps the jpeg compression (You need level 2 Postscript.). I got jpeg2ps from http://www.ifconnection.de/~tm/ As far as coverting bitmaps I often use Irfan View http://stud1.tuwien.ac.at/~e9227474/. It is a nice viewer, can convert between a number of formats (it can do multiple files in batch mode), can crop and cut and paste, and other stuff. It is freeware if you use it at home, the author would like a small fee if you use it at work. Psfrag package, at CTAN, /macros/latex/contrib/supported/psfrag. It will replace a label of your choice in your PostScript figure with properly typeset text of your specification. So you don't have to try to place formulas or symbols, etc. in your figures. 1) If you have the Cygwin package http://sourceware.cygnus.com/cygwin/ and an X server, you can compile xfig itself. 2) A very credible lookalike has been written in Java: http://tech-www.informatik.uni-hamburg.de/applets/javafig/ It lacks a few features of the full xfig(rotated text and alignment commands being the ones I miss most) but may do quite well for you. 3) Try SmartDraw Profession (v4), see www.smartdraw.com TkPaint is a very capaple vector drawing program with eps-export. It's available for both Windows and Unix. Windows: http://www.netanya.ac.il/~samy/tkpaint.html Unix: http://www.fe.msk.ru/~vitus/misc/tkpaint.html A good freeware graphics converter (.bmp, .jpg., .gif, etc. to .eps) for Win95/98? You can download the Viewer Pro from www.freewarehome.com . It doesn't convert to vector graphics so the files will be quite large. One program which can trace a vextor graphic from a bitmap format is Corel Draw. Try also, IrfanView: http://stud1.tuwien.ac.at/~e9227474/. Another one, http://www.wizards.dupont.com/cristy/ImageMagick.html. Use ImageMagic to convert Latex files to Html. Use it for converting equations from eps to gif format. Moreover, convert bitmaps to JPEG then use jpeg2ps to put an eps wrapping on it. That way the jpeg compression is retained and the file size is small. This works for Postscript level 2. Get it from: jpeg2ps http://www.ifconnection.de/~tm/ or from CTAN. LaTeX and TeX TUG Membership: http://www.tug.org. Introduction to TeX and LaTeX. You can find brilliant introductions both to TeX and LaTeX2e at CTAN:/tex-archive/info. Look for lshort2e for LaTeX (tex, dvi, ps, and pdf formats) and gentle.tex for TeX. If you need information concerning TeX and/or LaTeX, visit our page on TeX and LaTeX resources. For lshort2e go to: http://www.cdrom.com/pub/tex/ctan/info/lshort/ or any of the many mirror sites. For the Spanish speaking community, fonts for Una Descripción de LaTeX2e'' can be found at ftp://ftp.cma.ulpgc.es/pub/tex/latex2e/doc/ldesc2e, versions 0.3 and 0.4. As an alternative, if you use Adobe's Acrobat go to http://www.cma.ulpgc.es/users/bautista/other/ldesc2e.pdf. If you need "Una Descripción de LaTeX2e", write to : [email protected] or visit Tomas Bautista Home Page at: http://www.cma.ulpgc.es/users/bautista. Look also at the directory: ftp://ftp.cma.ulpgc.es/pub/tex/latex2e/doc/ldesc2e/mix, there you will find a Zip package, a Tar package and a gzip pack. Also there is a postscript document file, and a new pdf. Macros and packages for LaTeXftp://ftp.dante.de/tex-archive/macros/latex2e/contrib/other/comment TeX and Friends: http://www.stat.ucla.edu/develop/tex/ Information about LaTeX: http://users.cybercity.dk/~ccc7795/LaTeX.html LaTeX Textbooks Leslie Lamport's LaTeX User's Guide and Reference Manual The Latex Companion by Goossens, Mittelbach, and Samarin Goossens, Rahtz, Mittelbach. The LaTeX Graphics companion. A Guide to LaTex2e, by Helmut Kopka and Patrick W. Daly, Addison Wesley, 1997, ISBN 0-201-42777-X. All of them from Addison Wesley. There is also apparently a Latex Graphics Companion which is good if you use a lot of graphics. Further information on references can be found at http://www.tug.org/interest.html#dochttp://www.tug.org/interest.html#doc and http://tug2.cs.umb.edu/ctan/tex-archive/info/index.html. If you want to know about some of the best TeX and LaTeX books available for purchase online please visit: TeX and LaTeX Books: Our Favorites, or order your favorite from Cyber Math Virtual Bookstore. However, the real trick to learning TeX/LaTeX is getting someone in your field to give you copies of the TeX files for some nicely done papers. I can't imagine trying to learn TeX by any method other than seeing how someone has done it right for your own field. I am pretty sure that the xxx archive http://xxx.lanl.gov/ posts the TeX source files for the papers stored there. You might look there for some good samples. How to learn LaTeX. A piece of advice. 1.) Read. From CTAN: info/gentle.tex a gentle introduction to TeX info/epslatex.ps EPS graphics in LaTeX 2e documents. FILES.byname file (nearly 4 MB!) so you can see what is in CTAN. From CTAN or in your tex directory tree: tds.dvi "A Directory Structure for TeX Files" to help you figure a sane way to maintain your TeX tree. It comes with MiKTeX, texmf\doc\general\tds.dvi 2.) Buy 3 books: (Who said using LaTeX was going to be free?). See above. Be aware of where you can borrow Knuth. The TeXBook. Because sometimes you need a little TeX with your LaTeX. 3.) Get example LaTeX files from other people. 4.) Learn BibTeX IMMEDIATELY. As soon as you have to type in your very first reference in a bibliography, you will be farther ahead to learn BibTeX. It's easier to learn than most things in LaTeX. The first 2 books(above) I mentioned cover it, and there is documentation for it in your MiKTeX distribution. In fact... when you do literature searches on line, see if you can save them electronically. At least if you can save them in REFER format, then you can convert them with a perl script (or a mode in the emacs editor) to a BibTeX file. The converters are at CTAN (I think), or search for Nelson Beebe's BibTeX site (somewhere at math.utah.edu). Be aware of the makebst program to produce custom made bibliography styles (don't actually learn it until you have to, there's plenty of pre-defined bibliography styles.) 5.) Consider XEmacs for your editor if/when you are in Unix as it can display equations in a kind of wysiwyg mode with the x-symbol package (which you can find by web search). An emacs/Unix guru might need to help install that package. Be warned ... "Learning XEmacs is a lifelong activity." http://www.xemacs.org/faq/xemacs-faq.html#SEC2 It's more than an editor. It's a way of life. 6.) Use PSFrag for typesetting text in EPS graphics. You'll find it described in epslatex.ps 'This is a marvel. 7.) Read comp.text.tex newsgroup. Caution: high traffic. 8.) There's a Tex Users Group (TUG). You may consider join it. BibTeX and PSFrag are easy to put off, but they are gems. You save time and heartache by learning them up front. Latex by examples. Take a look at: http://www.ams.org/tex/author-info.html. If you are looking for books about LaTeX, go the site of Addison Wesley: http://www2.awl.com Spanish Babel now available: In the CTAN website you can now find version 1.1 of spanishb. As before look for it at macros/latex/packages/babel/contrib/spanishb. How to delete files associated with LaTeX and avoid lossing hard drive space. Set an association for .tex extension with a batch file, say texme.bat, which looks like this: me %1 del *.aux del *.log del *.bak del *.dvi rem -- optional where "me" is an editor (Multi-Edit), which I use as a shell (exactly as WinEdt or TeXshell or TeXed). To adjust margins look for the package vmargin. It is available on CTAN. For an APA style, search CTAN (http://tug2.cs.umb.edu/ctan/) for apa.sty. Help for LaTeX: http://sea.am.ub.es/Latex/ltx-2.html Macintosh TeX and LaTeX Web Site: http://www.esm.psu.edu/mac-tex/ Tex and PyTeX users page( in Spanish): http://www.ctv.es/USERS/irmina/pyttex.htm Python utility for doing Rumbaugh OO boxes in Tex: http://www.ctv.es/USERS/irmina/texpython.htm True Type in pdfLaTeX: http://quantum.bitp.kiev.ua/radamir/ttf-pdf.htm The "caption" package to be used in conjunction with LaTeX, can be find at: ftp://ftp.rediris.es/mirror/tex-archive/macros/latex/contrib/supported/caption If you need the style wrapfig: ftp://ftp.dante.de/pub/tex/macros/latex/contrib/other/misc/wrapfig.sty. A searchable index on a www page: http://www.dante.de/cgi-bin/ctan-index. The UK TeX Archive has an excellent alphabetically ordered catalogue. You can download the files needed for getting caption2.sty from there. The URL is: http://www.tex.ac.uk/tex-archive/help/Catalogue/ctindex.html. How to use the newcommand in order to generated natural integer, rational and real number sets symbols, respectively: \newcommand{\N}{\mbox{I\!\!N$}} \newcommand{\Z}{\mbox{$Z\!\!\!Z$}} \newcommand{\Q}{\mbox{$I\:\!\!\!\!\!Q$}} \newcommand{\R}{\mbox{$I\!\!R$}} If you have installed AMS packages, include the pack <<amsfonts>> thru \mathbb. You can define some macros as follows: > \usepackage{amsfonts} > \newcommand{\N}{\mathbb{N}} > \newcommand{\R}{\mathbb{R}} > \newcommand{\C}{\mathbb{C}} Thereafter, use \N, \R or \C, where you need them. You must use $$, to use them in equations or text. If you define your macros as \newcommand{\N}{\ensuremath{\mathbb{N}}} \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\C}{\ensuremath{\mathbb{C}}} you will be able to use them within the text without having to go to math mode. If you use AMSFonts package in Blackoard Bold, write \mathbb{N,Z,Q,R,C} If necessary you might load, \input amssym.def \input amssym (use, \usepackage if LaTeX). Another way, usepackage[T1]{fontenc} \usepackage{textcomp} \newcommand{\N}{\mbox{I\!\!N}} \newcommand{\Z}{\mbox{Z\!\!\!Z}} \newcommand{\Q}{\mbox{I\:\!\!\!\!\!Q}} \newcommand{\R}{\mbox{I\!\!R}} \newcommand{\C}{\mbox{I\:\!\!\!\!\!C}} \newcommand{\M}{\mbox{I\!\!I}} \newcommand{\D}{\mbox{I\!\!D}} \newcommand{\E}{\mbox{I\!\!E}} \newcommand{\F}{\mbox{I\!\!F}} \newcommand{\DE}{\mbox{D\;\!\!\!\!\!\!\!E}} % ---------------------------------------------------------------- \begin{document} \section{}\noindent This simbol represent the Natural numbers \N\\ This simbol represent the Whole numbers \Z\\ This simbol represent the Rational numbers \Q\\ This simbol represent the Irrationals numbers \M\\ This simbol represent the Real numbers \R\\ This simbol represent the Complex numbers \C\\ This simbol represent the operator \D, or \F or \E \\ This simbol represent the famous {\huge \DE}\\ %\usepackage[T1]{fontenc} %\usepackage{textcomp} This is the Euro sign: {\sffamily\texteuro\ }\\ \end{document} In Plain TeX: \font\msbmnormal=msbm10 \font\msbmpeq=msbm7 \font\msbmmuypeq=msbm5 \newfam\numeros \textfont\numeros=\msbmnormal \scriptfont\numeros=\msbmpeq \scriptscriptfont\numeros=\msbmmuypeq \def\num{\fam\numeros\msbmnormal} \def\N{{\num N}} \def\R{{\num R}} %If we leave a space between the last two braces " }" of the definition of the macro \R and \N it will not be necessary to add %a second \ in \R\, to have a space therafter. However, this have the inconvinience of leaving a space when followed by a %punctuation sign. \magnification=1200 \centerline {\bf COMMENTS:} \bigskip \This is the \R\ of the Real Numbers, and to denote the Natural Numbers we use \N. What we have defined works not only in math mode but outside it. For example, the letters \num ABCD are written in math mode, but {\num Z} is not, neither is {\num PQ}. Moreover, thanks to the definition as a "family of letters", in math mode they change size automatically when used as subscripts or subscripts-subscripts. This is better seem in the following example.$$\{p_n(x)\}_{n\in\N} .$$It does not matter that it is written within text,$\{p_n(x)\}_{n\in\N}$, it keep looking good. \end % If we use a much simpler definition % \font\msbm=msbm10 % \def\R{\hbox{\msbm R}} % \def\N{\hbox{\msbm N}} % then the R and the N of the Real and Natural Numbers does not becomes smaller when used as a subscript Another place where to find the Euro sign: ftp://ftp.dante.de/tex-archive/help/Catalogue/catalogue.html. If you go to http://www.ucc.ie/cgi-bin/ctan> you might end up with a big list, but somo of the links can be useful. For example, ftp://ftp.rediris.es/mirror/tex-archive/fonts/eurosym. For Postscript take a look at: ftp://ftp.rediris.es/mirror/tex-archive/fonts/psfonts/marvosym. The Euro symbol is in the package: textcomp.sty. This mean at the beginning of the document you must type: \usepackage{textcomp}. Then to print the symbol use \texteuro If you need to know what other symbols are contained in the package do as follows: latex textcomp.ins. You will get test.tex then, latex test.tex and finally, xdvi test.dvi. Fonts related to Euro from Adobe: ftp.adobe.com/pub/adobe/type/mac/all/eurofont.sea.hqx. This are for those using the Mac. If you use a PC, ftp.adobe.com/pub/adobe/type/pc/all/eurofont.sea.hqx To download the afms for Times, Helvetica, Courier, Palatina etc., go to 'some ctan' /systems/msdos/4alltex/diskp2 There are also tfms in the directory. For future reference, I recommend http://www.tex.ac.uk/cgi-bin/ctan-index to search the whole CTAN directory structure for any pattern. I have the following problem: I have a source code of program included in my document in verbatim environment and now I would like to add some comments to this program, that reference to the equations described elsewhere in the text. How to manage it (i.e. how to execute regular LaTex commands from verbatim env.)? There is a recent released package called fancyvrb at CTAN that can manage this. I am attaching an example that worked to me. This package can be found at ftp://ftp.dante.de/tex-archive/macros/latex/contrib/supported/fancyvrb/ A short example follows: %%%% example using fancyvrb %%%%%%%%%%%%%%%% \documentclass[12pt]{article} \usepackage{fancyvrb} \begin{document} A test with equation: \label{eq:01} 2*2=4 was typeset as \begin{Verbatim}[commandchars=+] \label{eq:01} 2*2=4 and the command \ref{eq:01} produces the number +ref[eq:01] \end{Verbatim} \end{document} %%%%%%%%% Example ends here %%%%%%%%%%%%%%%% Euro Sign. The following commands give you the euro symbol: ... \usepackage[T1]{fontenc} \usepackage{textcomp} ... bla {\sffamily\texteuro\ }blablabla ... If you need teTeX look for it aftp://sunsite.informatik.rwth-aachen.de/pub/comp/tex/teTeX. Is there a tool to convert MS-Word formatted files to LaTeX (at least the main part of the formating work...)? K-Talk has some commercial tools, http://www.ktalk.com/index.html Package to convert latex to html for WindowsNT-platforms. Try 'TechExplorer' in IBM web site. Just search it in yahoo, you will not miss it. See also, latex2html! The package is based on the Perl interpreter which is available for Win95/NT as well(see http://www.ActiveWare.com/ ). Latex2html is - compared to MikTeX - a bit hard to install, but the results (including formulas and pictures, is good. You can find it on CTAN (.../pub/tex/systems/win32/web2c). Perl installation is just running a setup program. latextohtml converter on a Win95 computer ftp://ftp.duke.edu/tex-archive/systems/win32/web2c/l2h-win32.tar.gz You'll find GDBM_File in sources and in precompiled binaries at http://www.roth.net/perl/GDBM WebEQ is a 100% pure Java system for creating and displaying interactive scientific Web documents. To gain access try: http://www.webeq.com. Seeking for a converter LaTeX to HTML. There is one quite simple and works under Solaris and Unix: http://hutchinson.belmont.ma.us/tth An article on "Using LaTeX to Create Quality PDF Documents for the WWW": look at www.math.uakron.edu/~dpstory/acrotex.html) The distribution of the french latex on ftp://ftp.univ-rennes1.fr/pub/GUTenberg/french/. To build up decision trees using latex. Look up the package called QobiTree, at http://www.dante.de/cgi-bin/ctan-index Chof’s TeX archives. Mostly in Korean: http://free.kaist.ac.kr/ChoF/ In http://www.germany.net/teilnehmer/100/122054/texwin.htm I have listed 17 shells or guis for tex/latex/bibtex. Listed programs are free- and shareware. See also http://www.gap.baynet.de/members/werdenfels.gym/tex/tex_engl.html for some useful hints. Where I can get "revtex.cls"? Look for it at ctan. There's an index at http://www.tex.ac.uk/tex-archive/help/Catalogue/ctindex.html revtex.cls, incidentally, would belong to the American Physical Society and they'd be a good starting point for the file if it's not at ctan. Look also at ftp://ftp.aps.org/pub/revtex How to get in latex command the 'registered' symbol (The capital R enclosed by a circle). A couple of ways:$\textcircled {R}$I have also used it as a superscript like this$^{\textcircled {\scriptsize R}}$Try also: \newcommand{\Rtrademark}{$^{\text{\textcircled{\tiny R}}}$} or, \textregistered The "L" symbol for a Laplace Transform. In order to create the Laplace L you need to install the mathrsfs.sty and mathrsfs.rme files as well as for metafont rsmf10.mf,rsfs5.mf and rsfs7.mf. All files are available on the CTAN. For more information see the Dante FAQ 8.1.2 Another solution: There are script fonts in AmS-TeX (as well as AmS-LaTeX);though, L isn't that fancy there (try it). In AmS-TeX it suffices to say \Cal L in math mode. Another option: Install the rsfs fonts (Ralph Smith's Formal Script Symbol Fonts), they do indeed look great. The file mathrsfs.rme (readme) and mathrsfs.sty are in macros/latex/contrib/supported/jknappen (and lots of other is in this directory as well) and the font files are in fonts/rsfs. There is a subdirectory of rsfs with type1 fonts (afm, pfm, and pfb files). Put all of this in a localtexmf directory and mimic tds (good idea? seems to be a lot of trouble). Another TeX and LaTeX engine, VTeX: The NTG board. WWW: http://www.ntg.nl; e-mail: [email protected] address: NTG / PO Box 394 / 1740 AJ Schagen / The Netherlands The release of VTeX 6.2 for the first time enabled TeX users to create quality Acrobat document mixing text and graphics, designing the new document compiler was only part of our work. To get the best out of TeX and PDF one also needs quality fonts. We therefore have developed two new font families: HV-math and TM-math HV-math supplements the standard Adobe Helvetica with all symbols required for TeX typesetting; TM-math does the same for the Times Roman. Both families include both regular and bold variants and are supplied in the Type 1 format. They can be used with either VTeX's PDF mode or with the PostScript output from the DVI mode. You can use them with other DVI drivers, provided that you have the ATM installed. The intended use for TM-math is a replacement for the Computer Modern for documents where the quality and traditional appearance is important. HV-math can be used as such replacement as well; in addition, you can use it for presentations/slides, or for the title lines within TM-text. More detailed descriptions and samples of the new families are posted on our Web site. http://www.micropress-inc.com/samples/hfonts.htm http://www.micropress-inc.com/samples/tfonts.htm Some additional URL's for sample documents are: http://www.micropress-inc.com/samples/paper-hv.pdf http://www.micropress-inc.com/samples/paper-tm.pdf http://www.micropress-inc.com/samples/math-hv.pdf http://www.micropress-inc.com/samples/math-tm.pdf For comparison, we also provide the same documents compiled with the CM fonts: http://www.micropress-inc.com/samples/math-cm.pdf http://www.micropress-inc.com/samples/paper-cm.pdf We will appreciate if you let us know what you think about these samples and what other fonts would be most useful for your work. The introductory price for either family is only US$100; you must have VTeX 6.2 to use them.
The up-to-date patches are now posted on our Web site. Notice that the current version of the software is 6.21 and if you
Version 6.21 has a couple of new features (page background in the PDF mode (the \pagecolor command in graphicx) and a workaround for a major bug in Adobe's Distiller) and fixes a couple of minor problems of EPS inclusion reported to us. Additional details on the changes in 6.21are in the current README.PDF file which can be downloaded from the
http://www.micropress-inc.com/patches/vpatch.htm page.
In addition, our web page now has the online description of 6.2/GeX: http://www.micropress-inc.com/gex/gex.htm
TeX and LaTeX resources for the spanish speaking peoples. For a list of members together with information about CervanTeX ( a TeX and LaTeX user groups for Spanish Speaking Peoples) go to: http://gordo.us.es/Actividades/GUTH or better, to the CervanTeX official page: http://gordo.us.es/Actividades/CervanTeX/CervanTeX.html. CervanTeX has been under creation since two years ago. To read their newly approved bylaws go to: ftp://tex.unirioja.es/pub/tex/estatutos.tex or better http://www.unirioja.es/dptos/dmc/jvarona/estatutos.txt. Page of spanish users of latex / pagina de usuarios en espanyol de latex: www.ctv.es/USERS/irmina/pyttex.htm. Page of drawing utility for tex / pagina de utilidad de dibujo para Latex: www.ctv.es/USERS/irmina/texpython.htm
There are two excellent web-resources for LaTeX. There is a newsgroup entirely devoted to TeX and LaTeX: news:comp.text.tex. Deja News archives the posts to this newsgroup, and offers a useful search engine:
http://www.dejanews.com/home_ps.shtml.
Chemistry in Tex/LaTeX: Check out chemtex in //ctan.tug.org/tex-archive/macros/latex209/supported and other 'chem' packages in //ctan.tug.org/tex-archive/macros/latex/supported. Try programs such as chemsketch or IsisDraw. They a free for
noncommercisl use. Formulas can be imported via eps into latex documents. Addresses can be found in Stevens chemistry software page for windows: http://nmr400a.mols.susx.ac.uk/~steven/mswin.html
Are 7pt, 8pt and 9pt options (size7.clo, size8.clo and size9.clo) for article.cls available somewhere? Take a look at:
http://www.informatik.uni-freiburg.de/~may/Extsizes/
. There are the options 8pt and 9pt in the AMS-classes amsart.cls, amsbook,cls, amsproc.cls.CTAN: macros/ latex /packages/amslatex/. In addition you can find a 9pt.clo on page http://www.informatik.uni-freiburg.de/~may/extsizes.html
I would like to ask is it possible to work with cyrrilic alphabet and where are location to download what is necessary. For cyrillic text and Russian is T2/X2 package. See: http://sunsite.oit.unc.edu/sergei/cy/tex.html
There is a CTAN Server with a search index where you can find all classes, styles, fonts and so on. Take a look at:
http://www.dante.de/cgi-bin/ctan-index
How to transfer lots of data between a msdos machine (win 3.X) and a win95/98:
http://wwwcip.informatik.uni-erlangen.de/user/orknorr/windcc.html
How to use TrueType fonts with TeX (pdfTeX) and LaTeX (pdfLaTeX): http://quantum.bitp.kiev.ua/radamir/ttf-tex.htm.
MSPS is a set of MS core fonts (free for non-commercial use, as far as I know)--- Times New Roman, Arial and Courier New in Postscript Type 1 format, reencoded so that only T2A+LCY symbols are left. This includes English characters. Find it at: http://quantum.bitp.kiev.ua/radamir/msps.htm.
I'm writing a proposal using LaTeX (MikeTeX) and I want to include some references. Can some one help me and show me how to write the references section!! I need the format of references to start with Name and Year; i.e., not with a number.
Solution: consider the natbib package found on CTAN. It can do exactly what you'd like. I also have used it together with the package makebib. The second package allows you to customize the order in which specific elements appear in each item on the reference list.
German umlaute in source code: If you need to use äöüß in the Source Code. Take a look at: http://www.gap.baynet.de/members/werdenfels.gym/tex/umlaute.html
Moreover, WinEdt can automatically convert your input from äöüß to \"a\"o\"u\ss or whatever... Here's a snippet from the Helpfile:
Spelling and International Characters [..]
For Example:
"\{\i}" -> "ì"
"\'{\i}" -> "í"
"\^{\i}" -> "î"
Write:
"ì" -> "\{\i}"
"í" -> "\'{\i}"
"î" -> "\^{\i}"
[..] while your sources remain 100% portable. [..] Please note that using "ansinew" codepage for LaTeX still allows you to explicitly translate a file when it is "imported" or before it is "exported". All this is configured in "Options - Settings - Translations" You can define that for a lot of strings/characters... e.g. that you type "©" and WinEdt writes "{\copyright}"...
Large letters: There's this short hack, using TeX commands:
\font\HHuge=cmr12 at 10cm
{\HHuge This is big!}
Also, take a look at http://www.informatik.uni-freiburg.de/~may/. There you will find files sizeXX.clo, where XX=14,17,20 to make large letters
Another solution. Since MiKTeX comes with LaTeX and the graphicx package, why not take advantage of the graphics commands? Something like
\usepackage{graphicx}
\scalebox{3}{Some big text}
Will increase the size of the text threefold. And I don't know of any (explicit) hardwired limits to the size of the text.
Python utility for doing Rumbaugh OO boxes in TeX and other TeX tools: http://www.ctv.es/USERS/irmina/texpython.htm
How I can get the style file psfrag.sty? Try, ftp://ftp.tut.ac.jp/.h1/TeX/macros/latex209/contrib/psfrag/
A descriptions for packages available for TeX and LaTeX: http://www.tex.ac.uk/
MiKTeX
Guidance for a complete reinstall: http://www.gap.baynet.de/members/werdenfels.gym/tex/tex_engl.html
FAQ or HowTo about MiKTeX : http://www.math.ku.dk/~jarl/miktex/faq.html
I will appreciate a lot if superusers of miktex will send (what they think are frequently asked) questions AND answers to me, Jarl Friis at: [email protected])
MikTeX in German: http://www.gap.baynet.de/members/werdenfels.gym/tex/.
To subscribe in the texk-win32 mailing list write to: [email protected]. Only send a mail with "Subject: subscribe".
To update MikTeX (v1.10) with the newest LaTeX2e release (6/98), look for a file called 'miktex.txt' within the base directory on ctan. In updating the LaTeX installation we noticed a few inaccuracies in the miktex.txt file that was in the base directory.
PREPARING THE INSTALLATION
The directions tell you to extract the base directory into /texmf/tex/latex. That won't work in the next step when you try to run initex on on unpack.ins. initex cannot find unpack.ins (even though you run it from a dos box in the same directory).
Unfortunately, the file miktex.txt was not updated by the latex-team. The distributed version was tested with miktex 1.09 and worked fine. Try using WinEdt, load unpack.ins and everything should ran smoothly.
What I did was to extract everything to a temp directory, run initex and everything else, and then move the new base directory into the texmf structure. I will check this as soon as the final version of miktex 1.11 is
available. Hope this will stay until the december-release of latex.
CREATING THE LATEX FORMAT FILE
There is a typo, type: C:\TEXMF\MIKTX\CONFIG\CONFIGURE --update-fndb. Should be type: C:\TEXMF\MIKTEX\CONFIG\CONFIGURE --update-fndb
HINTS AND TIPS
To check the installation, you should move ltxcheck.tex (misspelled as ltxcheck.ltx in the instructions) outside the texmf tree.
The final release of MikTeX 1.11 ftp://ftp.tex.ac.uk/pub/tex/systems/win32/miktex/1.11/
We've update our installation-guidiance for a TeX-System consisting of MiKTeX, GhostView/GhostScript and WinEdt16 up to MiKTeX 1.11. You could get it under http://www.gap.baynet.de/members/werdenfels.gym/tex/tex_engl.html
Information about MiKTeX, NTEmacs and AucTeX: http://www.esat.kuleuven.ac.be/%7Eminten/NTTeXing/NTTeXing.html
If you need information related to miktex, winedt, and gsview32 look at: http://www.gap.baynet.de/members/werdenfels.gym/tex/tex_engl.html
Zipped MikTeX without sources http://www.gap.baynet.de/members/werdenfels.gym/tex/tex_engl.html
MikTeX documentationhttp://www.inx.de/~cschenk/miktex/ and http://www.snafu.de/~cschenk/miktex/
Installation instructions for MikTeX in German.http://www.gap.baynet.de/members/werdenfels.gym/tex/tex.html and http://www.gap.baynet.de/members/werdenfels.gym/tex/
Translation of german guide to english: http://www.gap.baynet.de/members/werdenfels.gym/tex/tex_engl.html
If you need help with Windows NT and MikTeX take a look at: http://www.germany.net/teilnehmer/100/122054/texwin.htm
Installation instruction for MikTeX in English.ftp://tug2.cs.umb.edu/tex-archive/systems/win32/miktex/1.10/README.TXT and http://www.gap.baynet.de/members/m.linderer/tex/
A complement to MiKTex documentation. The article by Reckdahl is either of the files: ftp://tug2.cs.umb.edu/tex-archive/info/epslatex.ps (Postscript version) or ftp://tug2.cs.umb.edu/tex-archive/info/epslates.pdf (PDF version). It is an 85 page report, very well written, with index, etc.
If you want to install LateX2HTML and run it with MiKTeX take the following steps: You first need to get and install perl 5; latex2html is a set of perl scripts. Check out http://www.ActiveState.com/pw32/ for more info on how to get perl.
Next, download the latex2html distribution. A version for windows is in the web2c distribution in the CTAN archive, e.g.
ftp://ftp.duke.edu/tex-archive/systems/win32/web2c/l2h-win32.tar.gz
Note that you will need gunzip and tar to unpack this archive, if you don't have these utilities already. latex2html also uses some of the netpbm tools, so you need to download these executables as well. They allow you to convert virtually any image format to virtually any other image format. For latex people, one of the more useful conversions is conversion of gif, bmp, etc. to postscript. They are in the same directory as latex2html, in the file: netpbm-win32.zip. You need to unpack this archive and put the executables somewhere in you machine's search path. I did this by creating a directory c:\netpbm containing these executables and adding it to the path by editing my autoexec.bat. I unpacked all these files into the directory c:\texmf\l2h. Then, in the directory \texmf\l2h\users, there were several files that I needed to modify to get it to run. One was local.pm, which tells the locations of several tools that latex2html needs. The way you need to edit this file depends on where you have things installed on your computer.
You then need to add the environment variable L2HMODULE, setting the value to
c:/texmf/l2h. You can do this just before you run latex2html, or you can put this definition in autoexec.bat.
Then, to run it, I created a file called l2h.bat in the MiKTeX executables directory. This file contains the single line:
@perl \texmf\l2h\user\latex2html %1 %2 %3 %4 %5 %6 %7 %8 %9
You might have had to make some small modifications to the win32.config and pstoimg scripts so that they could find everything ok with MiKTeX.
Software to convert Latex to HTML: Note LaTeX2HTML was written originally for Unix machines, but there is a "port" for
Windows/DOS. It is a very powerful tool, wich converts your embedded .eps-files *and* your mathematical formulas into .gif-files (what is looking very good when you compare with the formula output of TTH), wich were automatically numbered and embedded into the generated .html-file(s). Also it generates navigation panels at the top of each page (when you like that).
For interested people. Here are my steps to run these fabulous perl-scripts. You will need the following files:
netpbm.zip: http://tug.ctan.org/cgi-bin/ctan-web-ftp?systems/win32/miktex/util/netpbm.zip&1026008
l2h98_2dos.tar.gz: http://saftsack.fs.uni-bayreuth.de/~latex2ht/l2h98_2dos.tar.gz
http://saftsack.fs.uni-bayreuth.de/~latex2ht/manual.ps.gz !!!!!
ActivePerl 509 for Win32: ftp://ftp.ActiveState.com/activeperl/APi509e.exe or ftp://ftp.linux.activestate.com/pub/ActivePerl/APi509e.exe or
Needed for Win95-Users (Perl-documenation says that) http://www.microsoft.com/com/dcom/dcom1_2/default.htm
Then the following steps:
0. install Perl
1. extract l2h98_2dos.tar.gz to C:\texmf
2. no changes from /usr/.../perl into C:/perl/bin in the recommended files (install-test ...)
3. add Path to gswin32c.exe to autoexec.bat (problem will be resolved in a next version of install-test)
4. C:\texmf\latex2html> perl install-test
5. minimal modifications at Iconserver 6. minimal modifications at latex2html.config and latex2html 7. write l2h-Batch; perl C:\texmf\latex2html\latex2html %1.tex %2 %3 >> %1_l2h.log 8. C:\texmf\latex2html> l2h.bat test 9. :-) A little problem is (still) there: you have to write the complete path to your .eps-files into the \includegraphics-command, otherwise dvips (still) can't find the files to convert. A site offering a Perl interpreter for Win32: http://www.ActiveState.com A very good conversion utility at the following url: http://www.snafu.de/~cschenk/miktex/downloads.html. It is called TtH and works on tables, equations etc. Equations are translated to HTML and not mapped in gifs as done by LaTeX2html Type 1 Fonts and MikTeX. In the document http://www.math.uakron.edu/~dpstory/acrotex.html there are some instructions on how to use Type1 fonts with MikTeX. A place for * pfb fonts: ftp://tug2.cs.umb.edu/tex-archive/fonts/cm/ps-type1/bakoma/pfb/ How to avoid printing page numbers in miktex? Do: \pagestyle{empty} A new help page for MikTeX from Pedro Aphalo: http://cc.joensuu.fi/~aphalo/tex.html Detailed instructions for getting Eitan Gurari's TeX4ht TeX/LaTeX-to-HTML translator working under MikTeX at: http://www.arch.ohio-state.edu/crp/faculty/pviton/support/tex4ht.html CTAN:/support/ttf2mf. ftp://ftp.tex.ac.uk/tex-archive/support/ttf2mf/ Eitan Gurari, author of the tex4ht tex/latex-to-html translator has just added a feature designed to minimize the time needed to generate gif files. I've updated my miktex document to describe the new feature. See http://archadserver.eng.ohio-state.edu/crp/faculty/pviton/tex4ht.html If you need help with MikTeX use: http://www.tu-ilmenau.de/~klug/index_e.html. The german version is greater. http://LaTeX.megapage.de Miscellaneous A Windows NT hints site: http://www.jsiinc.com/ GunZip for Win32: ftp://ftp.uni-bremen.de/pub/Systems/win32/archiver/gunzip-1.2.4-i386.exe. Archiplex Form a Search Tool: http://src.doc.ic.ac.uk/archieplexform.html If you need infozip: ftp://ftp.uu.net:/pub/archiving/zip or ftp://ftp.icce.rug.nl/infozip/WIN32/ X-client software: http://www.microimages.com/freestuf/mix/ Try PFE, a very good freeware that works very good with miktex and so on, at http://www.lancs.ac.uk/people/cpaap/pfe/ The VIM Home Page: http://www.vim.org/ A vi version that runs smoothly on win95. Try VIM, it has -sytax highlighting for lots of files, including tex - menus galore, someone even put in a tex/latex menu - very low cost (charity-ware) - source code so you can make it do whatever you want. - portablility from unix to win95, NT, VMS,... - great support in the comp.editors usenet news group Look at news://comp.editors or http://www.vim.org and go from there... To find a version of cygwinb19.dll that handles stdin/stdout from non-cygwin binaries go to Chris Faylor's Home Page: http://www.tiac.net/users/cgf/. Introductions to virtual fonts at TeX FAQ:http://www.cogs.susx.ac.uk/cgi-bin/texfaq2html?keyword=&question=28 A jpeg file converter to eps using a program called jpeg2ps by Thomas Merz. Downloaded it from http://www.muc.de/~tm. You can find a new version of dvips on the download page: http://www.snafu.de/user-cgi-bin/cschenk/download.cgi/dvips.zip dos-eps processing should be faster now. One of the best packages for making commutative diagrams is Xy-pic. Its home page is here: http://www.diku.dk/research-groups/topps/personal/kris/Xy-pic.html Gzip is a compression/decompression utility popular in the Unix world. The above site has executables for MSWindows. http://www.gzip.org/ To include a scanned photgraph into text. The most straightforward thing to do is convert/save the scanned image to eps format. The instructions you can get from here ftp://ctan.tug.org/tex-archive/info/epslatex.ps will tell you everything else you need to know. Some Fonts found on CTAN: RSFS (http://www.tex.ac.uk/tex-archive/fonts/rsfs/) and CALLIGRA (http://www.tex.ac.uk/tex-archive/fonts/calligra/) fonts. The Interactive TeX Editor: http://www.zib.de/Visual/software/ite The Home Page of Konstantin Vasil'ev: Programming in TeX. xfig-programs. If you have the Cygwin package (http://sourceware.cygnus.com/cygwin/) and an X server, you can compile xfig itself. A very credible lookalike has been written in Java: http://tech-www.informatik.uni-hamburg.de/applets/javafig/ It lacks a few features of the full xfig (rotated text and alignment commands being the ones I miss most) but otherwise, may do quite well. TkPaint is a very capaple vector drawing program with eps-export. It's available for both Windows and Unix. Windows: http://www.netanya.ac.il/~samy/tkpaint.html.Unix: http://www.fe.msk.ru/~vitus/misc/tkpaint.html Try SmartDraw Profession (v4), see www.smartdraw.com It can import and export as eps files + pdf + many others. As easy to use as xfig, and text lines can have multiple fonts, so you can write simple equations. To get version 0.7.3 of dvipdfm go to: http://odo.kettering.edu/dvipdfm/ From bitmap to eps. First convert a bitmap to JPEG and then use jpeg2ps to convert that to eps. For jpeg2ps convertor, look at http://www.muc.de/~tm. Tkpaint is a very nice object based drawing program supporting many of the standard objects (rectangles, circles, ellipses, polygons, splines, closed splines, arcs, pie-wedges, text, arrowheads, snapping-to-grid, included images, etc). It is freely available at http://www.netanya.ac.il/~samy/tkpaint.html. To convert wmf to eps. Get converter (wmf2eps) from ftp://ftp.tex.ac.uk/tex-archive/support/wmf2eps/ A software to typeset music: LilyPond. Take a look at http://www.realtime.net/~daboys/lilypond/ and http://www.cs.ruu.nl/people/hanwen/lilypond/index.html Pdf Using TrueType fonts with pdfTeX and pdfLaTeX: http://quantum.bitp.kiev.ua/radamir/ttf-pdf.htm The pdfTeX readme file ftp://ftp.cstug.cz/pub/tex/local/cstug/thanh/pdftex/README There is a mailing list for pdftex at [email protected]. Send a mail saying subscribe pdftex' to [email protected] to join. Mails posted to this list are also archived at news://news.muni.cz/cz.muni.redir.pdftex . If you need documentation for pdflatex / pdftex look at:http://www.tug.org/applications/pdftex/pdftexman.pdf There is also a pdftex FAQ in the works, not much but take a look: news: news.muni.cz where the group is cz.muni.redir.pdftex. There are also some useful references at http://www.tug.org/applications/pdftex/ Postscript If you need ghostscript, there is a newer version of GhostScript (5.5) available at http://www.cs.wisc.edu/~ghost/gsview/ See also, http://www.cs.wisc.edu/~ghost/. There is a web page devoted to printer compatibility and Ghostscript. You can find it at: http://www.cs.wisc.edu/~ghost/printer.html A program to put a Postscript wrapper on a JPEG image: Download it from http://www.muc.de/~tm/. If you need EPS, there is a shareware package on CTAN called WMF2EPS which works with Windows to make EPS. For more information write to [email protected]. Using EPS Graphics in Latex2e Documents has a good summary of includegraphics and its options. It is (or at least used to be) available as epslatex.ps from the ftp://ftp.tex.ac.uk/tex~archive/info/ directory or other CTAN sites. How to convert wmf files in eps files: look at ftp://ftp.dante.de/tex-archive/support/wmf2eps/ A list of PS tricks can be found at: http://www.tug.org/cgi-bin/lwgate/pstricks. The maintainer (Denis Girou) is very active and ready to help. If you need to render postscript figures try windvi44, using the ghostscript dll. It comes with web2c but a stand-alone version may be found here: ftp://ftp.dante.de/tex-archive/systems/win32/web2c/windvi44.zip A TIFF to EPS conversion utility:One is Image Magick, http://ftp.wizards.dupont.com/pub/ImageMagick/ImageMagick-4.0.4.tar.gz. it converts almost any format. You can also use the image tool "Paint Shop Pro" to convert "Tiff" files to "EPS". The latest version of the shareware can be downloaded from: http://www.digitalworkshop.co.uk/psp.htm A program that puts a PostScript wrapper around a JPEG file. This way it produces pretty compact EPS files. It can be found at http://www.muc.de/~tm/jpeg2ps/index.html. A new postscript driver from adobe. Pick it up at: http://www.adobe.com/supportservice/custsupport/LIBRARY/pdrvwin.htm In any windows application (like word, excel) you print the picture, diagram ... with that driver to a file. You have to change the property postscript option = EPS - and that created ps-file works well in your latex doc. More fonts: ftp://ftp.tex.ac.uk/tex-archive/fonts/cm/ps-type1/ To convert a prn file to a ps file (a "flatened" ps file), use pstoedit (look for the canonical archive at ftp://ftp.x.org/contrib/applications/pstoedit/pstoedit.2.60.tar.gz ) then use GSView to convert from ps to eps. This can be used by LaTeX. To Convert samp.* to the JPEG file samp.jpg, use AcdSee32, which lives at http://www.acdsystems.com. To Convert samp.jpg to samp.eps, use the program jpeg2ps, which lives at http://www.muc.de/~tm/jpeg2ps/index.html. Use AcdSee32 to generate * .jpg files from wmf, bmp, dcx, gif, iff, pcd, pcx, pic, png, psd, tga and tiff files. The jpeg2ps converter will finish the job . Find shareware at http://www.acdsystems.com PrintFile at: http://hem1.passagen.se/ptlerup/prfile.html. It will print text or PostScript files. It has "N-up" for printing multiple pages per side of paper, both for text and for PostScript files. It can also add headers and line numbers to text files, good for printing out program code. Supports drag and drop mode. If you use Excel and want to be able to edit the table afterwards and not just see and print (as is the case if you used postscript graphics) then go to: http://www.jam-software.com/software.html#X2L. IMHO its the best package for trees/decision trees/game. In the documentation you'll find examples of decision trees and game trees. Check out the examples under http://www.tug.org/applications/PSTricks/ and download it from CTAN. For more information about importing graphics in LaTeX, there's also a document (epslatex.ps or epslatex.pdf) available from CTAN (e.g. ftp://tug2.cs.umb.edu/tex-archive/) in the subdirectory info. If you want, a postscript compatible font go to Y&Y storefront at: http://www.yandy.com/. There is a website that will covert ps to pdf: http://www.babinszki.com/distiller/. Postscript Fonts: http://www.phys.washington.edu/~wright/texfonts/index.htm. pdfLaTeX Fonts: http://quantum.bitp.kiev.ua/radamir/ttf-pdf.htm Information from Adobe about creating PDF files from TeX with DVIPS. Point browser to: http://www.adobe.com/supportservice/custsupport/SOLUTIONS/543e.htm Adobe's web site about creating PDF from LaTeX. Links: http://www.adobe.com/supportservice/custsupport/SOLUTIONS/543e.htm and http://www.adobe.com/supportservice/custsupport/SOLUTIONS/12596.htm A free graphic converter for WIN95 which handles a lot of formats to open and save the pictures in colour eps format to include them in my file. Use Image Magick. It's a command line converter and it works fine. You can find pre-compiled binaries for Win95, for free, of course at www.wizards.dupont.com/cristy/ PS to PDF: If you need to convert a file from PS to PDF, try http://www.babinszki.com/distiller/ Computer Modern and AMSFonts in Type 1 (PostScript): http://www.ams.org/tex/type1-fonts.html. The PostScript Type 1 implementation of the Computer Modern and AMSFonts produced by and previously distributed by Blue Sky Research and Y&Y, Inc., are now freely available for general use. This has been accomplished through the cooperation of a consortium of scientific publishers with Blue Sky Research and Y&Y. A guide to PS fonts for TeX: http://www-theorie.physik.unizh.ch/~ichbin/texfonts/ How to install postscript fonts in MikTeX. There a way of getting emacs to compile Latex code using MikTeX? Finally, I don't get the colors that emacs is supposed to provide. I've even installed Auc-tex as described in W Minten's web page. You should be in "latex-mode" for latex files. M-x latex-mode (maybe just M-x tex-mode, try) More inforamtion: NT emacs home page: http://www.cs.washington.edu/homes/voelker/ntemacs.html There are example .emacs files linked to this page, some of them show how to change colors. They also show how to force emacs to auto-recognize your file types and go into the right mode automatically. Other people's helpful pages: http://www.gap.baynet.de/members/werdenfels.gym/tex/tex_engl.html and http://web.math.auc.dk/~dethlef/Tips/likehere.html How to feed compressed (zipped or gziped postcript files) to miktex´s dvips. Look at the document "epslatex.pdf" which you can get from CTAN.../info/. Note that zip and unzip freeware from InfoZIP ( ftp://ftp.cdrom.com/pub/infozip/win32/ ) work really well. Text Editors The MiKTeX project page has a link to a whole list of text editors. The list was started by a member of this newsgroup. look at: http://www.germany.net/teilnehmer/100/122054/texwin.htm Text editor to be used in conjunction with Windows 95 http://www.lancs.ac.uk/people/cpaap/pfe/ Winedt: (Another good editor for writing Tex, C and C++ files) ftp://ftp.dante.de/pub/tex/systems/win32/winedt/ and ftp://ftp.dante.de/tex-archive/systems/win32/winedt/ WinEdt32 is available from: ftp://ftp.tabu.uni-bonn.de/pub/tools/winedt/ or ftp://nt-newton.fho-emden.de/pub/software/uploads/winedt or http://www.wavenet.co.uk/ftp/pub/winedt/ WinEdt is shareware (40 per license), trial copy can be downloaded" from ftp://ftp.ctan.org/tex-archive/systems/win32
You may have a look at http://www.gap.baynet.de/members/werdenfels.gym/tex/
Wtex95: ftp://ftp.dante.de/tex-archive/systems/win32/wtex95/:
Text editor for windows 3.0: http://maitai.wsi.tu-muenchen.de/wheller/
Another Text Editor for windows 95: http://www.iridis.com/gwd
Another text editor, ultraedit: http://www.idmcomp.com
Text editors (GWD and UltraEdit). GWD: http://www.iridis.com/gwd/ and UltraEdit: http://www.idmcomp.com/
A good, free text editor for Win95/NT is PFE. You can find it at: http://www.lancs.ac.uk/people/cpaap/pfe
Wtex95. It colors latex expressions and the brackets, making it easily to recognize errors as you type. You can try it for free at, ftp://ftp.loria.fr/pub/unix/tex/ctan/systems/win32/wtex95
For all who needs a TeX-Editor try Dirk Struve's TeXShell32, available at
http://www.physik.uni-bielefeld.de/experi/d3/persons/struve/texshell05/index.html
Programmer's File Editor: http://www.lancs.ac.uk/people/cpaap/pfe. An editor for software programmer's.
In the spirit of friendly competition, I'd just like to note that I have written a WinEdt plug-in that provides many of the same functions as the LaTeX Wizard. You can find it at: http://home.istar.ca/~winedt/new.html.
Another editor: EDITEUR (www.studioware.com) . From inside the editor one can setup programs to execute; for example, compile, view with YAP, build a ps file (dvips) and view ps (ghost view).
YAP
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https://proofwiki.org/wiki/Definition:Rank_(Set_Theory)
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Definition:Rank (Set Theory)
Definition
Let $A$ be a set.
Let $V$ denote the von Neumann hierarchy.
Then the rank of $A$ is the smallest ordinal $x$ such that $A \in V \left({x+1}\right)$, given that $x$ exists.
Notation
The rank of the class $A$ is sometimes denoted as $\operatorname{rank} \left({A}\right)$.
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https://cstheory.stackexchange.com/tags/ds.data-structures/hot
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# Tag Info
37
The traditional analysis is fine. The "traditional" analysis is, if it is explained correctly, an approximation; it's based on calculating the expected number of cells that are 0/1 when you hash the keys into the filter, and then analyzing as though that was the actual number. The point is that the number of cells that are 0 (or 1) are tightly ...
32
Here's a lower bound from sorting. Given an input set $S$ of length $n$ to be sorted, create an input to your running median problem consisting of $n-1$ copies of a number smaller than the minimum of $S$, then $S$ itself, then $n-1$ copies of a number larger than the maximum of $S$, and set $k=2n-1$. The running medians of this input are the same as the ...
23
The cells in a $kD$-tree can have high aspect ratio, whereas octree cells are guaranteed to be cubical. Since this is a theory board, I'll give you the theoretical reason why high aspect ratio is a problem: it makes it impossible to use volume bounds to control the number of cells that you have to examine when solving approximate nearest neighbor queries. ...
21
The answer to this question is "no". To see why, we can think about a very extreme case, and how a regular bloom filter would work vs. a theoretical "Bizzaro World" bloom filter, which we can call a "gloom filter". What is great about a bloom filter is that you can do one-sided tests for membership of items (with false positives) using a data structure that ...
17
This is almost certainly impossible. Suppose you could solve your problem with preprocessing time $P(n)$ and query time $Q(n)$. Then there is a simple algorithm to solve the 3SUM problem—Given a set of $n$ real numbers, do any three elements sum to zero?—in $P(n)+n\cdot Q(n)$ time. We pre-process all the numbers, then for each number $a_k$, we find the ...
16
The communication complexity of the set disjointness problem is $\Omega(n)$. The communication complexity is a lower bound on the time complexity of testing whether the two instances are disjoint. Imagine Alice stores the data structure for the first set, and Bob stores the data structure for the second set; since they'll have to communicate $\Omega(n)$ ...
14
I wish I had a good answer for you. I use Book:Fundamental Data Structures (a collection of relevant Wikipedia articles) for my course on this subject but it's not really a complete textbook (for one thing, it has no exercises). CLRS is, I think, at a good level of detail for this sort of class but is missing too many of the important structures.
14
Probably not the best answer, but perhaps this is a useful starting point. If we wish to represent a non-negative integer, we can store it as a set of residues modulo sequential prime numbers starting from 2. In this form comparison is potentially hard, but multiplication and addition can be done pretty quickly. The product of the first $n$ primes is ...
14
A zipper, in general, is a data structure with a hole in it. Zippers are used for traversing/manipulating data structures, and the hole corresponds to the current focus of the traversal. Typically there is also an element of the data structure under consideration, so that one has a (list) zipper and a list or a (tree) zipper and a tree. The zipper allows the ...
14
Your problem is known in the learning literature as "learning monotone functions using membership queries". A class of monotone functions for which one can identify all minterms is known as "polynomially learnable using membership queries". It seems that the existence of a polynomial time algorithm is still open. Schmulevich et al. prove that "Almost all ...
13
No, this is not new; range searching with multilevel B-trees is completely standard. See, for example, the following surveys: Lars Arge. External memory data structures. Handbook of Massive Data Sets (James Abello, Panos M. Pardalos, and Mauricio G. C. Resende, eds.), 313-357. Kluwer Academic Publishers, 2002. See especially sections 5 and 6. Jeffrey ...
13
How to come up with sum-of-logs potential Let's consider the BST algorithm $A$ that for each access for element $x$, it rearranges only elements in the search path $P$ of $x$ called before-path, into some tree called after-tree. For any element $a$, let $s(a)$ and $s'(a)$ be the size of subtree rooted at $a$ before and after the rearrangement respectively. ...
13
There's a matching lower bound in the cell probe model (with a logarithmic number of bits per memory cell); see Fredman and Saks, "The cell probe complexity of dynamic data structures", STOC 1989.
12
There isn't really a single formalization of the kind of thing you are asking. There are many, many aspects to truth, trust, lies, and fallible reasoning, and this leads to an enormous variety of logical formalisms, each handling different aspects of this problem. If you want to account for uncertainty about your hypotheses, the traditional route is via ...
12
Let me add to Michael's answer that for split Bloom filters, where the hash functions have disjoint ranges, the traditional analysis is indeed correct without approximation or any concentration bounds. This is because the error probabilities for different hash functions become independent rather than correlated. The space/error trade-off for split Bloom ...
12
Let $A_k$ be the inverse of $\alpha_k$. $A_1(x) = 2x, A_2(x) = 2^x, \dots$. I claim that $k^{-1}(x) = A_x(x)$. Since $x = \alpha_x(A_x(x))$, and since $\forall z, \alpha_y(z) > \alpha_x(z)$, $\alpha_y(A_x(x)) > \alpha_x(A_x(x)) = x$. As a result $k(A_x(x)) = x$. Now consider the value of $\alpha(k^{-1}(n)) = \alpha(A_n(n))$. By definition of $\alpha$,...
10
I propose Reed-Solomon coding. The basic idea is that you can encode your data as a polynomial over a finite field. You can then evaluate this polynomial at several different points and these values become the messages that you will send. If the degree of the polynomial is N, then a receiving party only needs to receive N+1 messages in order to reconstruct ...
10
The only advanced data structures book that I'm aware of is the one by Peter Braß (Advanced Data Structures). It's not a bad book, but I'm not convinced that it's truly advanced at the graduate level.
10
The "offline" version of this question is addressed in my SODA 2014 paper with Huacheng Yu, Finding orthogonal vectors in discrete structures. For the case of $\mathbb F_2$, we give an $O(nd)$ time algorithm for determining, given two sets of $n$ vectors $A$ and $B$, whether there is a vector in $A$ and vector in $B$ with zero inner product. I'm sure you ...
10
You're quite right that the "queue = two lists" approaches don't give you the running time you want when you have the ability to re-use earlier versions. To get O(1) running time (amortized or worst case), you need a way to reverse the rear list and append it to the front list incrementally. Instead of doing the reverse&append all at once, taking O(N) ...
9
what are the advantages of octrees in spatial/temporal performance or otherwise, and in what situations are they most applicable (I've heard 3D graphics programming)? k-D trees are balanced binary trees and octrees are tries so the advantages and disadvantages are probably inherited from those more general data structures. Specifically: Rebalancing can be ...
9
The Handbook of Data Structures and Applications (Chapman & Hall/CRC Computer & Information Science Series) is mostly devoted to elementary data structures, but it also contains a few advanced materials that you may find useful for teaching a graduate level course. Given the huge size (1392 pages), this book may be classified as an encyclopedic ...
9
Edit: This algorithm is now presented here: http://arxiv.org/abs/1406.1717 Yes, to solve this problem it is sufficient to perform the following operations: Sort $n/k$ vectors, each with $k$ elements. Do linear-time post-processing. Very roughly, the idea is this: Consider two adjacent blocks of input, $a$ and $b$, both with $k$ elements; let the elements ...
9
Avishy Carmi and Daniel Moskovich have been developing tangle machines very recently, which is a topological model to describe information. There are two papers on the arXiv, as well as three introductory posts on the blog "Low Dimensional Topology" : http://ldtopology.wordpress.com/2014/05/04/low-dimensional-topology-of-information/
9
You will need to make some assumptions about what kinds of functions are allowed to get anywhere with this. The version of the problem where the elements of $S$ are linear functions from $\mathbb{R}$ to $\mathbb{R}$ has been studied, in a projectively dual form: if you think of each linear function $y=ax+b$ as being coordinatized by the pair of parameters $(... 9 The first one is average-case analysis, for sets of keys that are already somewhat randomly distributed (chosen either before or after the choice of hash function but with a probability distribution that is independent of the hash function). The second one is worst-case analysis, for sets of keys that are not random but are instead specially chosen to make ... 8 I'm pretty sure no such book exists. I drew up an annotated bibliography for my recent course, which was loosely based on Erik's course at MIT. It's definitely incomplete—I covered very few geometric data structures and no text data structures, for example—but you might still find it useful. 8 My recommendation is NOT to roll your own for now. There are two software packages that I'd recommend you try first. ANN (by Arya and Mount) is state-of-the-art for low dimensional near neighbor search and includes the "fixed radius" search that you're looking for. Nearpt3 (Wm Randolph Franklin) is another package that is specifically optimized for 3D, ... 8 No. Rotated-box queries and simplex queries are both generalizations of slab queries, where a slab is the volume between two parallel hyperplanes. Most lower bound proofs for simplex range searching actually assume that all query simplices are slabs of constant thickness. In particular, Chazelle [2] proved the following theorem. Let$P\$ be a random set ...
8
I'm not sure whether this qualifies as a purely topological computational model, but there is a topological approach to anyonic quantum computation within the framework of which Aharonov-Jones-Landau and Freedman-Kitaev-Wang proved that a quantum computer can "additively" approximate the Jones polynomial at a root of unity in polynomial time. Furthermore, by ...
Only top voted, non community-wiki answers of a minimum length are eligible
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http://www.mathworks.com/help/control/ref/balreal.html?nocookie=true
|
Accelerating the pace of engineering and science
# balreal
Gramian-based input/output balancing of state-space realizations
## Syntax
[sysb, g] = balreal(sys)
[sysb, g] = balreal(sys,'AbsTol',ATOL,'RelTol',RTOL,'Offset',ALPHA)
[sysb, g] = balreal(sys, condmax)
[sysb, g, T, Ti] = balreal(sys)
[sysb, g] = balreal(sys, opts)
## Description
[sysb, g] = balreal(sys) computes a balanced realization sysb for the stable portion of the LTI model sys. balreal handles both continuous and discrete systems. If sys is not a state-space model, it is first and automatically converted to state space using ss.
For stable systems, sysb is an equivalent realization for which the controllability and observability Gramians are equal and diagonal, their diagonal entries forming the vector G of Hankel singular values. Small entries in G indicate states that can be removed to simplify the model (use modred to reduce the model order).
If sys has unstable poles, its stable part is isolated, balanced, and added back to its unstable part to form sysb. The entries of g corresponding to unstable modes are set to Inf.
[sysb, g] = balreal(sys,'AbsTol',ATOL,'RelTol',RTOL,'Offset',ALPHA) specifies additional options for the stable/unstable decomposition. See the stabsep reference page for more information about these options. The default values are ATOL = 0, RTOL = 1e-8, and ALPHA = 1e-8.
[sysb, g] = balreal(sys, condmax) controls the condition number of the stable/unstable decomposition. Increasing condmax helps separate close by stable and unstable modes at the expense of accuracy. By default condmax=1e8.
[sysb, g, T, Ti] = balreal(sys) also returns the vector g containing the diagonal of the balanced gramian, the state similarity transformation xb = Tx used to convert sys to sysb, and the inverse transformation Ti = T-1.
If the system is normalized properly, the diagonal g of the joint gramian can be used to reduce the model order. Because g reflects the combined controllability and observability of individual states of the balanced model, you can delete those states with a small g(i) while retaining the most important input-output characteristics of the original system. Use modred to perform the state elimination.
[sysb, g] = balreal(sys, opts) computes the balanced realization using the options specified in the hsvdOptions object opts.
## Examples
### Balanced Realization of Stable System
Consider the following zero-pole-gain model, with near-canceling pole-zero pairs:
```sys = zpk([-10 -20.01],[-5 -9.9 -20.1],1)
```
```sys =
(s+10) (s+20.01)
----------------------
(s+5) (s+9.9) (s+20.1)
Continuous-time zero/pole/gain model.
```
A state-space realization with balanced gramians is obtained by
```[sysb,g] = balreal(sys);
```
The diagonal entries of the joint gramian are
```g'
```
```ans =
0.1006 0.0001 0.0000
```
This indicates that the last two states of sysb are weakly coupled to the input and output. You can then delete these states by
```sysr = modred(sysb,[2 3],'del');
```
This yields the following first-order approximation of the original system.
```zpk(sysr)
```
```ans =
1.0001
--------
(s+4.97)
Continuous-time zero/pole/gain model.
```
Compare the Bode responses of the original and reduced-order models.
```bodeplot(sys,sysr,'r--')
```
The plots shows that removing the second and third states does not have much effect on system dynamics.
### Balanced Realization of Unstable System
Create this unstable system:
```sys1=tf(1,[1 0 -1])
Transfer function:
1
-------
s^2 - 1
```
Apply balreal to create a balanced gramian realization.
```[sysb,g]=balreal(sys1)
a =
x1 x2
x1 1 0
x2 0 -1
b =
u1
x1 0.7071
x2 0.7071
c =
x1 x2
y1 0.7071 -0.7071
d =
u1
y1 0
Continuous-time model.
g =
Inf
0.2500
```
The unstable pole shows up as Inf in vector g.
expand all
### Algorithms
Consider the model
$\begin{array}{l}\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du\end{array}$
with controllability and observability gramians Wc and Wo. The state coordinate transformation $\overline{x}=Tx$ produces the equivalent model
$\begin{array}{l}\stackrel{˙}{\overline{x}}=TA{T}^{-1}\overline{x}+TBu\\ y=C{T}^{-1}\overline{x}+Du\end{array}$
and transforms the gramians to
$\begin{array}{cc}{\overline{W}}_{c}=T{W}_{c}{T}^{T},& {\overline{W}}_{o}={T}^{-T}{W}_{o}\end{array}{T}^{-1}$
The function balreal computes a particular similarity transformation T such that
${\overline{W}}_{c}={\overline{W}}_{o}=diag\left(g\right)$
See [1], [2] for details on the algorithm.
## References
[1] Laub, A.J., M.T. Heath, C.C. Paige, and R.C. Ward, "Computation of System Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms," IEEE® Trans. Automatic Control, AC-32 (1987), pp. 115-122.
[2] Moore, B., "Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction," IEEE Transactions on Automatic Control, AC-26 (1981), pp. 17-31.
[3] Laub, A.J., "Computation of Balancing Transformations," Proc. ACC, San Francisco, Vol.1, paper FA8-E, 1980.
|
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|
https://www.math.chapman.edu/~jipsen/structures/doku.php?id=commutative_bck-algebras
|
Commutative BCK-algebras
Abbreviation: ComBCK
Definition
A \emph{commutative BCK-algebra} is a structure $\mathbf{A}=\langle A,\cdot ,0\rangle$ of type $\langle 2,0\rangle$ such that
(1): $((x\cdot y)\cdot (x\cdot z))\cdot (z\cdot y) = 0$
(2): $x\cdot 0 = x$
(3): $0\cdot x = 0$
(4): $x\cdot y=y\cdot x= 0 \Longrightarrow x=y$
(5): $x\cdot (x\cdot y) = y\cdot (y\cdot x)$
Remark: Note that the commutativity does not refer to the operation $\cdot$, but rather to the term operation $x\wedge y=x\cdot (x\cdot y)$, which turns out to be a meet with respect to the following partial order:
$x\le y \iff x\cdot y=0$, with $0$ as least element.
Definition
A \emph{commutative BCK-algebra} is a BCK-algebra $\mathbf{A}=\langle A,\cdot ,0\rangle$ such that
$x\cdot (x\cdot y) = y\cdot (y\cdot x)$
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be commutative BCK-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x\cdot y)=h(x)\cdot h(y) \mbox{ and } h(0)=0$
Example 1:
Properties
Classtype variety no unbounded yes yes yes, $n=3$ no no
Finite members
$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &2 f(4)= &5 f(5)= &11 f(6)= &28 f(7)= &72 f(8)= &192 \end{array}$
|
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|
http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-2-1
|
PL EN
Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
Fundamenta Mathematicae
2014 | 227 | 2 | 97-128
Tytuł artykułu
Discrete homotopy theory and critical values of metric spaces
Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3/2. The latter two spectra are known to be discrete for compact geodesic spaces, and correspond to the values at which certain special covering maps, called δ-covers (Sormani-Wei) or ε-covers (Plaut-Wilkins), change equivalence type. In this paper we initiate the study of these ideas for non-geodesic spaces, motivated by the need to understand the extent to which the accompanying covering maps are topological invariants. We show that discreteness of the critical spectrum for general metric spaces can fail in several ways, which we classify. The, newcomer" critical values for compact non-geodesic spaces are completely determined by the homotopy critical values and the refinement critical values, the latter of which can, in many cases, be removed by changing the metric in a bi-Lipschitz way.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
97-128
Opis fizyczny
Daty
wydano
2014
Twórcy
autor
• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
autor
• 7078 W. Rainbow Rd., Sedalia, CO 80135, U.S.A.
autor
• Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A.
autor
• Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
autor
• Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, U.S.A.
autor
• 17 Sutherland Rd., Hicksville, NY 11801, U.S.A.
autor
• Department of Mathematics and, Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, U.S.A.
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory
|
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|
http://www.chemicalforums.com/index.php?topic=89531.0
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# Chemical Forums
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### AuthorTopic: Unknown isopropanol contaminant (Read 1775 times) !function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0];if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src="https://platform.twitter.com/widgets.js";fjs.parentNode.insertBefore(js,fjs);}}(document,"script","twitter-wjs"); (function() {var po = document.createElement("script"); po.type = "text/javascript"; po.async = true;po.src = "https://apis.google.com/js/plusone.js";var s = document.getElementsByTagName("script")[0]; s.parentNode.insertBefore(po, s);})();
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#### BROe
• Regular Member
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##### Unknown isopropanol contaminant
« on: January 03, 2017, 01:15:44 PM »
I recently bought some Isopropanol in the form of Iso-Heet to be used as a cleaning and reaction solvent. According to the MSDS sheet put out by the manufacturer, the product contains 99% IPA and 1% "proprietary additive". In order to separate the alcohol from the additive, a simple distillation has always been sufficient as there remains in the boiling flask a high-boiling syrupy amber liquid.
However, when I was cleaning the recently-bought Iso-Heet, it was distilling over at 45C rather than at its usual boiling point of 83C. Checking the distillation periodically I did not notice any of the "lines" normally observed when two liquids of different densities mix (I have heard them called Schlieren but I have no idea if this is an accurate term). Despite this one sticking point the liquid I distilled had a density within 0.01 of pure IPA, it was oxidized by acidic permanganate, it has the characteristic odor of IPA, and I observed no change in the solution density when I mixed a small amount of this liquid with some of my old stock that I am certain is (or rather was) pure Isopropanol.
I suspect that this impurity must have a density very similar to that of pure Isopropanol such that when the two are mixed there is little to no observable change, and that I have an azeotropic mixture of the two as when doing a second distillation I only collected one fraction at 43C. The other option is that the liquid in question simply isn't Isopropanol. Does anybody know of any further qualitative tests or separation methods I could try, maybe there is something I'm missing? I am tempted to see if I can't have a sample sent to a lab to have an IR and NMR done but that would likely be rather costly, so I view it as a last resort.
MSDS:
http://www.servicechamp.com/images/28202msds.pdf
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#### Borek
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##### Re: Unknown isopropanol contaminant
« Reply #1 on: January 03, 2017, 09:01:49 PM »
the product contains 99% IPA and 1% "proprietary additive"
Quote
it was distilling over at 45C rather than at its usual boiling point of 83C
Strange. Intuition tells me if the additive has a low boiling and is present in very small quantities it should be lost very fast. Also, if there is an azeotropic mixture and it contains below 1% of the other component (as would be in your case) I would expect its boiling point to be close to the BP of IPA. If the whole sample boils at so much lower temperature for a long period of time something is IMHO seriously off.
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#### Babcock_Hall
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##### Re: Unknown isopropanol contaminant
« Reply #2 on: January 04, 2017, 04:59:43 AM »
Are you confident that the thermometer was low enough in the distillation apparatus? If it sits too high, it gives a reading that is too low.
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#### BROe
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##### Re: Unknown isopropanol contaminant
« Reply #3 on: January 04, 2017, 06:31:55 AM »
The thermometer I use is jointed so it always sits in the same place in the stillhead, the last distillation I ran using it was of azeotropic nitric acid and I was getting appropriate readings there. A faulty thermometer was one of the first things I tested for after the first distillation, I checked it using boiling water as a standard and it did fine. Also part way through the distillation I swapped my analog thermometer for the digital thermocouple and was getting the same weird temperature readings.
*The picture is from the second distillation I'm currently running
*EDIT: In case the picture isn't showing up (as I can't see the image), the thermometer descends about two inches past the joint into the stillhead, the bottom being even with the midpoint of the arm that connects to the condenser
« Last Edit: January 04, 2017, 07:43:47 AM by Borek »
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#### P
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##### Re: Unknown isopropanol contaminant
« Reply #4 on: January 05, 2017, 03:58:58 AM »
Getting the obvious out of the way.... the pressure is the same yea? Reduced pressure is pretty good for distillation, maybe it has vac'd down too low? Probably not, as you would know this, but just checking. I used to reduce pressure regularly when trying to purify something by distillation.
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Tonight I’m going to party like it’s on sale for $19.99! - Apu Nahasapeemapetilon #### Babcock_Hall • Chemist • Sr. Member • Mole Snacks: +221/-16 • Offline • Posts: 3349 ##### Re: Unknown isopropanol contaminant « Reply #5 on: January 05, 2017, 09:44:55 AM » I am by far not the best organic chemists here, but I would have put the thermometer lower, if that is possible with your apparatus. I like to have the top of the bulb of liquid about even with the bottom of the elbow of glass. On the other hand, I don't have an explanation for why some liquids would boil at the correct temperature but not others. Logged #### BROe • Regular Member • Mole Snacks: +0/-0 • Offline • Posts: 17 ##### Re: Unknown isopropanol contaminant « Reply #6 on: January 06, 2017, 04:57:37 AM » Somewhat of a minor breakthrough, I was using that same isopropanol to clean some of my glassware and after a few hours of letting it air dry I came back to see a sort of oily film coating all the flasks I had rinsed with the IPA. The film had a very heavy, sweet, crude oil type of smell and on rinsing with tap water formed a milky emulsion that was quite difficult to remove with water. I ultimately ended up washing this out with a few small rinses of methanol. On the topic of thermometers, I just ran a distillation of methanol (also Heet brand, my backup solvent I suppose it could be called) and I was getting temperature readings that were pretty spot on, my digital thermocouple probe that sits about half an inch higher in the still head than my analog thermometer was getting a reading of 65.5, within acceptable levels of uncertainty for the thermocouple. Could it be that the greater volatility of methanol over IPA is negating the effects of a thermometer placed higher in the still head? Perhaps running tests with liquids of a low vapor pressure and higher boiling point would be more definitive. Getting the obvious out of the way.... the pressure is the same yea? Reduced pressure is pretty good for distillation, maybe it has vac'd down too low? Probably not, as you would know this, but just checking. I used to reduce pressure regularly when trying to purify something by distillation. No I wasn't pulling a vacuum, though that is something I would like to try in the future. Logged #### P • Full Member • Mole Snacks: +53/-15 • Offline • Gender: • Posts: 560 • I am what I am ##### Re: Unknown isopropanol contaminant « Reply #7 on: January 11, 2017, 12:00:33 AM » No I wasn't pulling a vacuum, though that is something I would like to try in the future. Makes distillation much easier.... and you can distil heat sensitive chemicals due to the greatly reduced temperatures you work at. You can get pressure/temperature curves which are east to read off with a ruler that will direct you as to your target temps and pressures for your system. Babcock hall might have a point with that thermometer placement - I didn't see it. Worth a try if possible to set it lower? Logged Tonight I’m going to party like it’s on sale for$19.99!
- Apu Nahasapeemapetilon
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# Nuclear Physics Consolidated Grant 2013
Lead Research Organisation: University of Manchester
Department Name: Physics and Astronomy
### Abstract
Nuclear Physics aims to understand the structure and dynamics of nuclear systems. It is key to understanding the Universe from the first microseconds of its inception when the quark-gluon plasma prevailed, through its history of star and galaxy formation where nuclear reactions play an essential role both in the generation of energy and the creation of elements. The field also has applications that benefit society in diverse areas, from medicine and security to power production, and a strong impact on other fields of science.
Our research interests are focused on fundamental aspects of nuclear physics ranging from hadrons to exotic nuclear systems. We use complementary methods to address key issues concerning: the structure and interactions of nucleons and light nuclei; new forms of collective motion; evolution of nuclear structure; and loosely bound systems.
The Manchester group is part of the UK nuclear community that has devised a mode of operation which enables it to make leading edge contributions at an international level in both experimental and theoretical nuclear physics. Experimental work is performed at specific overseas facilities with focussed investment in the necessary instrumentation to carry out this work.
Atomic nuclei provide a unique quantal laboratory in which microscopic as well as mesoscopic features, driven by effective two-body and three-body forces, can be studied. They are complex many-body systems, but often display unexpected regularities and simple excitation patterns that arise from underlying shell structure, pairing and collective modes of excitation. Such properties are also exhibited by simpler mesoscopic systems (for example, metallic clusters, quantum dots, and atomic condensates) the understanding of which draws heavily on techniques developed and honed in nuclear physics. A fundamental challenge is to understand nuclear properties ab-initio from the interplay of the strong, weak, and electromagnetic forces between individual nucleons. In recent years, enormous progress has been made with such programmes for light nuclei. For heavier nuclei, shell, cluster and other beyond mean field many-body techniques, based on effective interactions, provide essential frameworks for correlating experimental data, yet still lack the refinement to reliably predict nuclear properties as one moves more than a few nucleons from nuclei close to stability.
We also aim to make connections between the interactions of nucleons and the underlying theory that describes the strong force, Quantum Chromodynamics. Key quantities are the polarisabilities that describe how the structures of nucleons respond to external electric and magnetic fields. We are developing theoretical tools to determine these from experiments on the scattering of photons from hydrogen and other light nuclei. The latter are needed to learn about the the properties of the neutron since it is an unstable particle.
### Planned Impact
Trained manpower at postgraduate and postdoctoral levels is in great demand in nuclear, software and instrumentation industries. Young scientists trained within academic nuclear physics are the only source of independent expertise in areas concerning radioactivity and radiation detection. The importance of this expertise can only increase in the future as the UK moves into its new nuclear build programme, starting at Hinkley Point. The new Nuclear Industrial Strategy recognises the key enablers will be an increase in nuclear R&D and development of nuclear skills. Handling and disposal of nuclear wastes, reactor decommissioning and advanced reactor designs will become even more important issues in society. The research undertaken will also directly inform the teaching of undergraduates at Manchester who will benefit from advanced courses involving examples from topical, current research issues.
Since nuclear physics is the fundamental science underpinning the nuclear sector, our expertise developed in research projects such as these allows us to host for two major postgraduate training programmes: the Coordinating Centre for NTEC (Nuclear Technology Education Consortium involving 12 UK universities providing Masters-level courses to the nuclear industry) and the EPSRC Industrial Doctorate Centre for Nuclear Engineering (a consortium of 8 universities, see above). We deliver core and options modules for NTEC, and we are quickly expanding other KT activities (eg IAEA MSc course in nuclear security; leading involvement in a European project to design nuclear safety culture courses across the European nuclear sector; and nuclear codes training courses).
All members of the group, including academics, research fellows, PDRAs and PhD students, undertake public engagement activities. The members of the academic staff have a strong track record in outreach and have built up experience and a good reputation that can be used to good effect. They are regularly featured on local, national and foreign radio stations to address general issues, as well as for the direct promotion of their research to the general public. Research staff and students are less experienced, yet highly committed and training is encouraged. Through our participation in the Dalton Nuclear Institute, we collaborate with a number of local and national institutions as well. Dr John Roberts, a Nuclear Fellow within Dalton, is a member of the Nuclear Physics Group and coordinates our activities in this area. For example, we are running an annual course on nuclear energy for KS4 pupils. In collaboration with other UK nuclear physics groups, we organise an annual Teach the Teachers workshop that covers nuclear energy, nuclear medicine and nuclear physics. Members of the group are also active in various CERN-based public engagement activities.
Group members have also been able to influence UK and International Policy on nuclear related issues via participation in select committee activities and by representing the UK at a variety of international meetings related to the nuclear industry and skills.
Nuclear data and technological expertise in the group will be used to make measurements relevant to the nuclear industry (included in theme 6 of the proposal) by improving a variety of important nuclear cross sections. This will feed into the Joint European Fission-Fusion database, used throughout the nuclear industry to improve safety and economics of current and future operations, and of the design of advanced reactors and geological disposal facilities.
Group members are involved in an IPS project to improve SPECT imaging at the Christie hospital, with potential to commission commercial software. The group has supported medical research using short-lived positron emitters at the Wolfson Medical Imaging Centre, by joint supervision of MPhys and MSc students to help WMIC's research project work.
For more detailed information, see accompanying paperwork.
### Publications
10 25 50
Akber A (2015) Increased isomeric lifetime of hydrogen-like Os 192 m in Physical Review C
Andreyev A (2014) a decay of Au 176 in Physical Review C
Barber L (2019) Performing the differential decay curve method on ? -ray transitions with unresolved Doppler-shifted components in Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment
Baugher T (2016) In-beam ? -ray spectroscopy of Mn 63 in Physical Review C
Birse M (2016) New fixed points of the renormalisation group for two-body scattering in The European Physical Journal A
Birse M (2017) Potential problems with interpolating fields in The European Physical Journal A
Bissell M (2016) Cu charge radii reveal a weak sub-shell effect at N = 40 in Physical Review C
Brown R (2018) Edge Modes and Nonlocal Conductance in Graphene Superlattices. in Physical review letters
Butler P (2016) TSR: A Storage Ring for HIE-ISOLDE in Acta Physica Polonica B
Butler P (2016) TSR: A storage and cooling ring for HIE-ISOLDE in Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms
C Van Beveren (2016) a -decay study of ${}^{\mathrm{182,184}}$Tl in Journal of Physics G: Nuclear and Particle Physics
Campbell P (2016) Laser spectroscopy for nuclear structure physics in Progress in Particle and Nuclear Physics
Cocolios T (2016) High-resolution laser spectroscopy with the Collinear Resonance Ionisation Spectroscopy (CRIS) experiment at CERN-ISOLDE in Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms
Cosentino L (2016) Experimental setup and procedure for the measurement of the 7 Be(n,a)a reaction at n_TOF in Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment
Cullen D (2016) A dipole band above the I p = 31/2 - isomeric state in 189 Pb in EPJ Web of Conferences
Day Goodacre T (2017) RILIS-ionized mercury and tellurium beams at ISOLDE CERN in Hyperfine Interactions
Doncel M (2017) Spin-dependent evolution of collectivity in Te 112 in Physical Review C
Frost R (2016) A double-Bragg detector with digital signal processing for the event-by-event study of fission in actinide nuclei in International Journal of Modern Physics: Conference Series
Description Fundamental information about the structure of and reactions between atomic nuclei.
Exploitation Route Stimulated and fed information into other academic research.
Stimulated outreach activities to the public.
Sectors Energy,Culture, Heritage, Museums and Collections,Security and Diplomacy,Other
Description Fed into several outreach activities.
First Year Of Impact 2013
Sector Education,Culture, Heritage, Museums and Collections
Impact Types Cultural
Description Aizu
Organisation University of Aizu
Department Mathematical Science
Country Japan
Collaborator Contribution Theoretical calculations and support
Impact Research papers
Description Argonne
Organisation Argonne National Laboratory
Department Physics Division
Country United States
Sector Public
PI Contribution Intellectual input; performing experiments; analysing data.
Collaborator Contribution Facility and equipment provision; intellectual input.
Impact Forthcoming publications.
Description CERN-ISOLDE
Organisation European Organization for Nuclear Research (CERN)
Department ISOLDE Radioactive Ion Beam Facility
Country Switzerland
Sector Public
PI Contribution Research collaborations, equipment provision
Collaborator Contribution Research collaborations, equipment and facility provision
Impact Research outputs, collaborative equipment construction.
Description CRIS Collaboration
Organisation University of Leuven
Country Belgium
PI Contribution Manchester-led, Leuven is main partner
Collaborator Contribution Provides manpower and grant money for equipment
Impact See publication list
Start Year 2010
Description Laser spectroscopy at ISOLDE, CERN (optical detection)
Organisation European Organization for Nuclear Research (CERN)
Department ISOLDE Radioactive Ion Beam Facility
Country Switzerland
Sector Public
PI Contribution Leadership of some experimental proposals, new methodologies, data analysis, publication and dissemination.
Collaborator Contribution Personnel for experiments.
Impact Publications, conference talks and fellowships.
Start Year 2006
Description Laser spectroscopy at ISOLDE, CERN (optical detection)
Organisation Helmholtz Association of German Research Centres
Department GSI Helmholtz Centre for Heavy Ion Research
Country Germany
Sector Public
PI Contribution Leadership of some experimental proposals, new methodologies, data analysis, publication and dissemination.
Collaborator Contribution Personnel for experiments.
Impact Publications, conference talks and fellowships.
Start Year 2006
Description Laser spectroscopy at ISOLDE, CERN (optical detection)
Organisation Johannes Gutenberg University of Mainz
Department Institute for Nuclear Chemistry
Country Germany
PI Contribution Leadership of some experimental proposals, new methodologies, data analysis, publication and dissemination.
Collaborator Contribution Personnel for experiments.
Impact Publications, conference talks and fellowships.
Start Year 2006
Description Laser spectroscopy at ISOLDE, CERN (optical detection)
Organisation Max Planck Society
Department Max Planck Institute for Nuclear Physics
Country Germany
Sector Public
PI Contribution Leadership of some experimental proposals, new methodologies, data analysis, publication and dissemination.
Collaborator Contribution Personnel for experiments.
Impact Publications, conference talks and fellowships.
Start Year 2006
Description Laser spectroscopy at ISOLDE, CERN (optical detection)
Organisation University of Birmingham
Department School of Physics and Astronomy
Country United Kingdom
PI Contribution Leadership of some experimental proposals, new methodologies, data analysis, publication and dissemination.
Collaborator Contribution Personnel for experiments.
Impact Publications, conference talks and fellowships.
Start Year 2006
Description Laser spectroscopy at ISOLDE, CERN (optical detection)
Department Department of Physics
Country Finland
PI Contribution Leadership of some experimental proposals, new methodologies, data analysis, publication and dissemination.
Collaborator Contribution Personnel for experiments.
Impact Publications, conference talks and fellowships.
Start Year 2006
Description Laser spectroscopy at ISOLDE, CERN (optical detection)
Organisation University of Leuven
Department Institute for Nuclear and Radiation Physics
Country Belgium
PI Contribution Leadership of some experimental proposals, new methodologies, data analysis, publication and dissemination.
Collaborator Contribution Personnel for experiments.
Impact Publications, conference talks and fellowships.
Start Year 2006
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation European Organization for Nuclear Research (CERN)
Department ISOLDE Radioactive Ion Beam Facility
Country Switzerland
Sector Public
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation IPN Orsay - Nuclear structure
Country France
Sector Public
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation Johannes Gutenberg University of Mainz
Department Institute for Nuclear Physics
Country Germany
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation Max Planck Society
Department Max Planck Institute of Quantum Optics
Country Germany
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation New York University
Department Department of Physics
Country United States
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation University of Birmingham
Department School of Physics and Astronomy
Country United Kingdom
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation University of Leuven
Department Institute for Nuclear and Radiation Physics
Country Belgium
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at ISOLDE, CERN (particle detection)
Organisation University of Tokyo
Department Department of Nuclear Engineering and Management
Country Japan
PI Contribution Design and construction of a new beam line for collinear resonance ionisation spectroscopy of radioactive beams. Equipment, students and staff time contributed.
Collaborator Contribution Laboratory space, radioactive ion beams.
Impact Conference contributions, fellowship.
Start Year 2008
Description Laser spectroscopy at JYFL
Organisation University of Birmingham
Department School of Physics and Astronomy
Country United Kingdom
PI Contribution Leadership of the programme, experiment proposal creation, devising new experimental techniques, setting up and execution of experiments, data analysis, publication and dissemination of results.
Collaborator Contribution Shared responsibility for purchasing new equipment and renewal of old equipment. Provision of man power for running experiments.
Impact Peer reviewed publications, conference talks and fellowship awards.
Description Laser spectroscopy at JYFL
Department Department of Physics
Country Finland
PI Contribution Leadership of the programme, experiment proposal creation, devising new experimental techniques, setting up and execution of experiments, data analysis, publication and dissemination of results.
Collaborator Contribution Shared responsibility for purchasing new equipment and renewal of old equipment. Provision of man power for running experiments.
Impact Peer reviewed publications, conference talks and fellowship awards.
Description University of Tokyo
Organisation University of Tokyo
Department Centre for Nuclear Study
Country Japan
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http://dspace.library.daffodilvarsity.edu.bd:8080/browse?type=subject&value=Electric+Charge+Distribution
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Now showing items 1-1 of 1
• #### Anisotropic charged stellar models in Generalized Tolman IV spacetime
(Springer, 2015-01-12)
With the presence of electric charge and pressure anisotropy some anisotropic stellar models have been developed. An algorithm recently presented by Herrera et al. (Phys. Rev. D 77, 027502 (2008)) to generate static ...
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https://tex.stackexchange.com/questions/84460/tikz-callout-pointing-to-nested-align-text-math-environment-not-displayed?noredirect=1
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TikZ callout pointing to nested align/text/math environment not displayed
See the MWE below. Observation: In the second version the callout is displayed in the first it is not.
Can anyone tell me why this is so?
\documentclass[10pt,xcolor=dvipsnames]{beamer}
\usepackage{mathtools}
\usepackage{tikz}
\usetikzlibrary{shapes.callouts,shadows}
\usepackage{xparse}
\tikzset{
invisible/.style={opacity=0,text opacity=0},
visible on/.style={alt=#1{}{invisible}},
alt/.code args={<#1>#2#3}{%
\alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}} % \pgfkeysalso doesn't change the path
},
}
\NewDocumentCommand{\mycallout}{r<> O{opacity=0.8,text opacity=1} m m m m m m}{%
\tikz[remember picture, overlay]\node[drop shadow, rounded corners, align=left, fill=#8!30, text width=#5,
#2,visible on=<#1>,
draw,rectangle callout,anchor=pointer,callout relative pointer={(#6:#7cm)}]
at (#3) {#4};
}
\newcommand{\refbox}[3]{
\tikz[remember picture, baseline=(#1.base)] \node[fill=#3!30,anchor=base,rounded corners] (#1) {#2};
}
\begin{document}
\frame{\frametitle{ABC}
%%%%%%%%%%%%%%%%% with this version the callout is not displayed %%%%%%%%%%%%%%%%
\begin{align*}
&\text{ $g(x)\refbox{rb2711121055}{$\sim$}{yellow} x^\alpha$}
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% with this version the callout is displayed %%%%%%%%%%%%%%%%%%%%
% \begin{align*}
% \refbox{rb2711121055}{$2+2=4$}{yellow}
% \end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mycallout<1>{rb2711121055.north}{$1<2$}{1cm}{230}{1}{yellow}
}
\end{document}
PS: The code is based on https://tex.stackexchange.com/a/83786/16865. I don't think the problem has anything to do with the complicated stuff at the beginning of the preamble but, not being sure, I didn't further reduce the example.
1 Answer
It seems that your display math(text mode (inline math)) usage is affecting something. If you use a regular math environment it works after two runs.
\documentclass[10pt,xcolor=dvipsnames]{beamer}
\usepackage{mathtools,tikz,lmodern}
\usetikzlibrary{shapes.callouts,shadows}
\usepackage{xparse}
\tikzset{
invisible/.style={opacity=0,text opacity=0},
visible on/.style={alt=#1{}{invisible}},
alt/.code args={<#1>#2#3}{%
\alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}} % \pgfkeysalso doesn't change the path
},
}
\NewDocumentCommand{\mycallout}{r<> O{opacity=0.8,text opacity=1} m m m m m m}{%
\tikz[remember picture, overlay]
{\node[drop shadow, rounded corners, align=left, fill=#8!30, text width=#5,
#2,visible on=<#1>,
draw,rectangle callout,anchor=pointer,callout relative pointer={(#6:#7cm)}]
at (#3) {#4};}
}
\newcommand{\refbox}[3]{
\tikz[remember picture, baseline=(#1.base)]{
\node[fill=#3!30,anchor=base,rounded corners] (#1) {#2};}
}
\begin{document}
\begin{frame}{ABC}
$g(x)\refbox{rb}{\sim}{yellow} x^\alpha$
\mycallout<1>{rb.north}{$1<2$}{1cm}{230}{1}{yellow}
\end{frame}
\end{document}
• But why just in the first example? I mean, the second is also display style! – lpdbw Nov 27 '12 at 11:00
• @lpdbw I don't know (though you don't have text mode in the second example) but it's bad practice to use single lines with align anyway. Also beamer and align don't like each other that much. – percusse Nov 27 '12 at 11:03
• HHmm, single line in alignwas just for the MWE. Actually, I have many lines and it's not clear how to do it w/o align :-( – lpdbw Nov 27 '12 at 11:05
• @lpdbw You don't need to use $...$ inside \text just close one type the math and open another \text so it should work as in your second example. Might be a boxing issue. – percusse Nov 27 '12 at 11:09
• OK, yes ... \begin{align*} &\text{ abc } g(x)\refbox{rb2711121055}{$\sim$}{yellow} \text{ cde } x^\alpha \end{align*} works fine. Maybe you could put something like that in your answer so that I can accept it. – lpdbw Nov 27 '12 at 11:16
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http://sdss.kias.re.kr/astro/Horizon-Runs/Horizon-Run23.php
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## Scientific Purpose
We present two large cosmological N-body simulations, called Horizon Run 2 (HR2) and Horizon Run 3 (HR3), made using $6000^3$ = 216 billions and $7210^3$ = 374 billion particles, spanning a volume of $(7.200 h^{-1} \mathrm{Gpc})^3$ and $(10.815 h^{-1} \mathrm{Gpc})^3$, respectively. These simulations improve on our previous Horizon Run 1 (HR1) up to a factor of 4.4 in volume, and range from 2600 to over 8800 times the volume of the Millennium Run. In addition, they achieve a considerably finer mass resolution, down to $1.25\times 10^{11} h^{-1} M_\odot$, allowing to resolve galaxy-size halos with mean particle separations of $1.2h^{-1} \mathrm{Mpc}$ and $1.5h^{-1} \mathrm{Mpc}$, respectively. We have measured the power spectrum, correlation function, mass function and basic halo properties with percent level accuracy, and verified that they correctly reproduce the $\Lambda$CDM theoretical expectations, in excellent agreement with linear perturbation theory. Our unprecedentedly large-volume N-body simulations can be used for a variety of studies in cosmology and astrophysics, ranging from large-scale structure topology, baryon acoustic oscillations, dark energy and the characterization of the expansion history of the Universe, till galaxy formation science - in connection with the new SDSS-III. To this end, we made a total of 35 all-sky mock surveys along the past light cone out to z=0.7 (8 from the HR2 and 27 from the HR3), to simulate the BOSS geometry. The simulations and mock surveys are already publicly available on this page (to download the preprint, click here.)
## Authors
• Juhan Kim at CAC of Korea Institute for Advanced Study (KIAS; email contact: kjhan _at_ kias.re.kr)
• Changbom Park at KIAS
• Graziano Rossi at KIAS (corresponding author: graziano _at_ kias.re.kr)
• Sang Min Lee at Korea Institute of Science and Technology Information
• J.Richard Gott III at Princeton University
## Cosmological model of the Horizon Runs (HR's)
All the HR's share the same cosmology.
### Cosmological parameters of the HR's
Cosmology used for the HR's
Cosmological model $\Omega_{m,0}$ $\Omega_{b,0}$ $\Omega_{\Lambda,0}$ $n_\mathrm{s}$ $H_0$ (km/s/Mpc) $\sigma_8$
$\Lambda$CDM WMAP5 0.26 0.044 0.74 0.96 72 1/1.26
### Simulations specifics
Simulations Name Box Size ($h^{-1}\mathrm{Mpc}$) Number of CDM particles Starting redshift HR1 HR2 HR3 HR4 6592 7200 10815 3150 $4120^3$ $6000^3$ $7210^3$ $6300^3$ 23 32 27 100 Eisenstein & Hu (1998) CAMB Source CAMB Source CAMB Source Zel'dovich Zel'dovich Zel'dovich 2LPT
## Outputs from the simulation
• Snapshot data at z= 0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 1, and 4
• All-sky past lightcone data out to z=1.5
• Merger trees of FoF halos from z = 16 to 0, with their gravitationally most bound member particles
## Computers used
1. TACHYONII: SUN Blades B6275 at KISTI
• System resources used for the simulation(HR2/HR3)
• 1000/1030 nodes
• 8000/8240 CPU cores
• 9/21 TB main memory
• Infiniband interconnection
• about 200/400 TB disk storage of Lustre Filesystem
• total 20 days in wallclock time
2. QUEST: Linux Cluster at KIAS
• System resources used for the analysis
• 64 nodes
• 256 CPU cores
• 512 GB main memory
• Myrinet interconnection
• about 640 TB disk storage
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http://www.kaggle.com/forums/t/1064/users-ranking-method?page=6
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• Customer Solutions ▾
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Posts 1 Joined 14 Dec '11 Email user As a novice kaggler i had a doubt The current forumula is: $$\frac{100000}{\text{# Team Members}}\left(\text{Team Rank}\right)^{-0.75}\log_{10}\left(\text{# Teams}\right)\frac{\text{2 years - time elapsed since deadline}}{\text{2 years}}$$ What is the justification of this formula, when "time since deadline" > 2 years ? Competition points will be in negatives !! #76 / Posted 11 months ago
Ben Hamner Kaggle Admin Posts 754 Thanks 302 Joined 31 May '10 Email user shbahd wrote: As a novice kaggler i had a doubt The current forumula is: $$\frac{100000}{\text{# Team Members}}\left(\text{Team Rank}\right)^{-0.75}\log_{10}\left(\text{# Teams}\right)\frac{\text{2 years - time elapsed since deadline}}{\text{2 years}}$$ What is the justification of this formula, when "time since deadline" > 2 years ? Competition points will be in negatives !! Points can't be negative (that was in the text above the formula, but left out of the formula shown on the wiki for simplification). Also, the next update will include an exponential temporal decay, which has several desirable properties, as opposed to a linear one. Thanked by shbahd #77 / Posted 11 months ago
Posts 195 Thanks 46 Joined 12 Nov '10 Email user I agree with Chris H and Chris R: Divisor for multiple-person teams should be sqrt(N). Competitions that run longer should take longer to "fade out". #78 / Posted 11 months ago
Posts 304 Thanks 105 Joined 2 Dec '10 Email user Just one more data point to think about: one can make three random submissions to three current Research Competitions, take last place in each and still collect more points than member of three person team that takes 6th place in Biological Response competition. Thanked by DavidChudzicki , Bogdanovist , Christopher Hefele , Christian Stade-Schuldt , and B Yang #79 / Posted 11 months ago
Posts 83 Thanks 50 Joined 1 Jul '10 Email user Great point! In addition, I just did a calculation to see how much "perfect attendance" could trump skill. Let's say "Last-Place Larry" entered all 34 completed Kaggle competitions & came in dead last. By my calculations, he would have accumulated 105,000 points. That would place him 47th in the overall rankings. In other words, if you entered all the contests & never beat anybody, you'd still be ranked in the top 0.1% of all users. Also, 105,000 points is around the number of points the winners of the Heritage Health Prize will get. Each of the top HHP teams currently has 3 or 4 people on them. If one of those teams wins, their members would each get up to 101,000 points (using the current leaderboard). These numbers are a bit of a shock to me, really. I know others have said that there might be a dis-incentive to enter a contest at the last minute if it might drag down one's ranking. But on the other hand, knowing that a consistent random-submission strategy could significantly boost my overall ranking is a bit de-motivating, too. Ideally, I'd like rankings to roughly predict who would wind up at the top of any contest. On the other hand I understand that Kaggle wants to encourage participation, too. There's no reason why we couldn't have multiple rankings, though --- one exclusively for skill, and another for participation (or "most active"). Even if Kaggle wants just one ranking, seperating the problem into 2 pieces (skill-assessment vs participation) would allow one to explicitly weight how much each factor should matter in the overall rankings. In the current points equation & system, it's hard to disentangle the two. Thanked by William Cukierski , Last-Place Larry , and Chris Raimondi #80 / Posted 11 months ago / Edited 11 months ago
Posts 83 Thanks 50 Joined 1 Jul '10 Email user There's a lot of good discussion on this thread about user rankings, but crunching some real data might be helpful, too. Could Kagggle release a CSV file with the finishing-places of each team (& the team members) for all contests to date? Maybe this is something that would be good for a Kaggle Prospect Open Challenge. Is anyone else interested in this? Thanked by Ben Hamner #81 / Posted 11 months ago
Posts 194 Thanks 90 Joined 9 Jul '10 Email user it's hard to disentangle the two. Good points, but in theory - couldn't we make the optimum constant value entry (or something similar) the bottom of the scale? I think participation should be rewarded to some expent - but only skillful participation. I am not suggesting that anyone get negative points for scoring lower than the ocv entry - only that they should get zero points for that contest. I realize someone would only have to score slightly higher than that using my scheme - but some other similar system could be invoked that decays the scores to zero the closer they are to the benchmark or OCV entry. Thanked by Ben Hamner , and Christopher Hefele #82 / Posted 11 months ago
Ben Hamner Kaggle Admin Posts 754 Thanks 302 Joined 31 May '10 Email user Chris Raimondi wrote: it's hard to disentangle the two. Good points, but in theory - couldn't we make the optimum constant value entry (or something similar) the bottom of the scale? I think participation should be rewarded to some expent - but only skillful participation. I am not suggesting that anyone get negative points for scoring lower than the ocv entry - only that they should get zero points for that contest. I realize someone would only have to score slightly higher than that using my scheme - but some other similar system could be invoked that decays the scores to zero the closer they are to the benchmark or OCV entry. Thanks for the feedback! The challenge with making the ranking function dependent on benchmarks is that benchmarks mean different things for different contests / evaluation metrics and have a widey varying level of performance and complexity. (Compare the "optimized constant value" benchmark for some contests on the log loss metric to "human performance" on the gesture recognition challenge). #83 / Posted 11 months ago
Ben Hamner Kaggle Admin Posts 754 Thanks 302 Joined 31 May '10 Email user Christopher Hefele wrote: There's a lot of good discussion on this thread about user rankings, but crunching some real data might be helpful, too. Could Kagggle release a CSV file with the finishing-places of each team (& the team members) for all contests to date? Maybe this is something that would be good for a Kaggle Prospect Open Challenge. Is anyone else interested in this? Thanks for the suggestion! We're exploring this possibility Thanked by Christopher Hefele #84 / Posted 11 months ago
Posts 1 Thanks 5 Joined 12 Jun '12 Email user This thread is a disgrace. I take personal offense at what is a clear attack on my data mining abilities. "Eighty percent of success is showing up." -Woody Allen Thanked by Ben Hamner , Christopher Hefele , Bogdanovist , Jeff Moser , and F Bertrand #85 / Posted 11 months ago
Posts 194 Thanks 90 Joined 9 Jul '10 Email user There's no reason why we couldn't have multiple rankings, though --- one exclusively for skill, and another for participation. I also think it would be interesting to create a link graph with the "thanks" from the forum and see what everyone's ThankRank is - similar to PageRank, (and could easily be computed using that function from the igraph (or similar) package). For those unfamiliar with PageRank - it is based on the same concept of impact scores for journals (A citation from "Nature" counts more than a citation from "bob's journal of beer making".) #86 / Posted 11 months ago
Posts 83 Thanks 50 Joined 1 Jul '10 Email user Last-Place Larry wrote: This thread is a disgrace. I take personal offense at what is a clear attack on my data mining abilities. "Eighty percent of success is showing up." -Woody Allen Larry, woops, I should have known that you would show up here, too :) We can certainly try the Woody Allen weighting scheme (80% participation, 20% everything else). That's at least a little better than the Thomas Edison weighting scheme (1% inspiration, 99% perspiration)! #87 / Posted 11 months ago / Edited 11 months ago
Posts 47 Thanks 28 Joined 25 Dec '10 Email user I'm against the sqrt(#team members) suggestion, because many of the teams are opportunistic and do not actually imply good teamwork. There are some participants who team up from the beginning of a competition and don't add on members opportunistically, where dividing by sqrt(n) may be appropriate. Even so, collaborating increases the odds of winning, perhaps more than linearly, so why penalize in terms of points sub-linearly? As Martin has pointed out earlier in this thread, there are already a lot of motivations to team up and collaborate. Do we need a more generous point division scheme too? If the sqrt() is implemented, every competition should also have a cut off time for teaming up (i.e.; no change to teams in the last month of the competition). On the last place Larry issue, I think the user scores should be divided by the number of competitions they've participated in. That will encourage users to aim for a high batting average instead of a high total. The decay would take care of low frequency participants. #88 / Posted 11 months ago
Posts 304 Thanks 105 Joined 2 Dec '10 Email user Points for team members: Essentially, dividing by number of team members (1/N) or 1/sqrt(N) are both leaner dependences relative to leaderboard place. Coefficients are just different. For 1/N where N=2 coefficient is ~2.5 [ (1/2).^(-1/0.75)]. Two participants who are currently at 25 place should jump to place 10 by teaming up if they want to receive the same number of points as being alone. For team of three coefficient is 4.3. (jump from #22 to #5) If we use sqrt(N) , then coefficients are 1.6 and 2.1 respectively. Probably, everybody will have their own opinion on what coefficient is reasonable. And that opinion will definitely depend on how often member participates in teams. 2. Division by number of competition is equivalent to assigning negative points for poor performance. It will discourage top members from participation. (And best data scientist will be IRIG who participated once and was first on not extremely popular Eye motion competition) #89 / Posted 11 months ago
Posts 38 Thanks 22 Joined 26 Sep '11 Email user There seems to be an somewhat strange definition of how many competitions people have entered. According to the display in the ranking page, I have entered 9 competitions. In fact I have only made submissions to 2 competitions, which is correctly reported on my profile page as 'competitions completed'. I guess the other 7 are from competitions where I have accepted the terms and conditions in order to be able to download the data to have a look at for curiosity. In terms of making sense on the ranking page, the number of competitions you have actually competed in makes much more sense in my view. If the number of competitions entered was to ever be considered as part of the ranking calculation, that is certainly what should be used! In any case the current display is somewhat misleading and I think it would be much clearer if only competitions a user has actually entered in were displayed on the ranking page. Thanked by DavidChudzicki #90 / Posted 11 months ago
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http://ai-owl.blogspot.com/2010_06_01_archive.html
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## Tuesday, June 29, 2010
### log-sum-exp trick
when I implement models with discrete variables (which actually happens more than one can think), I always end up estimating this value:
$V = \log \left( \sum_i e^{b_i} \right)$
Why ? This usually happens at the denominator of a Bayes formula for example. I try to keep $$log$$-probabilities all the time so that not to have to deal with very small numbers and to do additions instead of multiplications. By the way, I was looking at the time and latency of floating-point instructions in the latest processors (like Intel Core i7 for example), and I realized that still in 2010, additions are faster than multiplications (even with SSEx and the like).
Therefore, use $$log$$
In this expression, $$b_i$$ are the log-probabilities and therefore $$e^{b_i}$$ are very small or very big yielding to overflow or underflow sometimes. A scaling trick can help using numbers in a better range without loss of accuracy and for a little extra cost as follows:
$\begin{array}{rcl} \log \left( \sum_i e^{b_i} \right)&=& \log \left( \sum_i e^{b_i}e^{-B}e^{B} \right)\\ ~ &=& \log \left( \left( \sum_i e^{b_i - B }\right)e^{B} \right)\\ ~ &=& \log \left( \sum_{i} e^{b_i - B} \right) + B \end{array}$
And that's it. For the value of $$B$$, take for instance $$B=\max_i b_i$$.
So the extra cost is to find the max value and to make a subtraction.
## Monday, June 28, 2010
Just for those of you who wants to know how to put formulas in Blogger, I used this link here : http://watchmath.com/vlog/?p=438
Pretty straighforward. It uses a public LaTeX server to render the formulas. Very pretty !
This is my first post on this blog. And to be honest, this is the first time I'm gonna try to blog my thoughts. So, I'll do it on what I like these days: Artificial Intelligence and Machine Learning.
The idea is to post thoughts, tricks, ideas, etc... In the hope people will read it and comment too.
And, oh yes, I just installed in function to include math formulas. I don't know if it works so let's try it now with a simple version of the Bayes formula:
$P(A|B) = \frac{P(B|A).P(A)}{P(B)}$
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http://physics.stackexchange.com/users/6456/frenchkheldar?tab=activity&sort=all&page=2
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# FrenchKheldar
less info
reputation
317
bio website ccl.gatech.edu location Atlanta, GA age 33 member for 2 years, 4 months seen Feb 11 at 21:57 profile views 53
Never-graduating PhD student in Computational Fluid Dynamics with Combustion. Main interest is in Liquid Rocket Engines and real gas flows.
# 69 Actions
Nov24 comment Mass diffusion: is $D_{AB} \neq D_{BA}$ at high pressures? If so, why? Nicely put ! That's what I think too, let's see what other answers we get. Nov24 asked Mass diffusion: is $D_{AB} \neq D_{BA}$ at high pressures? If so, why? Nov18 awarded Scholar Nov18 accepted Gas kinetic representation of trans-critical conditions Nov12 comment What velocity must an aircraft achieve for its shock wave to transform to plasma? Here it is, sort of. I can't link to the chart I was thinking about from Anderson, High Temperature Gas Dynamics, but this should do Check Fig. 1.1, they put the ionization limit at around 10 km/s which seems higher than what the Space Shuttle reaches, and yet I'm pretty sure there is a plasma cloud around the SS. I will research this further... Nov12 comment What velocity must an aircraft achieve for its shock wave to transform to plasma? Given that the Space Shuttle create a plasma during reentry, I doubt your calculation of 4E7 mph was correct... I'm looking for a chart that shows Cp and $\gamma$ of air vs Mach number and altitude but I can't find it... Nov12 answered How cold should it be outside for a hot coffee mug to break? Nov11 awarded Analytical Nov11 awarded Autobiographer Nov11 awarded Altruist Nov10 comment How do black holes accrete mass? I thought your question implied that it would take an infinite amount of time for the man to fall in the black hole. Which is not true. Also, I always thought the size of the black hole was smaller or equal to the Schwarzschild radius, which is the event horizon. Nov10 revised How do black holes accrete mass? added 1 characters in body Nov10 answered How do black holes accrete mass? Nov10 answered After what speed air friction starts to heat up an object? Nov9 awarded Quorum Nov9 comment Gas kinetic representation of trans-critical conditions Nov9 comment Gas kinetic representation of trans-critical conditions And then what I want to understand better is what happens at a molecular level for the macroscopic surface tension to disappear? Is this because the rate/energy of collisions becomes large enough? Nov9 comment Gas kinetic representation of trans-critical conditions So maybe I was not clear but the two drawings (left and right) are meant to illustrate the two paths (upper and lower) on the phase diagram. I'm considering a constant pressure system where I have a spatial temperature gradient. I'm considering a snapshot in time. Nov9 awarded Promoter Nov9 comment Gas kinetic representation of trans-critical conditions So no opinion whatsoever on this? Bounty coming up I guess...
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http://mathhelpforum.com/algebra/222034-simple-question.html
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# Math Help - Simple question
1. ## Simple question
I know this is kind of dumb but cos(-x) = - cos (x) ? Right?
2. ## Re: Simple question
Originally Posted by sakonpure6
I know this is kind of dumb but cos(-x) = - cos (x) ? Right?
NO! It is not correct. $\cos(x)$ is an even function, therefore $\cos(x)=\cos(-x)~.$
3. ## Re: Simple question
Omg thank you!!! I knew that sin(-x) = - sin x but when I put in a negative value for cos say, cos (-60) i got a positive number. Thanks!!!!
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http://mathhelpforum.com/number-theory/24853-gcd-proofs.html
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1. ## gcd proofs:
I'm kinda just starting out in this sort of stuff, and i'm rather confused. The professor assigned us to prove these. Not really sure how to start it to be honest. If I could see how these things get solved I think i'll understand it much better.
1. If a and b are 2 positive integers show that gcd(a,b) = gcd(a - b, b)
2. if a and b are 2 positive integers show taht gcd(a,b) = gcd(a, a +b)
3. show that gcd(fn,fn+1) = 1 for all natural numbers n.
2. Originally Posted by Mr.Obson
1. If a and b are 2 positive integers show that gcd(a,b) = gcd(a - b, b)
Say a > b >0 . Let gcd(a,b) = d. Now we compute gcd(a - b, b), we claim d is a common divisor. Indeed, d|(a-b) because d|a and d|b. Now we claim that if d' is a larger divisor then d'|(a-b) and d'|b implies d'|(a-b + b) thus d'|a, so d'|a, so d' is a common divisor to a and b so d' <= d.
2. if a and b are 2 positive integers show taht gcd(a,b) = gcd(a, a +b)
Same ideal
3. show that gcd(fn,fn+1) = 1 for all natural numbers n.
Use induction.
The inductive step if f_n+1 = f_n + f_n-1
But gcd(f_n,f_n-1) = 1 so gcd (f_n+1,f_n)=1 by Euclid's algorithm .
3. wow, thx it all makes a lot more sense now
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https://mrchasemath.com/2012/09/18/great-nctm-problem/?replytocom=797
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# Great NCTM problem
Yesterday I presented this problem from NCTM’s facebook page:
Solve for all real values of $x$:
$\frac{(x^2-13x+40)(x^2-13x+42)}{\sqrt{x^2-12x+35}}$
We’ve had an active discussion about this problem on their facebook page, and you should go check it out and join the conversation yourself. Go ahead and try it if you haven’t already.
Don’t read below until you’ve tried it for yourself.
Okay, here’s the work. Factor everything.
$\frac{(x-8)(x-5)(x-7)(x-6)}{\sqrt{(x-5)(x-7)}}=0$
Multiply both sides by the denominator.
$(x-8)(x-5)(x-7)(x-6)=0$
Use the zero-product property to find $x=5,6,7,8$. Now check for extraneous solutions and find that $x=5$ and $x=7$ give you $\frac{0}{0}\neq 0$ and $x=6$ gives $x=\frac{0}{\sqrt{-1}}=\frac{0}{i}=0$. This last statement DOES actually hold for $x=6$ but we exclude it because it’s not in the domain of the original expression.The original expression has domain $(-\infty,5)\cup(7,\infty)$. We could have started by identifying this, and right away we would know not to give any solutions outside this domain. The only solution is $x=8$.
Does this seem problematic? How can we exclude $x=6$ as a solution when it (a) satisfies the equation and (b) is a real solution? This is why we had such a lively discussion.
But this equation could be replaced with a simpler equation. Here’s one that raises the same issue:
Solve for all real values of x:
$\frac{x+5}{\sqrt{x}}=0$
Same question: Is $x=-5$ a solution? Again, notice that it DOES satisfy the equation and it IS a real solution. So why would we exclude it?
Of course a line is drawn in the sand and many people fall on one side and many fall on the other. It’s my impression that high-school math curriculum/textbooks would exclude $x=-5$ as a solution.
Here’s the big question: What does it mean to “solve for all real values of x“? Let’s consider the above equation within some other contexts:
Solve over $\mathbb{Z}$:
$\frac{x+5}{\sqrt{x}}=0$
Is $x=-5$ a solution? No, I think we must reject it. If we try to check it, we must evaluate $\frac{0}{\sqrt{5}}$ but this expression is undefined because $\sqrt{5}\notin\mathbb{Z}$. Here’s another one:
Solve over $\mathbb{Z}_5$:
$\frac{x+5}{\sqrt{x}}=0$
Is $x=-5$ a solution? No. Now when we try to check the solution we get $\frac{0}{\sqrt{5}}=\frac{0}{\sqrt{0}}=\frac{0}{0}$ which is undefined.
The point is that, if we go back to the same question and ask about the solutions of $\frac{x+5}{\sqrt{x}}=0$ over the reals, and we check the solution $x=-5$, we must evaluate $\frac{0}{\sqrt{-5}}$ which is undefined in the reals.[1]
So in the original NCTM question, we must exclude $x=6$ for the same reason. When you test this value, you get $\frac{0}{i}$ on the left side which YOU may think is 0. But this is news to the real numbers. The reals have no idea what $\frac{0}{i}$ evaluates to. It may as well be $\frac{0}{\text{moose}}$.
There’s a lot more to say here, so perhaps I’ll return to this topic another time. Special thanks to all the other folks on facebook who contributed to the discussion, especially my dad who helped me sort some of this out. Feel free to comment below, even if it means bringing a contrary viewpoint to the table.
________________________
[1] This last bit of work, where we fix the equation and change the domain of interest touches on the mathematical concept of algebraic varieties, which I claim to know *nothing* about. If someone comes across this post who can help us out, I’d be grateful! 🙂
## 7 thoughts on “Great NCTM problem”
1. -5 isn’t in Z_5, or rather it’s already 0 in Z_5, but we purposely confuse equivalence classes and representatives of the classes all the time, so No Big Deal: -5 = 5 = 0 in Z_5.
My Big Deal is why the Wolfram Alpha site refuses to see all of these obvious things that we’re talking about. It will make the domain of a function on Reals as large as possible whether we want it to or not. But then it doesn’t go far enough. It should allow us to say, “Please solve this in the Reals.” It did do this for me correctly:
solve 3x^2+x-7=4x in Z_11
Stepping out of a system and landing back in it reminds me of this following very mysterious fact that no mathematician understands. There are facts about integers whose only known proof uses analysis, often complex analysis. Is there a (meta-) theorem that guarantees that any proof of a statement about integers can be proven using only facts about integers, and not complex numbers? We call the first kind of proving doing “analytic number theory”; we call the second kind of proving “elementary number theory.” Elementary number theory is harder than analytic number theory, which shows you that sometimes going to a more encompassing system actually simplifies a problem. My intuition is that the statement “every theorem of analytic number theory is a theorem of elementary number theory” is unprovable, as is its negation.
2. As I posted to the NCTM Facebook page, I contend that x=5, x=6, x=7, and x=8 are all correct solutions to this problem. The equation can be simplified using algebra to (x-8)*sqrt(x-5)*sqrt(x-7)*(x-6)=0. For all real values of x, there are no indeterminate values. Or do you contend that (y*y)/y=0 –> y=0 is undefined at y=0?
• Right. y*y/y is undefined at zero, if we’re working in a field like the reals or complex numbers. More explicitly, 0/0 is undefined. Wouldn’t you agree that 0/0 is undefined? Check it on a calculator, too!
• Of course, 0/0 is undefined. According to Mathematica (a pretty decent calculator), y=0 *is* a solution:
Solve[y^2/y==0,y]
{{y->0}}
3. Same results even when we explicitly tell Mathematica that y is an element of the reals:
Solve[Assuming[y ∈ Reals, y^2/y] == 0, y]
{{y -> 0}}
4. Hmm.. that’s really interesting. After poking around a bit, I think this is an error in Mathematica/Wolfram Alpha.
If I ask for the value of 0/0, it correctly says “indeterminate”:
http://www.wolframalpha.com/input/?i=0%2F0
If I ask for the value of x/x evaluated at x=0, it says “1”, which I consider to be incorrect:
http://www.wolframalpha.com/input/?i=x%2Fx+evaluate+when+x%3D0
There’s a short discussion of this bug here:
http://community.wolframalpha.com/viewtopic.php?f=32&t=70878
Let me know if you discover anything else. I’m interested.
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http://mathhelpforum.com/pre-calculus/123145-solving-equations-matrix-help.html
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# Thread: Solving equations - Matrix Help!
1. ## Solving equations - Matrix Help!
I just started learning about matrices in class, so this question is supposed to be fairly simple.. but I've never been any good at math, so I'm having a bit of trouble with it.
Mark had $24,500 to invest. He divided the money into three different accounts. At the end of the year, he had made$1,300 in interest. The annual yield on each of the three accounts was 4%, 5.5%, and 6%. If the amount of money in the 4% account was four times the amount of money in the 5.5% account, how much had he placed in each account?
I started with -
0.06 0.055 0.04 1300
1 1 1 24500
0 422 -104 0
then I stopped at
0 1 4 34000
1 0 -4 -9500
0 0 792 7174000
I'm stuck!!
2. Originally Posted by Jiyongie
I just started learning about matrices in class, so this question is supposed to be fairly simple.. but I've never been any good at math, so I'm having a bit of trouble with it.
Mark had $24,500 to invest. He divided the money into three different accounts. At the end of the year, he had made$1,300 in interest. The annual yield on each of the three accounts was 4%, 5.5%, and 6%. If the amount of money in the 4% account was four times the amount of money in the 5.5% account, how much had he placed in each account?
I started with -
0.06 0.055 0.04 1300
1 1 1 24500
0 422 -104 0
then I stopped at
0 1 4 34000
1 0 -4 -9500
0 0 792 7174000
I'm stuck!!
It would make more sense if you would explain what you are doing and what equations/matrices you are using rather than just giving a list of numbers!
Since he had three accounts, let the amount of money he invested in each account be A, B, and C. He had $24,500 so A+ B+ C= 24500. He made 4%, 5.5%, and 6%, receiving$1300 in interest so .04A+ .055B+ .06C= 1300. Finally, " If the amount of money in the 4% account was four times the amount of money in the 5.5% account" so A= 4B. Your three equations for A, B, and C are A+ B+ C= 24500, .04A+ .055B+ .06C= 1300 and A- 4B= 0.
But what you have done is find and you are almost finished.
You can simplify by dividing that third row by 792 to get
0 0 1 9058.08
Add four times that new third row to the second row to get rid of the "-4":
1 0 (-4)+ 4(1) -9500+ 4(9058.08)
1 0 0 26732.32
And subtract four times that new third row to the first row to get rid of the "4":
0 1 4- 4(1) 34000- 4(9058.08)
0 1 0 -5658.08
That comes up negative but that is what I get too. It might not be possible to meet the conditions of the problem.
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https://overiq.com/c-examples/c-program-to-simulate-a-simple-calculator-using-switch-statement/
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# C Program to simulate a simple calculator using switch statement
Last updated on September 24, 2020
The following is a C Program to simulate a simple calculator using the switch statement:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 /*************************************************************** Program to simulate a simple calculator using switch statement * * Enter an expression: 100+50 * Result: 150 * * Enter an expression: 21*21 * Result: 441 **************************************************************/ #include // include stdio.h library int main(void) { int a, b, result; char op; // to store the operator printf("Enter an expression: "); scanf("%d%c%d", &a, &op, &b); switch(op) { case '+': result = a + b; break; case '-': result = a - b; break; case '*': result = a * b; break; case '/': result = a / b; break; } printf("Result = %d", result); return 0; // return 0 to operating system }
Expected Output:
1st run:
1 2 Enter an expression: 10+40 Result = 50
2nd run:
1 2 Enter an expression: 65*65 Result = 4225
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https://www.maplesoft.com/support/help/Maple/view.aspx?path=NumberTheory/SumOfDivisors
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NumberTheory - Maple Programming Help
Home : Support : Online Help : Mathematics : Group Theory : Numbers : NumberTheory/SumOfDivisors
NumberTheory
SumOfDivisors
sum of powers of the divisors
Calling Sequence SumOfDivisors(n, k) sigma[k](n) tau(n)
Parameters
n - non-zero integer k - (optional) non-negative integer; defaults to $1$
Description
• The SumOfDivisors command computes the sum of powers of the positive divisors of n.
• If n has divisors ${d}_{i}$ for $i$ from $1$ to $r$, then SumOfDivisors(n, k) is equal to $\sum _{i=1}^{r}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{d}_{i}^{k}$.
• sigma ($\mathrm{\sigma }$) is an alternate calling sequence for SumOfDivisors, where sigma[k](n) is equal to SumOfDivisors(n, k) and k defaults to $1$ if the index is omitted.
• tau ($\mathrm{\tau }$) counts the number of divisors of n, i.e. tau(n) is equal to SumOfDivisors(n, 0).
• If $\prod _{i=1}^{m}{p}_{i}^{{a}_{i}}$ is the prime factorization of the n, then SumOfDivisors is given by the formula $\prod _{i=1}^{m}\frac{{p}_{i}^{\left({a}_{i}+1\right)k}-1}{{p}_{i}^{k}-1}$ if k is non-zero and by the formula $\prod _{i=k}^{m}\left({a}_{i}+1\right)$ if k is zero.
Examples
> $\mathrm{with}\left(\mathrm{NumberTheory}\right):$
> $\mathrm{Divisors}\left(12\right)$
$\left\{{1}{,}{2}{,}{3}{,}{4}{,}{6}{,}{12}\right\}$ (1)
> $\mathrm{SumOfDivisors}\left(12\right)$
${28}$ (2)
> $\mathrm{τ}\left(12\right)$
${6}$ (3)
> $\mathrm{Divisors}\left(52\right)$
$\left\{{1}{,}{2}{,}{4}{,}{13}{,}{26}{,}{52}\right\}$ (4)
> $\mathrm{σ}[2]\left(52\right)$
${3570}$ (5)
> $\mathrm{SumOfDivisors}\left(52,2\right)$
${3570}$ (6)
Compatibility
• The NumberTheory[SumOfDivisors] command was introduced in Maple 2016.
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http://codeforces.com/problemset/problem/825/B
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B. Five-In-a-Row
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
Alice and Bob play 5-in-a-row game. They have a playing field of size 10 × 10. In turns they put either crosses or noughts, one at a time. Alice puts crosses and Bob puts noughts.
In current match they have made some turns and now it's Alice's turn. She wonders if she can put cross in such empty cell that she wins immediately.
Alice wins if some crosses in the field form line of length not smaller than 5. This line can be horizontal, vertical and diagonal.
Input
You are given matrix 10 × 10 (10 lines of 10 characters each) with capital Latin letters 'X' being a cross, letters 'O' being a nought and '.' being an empty cell. The number of 'X' cells is equal to the number of 'O' cells and there is at least one of each type. There is at least one empty cell.
It is guaranteed that in the current arrangement nobody has still won.
Output
Print 'YES' if it's possible for Alice to win in one turn by putting cross in some empty cell. Otherwise print 'NO'.
Examples
Input
XX.XX..........OOOO.................................................................................
Output
YES
Input
XXOXX.....OO.O......................................................................................
Output
NO
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https://global-sci.org/intro/article_detail/cicp/12481.html
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Volume 24, Issue 5
A Posteriori Error Estimates of Discontinuous Streamline Diffusion Methods for Transport Equations
Commun. Comput. Phys., 24 (2018), pp. 1355-1374.
Published online: 2018-06
Preview Full PDF 55 1602
Export citation
Cited by
• Abstract
Residual-based posteriori error estimates for discontinuous streamline diffusion methods for transport equations are studied in this paper. Computable upper bounds of the errors are measured based on mesh-dependent energy norm and negative norm. The estimates obtained are locally efficient, and thus suitable for adaptive mesh refinement applications. Numerical experiments are provided to illustrate underlying features of the estimators.
• Keywords
A posteriori error estimates, discontinuous streamline diffusion methods, transport equations.
65N30
• BibTex
• RIS
• TXT
@Article{CiCP-24-1355, author = {}, title = {A Posteriori Error Estimates of Discontinuous Streamline Diffusion Methods for Transport Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {5}, pages = {1355--1374}, abstract = {
Residual-based posteriori error estimates for discontinuous streamline diffusion methods for transport equations are studied in this paper. Computable upper bounds of the errors are measured based on mesh-dependent energy norm and negative norm. The estimates obtained are locally efficient, and thus suitable for adaptive mesh refinement applications. Numerical experiments are provided to illustrate underlying features of the estimators.
}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0120}, url = {http://global-sci.org/intro/article_detail/cicp/12481.html} }
TY - JOUR T1 - A Posteriori Error Estimates of Discontinuous Streamline Diffusion Methods for Transport Equations JO - Communications in Computational Physics VL - 5 SP - 1355 EP - 1374 PY - 2018 DA - 2018/06 SN - 24 DO - http://dor.org/10.4208/cicp.OA-2017-0120 UR - https://global-sci.org/intro/cicp/12481.html KW - A posteriori error estimates, discontinuous streamline diffusion methods, transport equations. AB -
Residual-based posteriori error estimates for discontinuous streamline diffusion methods for transport equations are studied in this paper. Computable upper bounds of the errors are measured based on mesh-dependent energy norm and negative norm. The estimates obtained are locally efficient, and thus suitable for adaptive mesh refinement applications. Numerical experiments are provided to illustrate underlying features of the estimators.
Juan Sun, Zhaojie Zhou & Huipo Liu. (2020). A Posteriori Error Estimates of Discontinuous Streamline Diffusion Methods for Transport Equations. Communications in Computational Physics. 24 (5). 1355-1374. doi:10.4208/cicp.OA-2017-0120
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J Mol Cell Cardiol. 2013 Mar;56:8-18. doi: 10.1016/j.yjmcc.2012.12.017. Epub 2013 Jan 3.
# Accessory subunits alter the temperature sensitivity of Kv4.3 channel complexes.
### Author information
1
Rudolf-Boehm-Institute of Pharmacology and Toxicology, University of Leipzig, Härtelstr.16-18, 04107 Leipzig, Germany. [email protected]
### Abstract
In human atrial myocytes the transient outward current I(to) develops a conspicuous faster inactivation with increasing temperatures. Since β-subunits are known to modulate I(to) current kinetics, we hypothesized that the temperature sensitivity of I(to) is not only determined by the property of the ion-passing α-subunit Kv4.3 but also by its interaction with accessory β-subunits. We therefore studied the influence of the transmembrane β-subunits KCNE1, KCNE2 and DPP6 on Kv4.3/KChIP2 channels in CHO cells at room temperature and at physiological temperature. Exposure to 37°C caused a significant acceleration of the channel kinetics, whereas current densities and voltage dependences remained unaltered at 37°C compared to 23°C. However, Kv4.3/KChIP2 channels without transmembrane β-subunits showed the strongest temperature sensitivity with considerably increased rates of activation and inactivation at 37°C. KCNE2 significantly slowed the current kinetics at 37°C compared to Kv4.3/KChIP2 channels, whereas KCNE1 did not influence the channel properties at both temperatures. Interestingly, the accelerating effects of DPP6 on current kinetics described at 23°C were diminished at physiological temperature, thus at 37°C current kinetics became remarkably similar for channel complexes Kv4.3/KChIP2 with and without DPP6 isoforms. A Markov state model was developed on the basis of experimental measurements to simulate the influence of β-subunits on Kv4.3 channel complex at both temperatures. In conclusion, the remarkably fast kinetics of the native I(to) at 37°C could be reproduced by co-expressing Kv4.3, KChIP2, KCNE2 and DPP6 in CHO cells, whereas the high temperature sensitivity of human I(to) could be not mimicked.
PMID:
23291429
DOI:
10.1016/j.yjmcc.2012.12.017
[Indexed for MEDLINE]
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http://slideplayer.com/slide/1664551/
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# The Calibrated Bayes Approach to Sample Survey Inference
## Presentation on theme: "The Calibrated Bayes Approach to Sample Survey Inference"— Presentation transcript:
The Calibrated Bayes Approach to Sample Survey Inference
Roderick Little Department of Biostatistics, University of Michigan Associate Director for Research & Methodology, Bureau of Census
Models for complex surveys 1: introduction
Learning Objectives Understand basic features of alternative modes of inference for sample survey data. Understand the mechanics of Bayesian inference for finite population quantitities under simple random sampling. Understand the role of the sampling mechanism in sample surveys and how it is incorporated in a Calibrated Bayesian analysis. More specifically, understand how survey design features, such as weighting, stratification, post-stratification and clustering, enter into a Bayesian analysis of sample survey data. Introduction to Bayesian tools for computing posterior distributions of finite population quantities. Models for complex surveys 1: introduction
Acknowledgement and Disclaimer
These slides are based in part on a short course on Bayesian methods in surveys presented by Dr. Trivellore Raghunathan and I at the 2010 Joint Statistical Meetings. While taking responsibility for errors, I’d like to acknowledge Dr. Raghunathan’s major contributions to this material Opinions are my own and not the official position of the U.S. Census Bureau Models for complex surveys 1: introduction
Models for complex surveys 1: introduction
Module 1: Introduction Distinguishing features of survey sample inference Alternative modes of survey inference Design-based, superpopulation models, Bayes Calibrated Bayes Models for complex surveys 1: introduction
Distinctive features of survey inference
1. Primary focus on descriptive finite population quantities, like overall or subgroup means or totals Bayes – which naturally concerns predictive distributions -- is particularly suited to inference about such quantities, since they require predicting the values of variables for non-sampled items This finite population perspective is useful even for analytic model parameters: Models for complex surveys 1: introduction
Distinctive features of survey inference
2. Analysis needs to account for "complex" sampling design features such as stratification, differential probabilities of selection, multistage sampling. Samplers reject theoretical arguments suggesting such design features can be ignored if the model is correctly specified. Models are always misspecified, and model answers are suspect even when model misspecification is not easily detected by model checks (Kish & Frankel 1974, Holt, Smith & Winter 1980, Hansen, Madow & Tepping 1983, Pfeffermann & Holmes (1985). Design features like clustering and stratification can and should be explicitly incorporated in the model to avoid sensitivity of inference to model misspecification. Models for complex surveys 1: introduction
Distinctive features of survey inference
3. A production environment that precludes detailed modeling. Careful modeling is often perceived as "too much work" in a production environment (e.g. Efron 1986). Some attention to model fit is needed to do any good statistics “Off-the-shelf" Bayesian models can be developed that incorporate survey sample design features, and for a given problem the computation of the posterior distribution is prescriptive, via Bayes Theorem. This aspect would be aided by a Bayesian software package focused on survey applications. Models for complex surveys 1: introduction
Distinctive features of survey inference
4. Antipathy towards methods/models that involve strong subjective elements or assumptions. Government agencies need to be viewed as objective and shielded from policy biases. Addressed by using models that make relatively weak assumptions, and noninformative priors that are dominated by the likelihood. The latter yields Bayesian inferences that are often similar to superpopulation modeling, with the usual differences of interpretation of probability statements. Bayes provides superior inference in small samples (e.g. small area estimation) Models for complex surveys 1: introduction
Distinctive features of survey inference
5. Concern about repeated sampling (frequentist) properties of the inference. Calibrated Bayes: models should be chosen to have good frequentist properties This requires incorporating design features in the model (Little 2004, 2006). Models for complex surveys 1: introduction
Approaches to Survey Inference
Design-based (Randomization) inference Superpopulation Modeling Specifies model conditional on fixed parameters Frequentist inference based on repeated samples from superpopulation and finite population (hybrid approach) Bayesian modeling Specifies full probability model (prior distributions on fixed parameters) Bayesian inference based on posterior distribution of finite population quantities argue that this is most satisfying approach Models for complex surveys 1: introduction
Design-Based Survey Inference
1 Models for complex surveys 1: introduction
Models for complex surveys 1: introduction
Random Sampling Random (probability) sampling characterized by: Every possible sample has known chance of being selected Every unit in the sample has a non-zero chance of being selected In particular, for simple random sampling with replacement: “All possible samples of size n have same chance of being selected” Models for complex surveys 1: introduction
Example 1: Mean for Simple Random Sample
Random variable Fixed quantity, not modeled Models for complex surveys 1: introduction
Example 2: Horvitz-Thompson estimator
Pro: unbiased under minimal assumptions Cons: variance estimator problematic for some designs (e.g. systematic sampling) can have poor confidence coverage and inefficiency Models for complex surveys 1: introduction
Role of Models in Classical Approach
Inference not based on model, but models are often used to motivate the choice of estimator. E.g.: Regression model regression estimator Ratio model ratio estimator Generalized Regression estimation: model estimates adjusted to protect against misspecification, e.g. HT estimation applied to residuals from the regression estimator (Cassel, Sarndal and Wretman book). Estimates of standard error are then based on the randomization distribution This approach is design-based, model-assisted Models for complex surveys 1: introduction
Model-Based Approaches
In our approach models are used as the basis for the entire inference: estimator, standard error, interval estimation This approach is more unified, but models need to be carefully tailored to features of the sample design such as stratification, clustering. One might call this model-based, design-assisted Two variants: Superpopulation Modeling Bayesian (full probability) modeling Common theme is “Infer” or “predict” about non-sampled portion of the population conditional on the sample and model Models for complex surveys 1: introduction
Superpopulation Modeling
Model distribution M: Predict non-sampled values : 1 In the modeling approach, prediction of nonsampled values is central In the design-based approach, weighting is central: “sample represents … units in the population” Models for complex surveys 1: introduction
Models for complex surveys 1: introduction
Bayesian Modeling Bayesian model adds a prior distribution for the parameters: 1 In the super-population modeling approach, parameters are considered fixed and estimated In the Bayesian approach, parameters are random and integrated out of posterior distribution – leads to better small-sample inference Models for complex surveys 1: introduction
Bayesian Point Estimates
Point estimate is often used as a single summary “best” value for the unknown Q Some choices are the mean, mode or the median of the posterior distribution of Q For symmetrical distributions an intuitive choice is the center of symmetry For asymmetrical distributions the choice is not clear. It depends upon the “loss” function. Models for complex surveys: simple random sampling
Bayesian Interval Estimation
Bayesian analog of confidence interval is posterior probability or credibility interval Large sample: posterior mean +/- z * posterior se Interval based on lower and upper percentiles of posterior distribution – 2.5% to 97.5% for 95% interval Optimal: fix the coverage rate 1-a in advance and determine the highest posterior density region C to include most likely values of Q totaling 1-a posterior probability Models for complex surveys: simple random sampling
Bayes for population quantities Q
Inferences about Q are conveniently obtained by first conditioning on and then averaging over posterior of . In particular, the posterior mean is: and the posterior variance is: Value of this technique will become clear in applications Finite population corrections are automatically obtained as differences in the posterior variances of Q and Inferences based on full posterior distribution useful in small samples (e.g. provides “t corrections”) Models for complex surveys: simple random sampling
Simulating Draws from Posterior Distribution
For many problems, particularly with high-dimensional it is often easier to draw values from the posterior distribution, and base inferences on these draws For example, if is a set of draws from the posterior distribution for a scalar parameter , then Models for complex surveys: simple random sampling
Models for complex surveys 1: introduction
Calibrated Bayes Any approach (including Bayes) has properties in repeated sampling We can study the properties of Bayes credibility intervals in repeated sampling – do 95% credibility intervals have 95% coverage? A Calibrated Bayes approach yields credibility intervals with close to nominal coverage Frequentist methods are useful for forming and assessing model, but the inference remains Bayesian See Little (2004) for more discussion Models for complex surveys 1: introduction
Models for complex surveys 1: introduction
Summary of approaches Design-based: Avoids need for models for survey outcomes Robust approach for large probability samples Less suited to small samples – inference basically assumes large samples Models needed for nonresponse, response errors, small areas – this leads to “inferential schizophrenia” Models for complex surveys 1: introduction
Models for complex surveys 1: introduction
Summary of approaches Superpopulation/Bayes models: Familiar: similar to modeling approaches to statistics in general Models needs to reflect the survey design Unified approach for large and small samples, nonresponse and response errors. Frequentist superpopulation modeling has the limitation that uncertainty in predicting parameters is not reflected in prediction inferences: Bayes propagates uncertainty about parameters, making it preferable for small samples – but needs specification of a prior distribution Models for complex surveys 1: introduction
Module 2: Bayesian models for simple random samples
2.1 Continuous outcome: normal model 2.2 Difference of two means 2.3 Regression models 2.4 Binary outcome: beta-binomial model 2.5 Nonparametric Bayes Models for complex surveys 1: introduction
Models for simple random samples
Consider Bayesian predictive inference for population quantities Focus here on the population mean, but other posterior distribution of more complex finite population quantities Q can be derived Key is to compute the posterior distribution of Q conditional on the data and model Summarize the posterior distribution using posterior mean, variance, HPD interval etc Modern Bayesian analysis uses simulation technique to study the posterior distribution Here consider simple random sampling: Module 3 considers complex design features Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
Diffuse priors In much practical analysis the prior information is diffuse, and the likelihood dominates the prior information. Jeffreys (1961) developed “noninformative priors” based on the notion of very little prior information relative to the information provided by the data. Jeffreys derived the noninformative prior requiring invariance under parameter transformation. In general, Models for complex surveys: simple random sampling
Examples of noninformative priors
In simple cases these noninformative priors result in numerically same answers as standard frequentist procedures Models for complex surveys: simple random sampling
2.1 Normal simple random sample
Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.1 Normal Example Posterior distribution of (m,s2) The above expressions imply that Models for complex surveys: simple random sampling
2.1 Posterior Distribution of Q
Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.1 HPD Interval for Q Note the posterior t distribution of Q is symmetric and unimodal -- values in the center of the distribution are more likely than those in the tails. Thus a (1-a)100% HPD interval is: Like frequentist confidence interval, but recovers the t correction Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.1 Some other Estimands Suppose Q=Median or some other percentile One is better off inferring about all non-sampled values As we will see later, simulating values of adds enormous flexibility for drawing inferences about any finite population quantity Modern Bayesian methods heavily rely on simulating values from the posterior distribution of the model parameters and predictive-posterior distribution of the nonsampled values Computationally, if the population size, N, is too large then choose any arbitrary value K large relative to n, the sample size National sample of size 2000 US population size 306 million For numerical approximation, we can choose K=2000/f, for some small f=0.01 or Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.1 Comments Even in this simple normal problem, Bayes is useful: t-inference is recovered for small samples by putting a prior distribution on the unknown variance Inference for other quantities, like Q=Median or some other percentile, is achieved very easily by simulating the nonsampled values (more on this below) Bayes is even more attractive for more complex problems, as discussed later. Models for complex surveys: simple random sampling
2.2 Comparison of Two Means
Population 1 Population 2 Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.2 Estimands Examples (Finite sample version of Behrens-Fisher Problem) Difference Difference in the population medians Ratio of the means or medians Ratio of Variances It is possible to analytically compute the posterior distribution of some these quantities It is a whole lot easier to simulate values of non-sampled in Population 1 and in Population 2 Models for complex surveys: simple random sampling
2.3 Ratio and Regression Estimates
Population: (yi,xi; i=1,2,…N) Sample: (yi, iinc, xi, i=1,2,…,N). For now assume SRS Objective: Infer about the population mean Excluded Y’s are missing values Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.3 Model Specification g=1/2: Classical Ratio estimator. Posterior variance equals randomization variance for large samples g=0: Regression through origin. The posterior variance is nearly the same as the randomization variance. g=1: HT model. Posterior variance equals randomization variance for large samples. Note that, no asymptotic arguments have been used in deriving Bayesian inferences. Makes small sample corrections and uses t-distributions. Models for complex surveys: simple random sampling
2.3 Posterior Draws for Normal Linear Regression g = 0
Easily extends to weighted regression Models for complex surveys: simple random sampling
2.4 Binary outcome: consulting example
In India, any person possessing a radio, transistor or television has to pay a license fee. In a densely populated area with mostly makeshift houses practically no one was paying these fees. Target enforcement in areas where the proportion of households possessing one or more of these devices exceeds 0.3, with high probability. Models for complex surveys: simple random sampling
2.4 Consulting example (continued)
Conduct a small scale survey to answer the question of interest Note that question only makes sense under Bayes paradigm Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.4 Consulting example Model for observable Prior distribution Estimand Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.4 Beta Binomial model The posterior distribution is Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.4 Infinite Population What is the maximum proportion of households in the population with devices that can be said with great certainty? Models for complex surveys: simple random sampling
2.5 Bayesian Nonparametric Inference
Population: All possible distinct values: Model: Prior: Mean and Variance: Models for complex surveys: simple random sampling
2.5 Bayesian Nonparametric Inference
SRS of size n with nk equal to number of dk in the sample Objective is to draw inference about the population mean: As before we need the posterior distribution of m and s2 Models for complex surveys: simple random sampling
2.5 Nonparametric Inference
Posterior distribution of q is Dirichlet: Posterior mean, variance and covariance of q Models for complex surveys: simple random sampling
Models for complex surveys: simple random sampling
2.5 Inference for Q Hence posterior mean and variance of Q are: Models for complex surveys: simple random sampling
Module 3: complex sample designs
Considered Bayesian predictive inference for population quantities Focused here on the population mean, but other posterior distribution of more complex finite population quantities Q can be derived Key is to compute the posterior distribution of Q conditional on the data and model Summarize the posterior distribution using posterior mean, variance, HPD interval etc Modern Bayesian analysis uses simulation technique to study the posterior distribution Models need to incorporate complex design features like unequal selection, stratification and clustering Models for complex surveys: simple random sampling
Modeling sample selection
Role of sample design in model-based (Bayesian) inference Key to understanding the role is to include the sample selection process as part of the model Modeling the sample selection process Simple and stratified random sampling Cluster sampling, other mechanisms See Chapter 7 of Bayesian Data Analysis (Gelman, Carlin, Stern and Rubin 1995) Models for complex sample designs
Models for complex sample designs
Full model for Y and I Observed data: Observed-data likelihood: Posterior distribution of parameters: Model for Population Model for Inclusion Models for complex sample designs
Ignoring the data collection process
The likelihood ignoring the data-collection process is based on the model for Y alone with likelihood: The corresponding posteriors for and are: When the full posterior reduces to this simpler posterior, the data collection mechanism is called ignorable for Bayesian inference about Posterior predictive distribution of Models for complex sample designs
Bayes inference for probability samples
A sufficient condition for ignoring the selection mechanism is that selection does not depend on values of Y, that is: This holds for probability sampling with design variables Z But the model needs to appropriately account for relationship of survey outcomes Y with the design variables Z. Consider how to do this for (a) unequal probability samples, and (b) clustered (multistage) samples Models for complex sample designs
Ex 1: stratified random sampling
Sample Population Population divided into J strata Z is set of stratum indicators: Stratified random sampling: simple random sample of units selected from population of units in stratum j. This design is ignorable providing model for outcomes conditions on the stratum variables Z. Same approach (conditioning on Z works for post-stratification, with extensions to more than one margin. Models for complex sample designs
Inference for a mean from a stratified sample
Consider a model that includes stratum effects: For simplicity assume is known and the flat prior: Standard Bayesian calculations lead to where: Models for complex sample designs
Bayes for stratified normal model
Bayes inference for this model is equivalent to standard classical inference for the population mean from a stratified random sample The posterior mean weights case by inverse of inclusion probability: With unknown variances, Bayes’ for this model with flat prior on log(variances) yields useful t-like corrections for small samples Models for complex sample designs
Suppose we ignore stratum effects?
Suppose we assume instead that: the previous model with no stratum effects. With a flat prior on the mean, the posterior mean of is then the unweighted mean This is potentially a very biased estimator if the selection rates vary across the strata The problem is that results from this model are highly sensitive violations of the assumption of no stratum effects … and stratum effects are likely in most realistic settings. Hence prudence dictates a model that allows for stratum effects, such as the model in the previous slide. Models for complex sample designs
Models for complex sample designs
Design consistency Loosely speaking, an estimator is design-consistent if (irrespective of the truth of the model) it converges to the true population quantity as the sample size increases, holding design features constant. For stratified sampling, the posterior mean based on the stratified normal model converges to , and hence is design-consistent For the normal model that ignores stratum effects, the posterior mean converges to and hence is not design consistent unless We generally advocate Bayesian models that yield design-consistent estimates, to limit effects of model misspecification Models for complex sample designs
Ex 2. A continuous (post)stratifier Z
Consider PPS sampling, Z = measure of size Sample Population Standard design-based estimator is weighted Horvitz-Thompson estimate When the relationship between Y and Z deviates a lot from the HT model, HT estimate is inefficient and CI’s can have poor coverage Models for complex sample designs
Ex 4. One continuous (post)stratifier Z
Sample Population Models for complex sample designs
Models for complex sample designs
Ex 3. Two stage sampling Most practical sample designs involve selecting a cluster of units and measure a subset of units within the selected cluster Two stage sample is very efficient and cost effective But outcome on subjects within a cluster may be correlated (typically, positively). Models can easily incorporate the correlation among observations Models for complex sample designs
Models for complex sample designs
Two-stage samples Sample design: Stage 1: Sample c clusters from C clusters Stage 2: Sample units from the selected cluster i=1,2,…,c Estimand of interest: Population mean Q Infer about excluded clusters and excluded units within the selected clusters Models for complex sample designs
Models for two-stage samples
Model for observables Prior distribution Models for complex sample designs
Estimand of interest and inference strategy
The population mean can be decomposed as Posterior mean given Yinc Models for complex sample designs
Models for complex sample designs
Posterior Variance Posterior variance can be easily computed Models for complex sample designs
Inference with unknown s and t
For unknown s and t Option 1: Plug in maximum likelihood estimates. These can be obtained using PROC MIXED in SAS. PROC MIXED actually gives estimates of q,s,t and E(mi|Yinc) (Empirical Bayes) Option 2: Fully Bayes with additional prior where b and v are small positive numbers Models for complex sample designs
Extensions and Applications
Relaxing equal variance assumption Incorporating covariates (generalization of ratio and regression estimates) Small Area estimation. An application of the hierarchical model. Here the quantity of interest is Models for complex sample designs
Models for complex sample designs
Extensions Relaxing normal assumptions Incorporate design features such as stratification and weighting by modeling explicitly the sampling mechanism. Models for complex sample designs
Models for complex sample designs
Summary Bayes inference for surveys must incorporate design features appropriately Stratification and clustering can be incorporated in Bayes inference through design variables Unlike design-based inference, Bayes inference is not asymptotic, and delivers good frequentist properties in small samples Models for complex sample designs
Module 4: Short introduction to Bayesian computation
A Bayesian analysis uses the entire posterior distribution of the parameter of interest. Summaries of the posterior distribution are used for statistical inferences Means, Median, Modes or measures of central tendency Standard deviation, mean absolute deviation or measures of spread Percentiles or intervals Conceptually, all these quantities can be expressed analytically in terms of integrals of functions of parameter with respect to its posterior distribution Computations Numerical integration routines Simulation techniques – outline here Models for Complex Surveys: Bayesian Computation
Models for Complex Surveys: Bayesian Computation
Types of Simulation Direct simulation (as for normal sample, regression) Approximate direct simulation Discrete approximation of the posterior density Rejection sampling Sampling Importance Resampling Iterative simulation techniques Metropolis Algorithm Gibbs sampler Software: WINBUGS Models for Complex Surveys: Bayesian Computation
Approximate Direct Simulation
Approximating the posterior distribution by a normal distribution by matching the posterior mean and variance. Posterior mean and variance computed using numerical integration techniques An alternative is to use the mode and a measure of curvature at the mode Mode and the curvature can be computed using many different methods Approximate the posterior distribution using a grid of values of the parameter and compute the posterior density at each grid and then draw values from the grid with probability proportional to the posterior density Models for Complex Surveys: Bayesian Computation
Models for Complex Surveys: Bayesian Computation
Normal Approximation Models for Complex Surveys: Bayesian Computation
Models for Complex Surveys: Bayesian Computation
Rejection Sampling Actual Density from which to draw from Candidate density from which it is easy to draw The importance ratio is bounded Sample q from g, accept q with probability p otherwise redraw from g Models for Complex Surveys: Bayesian Computation
Sampling Importance Resampling
Target density from which to draw Candidate density from which it is easy to draw The importance ratio Sample M values of q from g Compute the M importance ratios and resample with probability proportional to the importance ratios. Models for Complex Surveys: Bayesian Computation
Markov Chain Simulation
In real problems it may be hard to apply direct or approximate direct simulation techniques. The Markov chain methods involve a random walk in the parameter space which converges to a stationary distribution that is the target posterior distribution. Metropolis-Hastings algorithms Gibbs sampling Models for Complex Surveys: Bayesian Computation
Metropolis-Hastings algorithm
Try to find a Markov Chain whose stationary distribution is the desired posterior distribution. Metropolis et al (1953) showed how and the procedure was later generalized by Hastings (1970). This is called Metropolis-Hastings algorithm. Algorithm: Step 1 At iteration t, draw Models for Complex Surveys: Bayesian Computation
Models for Complex Surveys: Bayesian Computation
Step 2: Compute the ratio Step 3: Generate a uniform random number, u This Markov Chain has stationary distribution f(x). Any p(y|x) that has the same support as f(x) will work If p(y|x)=f(x) then we have independent samples Closer the proposal density p(y|x) to the actual density f(x), faster will be the convergence. Models for Complex Surveys: Bayesian Computation
Models for Complex Surveys: Bayesian Computation
Gibbs sampling Gibbs sampling a particular case of Markov Chain Monte Carlo method suitable for multivariate problems This is also a Markov Chain whose stationary Distribution is f(x) 2. This is an easier Algorithm, if the conditional densities are easy to work with If the conditionals are harder to sample from, then use MH or Rejection technique within the Gibbs sequence Models for Complex Surveys: Bayesian Computation
Models for complex surveys 1: introduction
Conclusion Design-based: limited, asymptotic Bayesian inference for surveys: flexible, unified, now feasible using modern computational methods Calibrated Bayes: build models that yield inferences with good frequentist properties – diffuse priors, strata and post-strata as covariates, clustering with mixed effects models Software: Winbugs, but software targeted to surveys would help. The future may be Calibrated Bayes! Models for complex surveys 1: introduction
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https://tex.stackexchange.com/questions/292012/detect-whether-running-on-pdftex-or-knuths-tex-for-conditional-include/292026
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# Detect whether running on pdfTeX or Knuth's TeX for conditional include
Does anyone know a way to detect whether one is running in pdfTeX or Knuth's TeX? I want to be able to conditionally include extra features when a document is being built using pdfTex, but also support original TeX as a fallback.
Something like
\ifdefined\pdfpagewidth
\pdfpagewidth 8.5 true in
\pdfpageheight 11 true in
\fi
would get the job done -- however the above relies on the e-TeX extensions and doesn't build on Knuth TeX.
Thanks
• Then use \ifx\pdfpagewidth\undefined \else ... \fi. This doesn't rely on e-TeX extensions. It will work also with tex executable. – user4686 Feb 10 '16 at 7:42
• @vacan1ty: Please unaccept my answer and choose another one – user31729 Feb 10 '16 at 15:39
Something like this?
Since \csname foo\endcsname expands to \relax if \foo is not defined, it's possible use \ifx.... to compare the command sequence to be equal to \relax. This does not need e-TeX at all.
\newif\ifknuthtex
\expandafter\ifx\csname pdfpagewidth\endcsname\relax
\knuthtextrue
\else
\pdfpagewidth 8.5 true in
\pdfpageheight 11 true in
\fi
This is \ifknuthtex Knuth's \TeX\else pdfTeX\fi
\bye
Compiling with tex gives the image below.
• @vancan1ty: You're welcome. Happy TeXing – user31729 Feb 9 '16 at 15:51
• This doesn't work. There is a missing \expandafter. Try with tex and you will get an error. I guess the image was generated before the update. Besides \ifx\pdfpagewidth\undefined works equally well. – user4686 Feb 9 '16 at 22:59
• @jfbu: I updated after a suggestion by Joseph Wright, but did not test. I reverted to the old version. Thanks. – user31729 Feb 10 '16 at 1:10
• @ChristianHupfer the problem with the reverted-to old version is that it leaves then control sequence \pdfpagewidth defined with meaning \relax if you run by Knuth's tex. This could fool later tests or force them to act like LaTeX @ifundefined (which does like your old code). The idea with the \begingroup...\endgroup and suitable \expandafter's is to avoid this after-effect. But \ifx\pdfpagewidth\undefined has no such after-effect and works. However it would be falsified if code like your reverted-to old version (now current) was executed earlier (or LaTeX \@ifundefined). – user4686 Feb 10 '16 at 7:02
• @jfbu: I'll investigate this. Apparently I am missing some point here I don't see at the moment. I asked the O.P. to unaccept and choose a better answer instead – user31729 Feb 10 '16 at 19:58
You can use the iftex "package". According to the documentation
This very simple package, for both Plain TeX and LaTeX, defines the \ifPDFTeX, \ifXeTeX, and \ifLuaTeX boolean for testing whether PDFTeX, or XeTeX, or LuaTeX is being used for typesetting.
If you only plan to distinguish between Knuth TeX and pdftex (or LuaTeX), this does it:
\begingroup\escapechar=-1
\edef\undefined{\string\undefined}%
\edef\test{\meaning\pdftexversion}%
\expandafter\endgroup\expandafter\let\expandafter\ifknuthtex
\csname if\ifx\undefined\test true\else false\fi\endcsname
\ifknuthtex
\message{Knuth TeX}
\else
\message{Not Knuth TeX}
\fi
One assumption is made: that no previous macro file or loaded format defines \pdftexversion. If you want that \ifknuthtex returns the correct truth value also with XeTeX, you can load ifxetex:
\input ifxetex.sty
\begingroup\escapechar=-1
\edef\undefined{\string\undefined}%
\edef\test{\meaning\pdftexversion}%
\expandafter\endgroup\expandafter\let\expandafter\ifknuthtex
\csname if%
\ifnum 0=%
\ifx\undefined\test 0\else 1\fi
\ifxetex 1\fi
true\else false\fi
\endcsname
\ifknuthtex
\message{Knuth TeX}
\else
\message{Not Knuth TeX}
\fi
• excuse me, what does mean Knuth TeX I think it's pdftex and etex now? – touhami Feb 9 '16 at 18:37
• In TeX Live (and also in MiKTeX, I guess), calling tex launches the original program with no extension and we use to call it “Knuth TeX” for better clarity. – egreg Feb 9 '16 at 18:38
• you're right for miktex I was ignore that – touhami Feb 9 '16 at 18:47
• \ifx\numexpr\undefined. – user4686 Feb 9 '16 at 23:05
• @jfbu Not really: you can make a format for pdftex that doesn't load e-TeX extensions, but it's not the same as Knuth TeX nonetheless. Try pdftex -ini '\show\numexpr' – egreg Feb 9 '16 at 23:35
For your problem, I strongly suspect that:
\ifx\pdfpagewidth\undefined
\else
\pdfpagewidth 8.5 true in
\pdfpageheight 11 true in
\fi
is, for all intents and purposes, quite fine enough. No need for complications.
Solutions based on a \csname..\endcsname approach have the drawback that they actually create a control sequence with meaning \relax, if the control sequence did not exist beforehand. Hence people do tricks with a bunch of \expandafter's to have this meaning exist only briefly and then disappear after the \ifx test. However the TeX memory will have a new entry, even for such a briefly existing control sequence, as also does the simple fact of writing \pdfpagewidth in your source code (if not commented out).
Unfortunately as soon as some code does the \csname..\endcsname approach without the alluded-to precautions, the simple \ifx\pdfpagewidth\undefined test above will become dysfunctional. And this is the main reason, I guess, that people are reluctant to recommend it. One should be aware that LaTeX's \@ifundefined creates precisely this problem. But on modern installations, latex executable is using pdftex binary (with in particular e-TeX extensions enabled), hence this is not Knuth TeX. On more ancient installations naturally you can have Knuth TeX + LaTeX format.
Note also that sometimes one may think (depending possibly on how your IDE presents things) that one is executing the Knuth TeX tex, whereas in fact it is pdftex in dvi output mode.
The \ifx\pdfpagewidth\undefined could also be compromised by some earlier code which has given a meaning to \undefined, but that would be very condemnable thing.
|
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|
http://archive.numdam.org/item/CM_1996__102_2_141_0/
|
An improved bound for the degree of smooth surfaces in ${𝐏}^{4}$ not of general type
Compositio Mathematica, Volume 102 (1996) no. 2, p. 141-145
@article{CM_1996__102_2_141_0,
author = {Cook, Michele},
title = {An improved bound for the degree of smooth surfaces in $\mathbf {P}^4$ not of general type},
journal = {Compositio Mathematica},
volume = {102},
number = {2},
year = {1996},
pages = {141-145},
zbl = {0871.14033},
mrnumber = {1394523},
language = {en},
url = {http://www.numdam.org/item/CM_1996__102_2_141_0}
}
Cook, Michele. An improved bound for the degree of smooth surfaces in $\mathbf {P}^4$ not of general type. Compositio Mathematica, Volume 102 (1996) no. 2, pp. 141-145. http://www.numdam.org/item/CM_1996__102_2_141_0/
[BS] Bayer, D. and Stillman, M.: "A criterion for detecting m-regularity", Invent. Math. 87 (1987) 1-11. | MR 862710 | Zbl 0625.13003
[BF] Braun, R. and Fløystad, G.: "A bound for the degree of smooth surfaces in P4 not of general type", Compositio Mathematica, Vol. 93, No. 2, September(I) 1994,211-229. | Numdam | MR 1287697 | Zbl 0823.14021
[EP] Ellingsrud, G. and Peskine, C.: "Sur les surfaces lisses de P4", Invent. Math. 95 (1989) 1-11. | MR 969410 | Zbl 0676.14009
[GLP] Gruson, L., Lazarsfeld, R. and Peskine, C.: "On a Theorem of Castelnuovo, and the equations defining space curves", Invent. Math. 72 (1983) 491-506. | MR 704401 | Zbl 0565.14014
[GP] Gruson, L. and Peskine, C.: "Genres des courbes de 1'espace projectif ", Lecture Notes in Mathematics, Algebraic Geometry, TromsØ 1977, 687 (1977) 31-59. | MR 527229 | Zbl 0412.14011
[H] Hartshorne, R.: Algebraic Geometry, Springer-Verlag (1977). | MR 463157 | Zbl 0367.14001
|
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http://spie.org/Publications/Proceedings/Paper/10.1117/12.164798
|
Share Email Print
### Proceedings Paper
Analysis of insertion device magnet measurements for the advanced light source
Author(s): Steve Marks; David E. Humphries; Brian M. Kincaid; Ross D. Schlueter; Chunxi Wang
Format Member Price Non-Member Price
PDF \$14.40 \$18.00
Paper Abstract
The Advanced Light Source (ALS), which is currently being commissioned at Lawrence Berkeley Laboratory, is a third generation light source designed to produce XUV radiation of unprecedented brightness. To meet the high brightness goal the storage ring has been designed for very small electron beam emittance and the undulators installed in the ALS are built to a high degree of precision. The allowable magnetic field errors are driven by electron beam and radiation requirements. Detailed magnetic measurements and adjustments are performed on each undulator to qualify it for installation in the ALS. The first two ALS undulators, IDA and IDB, have been installed. This paper describes the program of measurements, data analysis, and adjustments carried out for these two devices. Calculations of the radiation spectrum, based upon magnetic measurements, are included. Final field integral distributions are also shown. Good field integral uniformity has been achieved using a novel correction scheme, which is also described.
Paper Details
Date Published: 29 November 1993
PDF: 11 pages
Proc. SPIE 2013, Electron-Beam Sources of High-Brightness Radiation, (29 November 1993); doi: 10.1117/12.164798
Show Author Affiliations
Steve Marks, Lawrence Berkeley Lab. (United States)
David E. Humphries, Lawrence Berkeley Lab. (United States)
Brian M. Kincaid, Lawrence Berkeley Lab. (United States)
Ross D. Schlueter, Lawrence Berkeley Lab. (United States)
Chunxi Wang, Lawrence Berkeley Lab. (United States)
Published in SPIE Proceedings Vol. 2013:
|
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https://opentext.uleth.ca/apex-standard/sec_interp_deriv.html
|
## Section2.2Interpretations of the Derivative
Section 2.1 defined the derivative of a function and gave examples of how to compute it using its definition (i.e., using limits). The section also started with a brief motivation for this definition, that is, finding the instantaneous velocity of a falling object given its position function. Section 2.3 will give us more accessible tools for computing the derivative; tools that are easier to use than repeated use of limits.
This section falls in between the “What is the definition of the derivative?” and “How do I compute the derivative?” sections. Here we are concerned with “What does the derivative mean?”, or perhaps, when read with the right emphasis, “What is the derivative?” We offer two interconnected interpretations of the derivative, hopefully explaining why we care about it and why it is worthy of study.
### Subsection2.2.1Interpretation of the Derivative as Instantaneous Rate of Change
Section 2.1 started with an example of using the position of an object (in this case, a falling amusement park rider) to find the object's velocity. This type of example is often used when introducing the derivative because we tend to readily recognize that velocity is the instantaneous rate of change in position. In general, if $$f$$ is a function of $$x\text{,}$$ then $$\fp(x)$$ measures the instantaneous rate of change of $$f$$ with respect to $$x\text{.}$$ Put another way, the derivative answers “When $$x$$ changes, at what rate does $$f$$ change?” Thinking back to the amusement park ride, we asked “When time changed, at what rate did the height change?” and found the answer to be “By $$-64$$ feet per second.”
Now imagine driving a car and looking at the speedometer, which reads “60 mph.” Five minutes later, you wonder how far you have traveled. Certainly, lots of things could have happened in those $$5$$ minutes; you could have intentionally sped up significantly, you might have come to a complete stop, you might have slowed to 20 mph as you passed through construction. But suppose that you know, as the driver, none of these things happened. You know you maintained a fairly consistent speed over those $$5$$ minutes. What is a good approximation of the distance traveled?
One could argue the only good approximation, given the information provided, would be based on “$$\text{distance} = \text{rate}\times\text{time.}$$” In this case, we assume a constant rate of 60 mph with a time of $$5$$ minutes or $$5/60$$ of an hour. Hence we would approximate the distance traveled as $$5$$ miles.
Referring back to the falling amusement park ride, knowing that at $$t=2$$ the velocity was $$-64$$ ft/s, we could reasonably approximate that $$1$$ second later the riders' height would have dropped by about $$64$$ feet. Knowing that the riders were accelerating as they fell would inform us that this is an under-approximation. If all we knew was that $$f(2) = 86$$ and $$\fp(2) = -64\text{,}$$ we'd know that we'd have to stop the riders quickly otherwise they would hit the ground.
In both of these cases, we are using the instantaneous rate of change to predict future values of the output.
### Subsection2.2.2Units of the Derivative
It is useful to recognize the units of the derivative function. If $$y$$ is a function of $$x\text{,}$$ i.e., $$y=f(x)$$ for some function $$f\text{,}$$ and $$y$$ is measured in feet and $$x$$ in seconds, then the units of $$y' = \fp$$ are “feet per second,” commonly written as “ft/s.” In general, if $$y$$ is measured in units $$P$$ and $$x$$ is measured in units $$Q\text{,}$$ then $$y'$$ will be measured in units “$$P$$ per $$Q$$”, or “$$P/Q\text{.}$$” Here we see the fraction-like behavior of the derivative in the notation: the units of $$\frac{dy}{dx}$$are $$\frac{\text{units of }y}{\text{units of }x}\text{.}$$
###### Example2.2.1.The meaning of the derivative: World Population.
Let $$P(t)$$ represent the world population $$t$$ minutes after 12:00 a.m., January 1, 2012. It is fairly accurate to say that $$P(0) = 7{,}028{,}734{,}178$$ (www.prb.org). It is also fairly accurate to state that $$P'(0) = 156\text{;}$$ that is, at midnight on January 1, 2012, the population of the world was growing by about 156 people per minute (note the units). Twenty days later (or $$28{,}800$$ minutes later) we could reasonably assume the population grew by about $$28{,}800\cdot156 = 4{,}492{,}800$$ people.
###### Example2.2.2.The meaning of the derivative: Manufacturing.
The term widget is an economic term for a generic unit of manufacturing output. Suppose a company produces widgets and knows that the market supports a price of $$\10$$ per widget. Let $$P(n)$$ give the profit, in dollars, earned by manufacturing and selling $$n$$ widgets. The company likely cannot make a (positive) profit making just one widget; the start-up costs will likely exceed $$\10\text{.}$$ Mathematically, we would write this as $$P(1) \lt 0\text{.}$$
What do $$P(1000) = 500$$ and $$P'(1000)=0.25$$ mean? Approximate $$P(1100)\text{.}$$
Solution
The equation $$P(1000)=500$$ means that selling $$1000$$ widgets returns a profit of $$\500\text{.}$$ We interpret $$P'(1000) = 0.25$$ as meaning that when we are selling $$1000$$ widgets, the profit is increasing at rate of $$\0.25$$ per widget (the units are “dollars per widget.”) Since we have no other information to use, our best approximation for $$P(1100)$$ is:
\begin{align*} P(1100) \amp \approx P(1000) + P'(1000)\times100\\ \amp= \$500 + (100\text{ widgets })\cdot \$0.25/\text{widget}\\ \amp= \\$525\text{.} \end{align*}
We approximate that selling $$1100$$ widgets returns a profit of $$\525\text{.}$$
The previous examples made use of an important approximation tool that we first used in our previous “driving a car at 60 mph” example at the beginning of this section. Five minutes after looking at the speedometer, our best approximation for distance traveled assumed the rate of change was constant. In Examples 2.2.1 and Example 2.2.2 we made similar approximations. We were given rate of change information which we used to approximate total change. Notationally, we would say that
\begin{equation*} f(c+h) \approx f(c) + \fp(c)\cdot h\text{.} \end{equation*}
This approximation is best when $$h$$ is “small.” “Small” is a relative term; when dealing with the world population, $$h=22\text{ days} = 28{,}800\text{ minutes}$$ is small in comparison to years. When manufacturing widgets, $$100$$ widgets is small when one plans to manufacture thousands.
### Subsection2.2.3The Derivative and Motion
One of the most fundamental applications of the derivative is the study of motion. Let $$s(t)$$ be a position function, where $$t$$ is time and $$s(t)$$ is distance. For instance, $$s$$ could measure the height of a projectile or the distance an object has traveled.
Let's let $$s(t)$$ measure the distance traveled, in feet, of an object after $$t$$ seconds of travel. Then $$s'(t)$$ has units “feet per second,” and $$s'(t)$$ measures the instantaneous rate of distance change with repsect to time — it measures velocity.
Now consider $$v(t)\text{,}$$ a velocity function. That is, at time $$t\text{,}$$ $$v(t)$$ gives the velocity of an object. The derivative of $$v\text{,}$$ $$v'(t)\text{,}$$ gives the instantaneous rate of velocity change with respect to timeacceleration. (We often think of acceleration in terms of cars: a car may “go from $$0$$ to $$60$$ in $$4.8$$ seconds.” This is an average acceleration, a measurement of how quickly the velocity changed.) If velocity is measured in feet per second, and time is measured in seconds, then the units of acceleration (i.e., the units of $$v'(t)$$) are “feet per second per second,” or $$($$ft/s$$)$$/s. We often shorten this to “feet per second squared,” or fts2, but this tends to obscure the meaning of the units.
Perhaps the most well known acceleration is that of gravity. In this text, we use $$g=32\,\text{ft}/\text{s}^2$$ or $$g=9.8\,\text{m}/\text{s}^2\text{.}$$ What do these numbers mean?
A constant acceleration of $$32\,\frac{\text{ft}/\text{s}}{\text{s}}$$ means that the velocity changes by $$32\,\text{ft}/\text{s}$$ each second. For instance, let $$v(t)$$ measure the velocity of a ball thrown straight up into the air, where $$v$$ has units ft/s and $$t$$ is measured in seconds. The ball will have a positive velocity while traveling upwards and a negative velocity while falling down. The acceleration is thus $$-32\,\text{ft}/\text{s}^2\text{.}$$ If $$v(1) = 20\,\text{ft}/\text{s}\text{,}$$ then $$1$$ second later, the velocity will have decreased by $$32\,\text{ft}/\text{s}\text{;}$$ that is, $$v(2) = -12\,\text{ft/s}\text{.}$$ We can continue: $$v(3) = -44\,\text{ft/s}\text{.}$$ Working backward, we can also figure that $$v(0) = 52\,\text{ft}/\text{s}\text{.}$$
These ideas are so important we write them out as a Key Idea.
###### Key Idea2.2.3.The Derivative and Motion.
1. Let $$s(t)$$ be the position function of an object. Then $$s'(t)=v(t)$$ is the velocity function of the object.
2. Let $$v(t)$$ be the velocity function of an object. Then $$v'(t)=a(t)$$ is the acceleration function of the object.
### Subsection2.2.4Interpretation of the Derivative as the Slope of the Tangent Line
We now consider the second interpretation of the derivative given in this section. This interpretation is not independent from the first by any means; many of the same concepts will be stressed, just from a slightly different perspective.
Given a function $$y=f(x)\text{,}$$ the difference quotient $$\frac{f(c+h)-f(c)}{h}$$ gives a change in $$y$$ values divided by a change in $$x$$ values; i.e., it is a measure of the “rise over run,” or “slope,” of the secant line that goes through two points on the graph of $$f\text{:}$$ $$(c, f(c))$$ and $$(c+h,f(c+h))\text{.}$$ As $$h$$ shrinks to $$0\text{,}$$ these two points come close together; in the limit we find $$\fp(c)\text{,}$$ the slope of a special line called the tangent line that intersects $$f$$ only once near $$x=c\text{.}$$
Lines have a constant rate of change, their slope. Nonlinear functions do not have a constant rate of change, but we can measure their instantaneous rate of change at a given $$x$$ value $$c$$ by computing $$\fp(c)\text{.}$$ We can get an idea of how $$f$$ is behaving by looking at the slopes of its tangent lines. We explore this idea in the following example.
###### Example2.2.4.Understanding the derivative: the rate of change.
Consider $$f(x) = x^2$$ as shown in Figure 2.2.5. It is clear that at $$x=3$$ the function is growing faster than at $$x=1\text{,}$$ as it is steeper at $$x=3\text{.}$$ How much faster is it growing at $$3$$ compared to $$1\text{?}$$
Solution
We can answer this exactly (and quickly) after Section 2.3, where we learn to quickly compute derivatives. For now, we will answer graphically, by considering the slopes of the respective tangent lines.
With practice, one can fairly effectively sketch tangent lines to a curve at a particular point. In Figure 2.2.6, we have sketched the tangent lines to $$f$$ at $$x=1$$ and $$x=3\text{,}$$ along with a grid to help us measure the slopes of these lines. At $$x=1\text{,}$$ the slope is $$2\text{;}$$ at $$x=3\text{,}$$ the slope is $$6\text{.}$$ Thus we can say not only is $$f$$ growing faster at $$x=3$$ than at $$x=1\text{,}$$ it is growing three times as fast.
###### Example2.2.7.Understanding the graph of the derivative.
Consider the graph of $$f(x)$$ and its derivative, $$\fp(x)\text{,}$$ in Figure 2.2.8. Use these graphs to find the slopes of the tangent lines to the graph of $$f$$ at $$x=1\text{,}$$ $$x=2\text{,}$$ and $$x=3\text{.}$$
Solution
To find the appropriate slopes of tangent lines to the graph of $$f\text{,}$$ we need to look at the corresponding values of $$\fp\text{.}$$
• The slope of the tangent line to $$f$$ at $$x=1$$ is $$\fp(1)\text{;}$$ this looks to be about $$-1\text{.}$$
• The slope of the tangent line to $$f$$ at $$x=2$$ is $$\fp(2)\text{;}$$ this looks to be about $$4\text{.}$$
• The slope of the tangent line to $$f$$ at $$x=3$$ is $$\fp(3)\text{;}$$ this looks to be about $$3\text{.}$$
Using these slopes, tangent line segments to $$f$$ are sketched in Figure 2.2.9. Included on the graph of $$\fp$$ in this figure are points where $$x=1\text{,}$$ $$x=2$$ and $$x=3$$ to help better visualize the $$y$$ value of $$\fp$$ at those points.
###### Example2.2.10.Approximation with the derivative.
Consider again the graph of $$f(x)$$ and its derivative $$\fp(x)$$ in Example 2.2.7. Use the tangent line to $$f$$ at $$x=3$$ to approximate the value of $$f(3.1)\text{.}$$
Solution
Figure 2.2.11 shows the graph of $$f$$ along with its tangent line, zoomed in at $$x=3\text{.}$$ Notice that near $$x=3\text{,}$$ the tangent line makes an excellent approximation of $$f\text{.}$$ Since lines are easy to deal with, often it works well to approximate a function with its tangent line. (This is especially true when you don't actually know much about the function at hand, as we don't in this example.)
While the tangent line to $$f$$ was drawn in Example 2.2.7, it was not explicitly computed. Recall that the tangent line to $$f$$ at $$x=c$$ is $$y = \fp(c)(x-c)+f(c)\text{.}$$ While $$f$$ is not explicitly given, by the graph it looks like $$f(3) = 4\text{.}$$ Recalling that $$\fp(3) = 3\text{,}$$ we can compute the tangent line to be approximately $$y = 3(x-3)+4\text{.}$$ It is often useful to leave the tangent line in point-slope form.
To use the tangent line to approximate $$f(3.1)\text{,}$$ we simply evaluate $$y$$ at $$3.1$$ instead of $$f\text{.}$$
\begin{align*} f(3.1) \amp \approx y(3.1)\\ \amp= 3(3.1-3)+4\\ \amp= 0.1\cdot3+4\\ \amp = 4.3\text{.} \end{align*}
We approximate $$f(3.1) \approx 4.3\text{.}$$
To demonstrate the accuracy of the tangent line approximation, we now state that in Example 2.2.10, $$f(x) = -x^3+7x^2-12x+4\text{.}$$ We can evaluate $$f(3.1) = 4.279\text{.}$$ Had we known $$f$$ all along, certainly we could have just made this computation. In reality, we often only know two things:
1. what $$f(c)$$ is, for some value of $$c\text{,}$$ and
2. what $$\fp(c)$$ is.
For instance, we can easily observe the location of an object and its instantaneous velocity at a particular point in time. We do not have a “function $$f$$” for the location, just an observation. This is enough to create an approximating function for $$f\text{.}$$
This last example has a direct connection to our approximation method explained above after Example 2.2.2. We stated there that
\begin{equation*} f(c+h) \approx f(c)+\fp(c)\cdot h\text{.} \end{equation*}
If we know $$f(c)$$ and $$\fp(c)$$ for some value $$x=c\text{,}$$ then computing the tangent line at $$(c,f(c))$$ is easy: $$y(x) = \fp(c)(x-c)+f(c)\text{.}$$ In Example 2.2.10, we used the tangent line to approximate a value of $$f\text{.}$$ Let's use the tangent line at $$x=c$$ to approximate a value of $$f$$ near $$x=c\text{;}$$ i.e., compute $$y(c+h)$$ to approximate $$f(c+h)\text{,}$$ assuming again that $$h$$ is “small.” Note:
\begin{align*} y(c+h) \amp = \fp(c)\left((c+h)-c\right)+f(c)\\ \amp = \fp(c)\cdot h + f(c)\text{.} \end{align*}
This is the exact same approximation method used above! Not only does it make intuitive sense, as explained above, it makes analytical sense, as this approximation method is simply using a tangent line to approximate a function's value.
The importance of understanding the derivative cannot be understated. When $$f$$ is a function of $$x\text{,}$$ $$\fp(x)$$ measures the instantaneous rate of change of $$f$$ with respect to $$x$$ and gives the slope of the tangent line to $$f$$ at $$x\text{.}$$
### Exercises2.2.5Exercises
###### 1.
What is the instantaneous rate of change of position called?
###### 2.
Given a function $$y=f(x)\text{,}$$ in your own words describe how to find the units of $$\fp(x)\text{.}$$
###### 3.
What functions have a constant rate of change?
###### 4.
Given $$f(2)=12$$ and $$\fp(2) = -1\text{,}$$ approximate $$f(3)\text{.}$$
###### 5.
Given $$P(70)=68$$ and $$P'(70) = 6\text{,}$$ approximate $$P(75)\text{.}$$
###### 6.
Given $$z(40)=150$$ and $$z'(40) = -11\text{,}$$ approximate $$z(25)\text{.}$$
###### 7.
Knowing $$f(10)=25$$ and $$\fp(10) = 5$$ and the methods described in this section, which approximation is likely to be most accurate?
• f(10.1)
• f(11)
• f(20)
###### 8.
Given $$f(6)=82$$ and $$f(7) = 73\text{,}$$ approximate $$\fp(6)\text{.}$$
###### 9.
Given $$H(2)=51$$ and $$H(8) = 99\text{,}$$ approximate $$H'(2)\text{.}$$
###### 10.
Let $$V(x)$$ measure the volume, in decibels, measured inside a restaurant with $$x$$ customers. What are the units of $$V'(x)\text{?}$$
###### 11.
Let $$v(t)$$ measure the velocity, in ft/s, of a car moving in a straight line $$t$$ seconds after starting. What are the units of $$v'(t)\text{?}$$
###### 12.
The height $$H\text{,}$$ in feet, of a river is recorded $$t$$ hours after midnight, April 1. What are the units of $$H'(t)\text{?}$$
###### 13.
$$P$$ is the profit, in thousands of dollars, of producing and selling $$c$$ cars.
1. What are the units of $$P'(c)\text{?}$$
2. What is likely true of $$P(0)\text{?}$$
###### 14.
$$T$$ is the temperature in degrees Fahrenheit, $$h$$ hours after midnight on July 4 in Sidney, NE.
1. What are the units of $$T'(h)\text{?}$$
2. Is $$T'(8)$$ likely greater than or less than 0? Why?
3. Is $$T(8)$$ likely greater than or less than 0? Why?
Graphs of functions $$f$$ and $$g$$ are given. Identify which function is the derivative of the other.
###### 15.
• $$f$$ is the derivative of $$g\text{.}$$
• $$g$$ is the derivative of $$f\text{.}$$
###### 16.
• $$f$$ is the derivative of $$g\text{.}$$
• $$g$$ is the derivative of $$f\text{.}$$
###### 17.
• $$f$$ is the derivative of $$g\text{.}$$
• $$g$$ is the derivative of $$f\text{.}$$
###### 18.
• $$f$$ is the derivative of $$g\text{.}$$
• $$g$$ is the derivative of $$f\text{.}$$
###### Review
Use the definition of the derivative to compute the derivative of $$f\text{.}$$
###### 19.
$$f(x)=5x^2$$
###### 20.
$$f(x)=(x-2)^3$$
Numerically approximate the derivative.
###### 21.
$$f'(\pi)$$ where $$f(x) = \cos(x)$$
###### 22.
$$f'(9)$$ where $$f(x) = \sqrt{x}$$
|
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|
https://typeset.io/papers/design-and-analysis-of-hall-effect-probe-based-pressure-2xzxtxt54a
|
Journal ArticleDOI
# Design and Analysis of Hall Effect Probe-Based Pressure Transmitter Using Bellows as Sensor
04 Mar 2015--Vol. 64, Iss: 9, pp 2548-2556
TL;DR: The design of a noncontact pressure transducer along with a transmitting unit using bellows as the primary sensing element and a Hall sensor as secondary sensing element has been described and theoretical equations describing the operation of the proposed transducers and transmitter have been derived.
AbstractBellows is an elastic-type pressure sensor used as a local indicator in industry. Transmission of bellows reading to a remote location in control room is very important in pressure measurement and control system in industry. In this paper, the design of a noncontact pressure transducer along with a transmitting unit using bellows as the primary sensing element and a Hall sensor as secondary sensing element has been described. The theoretical equations describing the operation of the proposed transducer and transmitter have been derived. The function of the transducer and transmitter has been experimentally tested and the experimental results are reported in the paper. Both transducer and transmitter characteristics have been found to be linear with good repeatability. The graphical abstract is shown in Fig. 1 . Fig. 1. Graphical abstract.
##### Citations
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Journal ArticleDOI
, Yu Zhu1
TL;DR: An on-line simultaneous computation of model parameters and position method is proposed to measure position for a permanent magnet synchronous linear motor and results indicate that the standard deviation of measurement error is less than less than 10% with the comparison of a reference optical encoder.
Abstract: With the advantage of low cost and small size, a hall sensor has been exploited to detect motor position, and this strategy has attracted much attention. A challenge is that measurement model parameters vary with position and vary from motor to motor because of the machining errors of permanent magnets. While, most existing methods use other auxiliary displacement sensors to calibrate model parameters off-line, which require a lot of preliminary work and lose adaptation when they are applied in industry. In this paper, an on-line simultaneous computation of model parameters and position method is proposed to measure position for a permanent magnet synchronous linear motor. The measurement model with Fourier series is employed to approximate the mapping relationship between magnetic field strength and mover position. Based on the continuous variation characteristics of model parameters and position, a simultaneous computational algorithm is proposed to real-timely compute model parameters and position without other sensors. In the experiments of one-way and reciprocating motion, magnetic field information is detected by a self-designed hall sensor array. Experimental results indicate that the standard deviation of measurement error is less than $10~\mu \text{m}$ with the comparison of a reference optical encoder.
19 citations
### Cites methods from "Design and Analysis of Hall Effect ..."
• ...Hall sensors also have been applied in others motion systems to measure position [9], [10]....
[...]
Journal ArticleDOI
27 Jul 2020-Sensors
TL;DR: An up-to-date review of the main existing devices, based on the classic and new related Hall Effects, to serve the scientific community as a basis for novel research oriented to new nanoscale devices, modules, and Process Development Kit (PDK) markets.
Abstract: A comprehensive review of the main existing devices, based on the classic and new related Hall Effects is hereby presented. The review is divided into sub-categories presenting existing macro-, micro-, nanoscales, and quantum-based components and circuitry applications. Since Hall Effect-based devices use current and magnetic field as an input and voltage as output. researchers and engineers looked for decades to take advantage and integrate these devices into tiny circuitry, aiming to enable new functions such as high-speed switches, in particular at the nanoscale technology. This review paper presents not only an historical overview of past endeavors, but also the remaining challenges to overcome. As part of these trials, one can mention complex design, fabrication, and characterization of smart nanoscale devices such as sensors and amplifiers, towards the next generations of circuitry and modules in nanotechnology. When compared to previous domain-limited text books, specialized technical manuals and focused scientific reviews, all published several decades ago, this up-to-date review paper presents important advantages and novelties: Large coverage of all domains and applications, clear orientation to the nanoscale dimensions, extended bibliography of almost one hundred fifty recent references, review of selected analytical models, summary tables and phenomena schematics. Moreover, the review includes a lateral examination of the integrated Hall Effect per sub-classification of subjects. Among others, the following sub-reviews are presented: Main existing macro/micro/nanoscale devices, materials and elements used for the fabrication, analytical models, numerical complementary models and tools used for simulations, and technological challenges to overcome in order to implement the effect in nanotechnology. Such an up-to-date review may serve the scientific community as a basis for novel research oriented to new nanoscale devices, modules, and Process Development Kit (PDK) markets.
6 citations
### Cites background from "Design and Analysis of Hall Effect ..."
• ...Robotics, Office 2015, 2017 Pressure sensors [95,96] Pressure measurement, piston position in a high-pressure....
[...]
Journal ArticleDOI
TL;DR: In this article, a high-speed rotating spindle has been designed and installed to the self-developed precise micro-wire electrical discharge machining (micro-WEDM) equipment with two axis to enable the processing of complex rotary structures.
Abstract: In this paper, a method of complex rotary structures machined by micro wire electrical discharge machining (micro-WEDM) is presented. A high-speed rotating spindle has been designed and installed to the self-developed precise micro-WEDM equipment with two axis to enable the processing of complex rotary structures. The feasibility of the device for machining rotary structures is verified by performing many preliminary experiments. Meanwhile, the effects of open voltage and revolving speed on material removing rate (MRR) and average surface roughness (Ra) are investigated. A multiple cutting method for micro ball-ended probe using micro-WEDM is proposed, which includes rough machining with high machining parameters and finish machining with low machining parameters. What's more, the reuse of the reciprocated fine wire electrode with 30 μm diameter was realized in both process steps. Ultimately, a typical experimental sample, micro ball-ended probe with 97.6 μm in diameter and 0.7 μm in Ra was successfully fabricated.
5 citations
Journal ArticleDOI
TL;DR: In this article, the authors presented a smart pressure transmitter using bellow as primary sensor and the deflection of bellow is converted into electrical output using hall probe sensor as secondary sensor.
Abstract: This paper presents a smart pressure transmitter using bellow as primary sensor. The deflection of bellow is converted into electrical output using hall probe sensor as secondary sensor. The output Hall voltage is affected by change in input parameters like temperature. So firstly the effect of temperature on Hall voltage is derived mathematically and then experimentally analyzed. This effect of temperature on output Hall voltage is then compensated using artificial neural network. The compensated output Hall voltage is then converted into (4–20) mA current signal using signal conditioning circuit. The proposed design, experimental and testing results are reported in this paper.
4 citations
Journal ArticleDOI
TL;DR: In this paper, a combination of a Bellows and a Mach-Zehnder Interferometer (MZI)-based pressure transmitting system has been proposed, which is capable to transmit pressure reading to a remote location using an electro-optic modulation in the MZI.
Abstract: In process industries, an optical transmitting system has a great impact especially for long distance transmission through inflammable and hazardous areas. A combination of bellows and Mach-Zehnder Interferometer (MZI)-based pressure transmitting system has been proposed in this paper. This novel approach is capable to transmit pressure reading to a remote location using an electro-optic modulation in the MZI. The proposed system is the hybrid type and consists of two sections. First part is the electrical type which is responsible for transmission of measured information in an electrical domain (primary) and a second section is designed for transmission of measured information in an optical domain (secondary). Primary part is the combination of bellows, a Hall probe sensor, and a signal conditioning circuit, and the second part consists of the MZI with three electrodes. The pressure is directly applied to the bellows. The magnet is placed at the top of the Bellows and the variation of the magnet positions is sensed by the Hall probe sensor. The response of the Hall probe sensor, in terms of millivolt range, is not capable to perform the electro-optic modulation in the MZI, so it is further amplified by an instrumentation amplifier and a calibrating circuit. The output from the signal conditioning circuit is fed to the MZI so that the intensity of light which is propagating in the MZI is modulated due to the electro-optic effect. Transmission of the optical signal is very much beneficial for the inflammable industry to prevent the sparking like situation. The operation of the system has been explained with the help of derived theoretical equations. The experiment has been done and the performance of the system along with the experimental results is reported in this paper. The proposed system is linear and provides good repeatability with the applied pressure.
3 citations
### Cites methods from "Design and Analysis of Hall Effect ..."
• ...[16] have produced a Hall effect pressure transmitter using bellows as a primary sensor....
[...]
##### References
More filters
Book
01 Jan 1966
TL;DR: This paper aims to provide a history of ecoulement and mesures used in this discipline over a 25-year period and aims to establish a chronology of events leading up to and including the invention of EMT.
Abstract: Part 1 General Concepts 1 Types of Applications of Measurement Instrumentation 2 Generalized Configurations and Functional Descriptions of Measuring Instruments 3 Generalized Performance Characteristics of Instruments Part 2 Measuring Devices 4 Motion and Dimensional Measurement 5 Force, Torque, and Shaft Power Measurement 6 Pressure and Sound Measurement 7 Flow Measurement 8 Temperature and Heat-Flux Measurement 9 Miscellaneous Measurements Part 3 Manipulation, Transmission, and Recording of Data 10 Manipulating, Computing, and Compensating Devices 11 Data Transmission and Instrument Connectivity 12 Voltage-Indicating and -Recording Devices 13 Data-Acquisition Systems for Personal Computers 14 Measurement Systems Applied to Micro- and Nanotechnology
904 citations
### "Design and Analysis of Hall Effect ..." refers methods in this paper
• ...There are different techniques [1]–[4] available in industry to measure the absolute and gauge pressures....
[...]
• ...Secondary sensors like strain gauge, piezoelectric transducer, linear variable differential transformer and capacitive element [1]–[4] are used for that conversion....
[...]
Book
01 Jan 1983
TL;DR: In this paper, the authors present general principles of measurement systems, including reliability, choice and economics of measurement system elements, as well as the accuracy and reliability of the measurement system in the steady state.
Abstract: Part I: General Principles 1. The general measurement system. 2. Static characteristics of measurement system elements. 3. The accuracy of measurement systems in the steady state. 4. Dynamic characteristics of measurement systems. 5. Loading effects and two port networks. 6. Signals and noise in measurement systems. 7. Reliability, choice and economics of measurement systems. Part II: Typical Measurement System elements. 8. Sensing elements. 9. Signal conditioning elements. 10. Signal processing elements. 11. Data presentation elements. Part III: Speciaised Measurement Systems 12. Flow measurement systems. 13. Intrinsically safe measurement systems. 14. Heat transfer effects in measurement systems. 15. Optical measurement systems. 16. Ultrasonic measurement systems. 17. Gas chromatography. 18. Data acquisition. Answers to numerical problems. Index.
334 citations
Journal ArticleDOI
TL;DR: In this review paper, the performance (in particular the magnetic field resolution), micro-fabrication technologies and applications of micrometer sized Hall effect devices are summarized.
Abstract: In this review paper, we summarize the performance (in particular the magnetic field resolution), micro-fabrication technologies and applications of micrometer sized Hall effect devices. Additionally, our activities in this domain are briefly described.
141 citations
Book
01 Jan 2003
TL;DR: In this article, the authors present a detailed description of the characteristics of a flowmetering system and its application in a variety of applications, including the following: anemometers BTU Flowmeters for Heat Exchangers BTUs for Gaseous Fuels Cross-Correlation Flow Metering Elbow Taps Flow Switches Jet Deflection Flow Detectors Laminar Flow Meters, Magnetic FlowMeters, Coriolis Mass Flow-meters-Miscellaneous Mass Flowmetmers-Thermal Metering Pumps Orifices Pitot Tubes and
Abstract: GENERAL CONSIDERATIONS Flowsheet Symbols and P&I Diagrams Functional Diagrams and Function Symbols Instrument Terminology and Performance System Accuracy Uncertainty Calculations Configuring Intelligent Devices Instrument Installation Instrument Calibration Response Time and Drift Testing Redundant and Voting Systems Instrument Evaluation Binary Logic Diagrams FLOW MEASUREMENT Application and Selection Anemometers BTU Flowmeters for Heat Exchangers BTU Flowmeters for Gaseous Fuels Cross-Correlation Flow Metering Elbow Taps Flow Switches Jet Deflection Flow Detectors Laminar Flowmeters Magnetic Flowmeters Mass Flowmeters, Coriolis Mass Flowmeters-Miscellaneous Mass Flowmeters-Thermal Metering Pumps Orifices Pitot Tubes and Area Averaging Units Polyphase (Oil/Water/Gas) Flowmeters Positive-Displacement Gas Flowmeters Positive-Displacement Liquid Meters and Provers Purge Flow Regulators Segmental Wedge Flowmeter Sight Flow Indicators Solids Flowmeters and Feeders Target Meters Turbine and Other Rotary Element Flowmeters Ultrasonic Flowmeters Variable-Area, Gap, and Vane Flowmeters V-Cone Flowmeter Venturi Tubes, Flow Tubes, and Flow Nozzles Vortex and Fluidic Flowmeters Weirs and Flumes LEVEL MEASUREMENT Application and Selection Bubblers Capacitance and Radio Frequency (RF) Admittance Probes Conductivity and Field Effect Level Switches Diaphragm Level Detectors Differential Pressure Level Detectors Displacer Level Detectors Float Level Devices Laser Level Sensors Level Gauges, Including Magnetic Microwave Level Switches Optical Level Devices Radar, Noncontacting Level Sensors Radar, Contact Level Sensors (TDR, GWR, PDS) Radiation Level Sensors Resistance Tapes Rotating Paddle Switches Tank Gauges Including Float-Type Tape Gauges Thermal Level Sensors Time Domain Reflectometry and Phase Difference Sensors Ultrasonic Level Detectors Vibrating Level Switches TEMPERATURE MEASUREMENT Application and Selection Bimetallic Thermometers Calibrators and Simulators Color Indicators, Crayons, Pellets Fiber-Optic Thermometers Filled-Bulb and Glass-Stem Thermometers Integrated Circuitry (IC) Transistors and Diodes Miscellaneous Temperature Sensors Pneumatic and Suction Pyrometers Pyrometric Cones Radiation and Infrared Pyrometers Quartz Crystal Thermometry Resistance Temperature Detectors (RTDs) Temperature Switches and Thermostats Thermistors Thermocouples Thermowells Ultrasonic Thermometers PRESSURE MEASUREMENT Selection and Application Accessories: Seals, Snubbers, Calibrators, and Manifolds Bellows-Type Pressure Sensors Bourdon and Helical Pressure Sensors Diaphragm or Capsule-Type Sensors Differential Pressure Instruments Electronic Pressure Sensors High-Pressure Sensors Manometers Multiple Pressure Scanners Multiple Pressure Scanners Pressure Gauges Pressure Repeaters Pressure and Differential Pressure Switches Vacuum Sensors DENSITY MEASUREMENT Density: Applications and Selection Displacement- and Float-Type Densitometers Hydrometers Hydrostatic Densitometers Oscillating Coriolis Densitometer (Gas, Liquid, and Slurry Services) Radiation Densitometers Ultrasonic Sludge and Slurry Densitometers Liquid/Slurry/Gas Density-Vibrating Densitometers Weight-Based and Miscellaneous Densitometers Gas Densitometers SAFETY AND MISCELLANEOUS SENSORS Boroscopes Electrical and Intrinsic Safety Electrical Meters and Sensors Energy Management Devices (Peak Load Shedding) Excess Flow and Regular Check Valves Explosion Suppression and Deluge Systems Flame Arresters, Conservation Vents, and Emergency Vents Flame, Fire, and Smoke Detectors Leak Detectors Linear and Angular Position Detection Machine Vision Technology Metal Detectors Noise Sensors Proximity Sensors and Limit Switches Relief Valves-Determination of Required Capacity Relief Valves-Sizing, Specification, and Installation Rupture Discs Soft Sensors Tachometers and Angular Speed Detectors Thickness and Dimension Measurement Torque and Force Transducers Vibration, Shock, and Acceleration Weather Stations Weighing Systems: General Considerations Weight Sensors ANALYTICAL INSTRUMENTATION Analyzer Application and Selection Analyzer Sampling: Process Samples Analyzer Sampling: Stack Particulates Analyzers Operating on Electrochemical Principles Air Quality Monitoring Biometers Biochemical Oxygen Demand, Chemical Oxygen Demand, and Total Oxygen Demand Calorimeters Carbon Dioxide Carbon Monoxide Chlorine Chromatographs: Gas Chromatographs: Liquid Coal Analyzers Colorimeters Combustibles Conductivity Analyzers Consistency Analyzers Corrosion Monitoring Differential Vapor Pressure Sensor Dioxin Analysis Elemental Monitors Fiber-Optic Probes Fluoride Analyzers Hydrocarbon Analyzers Hydrogen Sulfide Infrared Analyzers Ion-Selective Electrodes Mass Spectrometers Mercury in Air Mercury in Water Moisture in Air: Humidity and Dew Point Moisture in Gases and Liquids Moisture in Solids Molecular Weight Nitrate, Ammonia, and Total Nitrogen Nitrogen Oxide Analyzers Odor Detection Oil in or on Water Open Path Spectrometry Oxidation-Reduction Potential (ORP) Oxygen in Gases Oxygen in Liquids (Dissolved Oxygen) Ozone in Gas Ozone in Water Particulates, Opacity, Dust, and Smoke Particle Size and Distribution Monitors pH Measurement Phosphorus Analyzer Physical Properties Analyzers - ASTM Methods Raman Analyzers Refractometers Rheometers Streaming Current or Particle Charge Analyzer Sulfur-in-Oil Analyzers Sulfur Oxide Analyzers Thermal Conductivity Detectors Total Carbon Analyzers Toxic Gas Monitoring Turbidity, Sludge, and Suspended Solids Ultraviolet and Visible Analyzers Viscometers-Application and Selection Viscometers-Laboratory Viscometers-Industrial Water Quality Monitoring Wet Chemistry and Autotitrator Analyzers APPENDIX International System of Units Engineering Conversion Factors Chemical Resistance of Materials Composition of Metallic and Other Materials Steam and Water Tables Friction Loss in Pipes Tank Volumes Directory of "Lost" Companies INDEX
91 citations
### "Design and Analysis of Hall Effect ..." refers methods in this paper
• ...There are different techniques [1]–[4] available in industry to measure the absolute and gauge pressures....
[...]
• ...Secondary sensors like strain gauge, piezoelectric transducer, linear variable differential transformer and capacitive element [1]–[4] are used for that conversion....
[...]
Book
01 Oct 1999
TL;DR: The "Process Instruments and Controls Handbook" as discussed by the authors has been used by designers, engineers, and technicians for dealing with process design problems commonly encountered in the chemical process industries (CPI) and has been expanded to include detailed coverage of instruments and control devices as they are used in the manufacturing industries.
Abstract: Through three editions, "Process Instruments and Controls Handbook" has set the standard for authoritative information on instruments and controls used in the chemical process industries (CPI). Broad in scope, thorough, and practical, it has been used by designers, engineers, and technicians for dealing with process design problems commonly encountered in the CPI. The new fourth edition, now titled "Process/Industrial Instruments and Controls Handbook", has been expanded to include detailed coverage of instruments and control devices as they are used in the manufacturing industries. Extensively revised and updated, this monumental handbook reflects all of the latest developments on: control and data objectives; control and data systems, discrete-piece manufacturing control, and data acquisition and communication system architectures; measurement, sensors, transducers and transmitters; automatic control fundamentals and theory; computing elements in control settings; valves, actuators, and manipulators in control systems; operators' control system interface; control system design, selection, and specification. This reference source of measurement and control data should help design, control, instrumentation, process, manufacturing, and plant engineers design a system right the very first time and avoid costly downtime.
78 citations
|
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