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Topological monoid
In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's binary operation is continuous. Every topological group is a topological monoid.
See also
• H-space
References
• Steenrod, N.E. (1968). "Milgram's classifying space of a topological group". Topology. 7 (4): 349–368. doi:10.1016/0040-9383(68)90012-8. ISSN 0040-9383.
• Fiedorowicz, Z. (1984). "Classifying Spaces of Topological Monoids and Categories". American Journal of Mathematics. 106 (2): 301. doi:10.2307/2374307. ISSN 0002-9327.
External links
• topological monoid from symmetric monoidal category
| Wikipedia |
Product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.
Definition
Throughout, $I$ will be some non-empty index set and for every index $i\in I,$ let $X_{i}$ be a topological space. Denote the Cartesian product of the sets $X_{i}$ by
$X:=\prod X_{\bullet }:=\prod _{i\in I}X_{i}$
and for every index $i\in I,$ denote the $i$-th canonical projection by
${\begin{alignedat}{4}p_{i}:\;&&\prod _{j\in I}X_{j}&&\;\to \;&X_{i}\\[0.3ex]&&\left(x_{j}\right)_{j\in I}&&\;\mapsto \;&x_{i}\\\end{alignedat}}$
The product topology, sometimes called the Tychonoff topology, on $ \prod _{i\in I}X_{i}$ is defined to be the coarsest topology (that is, the topology with the fewest open sets) for which all the projections $ p_{i}:\prod X_{\bullet }\to X_{i}$ are continuous. The Cartesian product $ X:=\prod _{i\in I}X_{i}$ endowed with the product topology is called the product space. The open sets in the product topology are arbitrary unions (finite or infinite) of sets of the form $ \prod _{i\in I}U_{i},$ where each $U_{i}$ is open in $X_{i}$ and $U_{i}\neq X_{i}$ for only finitely many $i.$ In particular, for a finite product (in particular, for the product of two topological spaces), the set of all Cartesian products between one basis element from each $X_{i}$ gives a basis for the product topology of $ \prod _{i\in I}X_{i}.$ That is, for a finite product, the set of all $ \prod _{i\in I}U_{i},$ where $U_{i}$ is an element of the (chosen) basis of $X_{i},$ is a basis for the product topology of $ \prod _{i\in I}X_{i}.$
The product topology on $ \prod _{i\in I}X_{i}$ is the topology generated by sets of the form $p_{i}^{-1}\left(U_{i}\right),$ where $i\in I$ and $U_{i}$ is an open subset of $X_{i}.$ In other words, the sets
$\left\{p_{i}^{-1}\left(U_{i}\right)~:~i\in I{\text{ and }}U_{i}\subseteq X_{i}{\text{ is open in }}X_{i}\right\}$
form a subbase for the topology on $X.$ A subset of $X$ is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form $p_{i}^{-1}\left(U_{i}\right).$ The $p_{i}^{-1}\left(U_{i}\right)$ are sometimes called open cylinders, and their intersections are cylinder sets.
The product topology is also called the topology of pointwise convergence because a sequence (or more generally, a net) in $ \prod _{i\in I}X_{i}$ converges if and only if all its projections to the spaces $X_{i}$ converge. Explicitly, a sequence $ s_{\bullet }=\left(s_{n}\right)_{n=1}^{\infty }$ (respectively, a net $ s_{\bullet }=\left(s_{a}\right)_{a\in A}$) converges to a given point $ x\in \prod _{i\in I}X_{i}$ if and only if $p_{i}\left(s_{\bullet }\right)\to p_{i}(x)$ in $X_{i}$ for every index $i\in I,$ where $p_{i}\left(s_{\bullet }\right):=p_{i}\circ s_{\bullet }$ denotes $\left(p_{i}\left(s_{n}\right)\right)_{n=1}^{\infty }$ (respectively, denotes $\left(p_{i}\left(s_{a}\right)\right)_{a\in A}$). In particular, if $X_{i}=\mathbb {R} $ is used for all $i$ then the Cartesian product is the space $ \prod _{i\in I}\mathbb {R} =\mathbb {R} ^{I}$ of all real-valued functions on $I,$ and convergence in the product topology is the same as pointwise convergence of functions.
Examples
If the real line $\mathbb {R} $ is endowed with its standard topology then the product topology on the product of $n$ copies of $\mathbb {R} $ is equal to the ordinary Euclidean topology on $\mathbb {R} ^{n}.$ (Because $n$ is finite, this is also equivalent to the box topology on $\mathbb {R} ^{n}.$)
The Cantor set is homeomorphic to the product of countably many copies of the discrete space $\{0,1\}$ and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology.
Several additional examples are given in the article on the initial topology.
Properties
The set of Cartesian products between the open sets of the topologies of each $X_{i}$ forms a basis for what is called the box topology on $X.$ In general, the box topology is finer than the product topology, but for finite products they coincide.
The product space $X,$ together with the canonical projections, can be characterized by the following universal property: if $Y$ is a topological space, and for every $i\in I,$ $f_{i}:Y\to X_{i}$ is a continuous map, then there exists precisely one continuous map $f:Y\to X$ such that for each $i\in I$ the following diagram commutes.
This shows that the product space is a product in the category of topological spaces. It follows from the above universal property that a map $f:Y\to X$ is continuous if and only if $f_{i}=p_{i}\circ f$ is continuous for all $i\in I.$ In many cases it is easier to check that the component functions $f_{i}$ are continuous. Checking whether a map $X\to Y$ is continuous is usually more difficult; one tries to use the fact that the $p_{i}$ are continuous in some way.
In addition to being continuous, the canonical projections $p_{i}:X\to X_{i}$ are open maps. This means that any open subset of the product space remains open when projected down to the $X_{i}.$ The converse is not true: if $W$ is a subspace of the product space whose projections down to all the $X_{i}$ are open, then $W$ need not be open in $X$ (consider for instance $ W=\mathbb {R} ^{2}\setminus (0,1)^{2}.$) The canonical projections are not generally closed maps (consider for example the closed set $ \left\{(x,y)\in \mathbb {R} ^{2}:xy=1\right\},$ whose projections onto both axes are $\mathbb {R} \setminus \{0\}$).
Suppose $ \prod _{i\in I}S_{i}$ is a product of arbitrary subsets, where $S_{i}\subseteq X_{i}$ for every $i\in I.$ If all $S_{i}$ are non-empty then $ \prod _{i\in I}S_{i}$ is a closed subset of the product space $X$ if and only if every $S_{i}$ is a closed subset of $X_{i}.$ More generally, the closure of the product $ \prod _{i\in I}S_{i}$ of arbitrary subsets in the product space $X$ is equal to the product of the closures:[1]
$\operatorname {Cl} _{X}\left(\prod _{i\in I}S_{i}\right)=\prod _{i\in I}\left(\operatorname {Cl} _{X_{i}}S_{i}\right).$
Any product of Hausdorff spaces is again a Hausdorff space.
Tychonoff's theorem, which is equivalent to the axiom of choice, states that any product of compact spaces is a compact space. A specialization of Tychonoff's theorem that requires only the ultrafilter lemma (and not the full strength of the axiom of choice) states that any product of compact Hausdorff spaces is a compact space.
If $ z=\left(z_{i}\right)_{i\in I}\in X$ is fixed then the set
$\left\{x=\left(x_{i}\right)_{i\in I}\in X\colon x_{i}=z_{i}{\text{ for all except at most finitely many }}i\right\}$
is a dense subset of the product space $X$.[1]
Relation to other topological notions
Separation
• Every product of T0 spaces is T0.
• Every product of T1 spaces is T1.
• Every product of Hausdorff spaces is Hausdorff.
• Every product of regular spaces is regular.
• Every product of Tychonoff spaces is Tychonoff.
• A product of normal spaces need not be normal.
Compactness
• Every product of compact spaces is compact (Tychonoff's theorem).
• A product of locally compact spaces need not be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact is locally compact (This condition is sufficient and necessary).
Connectedness
• Every product of connected (resp. path-connected) spaces is connected (resp. path-connected).
• Every product of hereditarily disconnected spaces is hereditarily disconnected.
Metric spaces
• Countable products of metric spaces are metrizable spaces.
Axiom of choice
One of many ways to express the axiom of choice is to say that it is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.[2] The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.
The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that requires the axiom of choice and is equivalent to it in its most general formulation,[3] and shows why the product topology may be considered the more useful topology to put on a Cartesian product.
See also
• Disjoint union (topology) – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topologyPages displaying wikidata descriptions as a fallback
• Final topology – Finest topology making some functions continuous
• Initial topology – Coarsest topology making certain functions continuous - Sometimes called the projective limit topology
• Inverse limit – Construction in category theory
• Pointwise convergence – A notion of convergence in mathematics
• Quotient space (topology) – Topological space construction
• Subspace (topology) – Inherited topologyPages displaying short descriptions of redirect targets
• Weak topology – Mathematical term
Notes
1. Bourbaki 1989, pp. 43–50.
2. Pervin, William J. (1964), Foundations of General Topology, Academic Press, p. 33
3. Hocking, John G.; Young, Gail S. (1988) [1961], Topology, Dover, p. 28, ISBN 978-0-486-65676-2
References
• Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
• Willard, Stephen (1970). General Topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0486434796. Retrieved 13 February 2013.
| Wikipedia |
Topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them.
Properties of topological properties
A property $P$ is:
• Hereditary, if for every topological space $(X,{\mathcal {T}})$ and subset $S\subseteq X,$ the subspace $\left(S,{\mathcal {T}}|_{S}\right)$ has property $P.$
• Weakly hereditary, if for every topological space $(X,{\mathcal {T}})$ and closed subset $S\subseteq X,$ the subspace $\left(S,{\mathcal {T}}|_{S}\right)$ has property $P.$
Common topological properties
Cardinal functions
Main article: Cardinal function § Cardinal functions in topology
• The cardinality |X| of the space X.
• The cardinality $\vert $τ(X)$\vert $ of the topology (the set of open subsets) of the space X.
• Weight w(X), the least cardinality of a basis of the topology of the space X.
• Density d(X), the least cardinality of a subset of X whose closure is X.
Separation
Main article: Separation axiom
Note that some of these terms are defined differently in older mathematical literature; see history of the separation axioms.
• T0 or Kolmogorov. A space is Kolmogorov if for every pair of distinct points x and y in the space, there is at least either an open set containing x but not y, or an open set containing y but not x.
• T1 or Fréchet. A space is Fréchet if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its singletons are closed. T1 spaces are always T0.
• Sober. A space is sober if every irreducible closed set C has a unique generic point p. In other words, if C is not the (possibly nondisjoint) union of two smaller closed non-empty subsets, then there is a p such that the closure of {p} equals C, and p is the only point with this property.
• T2 or Hausdorff. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. T2 spaces are always T1.
• T2½ or Urysohn. A space is Urysohn if every two distinct points have disjoint closed neighbourhoods. T2½ spaces are always T2.
• Completely T2 or completely Hausdorff. A space is completely T2 if every two distinct points are separated by a function. Every completely Hausdorff space is Urysohn.
• Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
• T3 or Regular Hausdorff. A space is regular Hausdorff if it is a regular T0 space. (A regular space is Hausdorff if and only if it is T0, so the terminology is consistent.)
• Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are separated by a function.
• T3½, Tychonoff, Completely regular Hausdorff or Completely T3. A Tychonoff space is a completely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
• Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.
• T4 or Normal Hausdorff. A normal space is Hausdorff if and only if it is T1. Normal Hausdorff spaces are always Tychonoff.
• Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods.
• T5 or Completely normal Hausdorff. A completely normal space is Hausdorff if and only if it is T1. Completely normal Hausdorff spaces are always normal Hausdorff.
• Perfectly normal. A space is perfectly normal if any two disjoint closed sets are precisely separated by a function. A perfectly normal space must also be completely normal.
• T6 or Perfectly normal Hausdorff, or perfectly T4. A space is perfectly normal Hausdorff, if it is both perfectly normal and T1. A perfectly normal Hausdorff space must also be completely normal Hausdorff.
• Discrete space. A space is discrete if all of its points are completely isolated, i.e. if any subset is open.
• Number of isolated points. The number of isolated points of a topological space.
Countability conditions
See also: Axiom of countability
• Separable. A space is separable if it has a countable dense subset.
• First-countable. A space is first-countable if every point has a countable local base.
• Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.
Connectedness
• Connected. A space is connected if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only clopen sets are the empty set and itself.
• Locally connected. A space is locally connected if every point has a local base consisting of connected sets.
• Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.
• Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected.
• Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.
• Arc-connected. A space X is arc-connected if for every two points x, y in X, there is an arc f from x to y, i.e., an injective continuous map f: [0,1] → X with p(0) = x and p(1) = y. Arc-connected spaces are path-connected.
• Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S1 → X is homotopic to a constant map.
• Locally simply connected. A space X is locally simply connected if every point x in X has a local base of neighborhoods U that is simply connected.
• Semi-locally simply connected. A space X is semi-locally simply connected if every point has a local base of neighborhoods U such that every loop in U is contractible in X. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a universal cover.
• Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.
• Hyperconnected. A space is hyperconnected if no two non-empty open sets are disjoint. Every hyperconnected space is connected.
• Ultraconnected. A space is ultraconnected if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected.
• Indiscrete or trivial. A space is indiscrete if the only open sets are the empty set and itself. Such a space is said to have the trivial topology.
Compactness
• Compact. A space is compact if every open cover has a finite subcover. Some authors call these spaces quasicompact and reserve compact for Hausdorff spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.
• Sequentially compact. A space is sequentially compact if every sequence has a convergent subsequence.
• Countably compact. A space is countably compact if every countable open cover has a finite subcover.
• Pseudocompact. A space is pseudocompact if every continuous real-valued function on the space is bounded.
• σ-compact. A space is σ-compact if it is the union of countably many compact subsets.
• Lindelöf. A space is Lindelöf if every open cover has a countable subcover.
• Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.
• Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff.
• Ultraconnected compact. In an ultra-connected compact space X every open cover must contain X itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a monolith.
Metrizability
• Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover, a topological space (X,T) is said to be metrizable if there exists a metric for X such that the metric topology T(d) is identical with the topology T.
• Polish. A space is called Polish if it is metrizable with a separable and complete metric.
• Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.
Miscellaneous
• Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.
• Door space. A topological space is a door space if every subset is open or closed (or both).
• Topological Homogeneity. A space X is (topologically) homogeneous if for every x and y in X there is a homeomorphism $f:X\to X$ such that $f(x)=y.$ Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.
• Finitely generated or Alexandrov. A space X is Alexandrov if arbitrary intersections of open sets in X are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the finitely generated members of the category of topological spaces and continuous maps.
• Zero-dimensional. A space is zero-dimensional if it has a base of clopen sets. These are precisely the spaces with a small inductive dimension of 0.
• Almost discrete. A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
• Boolean. A space is Boolean if it is zero-dimensional, compact and Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the Stone spaces of Boolean algebras.
• Reidemeister torsion
• $\kappa $-resolvable. A space is said to be κ-resolvable[1] (respectively: almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (respectively: almost disjoint over the ideal of nowhere dense subsets). If the space is not $\kappa $-resolvable then it is called $\kappa $-irresolvable.
• Maximally resolvable. Space $X$ is maximally resolvable if it is $\Delta (X)$-resolvable, where $\Delta (X)=\min\{|G|:G\neq \varnothing ,G{\mbox{ is open}}\}.$ Number $\Delta (X)$ is called dispersion character of $X.$
• Strongly discrete. Set $D$ is strongly discrete subset of the space $X$ if the points in $D$ may be separated by pairwise disjoint neighborhoods. Space $X$ is said to be strongly discrete if every non-isolated point of $X$ is the accumulation point of some strongly discrete set.
Non-topological properties
There are many examples of properties of metric spaces, etc, which are not topological properties. To show a property $P$ is not topological, it is sufficient to find two homeomorphic topological spaces $X\cong Y$ such that $X$ has $P$, but $Y$ does not have $P$.
For example, the metric space properties of boundedness and completeness are not topological properties. Let $X=\mathbb {R} $ and $Y=(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}})$ be metric spaces with the standard metric. Then, $X\cong Y$ via the homeomorphism $\operatorname {arctan} \colon X\to Y$. However, $X$ is complete but not bounded, while $Y$ is bounded but not complete.
See also
• Characteristic class – Association of cohomology classes to principal bundles
• Characteristic numbers – DimensionPages displaying short descriptions of redirect targetsPages displaying short descriptions with no spaces
• Chern class – Characteristic classes on algebraic vector bundles
• Euler characteristic – Topological invariant in mathematics
• Fixed-point property – Mathematical property
• Homology and cohomology
• Homotopy group and Cohomotopy group
• Knot invariant – Function of a knot that takes the same value for equivalent knots
• Linking number – Numerical invariant that describes the linking of two closed curves in three-dimensional space
• List of topologies – List of concrete topologies and topological spaces
• Quantum invariant – Concept in mathematical knot theory
• Topological quantum number – Physical quantities that take discrete values because of topological quantum physical effects
• Winding number – Number of times a curve wraps around a point in the plane
Citations
1. Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán (2008). "Resolvability and monotone normality". Israel Journal of Mathematics. 166 (1): 1–16. arXiv:math/0609092. doi:10.1007/s11856-008-1017-y. ISSN 0021-2172. S2CID 14743623.
References
• Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. p. 369. ISBN 9780486434797.
• Munkres, James R. (2000). Topology. Prentice-Hall. ISBN 0-13-181629-2.
[2] Simon Moulieras, Maciej Lewenstein and Graciana Puentes, Entanglement engineering and topological protection by discrete-time quantum walks, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (10), 104005 (2013). https://iopscience.iop.org/article/10.1088/0953-4075/46/10/104005/pdf
| Wikipedia |
Topological quantum computer
A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997.[1] It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an ordinary quantum particle in four-dimensional spacetime) bumping into a wall.
While the elements of a topological quantum computer originate in a purely mathematical realm, experiments in fractional quantum Hall systems indicate these elements may be created in the real world using semiconductors made of gallium arsenide at a temperature of near absolute zero and subjected to strong magnetic fields.
Introduction
Anyons are quasiparticles in a two-dimensional space. Anyons are neither fermions nor bosons, but like fermions, they cannot occupy the same state. Thus, the world lines of two anyons cannot intersect or merge, which allows their paths to form stable braids in space-time. Anyons can form from excitations in a cold, two-dimensional electron gas in a very strong magnetic field, and carry fractional units of magnetic flux. This phenomenon is called the fractional quantum Hall effect. In typical laboratory systems, the electron gas occupies a thin semiconducting layer sandwiched between layers of aluminium gallium arsenide.
When anyons are braided, the transformation of the quantum state of the system depends only on the topological class of the anyons' trajectories (which are classified according to the braid group). Therefore, the quantum information which is stored in the state of the system is impervious to small errors in the trajectories.[2] In 2005, Sankar Das Sarma, Michael Freedman, and Chetan Nayak proposed a quantum Hall device that would realize a topological qubit. In 2005 Vladimir J. Goldman, Fernando E. Camino, and Wei Zhou[3] claimed to have created and observed the first experimental evidence for using a fractional quantum Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena not involving anyons. Non-abelian anyons, a species required for topological quantum computers, have yet to be experimentally confirmed. Possible experimental evidence has been found,[4] but the conclusions remain contested.[5] In 2018, scientists again claimed to have isolated the required Majorana particles, but the finding was retracted in 2021. Quanta Magazine stated in 2021 that "no one has convincingly shown the existence of even a single (Majorana zero-mode) quasiparticle",[6] although in 2023 a new article[7] by the magazine has covered some preprints by Google[8] and Quantinuum[9] claiming the realization of non-abelian anyons on quantum processors, the first used a toric code with twist defects as a topological degenerancy (or topological defect) while the second used a different but related protocol both of which can be understood as Majorana bound states in quantum error correction
Topological vs. standard quantum computer
Topological quantum computers are equivalent in computational power to other standard models of quantum computation, in particular to the quantum circuit model and to the quantum Turing machine model.[10] That is, any of these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.
Computations
To live up to its name, a topological quantum computer must provide the unique computation properties promised by a conventional quantum computer design, which uses trapped quantum particles. In 2000, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang proved that a topological quantum computer can, in principle, perform any computation that a conventional quantum computer can do, and vice versa.[10][11][12]
They found that a conventional quantum computer device, given an error-free operation of its logic circuits, will give a solution with an absolute level of accuracy, whereas a topological quantum computing device with flawless operation will give the solution with only a finite level of accuracy. However, any level of precision for the answer can be obtained by adding more braid twists (logic circuits) to the topological quantum computer, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer. Actual computation [gates] are done by the edge states of a fractional quantum Hall effect. This makes models of one-dimensional anyons important. In one space dimension, anyons are defined algebraically.
Error correction and control
Even though quantum braids are inherently more stable than trapped quantum particles, there is still a need to control for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with adjoining braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering strays drops to near zero. Simulating the dynamics of a topological quantum computer may be a promising method of implementing fault-tolerant quantum computation even with a standard quantum information processing scheme. Raussendorf, Harrington, and Goyal have studied one model, with promising simulation results.[13]
Example: Computing with Fibonacci anyons
One of the prominent examples in topological quantum computing is with a system of Fibonacci anyons. In the context of conformal field theory, fibonacci anyons are described by the Yang–Lee model, the SU(2) special case of the Chern–Simons theory and Wess–Zumino–Witten models.[14] These anyons can be used to create generic gates for topological quantum computing. There are three main steps for creating a model:
• Choose our basis and restrict our Hilbert space
• Braid the anyons together
• Fuse the anyons at the end, and detect how they fuse in order to read the output of the system.
State preparation
Fibonacci anyons are defined by three qualities:
1. They have a topological charge of $\tau $. In this discussion, we consider another charge called $1$ which is the ‘vacuum’ charge if anyons are annihilated with each-other.
2. Each of these anyons are their own antiparticle. $\tau =\tau ^{*}$ and $1=1^{*}$.
3. If brought close to each-other, they will ‘fuse’ together in a nontrivial fashion. Specifically, the ‘fusion’ rules are:
1. $1\otimes 1=1$
2. $1\otimes \tau =\tau \otimes 1=\tau $
3. $\tau \otimes \tau =1\oplus \tau $
4. Many of the properties of this system can be explained similarly to that of two spin 1/2 particles. Particularly, we use the same tensor product $\otimes $ and direct sum $\oplus $ operators.
The last ‘fusion’ rule can be extended this to a system of three anyons:
$\tau \otimes \tau \otimes \tau =\tau \otimes (1\oplus \tau )=\tau \otimes 1\oplus \tau \otimes \tau =\tau \oplus 1\oplus \tau =1\oplus 2\cdot \tau $
Thus, fusing three anyons will yield a final state of total charge $\tau $ in 2 ways, or a charge of $1$ in exactly one way. We use three states to define our basis.[15] However, because we wish to encode these three anyon states as superpositions of 0 and 1, we need to limit the basis to a two-dimensional Hilbert space. Thus, we consider only two states with a total charge of $\tau $. This choice is purely phenomenological. In these states, we group the two leftmost anyons into a 'control group', and leave the rightmost as a 'non-computational anyon'. We classify a $|0\rangle $ state as one where the control group has total 'fused' charge of $1$, and a state of $|1\rangle $ has a control group with a total 'fused' charge of $\tau $. For a more complete description, see Nayak.[15]
Gates
Following the ideas above, adiabatically braiding these anyons around each-other will result in a unitary transformation. These braid operators are a result of two subclasses of operators:
• The F matrix
• The R matrix
The R matrix can be conceptually thought of as the topological phase that is imparted onto the anyons during the braid. As the anyons wind around each-other, they pick up some phase due to the Aharonov–Bohm effect.
The F matrix is a result of the physical rotations of the anyons. As they braid between each-other, it is important to realize that the bottom two anyons—the control group—will still distinguish the state of the qubit. Thus, braiding the anyons will change which anyons are in the control group, and therefore change the basis. We evaluate the anyons by always fusing the control group (the bottom anyons) together first, so exchanging which anyons these are will rotate the system. Because these anyons are non-abelian, the order of the anyons (which ones are within the control group) will matter, and as such they will transform the system.
The complete braid operator can be derived as:
$B=F^{-1}RF$
In order to mathematically construct the F and R operators, we can consider permutations of these F and R operators. We know that if we sequentially change the basis that we are operating on, this will eventually lead us back to the same basis. Similarly, we know that if we braid anyons around each-other a certain number of times, this will lead back to the same state. These axioms are called the pentagonal and hexagonal axioms respectively as performing the operation can be visualized with a pentagon/hexagon of state transformations. Although mathematically difficult,[16] these can be approached much more successfully visually.
With these braid operators, we can finally formalize the notion of braids in terms of how they act on our Hilbert space and construct arbitrary universal quantum gates.[17]
See also
• Topological order
• Symmetry-protected topological order
• Ginzburg–Landau theory
• Husimi Q representation
• Random matrix
References
1. Kitaev, Alexei (9 Jul 1997). "Fault-tolerant quantum computation by anyons". Annals of Physics. 303: 2–30. arXiv:quant-ph/9707021v1. doi:10.1016/S0003-4916(02)00018-0. S2CID 11199664.
2. Castelvecchi, Davide (July 3, 2020). "Welcome anyons! Physicists find best evidence yet for long-sought 2D structures". Nature. 583 (7815): 176–177. Bibcode:2020Natur.583..176C. doi:10.1038/d41586-020-01988-0. PMID 32620884. S2CID 220336025. Simon and others have developed elaborate theories that use anyons as the platform for quantum computers. Pairs of the quasiparticle could encode information in their memory of how they have circled around one another. And because the fractional statistics is 'topological' — it depends on the number of times one anyon went around another, and not on slight changes to its path — it is unaffected by tiny perturbations. This robustness could make topological quantum computers easier to scale up than are current quantum-computing technologies, which are error-prone.
3. Camino, Fernando E.; Zhou, Wei; Goldman, Vladimir J. (December 6, 2005). "Aharonov–Bohm superperiod in a Laughlin quasiparticle interferometer". Phys. Rev. Lett. 95 (24): 246802. arXiv:cond-mat/0504341. Bibcode:2005PhRvL..95x6802C. doi:10.1103/PhysRevLett.95.246802. PMID 16384405.
4. Willet, R. L. (January 15, 2013). "Magnetic field-tuned Aharonov–Bohm oscillations and evidence for non-Abelian anyons at ν = 5/2". Physical Review Letters. 111 (18): 186401. arXiv:1301.2639. Bibcode:2013PhRvL.111r6401W. doi:10.1103/PhysRevLett.111.186401. PMID 24237543. S2CID 22780228.
5. von Keyserling, Curt; Simon, S. H.; Bernd, Rosenow (2015). "Enhanced Bulk-Edge Coulomb Coupling in Fractional Fabry-Perot Interferometers". Physical Review Letters. 115 (12): 126807. arXiv:1411.4654. Bibcode:2015PhRvL.115l6807V. doi:10.1103/PhysRevLett.115.126807. PMID 26431008. S2CID 20103218.
6. Ball, Philip (29 September 2021). "Major Quantum Computing Strategy Suffers Serious Setbacks". Quanta Magazine. Retrieved 30 September 2021.
7. Wood, Charlie (9 May 2023). "Physicists Create Elusive Particles That Remember Their Pasts". Quanta Magazine.
8. Andersen, Trond and more (9 Oct 2023). "Observation of non-Abelian exchange statistics on a superconducting processor". arXiv:2210.10255. {{cite journal}}: Cite journal requires |journal= (help)
9. Iqbal, Mohsin and more (5 May 2023). "Creation of Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor". arXiv:2305.03766. {{cite journal}}: Cite journal requires |journal= (help)
10. Freedman, Michael H.; Larsen, Michael; Wang, Zhenghan (2002-06-01). "A Modular Functor Which is Universal for Quantum Computation". Communications in Mathematical Physics. 227 (3): 605–622. arXiv:quant-ph/0001108. Bibcode:2002CMaPh.227..605F. doi:10.1007/s002200200645. ISSN 0010-3616. S2CID 8990600.
11. Freedman, Michael H.; Kitaev, Alexei; Wang, Zhenghan (2002-06-01). "Simulation of Topological Field Theories by Quantum Computers". Communications in Mathematical Physics. 227 (3): 587–603. arXiv:quant-ph/0001071. Bibcode:2002CMaPh.227..587F. doi:10.1007/s002200200635. ISSN 0010-3616. S2CID 449219.
12. Freedman, Michael; Kitaev, Alexei; Larsen, Michael; Wang, Zhenghan (2003-01-01). "Topological quantum computation". Bulletin of the American Mathematical Society. 40 (1): 31–38. arXiv:quant-ph/0101025. doi:10.1090/S0273-0979-02-00964-3. ISSN 0273-0979.
13. Raussendorf, R.; Harrington, J.; Goyal, K. (2007-01-01). "Topological fault-tolerance in cluster state quantum computation". New Journal of Physics. 9 (6): 199. arXiv:quant-ph/0703143. Bibcode:2007NJPh....9..199R. doi:10.1088/1367-2630/9/6/199. ISSN 1367-2630. S2CID 13811487.
14. Trebst, Simon; Troyer, Matthias; Wang, Zhenghan; Ludwig, Andreas W. W. (2008). "A Short Introduction to Fibonacci Anyon Models". Progress of Theoretical Physics Supplement. 176: 384–407. arXiv:0902.3275. Bibcode:2008PThPS.176..384T. doi:10.1143/PTPS.176.384. S2CID 16880657.
15. Nayak, Chetan (2008). "Non-Abelian Anyons and Topological Quantum Computation". Reviews of Modern Physics. 80 (3): 1083–1159. arXiv:0707.1889. Bibcode:2008RvMP...80.1083N. doi:10.1103/RevModPhys.80.1083. S2CID 119628297.
16. Eric Paquette. Topological quantum computing with anyons, 1 2009. Categories, Logic and Foundations of Physics IV.
17. Explicit braids that perform particular quantum computations with Fibonacci anyons have been given by Bonesteel, N. E.; Hormozi, L.; Zikos, G.; Simon, S. H.; West, K. W. (2005). "Braid Topologies for Quantum Computation". Physical Review Letters. 95 (14): 140503. arXiv:quant-ph/0505065. Bibcode:2005PhRvL..95n0503B. doi:10.1103/PhysRevLett.95.140503. PMID 16241636. S2CID 1246885.
Further reading
• Collins, Graham P. (April 2006). "Computing with Quantum Knots" (PDF). Scientific American.
• Sarma, Sankar Das; Freedman, Michael; Nayak, Chetan (2005). "Topologically Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State". Physical Review Letters. 94 (16): 166802. arXiv:cond-mat/0412343. Bibcode:2005PhRvL..94p6802D. doi:10.1103/PhysRevLett.94.166802. PMID 15904258. S2CID 8773427.
• Nayak, Chetan; Simon, Steven H.; Stern, Ady; Freedman, Michael; Sarma, Sankar Das (2008). "Non-Abelian Anyons and Topological Quantum Computation". Reviews of Modern Physics. 80 (3): 1083–1159. arXiv:0707.1889. Bibcode:2008RvMP...80.1083N. doi:10.1103/RevModPhys.80.1083. S2CID 119628297.
• Kundu, A. (1999). "Exact solution of double-delta function Bose gas through interacting anyon gas". Phys. Rev. Lett. 83 (7): 1275–1278. arXiv:hep-th/9811247. Bibcode:1999PhRvL..83.1275K. doi:10.1103/physrevlett.83.1275. S2CID 119329417.
• Batchelor, M.T.; Guan, X. W..; Oelkers, N.. (2006). "One-dimensional interacting anyon gas: low energy properties and Haldane exclusion statistics" (PDF). Phys. Rev. Lett. 96 (21): 210402. arXiv:cond-mat/0603643. Bibcode:2006PhRvL..96u0402B. doi:10.1103/physrevlett.96.210402. PMID 16803221. S2CID 28378363.
• Girardeau, M. D. (2006). "Anyon-fermion mapping and applications to ultracold gasses in tight waveguides". Phys. Rev. Lett. 97 (10): 100402. arXiv:cond-mat/0604357. Bibcode:2006PhRvL..97j0402G. doi:10.1103/physrevlett.97.100402. PMID 17025794. S2CID 206330101.
• Averin, D. V.; Nesteroff, J. A. (2007). "Coulomb blockade of anyons in quantum antidots". Phys. Rev. Lett. 99 (9): 096801. arXiv:0704.0439. Bibcode:2007PhRvL..99i6801A. doi:10.1103/physrevlett.99.096801. PMID 17931025. S2CID 41119577.
• Simon, Steven H. "Quantum Computing with a Twist".
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| Wikipedia |
Topological recursion
In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory.
Introduction
The topological recursion is a construction in algebraic geometry.[1] It takes as initial data a spectral curve: the data of $\left(\Sigma ,\Sigma _{0},x,\omega _{0,1},\omega _{0,2}\right)$, where: $x:\Sigma \to \Sigma _{0}$ is a covering of Riemann surfaces with ramification points; $\omega _{0,1}$ is a meromorphic differential 1-form on $\Sigma $, regular at the ramification points; $\omega _{0,2}$ is a symmetric meromorphic bilinear differential form on $\Sigma ^{2}$ having a double pole on the diagonal and no residue.
The topological recursion is then a recursive definition of infinite sequences of symmetric meromorphic n-forms $\omega _{g,n}$ on $\Sigma ^{n}$, with poles at ramification points only, for integers g≥0 such that 2g-2+n>0. The definition is a recursion on the integer 2g-2+n.
In many applications, the n-form $\omega _{g,n}$ is interpreted as a generating function that measures a set of surfaces of genus g and with n boundaries. The recursion is on 2-2g+n the Euler characteristics, whence the name "topological recursion".
Origin
The topological recursion was first discovered in random matrices. One main goal of random matrix theory, is to find the large size asymptotic expansion of n-point correlation functions, and in some suitable cases, the asymptotic expansion takes the form of a power series. The n-form $\omega _{g,n}$ is then the gth coefficient in the asymptotic expansion of the n-point correlation function. It was found[2][3][4] that the coefficients $\omega _{g,n}$ always obey a same recursion on 2g-2+n. The idea to consider this universal recursion relation beyond random matrix theory, and to promote it as a definition of algebraic curves invariants, occurred in Eynard-Orantin 2007[1] who studied the main properties of those invariants.
An important application of topological recursion was to Gromov–Witten invariants. Marino and BKMP[5] conjectured that Gromov–Witten invariants of a toric Calabi–Yau 3-fold ${\mathfrak {X}}$ are the TR invariants of a spectral curve that is the mirror of ${\mathfrak {X}}$.
Since then, topological recursion has generated a lot of activity in particular in enumerative geometry. The link to Givental formalism and Frobenius manifolds has been established.[6]
Definition
(Case of simple branch points. For higher order branchpoints, see the section Higher order ramifications below)
• For $n\geq 1$ and $2g-2+n>0$:
${\begin{aligned}\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})&=\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}K(z_{1},z,\sigma _{a}(z)){\Big (}\omega _{g-1,n+1}(z,\sigma _{a}(z),z_{2},\dots ,z_{n})\\&\qquad \qquad \qquad +\sum '_{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},1+\#I_{1}}(z,I_{1})\omega _{g_{2},1+\#I_{2}}(\sigma _{a}(z),I_{2}){\Big )}\end{aligned}}$
where $K(z_{1},z_{2},z_{3})$ is called the recursion kernel: $K(z_{1},z_{2},z_{3})={\frac {{\frac {1}{2}}\int _{z'=z_{3}}^{z_{2}}\omega _{0,2}(z_{1},z')}{\omega _{0,1}(z_{2})-\omega _{0,1}(z_{3})}}$
and $\sigma _{a}$ is the local Galois involution near a branch point $a$, it is such that $x(\sigma _{a}(z))=x(z)$. The primed sum $\sum '$ means excluding the two terms $(g_{1},I_{1})=(0,\emptyset )$ and $(g_{2},I_{2})=(0,\emptyset )$.
• For $n=0$ and $2g-2>0$:
$F_{g}=\omega _{g,0}={\frac {1}{2-2g}}\ \sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\omega _{g,1}(z)$
with $dF_{0,1}=\omega _{0,1}$ any antiderivative of $\omega _{0,1}$.
• The definition of $F_{0}=\omega _{0,0}$ and $F_{1}=\omega _{1,0}$ is more involved and can be found in the original article of Eynard-Orantin.[1]
Main properties
• Symmetry: each $\omega _{g,n}$ is a symmetric $n$-form on $\Sigma ^{n}$.
• poles: each $\omega _{g,n}$ is meromorphic, it has poles only at branchpoints, with vanishing residues.
• Homogeneity: $\omega _{g,n}$ is homogeneous of degree $2-2g-n$. Under the change $\omega _{0,1}\to \lambda \omega _{0,1}$, we have $\omega _{g,n}\to \lambda ^{2-2g-n}\omega _{g,n}$.
• Dilaton equation:
$\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}F_{0,1}(z)\ \omega _{g,n+1}(z_{1},\dots ,z_{n},z)=(2-2g-n)\omega _{g,n}(z_{1},\dots ,z_{n})$
where $dF_{0,1}=\omega _{0,1}$.
• Loop equations: The following forms have no poles at branchpoints
$\sum _{z\in x^{-1}(x)}\omega _{g,n+1}(z,z_{1},\dots ,z_{n})$
$\sum _{\{z\neq z'\}\subset x^{-1}(x)}{\Big (}\omega _{g,n+1}(z,z',z_{2},\dots ,z_{n})+\sum _{\overset {g_{1}+g_{2}=g}{I_{1}\uplus I_{2}=\{z_{2},\dots ,z_{n}\}}}\omega _{g_{1},1+\#I_{1}}(z,I_{1})\omega _{g_{2},1+\#I_{2}}(z',I_{2}){\Big )}$ where the sum has no prime, i.e. no term excluded.
• Deformations: The $\omega _{g,n}$ satisfy deformation equations
• Limits: given a family of spectral curves ${\mathcal {S}}_{t}$, whose limit as $t\to 0$ is a singular curve, resolved by rescaling by a power of $t^{\mu }$, then $\lim _{t\to 0}t^{(2-2g-n)\mu }\omega _{g,n}({\mathcal {S}}_{t})=\omega _{g,n}(\lim _{t\to 0}t^{\mu }{\mathcal {S}}_{t})$.
• Symplectic invariance: In the case where $\Sigma $ is a compact algebraic curve with a marking of a symplectic basis of cycles, $x$ is meromorphic and $\omega _{0,1}=ydx$ is meromorphic and $\omega _{0,2}=B$ is the fundamental second kind differential normalized on the marking, then the spectral curve ${\mathcal {S}}=(\Sigma ,\mathbb {C} ,x,ydx,B)$ and ${\tilde {\mathcal {S}}}=(\Sigma ,\mathbb {C} ,y,-xdy,B)$, have the same $F_{g}$ shifted by some terms.
• Modular properties: In the case where $\Sigma $ is a compact algebraic curve with a marking of a symplectic basis of cycles, and $\omega _{0,2}=B$ is the fundamental second kind differential normalized on the marking, then the invariants $\omega _{g,n}$ are quasi-modular forms under the modular group of marking changes. The invariants $\omega _{g,n}$ satisfy BCOV equations.
Generalizations
Higher order ramifications
In case the branchpoints are not simple, the definition is amended as follows[7] (simple branchpoints correspond to k=2):
$\omega _{g,n}(z_{1},z_{2},\dots ,z_{n})=\sum _{a={\text{branchpoints}}}\operatorname {Res} _{z\to a}\sum _{k=2}^{{\rm {order}}_{x}(a)}\sum _{J\subset x^{-1}(x(z))\setminus \{z\},\,\#J=k-1}K_{k}(z_{1},z,J)\sum _{J_{1},\dots ,J_{\ell }\vdash J\cup \{z\}}\sum '_{\overset {g_{1}+\dots +g_{\ell }=g+\ell -k}{I_{1}\uplus \dots I_{\ell }=\{z_{2},\dots ,z_{n}\}}}\prod _{i=1}^{l}\omega _{g_{i},\#J_{i}+\#I_{i}}(J_{i},I_{i})$
The first sum is over partitions $J_{1},\dots ,J_{\ell }$ of $J\cup \{z\}$ with non empty parts $J_{i}\neq \emptyset $, and in the second sum, the prime means excluding all terms such that $(g_{i},\#J_{i}+\#I_{i})=(0,1)$.
$K_{k}$ is called the recursion kernel:
$K_{k}(z_{0},z_{1},\dots ,z_{k})={\frac {\int _{z'=*}^{z_{1}}\omega _{0,2}(z_{0},z')}{\prod _{i=2}^{k}(\omega _{0,1}(z_{1})-\omega _{0,1}(z_{i}))}}$
The base point * of the integral in the numerator can be chosen arbitrarily in a vicinity of the branchpoint, the invariants $\omega _{g,n}$ will not depend on it.
Topological recursion invariants and intersection numbers
The invariants $\omega _{g,n}$ can be written in terms of intersection numbers of tautological classes
[8]
(*) $\omega _{g,n}(z_{1},\dots ,z_{n})=2^{3g-3+n}\sum _{G={\text{Graphs}}}{\frac {1}{\#{\text{Aut}}(G)}}\int _{\left(\prod _{v={\text{vertices}}}{\overline {\mathcal {M}}}_{g_{v},n_{v}}\right)}\,\,\prod _{v={\text{vertices}}}e^{\sum _{k}{\hat {t}}_{\sigma (v),k}\kappa _{k}}\prod _{(p,p')={\text{nodal points}}}\left(\sum _{d,d'}B_{\sigma (p),2d;\sigma (p'),2d'}\psi _{p}^{d}\psi _{p'}^{d'}\right)\prod _{p_{i}={\text{marked points}}\,i=1,\dots ,n}\left(\sum _{d_{i}}\psi _{p_{i}}^{d_{i}}d\xi _{\sigma (p_{i}),d_{i}}(z_{i})\right)$
where the sum is over dual graphs of stable nodal Riemann surfaces of total arithmetic genus $g$, and $n$ smooth labeled marked points $p_{1},\dots ,p_{n}$, and equipped with a map $\sigma :\{{\text{vertices}}\}\to \{{\text{branchpoints}}\}$ :\{{\text{vertices}}\}\to \{{\text{branchpoints}}\}} . $\psi _{p}=c_{1}({\mathcal {L}}_{p})$ is the Chern class of the cotangent line bundle ${\mathcal {L}}_{p}$ whose fiber is the cotangent plane at $p$. $\kappa _{k}$ is the $k$th Mumford's kappa class. The coefficients ${\hat {t}}_{a,k}$, $B_{a,k;a',k'}$, $d\xi _{a,k}(z)$, are the Taylor expansion coefficients of $\omega _{0,1}$ and $\omega _{0,2}$ in the vicinity of branchpoints as follows: in the vicinity of a branchpoint $a$ (assumed simple), a local coordinate is $\zeta _{a}(z)={\sqrt {x(z)-a}}$. The Taylor expansion of $\omega _{0,2}(z,z')$ near branchpoints $z\to a$, $z'\to a'$ defines the coefficients $B_{a,d;a',d'}$
$\omega _{0,2}(z,z')\mathop {\sim } _{z\to a,\ z'\to a'}\left({\frac {\delta _{a,a'}}{(\zeta _{a}(z)-\zeta _{a'}(z'))^{2}}}+2\pi \sum _{d,d'=0}^{\infty }{\frac {B_{a,d;a',d'}}{\Gamma ({\frac {d+1}{2}})\Gamma ({\frac {d'+1}{2}})}}\,\zeta _{a}(z)^{d}\zeta _{a'}(z')^{d'}\right)d\zeta _{a}(z)d\zeta _{a'}(z')$.
The Taylor expansion at $z'\to a$, defines the 1-forms coefficients $d\xi _{a,d}(z)$
$d\xi _{a,d}(z)={\frac {-\Gamma (d+{\frac {1}{2}})}{\Gamma ({\frac {1}{2}})}}\operatorname {Res} _{z'\to a}(x(z')-a)^{-d-{\frac {1}{2}}}\omega _{0,2}(z,z')$ whose Taylor expansion near a branchpoint $a'$ is
$d\xi _{a,d}(z)\mathop {\sim } _{z\to a'}{\frac {-\delta _{a,a'}(2d+1)!!d\zeta _{a}(z)}{2^{d}\zeta _{a}(z)^{2d+2}}}+\sum _{k=0}^{\infty }{\frac {B_{a,2d;a',2k}2^{k+1}}{(2k-1)!!}}\zeta _{a'}(z)^{2k}d\zeta _{a'}(z)$.
Write also the Taylor expansion of $\omega _{0,1}$
$\omega _{0,1}(z)\mathop {\sim } _{z\to a}\sum _{k=0}^{\infty }t_{a,k}\ {\frac {\Gamma ({\frac {1}{2}})}{(k+1)\Gamma ({\frac {k+1}{2}})}}\ \zeta _{a}(z)^{k}d\zeta _{a}(z)$.
Equivalently, the coefficients $t_{a,k}$ can be found from expansion coefficients of the Laplace transform, and the coefficients ${\hat {t}}_{a,k}$ are the expansion coefficients of the log of the Laplace transform
$\int _{x(z)-x(a)\in \mathbb {R} _{+}}\omega _{0,1}(z)e^{-ux(z)}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}\sum _{k=0}^{\infty }t_{a,k}u^{-k}={\frac {e^{-ux(a)}{\sqrt {\pi }}}{2u^{3/2}}}e^{-\sum _{k=0}^{\infty }{\hat {t}}_{a,k}u^{-k}}$ .
For example, we have
$\omega _{0,3}(z_{1},z_{2},z_{3})=\sum _{a}e^{{\hat {t}}_{a,0}}d\xi _{a,0}(z_{1})d\xi _{a,0}(z_{2})d\xi _{a,0}(z_{3}).$
$\omega _{1,1}(z)=2\sum _{a}e^{{\hat {t}}_{a,0}}\left({\frac {1}{24}}d\xi _{a,1}(z)+{\frac {{\hat {t}}_{a,1}}{24}}d\xi _{a,0}(z)+{\frac {1}{2}}B_{a,0;a,0}d\xi _{a,0}(z)\right).$
The formula (*) generalizes ELSV formula as well as Mumford's formula and Mariño-Vafa formula.
Some applications in enumerative geometry
Mirzakhani's recursion
M. Mirzakhani's recursion for hyperbolic volumes of moduli spaces is an instance of topological recursion. For the choice of spectral curve $\left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)={\frac {4}{\pi }}z\sin {(\pi z)}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)$ ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)={\frac {4}{\pi }}z\sin {(\pi z)}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)}
the n-form $\omega _{g,n}=d_{1}\dots d_{n}F_{g,n}$ is the Laplace transform of the Weil-Petersson volume
$F_{g,n}(z_{1},\dots ,z_{n})=\int _{0}^{\infty }e^{-z_{1}L_{1}}dL_{1}\dots \int _{0}^{\infty }e^{-z_{n}L_{n}}dL_{n}\quad \int _{{\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})}w$
where ${\mathcal {M}}_{g,n}(L_{1},\dots ,L_{n})$ is the moduli space of hyperbolic surfaces of genus g with n geodesic boundaries of respective lengths $L_{1},\dots ,L_{n}$, and $w$ is the Weil-Petersson volume form.
The topological recursion for the n-forms $\omega _{g,n}(z_{1},\dots ,z_{n})$, is then equivalent to Mirzakhani's recursion.
Witten–Kontsevich intersection numbers
For the choice of spectral curve $\left(\mathbb {C} ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)=2z^{2}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)$ ;\ \mathbb {C} ;\ x:z\mapsto z^{2};\ \omega _{0,1}(z)=2z^{2}dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)}
the n-form $\omega _{g,n}=d_{1}\dots d_{n}F_{g,n}$ is
$F_{g,n}(z_{1},\dots ,z_{n})=2^{2-2g-n}\sum _{d_{1}+\dots +d_{n}=3g-3+n}\prod _{i=1}^{n}{\frac {(2d_{i}-1)!!}{z_{i}^{2d_{i}+1}}}\quad \left\langle \tau _{d_{1}}\dots \tau _{d_{n}}\right\rangle _{g}$
where $\left\langle \tau _{d_{1}}\dots \tau _{d_{n}}\right\rangle _{g}$ is the Witten-Kontsevich intersection number of Chern classes of cotangent line bundles in the compactified moduli space of Riemann surfaces of genus g with n smooth marked points.
Hurwitz numbers
For the choice of spectral curve $\left(\mathbb {C} ;\ \mathbb {C} ;\ x:-z+\ln {z};\ \omega _{0,1}(z)=(1-z)dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)$ ;\ \mathbb {C} ;\ x:-z+\ln {z};\ \omega _{0,1}(z)=(1-z)dz;\,\omega _{0,2}(z_{1},z_{2})={\frac {dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}}\right)}
the n-form $\omega _{g,n}=d_{1}\dots d_{n}F_{g,n}$ is
$F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\ell (\mu )\leq n}m_{\mu }(e^{x(z_{1})},\dots ,e^{x(z_{n})})\quad h_{g,\mu _{1},\dots ,\mu _{n}}$
where $h_{g,\mu }$ is the connected simple Hurwitz number of genus g with ramification $\mu =(\mu _{1},\dots ,\mu _{n})$: the number of branch covers of the Riemann sphere by a genus g connected surface, with 2g-2+n simple ramification points, and one point with ramification profile given by the partition $\mu $.
Gromov–Witten numbers and the BKMP conjecture
Let ${\mathfrak {X}}$ a toric Calabi–Yau 3-fold, with Kähler moduli $t_{1},\dots ,t_{b_{2}({\mathfrak {X}})}$. Its mirror manifold is singular over a complex plane curve $\Sigma $ given by a polynomial equation $P(e^{x},e^{y})=0$, whose coefficients are functions of the Kähler moduli. For the choice of spectral curve $\left(\Sigma ;\ \mathbb {C} ^{*};\ x;\ \omega _{0,1}=ydx;\,\omega _{0,2}\right)$ ;\ \mathbb {C} ^{*};\ x;\ \omega _{0,1}=ydx;\,\omega _{0,2}\right)} with $\omega _{0,2}$ the fundamental second kind differential on $\Sigma $,
According to the BKMP[5] conjecture, the n-form $\omega _{g,n}=d_{1}\dots d_{n}F_{g,n}$ is
$F_{g,n}(z_{1},\dots ,z_{n})=\sum _{\mathbf {d} \in H_{2}({\mathfrak {X}},\mathbb {Z} )}\sum _{\mu _{1},\dots ,\mu _{n}\in H_{1}({\mathcal {L}},\mathbb {Z} )}t^{d}\prod _{i=1}^{n}e^{x(z_{i})}{\mathcal {N}}_{g}({\mathfrak {X}},{\mathcal {L}};\mathbf {d} ,\mu _{1},\dots ,\mu _{n})$
where ${\mathcal {N}}_{g}({\mathfrak {X}},{\mathcal {L}};\mathbf {d} ,\mu _{1},\dots ,\mu _{n})=\int _{[{\overline {\mathcal {M}}}_{g,n}({\mathfrak {X}},{\mathcal {L}},\mathbf {d} ,\mu _{1},\dots ,\mu _{n})]^{\rm {vir}}}1$
is the genus g Gromov–Witten number, representing the number of holomorphic maps of a surface of genus g into ${\mathfrak {X}}$, with n boundaries mapped to a special Lagrangian submanifold ${\mathcal {L}}$. $\mathbf {d} =(d_{1},\dots ,d_{b_{2}({\mathfrak {X}})})$ is the 2nd relative homology class of the surface's image, and $\mu _{i}\in H_{1}({\mathcal {L}},\mathbb {Z} )$ are homology classes (winding number) of the boundary images.
The BKMP[5] conjecture has since then been proven.
Notes
1. Invariants of algebraic curves and topological expansion, B. Eynard, N. Orantin, math-ph/0702045, ccsd-hal-00130963, Communications in Number Theory and Physics, Vol 1, Number 2, p347-452.
2. B. Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, JHEP/024A/0904, hep-th/0407261 A short overview of the ”Topological recursion”, math-ph/arXiv:1412.3286
3. A. Alexandrov, A. Mironov, A. Morozov, Solving Virasoro Constraints in Matrix Models, Fortsch.Phys.53:512-521,2005, arXiv:hep-th/0412205
4. L. Chekhov, B. Eynard, N. Orantin, Free energy topological expansion for the 2-matrix model, JHEP 0612 (2006) 053, math-ph/0603003
5. Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178,2009
6. P. Dunin-Barkowski, N. Orantin, S. Shadrin, L. Spitz, "Identification of the Givental formula with the spectral curve topological recursion procedure", Commun.Math.Phys. 328 (2014) 669-700.
7. V. Bouchard, B. Eynard, "Think globally, compute locally", JHEP02(2013)143.
8. B. Eynard, Invariants of spectral curves and intersection theory of moduli spaces of complex curves, math-ph: arxiv.1110.2949, Journal Communications in Number Theory and Physics, Volume 8, Number 3.
References
[1]
1. O. Dumitrescu and M. Mulase, Lectures on the topological recursion for Higgs bindles and quantum curves, https://www.math.ucdavis.edu/~mulase/texfiles/OMLectures.pdf
| Wikipedia |
Topological rigidity
In the mathematical field of topology, a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M.[1]
Motivation
A central problem in topology is determining when two spaces are the same i.e. homeomorphic or diffeomorphic. Constructing a morphism explicitly is almost always impractical. If we put further condition on one or both spaces (manifolds) we can exploit this additional structure in order to show that the desired morphism must exist.
Rigidity theorem is about when a fairly weak equivalence between two manifolds (usually a homotopy equivalence) implies the existence of stronger equivalence homeomorphism, diffeomorphism or isometry.
Definition.
A closed topological manifold M is called topological rigid if any homotopy equivalence f : N → M with some manifold N as source and M as target is homotopic to a homeomorphism.
Examples
Example 1.
If closed 2-manifolds M and N are homotopically equivalent then they are homeomorphic. Moreover, any homotopy equivalence of closed surfaces deforms to a homeomorphism.
Example 2.
If a closed manifold Mn (n ≠ 3) is homotopy-equivalent to Sn then Mn is homeomorphic to Sn.
Rigidity theorem in geometry
Definition.
A diffeomorphism of flat-Riemannian manifolds is said to be affine iff it carries geodesics to geodesic.
Theorem (Bieberbach)
If f : M → N is a homotopy equivalence between flat closed connected Riemannian manifolds then f is homotopic to an affine homeomorphism.
Mostow's rigidity theorem
Theorem: Let M and N be compact, locally symmetric Riemannian manifolds with everywhere non-positive curvature having no closed one or two dimensional geodesic subspace which are direct factor locally. If f : M → N is a homotopy equivalence then f is homotopic to an isometry.
Theorem (Mostow's theorem for hyperbolic n-manifolds, n ≥ 3): If M and N are complete hyperbolic n-manifolds, n ≥ 3 with finite volume and f : M → N is a homotopy equivalence then f is homotopic to an isometry.
These results are named after George Mostow.
Algebraic form
Let Γ and Δ be discrete subgroups of the isometry group of hyperbolic n-space H, where n ≥ 3, whose quotients H/Γ and H/Δ have finite volume. If Γ and Δ are isomorphic as discrete groups then they are conjugate.
Remarks
(1) In the 2-dimensional case any manifold of genus at least two has a hyperbolic structure. Mostow's rigidity theorem does not apply in this case. In fact, there are many hyperbolic structures on any such manifold; each such structure corresponds to a point in Teichmuller space.
(2) On the other hand, if M and N are 2-manifolds of finite volume then it is easy to show that they are homeomorphic exactly when their fundamental groups are the same.
Application
The group of isometries of a finite-volume hyperbolic n-manifold M (for n ≥ 3) is finitely generated[2] and isomorphic to π1(M).
References
1. Martin, Alexandre. "The topological rigidity of the torus (thesis)" (PDF). University of Edinburgh. Retrieved 10 October 2013.
2. http://www.macs.hw.ac.uk/~jim/samos.pdf
| Wikipedia |
Topological ring
In mathematics, a topological ring is a ring $R$ that is also a topological space such that both the addition and the multiplication are continuous as maps:[1]
$R\times R\to R$
where $R\times R$ carries the product topology. That means $R$ is an additive topological group and a multiplicative topological semigroup.
Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field.[2]
General comments
The group of units $R^{\times }$ of a topological ring $R$ is a topological group when endowed with the topology coming from the embedding of $R^{\times }$ into the product $R\times R$ as $\left(x,x^{-1}\right).$ However, if the unit group is endowed with the subspace topology as a subspace of $R,$ it may not be a topological group, because inversion on $R^{\times }$ need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on $R^{\times }$ is continuous in the subspace topology of $R$ then these two topologies on $R^{\times }$ are the same.
If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring that is a topological group (for $+$) in which multiplication is continuous, too.
Examples
Topological rings occur in mathematical analysis, for example as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras are topological rings. The rational, real, complex and $p$-adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples.
In algebra, the following construction is common: one starts with a commutative ring $R$ containing an ideal $I,$ and then considers the $I$-adic topology on $R$: a subset $R=U$ of $R$ is open if and only if for every $x\in U$ there exists a natural number $n$ such that $x+I^{n}\subseteq U.$ This turns $R$ into a topological ring. The $I$-adic topology is Hausdorff if and only if the intersection of all powers of $I$ is the zero ideal $(0).$
The $p$-adic topology on the integers is an example of an $I$-adic topology (with $I=(p)$).
Completion
Main article: Completion (algebra)
Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ring$R$ is complete. If it is not, then it can be completed: one can find an essentially unique complete topological ring $S$ that contains $R$ as a dense subring such that the given topology on $R$ equals the subspace topology arising from $S.$ If the starting ring $R$ is metric, the ring $S$ can be constructed as a set of equivalence classes of Cauchy sequences in $R,$ this equivalence relation makes the ring $S$ Hausdorff and using constant sequences (which are Cauchy) one realizes a (uniformly) continuous morphism (CM in the sequel) $c:R\to S$ such that, for all CM $f:R\to T$ where $T$ is Hausdorff and complete, there exists a unique CM $g:S\to T$ such that $f=g\circ c.$ If $R$ is not metric (as, for instance, the ring of all real-variable rational valued functions, that is, all functions $f:\mathbb {R} \to \mathbb {Q} $ endowed with the topology of pointwise convergence) the standard construction uses minimal Cauchy filters and satisfies the same universal property as above (see Bourbaki, General Topology, III.6.5).
The rings of formal power series and the $p$-adic integers are most naturally defined as completions of certain topological rings carrying $I$-adic topologies.
Topological fields
Some of the most important examples are topological fields. A topological field is a topological ring that is also a field, and such that inversion of non zero elements is a continuous function. The most common examples are the complex numbers and all its subfields, and the valued fields, which include the $p$-adic fields.
See also
• Compact group – Topological group with compact topology
• Complete field – Algebraic structure that is complete relative to a metricPages displaying wikidata descriptions as a fallback
• Locally compact field
• Locally compact quantum group – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback
• Locally compact group – topological group G for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback
• Ordered topological vector space
• Strongly continuous semigroup – Generalization of the exponential functionPages displaying short descriptions of redirect targets
• Topological abelian group – concept in mathematicsPages displaying wikidata descriptions as a fallback
• Topological field – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets
• Topological group – Group that is a topological space with continuous group action
• Topological module
• Topological semigroup – semigroup with continuous operationPages displaying wikidata descriptions as a fallback
• Topological vector space – Vector space with a notion of nearness
Citations
1. Warner 1993, pp. 1–2, Def. 1.1.
2. Warner 1989, p. 77, Ch. II.
References
• L. V. Kuzmin (2001) [1994], "Topological ring", Encyclopedia of Mathematics, EMS Press
• D. B. Shakhmatov (2001) [1994], "Topological field", Encyclopedia of Mathematics, EMS Press
• Warner, Seth (1989). Topological Fields. Elsevier. ISBN 9780080872681.
• Warner, Seth (1993). Topological Rings. Elsevier. ISBN 9780080872896.
• Vladimir I. Arnautov, Sergei T. Glavatsky and Aleksandr V. Michalev: Introduction to the Theory of Topological Rings and Modules. Marcel Dekker Inc, February 1996, ISBN 0-8247-9323-4.
• N. Bourbaki, Éléments de Mathématique. Topologie Générale. Hermann, Paris 1971, ch. III §6
| Wikipedia |
Topological semigroup
In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.[1]
Every topological group is a topological semigroup.
See also
• Analytic semigroup
• Compact group – Topological group with compact topology
• Complete field – Algebraic structure that is complete relative to a metricPages displaying wikidata descriptions as a fallback
• Ellis–Numakura lemma – A compact topological semigroup with a semicontinuous product has an idempotent element
• Locally compact group – topological group G for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be definedPages displaying wikidata descriptions as a fallback
• Locally compact quantum group – relatively new C*-algebraic approach toward quantum groupsPages displaying wikidata descriptions as a fallback
• Ordered topological vector space
• Strongly continuous semigroup – Generalization of the exponential functionPages displaying short descriptions of redirect targets
• Topological abelian group – concept in mathematicsPages displaying wikidata descriptions as a fallback
• Topological field – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets
• Topological group – Group that is a topological space with continuous group action
• Topological module
• Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback
• Topological vector lattice
• Topological vector space – Vector space with a notion of nearness
References
1. Artur Hideyuki Tomita. On sequentially compact both-sides cancellative semigroups with sequentially continuous addition.
| Wikipedia |
Topological vector lattice
In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) $X$ that has a partial order $\,\leq \,$ making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets.[1] Ordered vector lattices have important applications in spectral theory.
Definition
If $X$ is a vector lattice then by the vector lattice operations we mean the following maps:
1. the three maps $X$ to itself defined by $x\mapsto |x|$, $x\mapsto x^{+}$, $x\mapsto x^{-}$, and
2. the two maps from $X\times X$ into $X$ defined by $(x,y)\mapsto \sup _{}\{x,y\}$ and$(x,y)\mapsto \inf _{}\{x,y\}$.
If $X$ is a TVS over the reals and a vector lattice, then $X$ is locally solid if and only if (1) its positive cone is a normal cone, and (2) the vector lattice operations are continuous.[1]
If $X$ is a vector lattice and an ordered topological vector space that is a Fréchet space in which the positive cone is a normal cone, then the lattice operations are continuous.[1]
If $X$ is a topological vector space (TVS) and an ordered vector space then $X$ is called locally solid if $X$ possesses a neighborhood base at the origin consisting of solid sets.[1] A topological vector lattice is a Hausdorff TVS $X$ that has a partial order $\,\leq \,$ making it into vector lattice that is locally solid.[1]
Properties
Every topological vector lattice has a closed positive cone and is thus an ordered topological vector space.[1] Let ${\mathcal {B}}$ denote the set of all bounded subsets of a topological vector lattice with positive cone $C$ and for any subset $S$, let $[S]_{C}:=(S+C)\cap (S-C)$ be the $C$-saturated hull of $S$. Then the topological vector lattice's positive cone $C$ is a strict ${\mathcal {B}}$-cone,[1] where $C$ is a strict ${\mathcal {B}}$-cone means that $\left\{[B]_{C}:B\in {\mathcal {B}}\right\}$ is a fundamental subfamily of ${\mathcal {B}}$ that is, every $B\in {\mathcal {B}}$ is contained as a subset of some element of $\left\{[B]_{C}:B\in {\mathcal {B}}\right\}$).[2]
If a topological vector lattice $X$ is order complete then every band is closed in $X$.[1]
Examples
The Banach spaces $L^{p}(\mu )$ ($1\leq p\leq \infty $) are Banach lattices under their canonical orderings. These spaces are order complete for $p<\infty $.
See also
• Banach lattice – Banach space with a compatible structure of a latticePages displaying wikidata descriptions as a fallback
• Complemented lattice
• Fréchet lattice
• Locally convex vector lattice
• Normed lattice
• Ordered vector space – Vector space with a partial order
• Pseudocomplement
• Riesz space – Partially ordered vector space, ordered as a lattice
References
1. Schaefer & Wolff 1999, pp. 234–242.
2. Schaefer & Wolff 1999, pp. 215–222.
Bibliography
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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| Wikipedia |
Topological homomorphism
In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.
Definitions
A topological homomorphism or simply homomorphism (if no confusion will arise) is a continuous linear map $u:X\to Y$ between topological vector spaces (TVSs) such that the induced map $u:X\to \operatorname {Im} u$ is an open mapping when $\operatorname {Im} u:=u(X),$ which is the image of $u,$ is given the subspace topology induced by $Y.$[1] This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.
A TVS embedding or a topological monomorphism[2] is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.
Characterizations
Suppose that $u:X\to Y$ is a linear map between TVSs and note that $u$ can be decomposed into the composition of the following canonical linear maps:
$X~{\overset {\pi }{\rightarrow }}~X/\operatorname {ker} u~{\overset {u_{0}}{\rightarrow }}~\operatorname {Im} u~{\overset {\operatorname {In} }{\rightarrow }}~Y$
where $\pi :X\to X/\operatorname {ker} u$ is the canonical quotient map and $\operatorname {In} :\operatorname {Im} u\to Y$ :\operatorname {Im} u\to Y} is the inclusion map.
The following are equivalent:
1. $u$ is a topological homomorphism
2. for every neighborhood base ${\mathcal {U}}$ of the origin in $X,$ $u\left({\mathcal {U}}\right)$ is a neighborhood base of the origin in $Y$[1]
3. the induced map $u_{0}:X/\operatorname {ker} u\to \operatorname {Im} u$ is an isomorphism of TVSs[1]
If in addition the range of $u$ is a finite-dimensional Hausdorff space then the following are equivalent:
1. $u$ is a topological homomorphism
2. $u$ is continuous[1]
3. $u$ is continuous at the origin[1]
4. $u^{-1}(0)$ is closed in $X$[1]
Sufficient conditions
Theorem[1] — Let $u:X\to Y$ be a surjective continuous linear map from an LF-space $X$ into a TVS $Y.$ If $Y$ is also an LF-space or if $Y$ is a Fréchet space then $u:X\to Y$ is a topological homomorphism.
Theorem[3] — Suppose $f:X\to Y$ be a continuous linear operator between two Hausdorff TVSs. If $M$ is a dense vector subspace of $X$ and if the restriction $f{\big \vert }_{M}:M\to Y$ to $M$ is a topological homomorphism then $f:X\to Y$ is also a topological homomorphism.[3]
So if $C$ and $D$ are Hausdorff completions of $X$ and $Y,$ respectively, and if $f:X\to Y$ is a topological homomorphism, then $f$'s unique continuous linear extension $F:C\to D$ is a topological homomorphism. (However, it is possible for $f:X\to Y$ to be surjective but for $F:C\to D$ to not be injective.)
Open mapping theorem
The open mapping theorem, also known as Banach's homomorphism theorem, gives a sufficient condition for a continuous linear operator between complete metrizable TVSs to be a topological homomorphism.
Theorem[4] — Let $u:X\to Y$ be a continuous linear map between two complete metrizable TVSs. If $\operatorname {Im} u,$ which is the range of $u,$ is a dense subset of $Y$ then either $\operatorname {Im} u$ is meager (that is, of the first category) in $Y$ or else $u:X\to Y$ is a surjective topological homomorphism. In particular, $u:X\to Y$ is a topological homomorphism if and only if $\operatorname {Im} u$ is a closed subset of $Y.$
Corollary[4] — Let $\sigma $ and $\tau $ be TVS topologies on a vector space $X$ such that each topology makes $X$ into a complete metrizable TVSs. If either $\sigma \subseteq \tau $ or $\tau \subseteq \sigma $ then $\sigma =\tau .$
Corollary[4] — If $X$ is a complete metrizable TVS, $M$ and $N$ are two closed vector subspaces of $X,$ and if $X$ is the algebraic direct sum of $M$ and $N$ (i.e. the direct sum in the category of vector spaces), then $X$ is the direct sum of $M$ and $N$ in the category of topological vector spaces.
Examples
Every continuous linear functional on a TVS is a topological homomorphism.[1]
Let $X$ be a $1$-dimensional TVS over the field $\mathbb {K} $ and let $x\in X$ be non-zero. Let $L:\mathbb {K} \to X$ be defined by $L(s):=sx.$ If $\mathbb {K} $ has it usual Euclidean topology and if $X$ is Hausdorff then $L:\mathbb {K} \to X$ is a TVS-isomorphism.
See also
• Homomorphism – Structure-preserving map between two algebraic structures of the same type
• Open mapping – A function that sends open (resp. closed) subsets to open (resp. closed) subsetsPages displaying short descriptions of redirect targets
• Surjection of Fréchet spaces – Characterization of surjectivity
• Topological vector space – Vector space with a notion of nearness
References
1. Schaefer & Wolff 1999, pp. 74–78.
2. Köthe 1969, p. 91.
3. Schaefer & Wolff 1999, p. 116.
4. Schaefer & Wolff 1999, p. 78.
Bibliography
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Valdivia, Manuel (1982). Nachbin, Leopoldo (ed.). Topics in Locally Convex Spaces. Vol. 67. Amsterdam New York, N.Y.: Elsevier Science Pub. Co. ISBN 978-0-08-087178-3. OCLC 316568534.
• Voigt, Jürgen (2020). A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Cham: Birkhäuser Basel. ISBN 978-3-030-32945-7. OCLC 1145563701.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
Topological vector spaces (TVSs)
Basic concepts
• Banach space
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Main results
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• Mathematics portal
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| Wikipedia |
Complemented subspace
In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space $X,$ is a vector subspace $M$ for which there exists some other vector subspace $N$ of $X,$ called its (topological) complement in $X$, such that $X$ is the direct sum $M\oplus N$ in the category of topological vector spaces. Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional vector spaces.
Every finite-dimensional subspace of a Banach space is complemented, but other subspaces may not. In general, classifying all complemented subspaces is a difficult problem, which has been solved only for some well-known Banach spaces.
The concept of a complemented subspace is analogous to, but distinct from, that of a set complement. The set-theoretic complement of a vector subspace is never a complementary subspace.
Preliminaries: definitions and notation
If $X$ is a vector space and $M$ and $N$ are vector subspaces of $X$ then there is a well-defined addition map
${\begin{alignedat}{4}S:\;&&M\times N&&\;\to \;&X\\&&(m,n)&&\;\mapsto \;&m+n\\\end{alignedat}}$
The map $S$ is a morphism in the category of vector spaces — that is to say, linear.
Algebraic direct sum
Main articles: Direct sum and Direct sum of modules
The vector space $X$ is said to be the algebraic direct sum (or direct sum in the category of vector spaces) $M\oplus N$ when any of the following equivalent conditions are satisfied:
1. The addition map $S:M\times N\to X$ is a vector space isomorphism.[1][2]
2. The addition map is bijective.
3. $M\cap N=\{0\}$ and $M+N=X$; in this case $N$ is called an algebraic complement or supplement to $M$ in $X$ and the two subspaces are said to be complementary or supplementary.[2][3]
When these conditions hold, the inverse $S^{-1}:X\to M\times N$ is well-defined and can be written in terms of coordinates as
$S^{-1}=\left(P_{M},P_{N}\right){\text{.}}$
The first coordinate $P_{M}:X\to M$ is called the canonical projection of $X$ onto $M$; likewise the second coordinate is the canonical projection onto $N.$[4]
Equivalently, $P_{M}(x)$ and $P_{N}(x)$ are the unique vectors in $M$ and $N,$ respectively, that satisfy
$x=P_{M}(x)+P_{N}(x){\text{.}}$
As maps,
$P_{M}+P_{N}=\operatorname {Id} _{X},\qquad \ker P_{M}=N,\qquad {\text{ and }}\qquad \ker P_{N}=M$
where $\operatorname {Id} _{X}$ denotes the identity map on $X$.[2]
Motivation
See also: Coproduct, Direct sum § Direct sum in categories, and Direct sum of topological groups
Suppose that the vector space $X$ is the algebraic direct sum of $M\oplus N$. In the category of vector spaces, finite products and coproducts coincide: algebraically, $M\oplus N$ and $M\times N$ are indistinguishable. Given a problem involving elements of $X$, one can break the elements down into their components in $M$ and $N$, because the projection maps defined above act as inverses to the natural inclusion of $M$ and $N$ into $X$. Then one can solve the problem in the vector subspaces and recombine to form an element of $X$.
In the category of topological vector spaces, that algebraic decomposition becomes less useful. The definition of a topological vector space requires the addition map $S$ to be continuous; its inverse $S^{-1}:X\to M\times N$ may not be.[1] The categorical definition of direct sum, however, requires $P_{M}$ and $P_{N}$ to be morphisms — that is, continuous linear maps.
The space $X$ is the topological direct sum of $M$ and $N$ if (and only if) any of the following equivalent conditions hold:
1. The addition map $S:M\times N\to X$ is a TVS-isomorphism (that is, a surjective linear homeomorphism).[1]
2. $X$ is the algebraic direct sum of $M$ and $N$ and also any of the following equivalent conditions:
1. The inverse of the addition map $S^{-1}:X\to M\times N$ is continuous.
2. Both canonical projections $P_{M}:X\to M$ and $P_{N}:X\to N$ are continuous.
3. At least one of the canonical projections $P_{M}$ and $P_{N}$ is continuous.
4. The canonical quotient map $p:N\to X/M;p(n)=n+M$ is an isomorphism of topological vector spaces (i.e. a linear homeomorphism).[2]
3. $X$ is the direct sum of $M$ and $N$ in the category of topological vector spaces.
4. The map $S$ is bijective and open.
5. When considered as additive topological groups, $X$ is the topological direct sum of the subgroups $M$ and $N.$
The topological direct sum is also written $X=M\oplus N$; whether the sum is in the topological or algebraic sense is usually clarified through context.
Definition
Every topological direct sum is an algebraic direct sum $X=M\oplus N$; the converse is not guaranteed. Even if both $M$ and $N$ are closed in $X$, $S^{-1}$ may still fail to be continuous. $N$ is a (topological) complement or supplement to $M$ if it avoids that pathology — that is, if, topologically, $X=M\oplus N$. (Then $M$ is likewise complementary to $N$.)[1] Condition 1(d) above implies that any topological complement of $M$ is isomorphic, as a topological vector space, to the quotient vector space $X/M$.
$M$ is called complemented if it has a topological complement $N$ (and uncomplemented if not). The choice of $N$ can matter quite strongly: every complemented vector subspace $M$ has algebraic complements that do not complement $M$ topologically.
Because a linear map between two normed (or Banach) spaces is bounded if and only if it is continuous, the definition in the categories of normed (resp. Banach) spaces is the same as in topological vector spaces.
Equivalent characterizations
The vector subspace $M$ is complemented in $X$ if and only if any of the following holds:[1]
• There exists a continuous linear map $P_{M}:X\to X$ with image $P_{M}(X)=M$ such that $P\circ P=P$;
• There exists a continuous linear projection $P_{M}:X\to X$ with image $P_{M}(X)=M$ such that algebraically $X=M\oplus \ker {P}$.
• For every TVS $Y,$ the restriction map $R:L(X;Y)\to L(M;Y);R(u)=u|_{M}$ is surjective.[5]
If in addition $X$ is Banach, then an equivalent condition is
• $M$ is closed in $X$, there exists another closed subspace $N\subseteq X$, and $S$ is an isomorphism from the abstract direct sum $M\oplus N$ to $X$.
Examples
• If $Y$ is a measure space and $X\subseteq Y$ has positive measure, then $L^{p}(X)$ is complemented in $L^{p}(Y)$.
• $c_{0}$, the space of sequences converging to $0$, is complemented in $c$, the space of convergent sequences.
• By Lebesgue decomposition, $L^{1}([0,1])$ is complemented in $\mathrm {rca} ([0,1])\cong C([0,1])^{*}$.
Sufficient conditions
For any two topological vector spaces $X$ and $Y$, the subspaces $X\times \{0\}$ and $\{0\}\times Y$ are topological complements in $X\times Y$.
Every algebraic complement of ${\overline {\{0\}}}$, the closure of $0$, is also a topological complement. This is because ${\overline {\{0\}}}$ has the indiscrete topology, and so the algebraic projection is continuous.[6]
If $X=M\oplus N$ and $A:X\to Y$ is surjective, then $Y=AM\oplus AN$.[2]
Finite dimension
Suppose $X$ is Hausdorff and locally convex and $Y$ a free topological vector subspace: for some set $I$, we have $Y\cong \mathbb {K} ^{I}$ (as a t.v.s.). Then $Y$ is a closed and complemented vector subspace of $X$.[proof 1] In particular, any finite-dimensional subspace of $X$ is complemented.[7]
In arbitrary topological vector spaces, a finite-dimensional vector subspace $Y$ is topologically complemented if and only if for every non-zero $y\in Y$, there exists a continuous linear functional on $X$ that separates $y$ from $0$.[1] For an example in which this fails, see § Fréchet spaces.
Finite codimension
Not all finite-codimensional vector subspaces of a TVS are closed, but those that are, do have complements.[7][8]
Hilbert spaces
In a Hilbert space, the orthogonal complement $M^{\bot }$ of any closed vector subspace $M$ is always a topological complement of $M$. This property characterizes Hilbert spaces within the class of Banach spaces: every infinite dimensional, non-Hilbert Banach space contains a closed uncomplemented subspace.[3]
Fréchet spaces
Let $X$ be a Fréchet space over the field $\mathbb {K} $. Then the following are equivalent:[9]
1. $X$ is not normable (that is, any continuous norm does not generate the topology)
2. $X$ contains a vector subspace TVS-isomorphic to $\mathbb {K} ^{\mathbb {N} }.$
3. $X$ contains a complemented vector subspace TVS-isomorphic to $\mathbb {K} ^{\mathbb {N} }$.
Properties; examples of uncomplemented subspaces
A complemented (vector) subspace of a Hausdorff space $X$ is necessarily a closed subset of $X$, as is its complement.[1][proof 2]
From the existence of Hamel bases, every infinite-dimensional Banach space contains unclosed linear subspaces.[proof 3] Since any complemented subspace is closed, none of those subspaces is complemented.
Likewise, if $X$ is a complete TVS and $X/M$ is not complete, then $M$ has no topological complement in $X.$[10]
Applications
If $A:X\to Y$ is a continuous linear surjection, then the following conditions are equivalent:
1. The kernel of $A$ has a topological complement.
2. There exists a "right inverse": a continuous linear map $B:Y\to X$ such that $AB=\mathrm {Id} _{Y}$, where $\operatorname {Id} _{Y}:Y\to Y$ is the identity map.[5]
The Method of Decomposition
Topological vector spaces admit the following Cantor-Schröder-Bernstein–type theorem:
Let $X$ and $Y$ be TVSs such that $X=X\oplus X$ and $Y=Y\oplus Y.$ Suppose that $Y$ contains a complemented copy of $X$ and $X$ contains a complemented copy of $Y.$ Then $X$ is TVS-isomorphic to $Y.$
The "self-splitting" assumptions that $X=X\oplus X$ and $Y=Y\oplus Y$ cannot be removed: Tim Gowers showed in 1996 that there exist non-isomorphic Banach spaces $X$ and $Y$, each complemented in the other.[11]
In classical Banach spaces
Understanding the complemented subspaces of an arbitrary Banach space $X$ up to isomorphism is a classical problem that has motivated much work in basis theory, particularly the development of absolutely summing operators. The problem remains open for a variety of important Banach spaces, most notably the space $L_{1}[0,1]$.[12]
For some Banach spaces the question is closed. Most famously, if $1\leq p\leq \infty $ then the only complemented subspaces of $\ell _{p}$ are isomorphic to $\ell _{p},$ and the same goes for $c_{0}.$ Such spaces are called prime (when their only infinite-dimensional complemented subspaces are isomorphic to the original). These are not the only prime spaces, however.[12]
The spaces $L_{p}[0,1]$ are not prime whenever $p\in (1,2)\cup (2,\infty );$ in fact, they admit uncountably many non-isomorphic complemented subspaces.[12]
The spaces $L_{2}[0,1]$ and $L_{\infty }[0,1]$ are isomorphic to $\ell _{2}$ and $\ell _{\infty },$ respectively, so they are indeed prime.[12]
The space $L_{1}[0,1]$ is not prime, because it contains a complemented copy of $\ell _{1}$. No other complemented subspaces of $L_{1}[0,1]$ are currently known.[12]
Indecomposable Banach spaces
An infinite-dimensional Banach space is called indecomposable whenever its only complemented subspaces are either finite-dimensional or -codimensional. Because a finite-codimensional subspace of a Banach space $X$ is always isomorphic to $X,$ indecomposable Banach spaces are prime.
The most well-known example of indecomposable spaces are in fact hereditarily indecomposable, which means every infinite-dimensional subspace is also indecomposable.[13]
See also
• Direct sum – Operation in abstract algebra composing objects into "more complicated" objects
• Direct sum of modules – Operation in abstract algebra
• Direct sum of topological groups
Proofs
1. $Y$ is closed because $\mathbb {K} ^{I}$ is complete and $X$ is Hausdorff.
Let $f=\left(f_{i}\right)_{i\in I}:Y\to \mathbb {K} ^{I}$ be a TVS-isomorphism; each $f_{i}:Y\to \mathbb {K} $ is a continuous linear functional. By the Hahn–Banach theorem, we may extend each $f_{i}$ to a continuous linear functional $F_{i}:X\to \mathbb {K} $ on $X.$ The joint map $F:X\to \mathbb {K} ^{I}$ is a continuous linear surjection whose restriction to $Y$ is $f$. The composition $P=f^{-1}\circ F:X\to Y$ is then a continuous continuous projection onto $Y$.
Q.E.D.
2. In a Hausdorff space, $\{0\}$ is closed. A complemented space is the kernel of the (continuous) projection onto its complement. Thus it is the preimage of $\{0\}$ under a continuous map, and so closed.
Q.E.D.
3. Any sequence $\{e_{j}\}_{j=0}^{\infty }\in X^{\omega }$ defines a summation map $T:l^{1}\to X;T(\{x_{j}\}_{j})=\sum _{j}{x_{j}e_{j}}$. But if $\{e_{j}\}_{j}$ are (algebraically) linearly independent and $\{x_{j}\}_{j}$ has full support, then $T(x)\in {\overline {\operatorname {span} {\{e_{j}\}_{j}}}}\setminus \operatorname {span} {\{e_{j}\}_{j}}$.
Q.E.D.
References
1. Grothendieck 1973, pp. 34–36.
2. Fabian, Marián J.; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Zizler, Václav (2011). Banach Space Theory: The Basis for Linear and Nonlinear Analysis (PDF). New York: Springer. pp. 179–181. doi:10.1007/978-1-4419-7515-7. ISBN 978-1-4419-7515-7.
3. Brezis, Haim (2011). Functional Analysis, Sobolev Spaces, and Partial Differential Equations. Universitext. New York: Springer. pp. 38–39. ISBN 978-0-387-70913-0.
4. Schaefer & Wolff 1999, pp. 19–24.
5. Trèves 2006, p. 36.
6. Wilansky 2013, p. 63.
7. Rudin 1991, p. 106.
8. Serre, Jean-Pierre (1955). "Un théoreme de dualité". Commentarii Mathematici Helvetici. 29 (1): 9–26. doi:10.1007/BF02564268. S2CID 123643759.
9. Jarchow 1981, pp. 129–130.
10. Schaefer & Wolff 1999, pp. 190–202.
11. Narici & Beckenstein 2011, pp. 100–101.
12. Albiac, Fernando; Kalton, Nigel J. (2006). Topics in Banach Space Theory. GTM 233 (2nd ed.). Switzerland: Springer (published 2016). pp. 29–232. doi:10.1007/978-3-319-31557-7. ISBN 978-3-319-31557-7.
13. Argyros, Spiros; Tolias, Andreas (2004). Methods in the Theory of Hereditarily Indecomposable Banach Spaces. American Mathematical Soc. ISBN 978-0-8218-3521-0.
Bibliography
• Bachman, George; Narici, Lawrence (2000). Functional Analysis (Second ed.). Mineola, New York: Dover Publications. ISBN 978-0486402512. OCLC 829157984.
• Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
Topological vector spaces (TVSs)
Basic concepts
• Banach space
• Completeness
• Continuous linear operator
• Linear functional
• Fréchet space
• Linear map
• Locally convex space
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| Wikipedia |
Topological indistinguishability
In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and Nx is the set of all neighborhoods that contain x, and Ny is the set of all neighborhoods that contain y, then x and y are "topologically indistinguishable" if and only if Nx = Ny. (See Hausdorff's axiomatic neighborhood systems.)
Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T3½(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
Hausdorff)
T6 (perfectly normal
Hausdorff)
• History
Intuitively, two points are topologically indistinguishable if the topology of X is unable to discern between the points.
Two points of X are topologically distinguishable if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points). This open set can then be used to distinguish between the two points. A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms.
Topological indistinguishability defines an equivalence relation on any topological space X. If x and y are points of X we write x ≡ y for "x and y are topologically indistinguishable". The equivalence class of x will be denoted by [x].
Examples
For T0 spaces (in particular, for Hausdorff spaces) the notion of topological indistinguishability is trivial, so one must look to non-T0 spaces to find interesting examples. On the other hand, regularity and normality do not imply T0, so we can find examples with these properties. In fact, almost all of the examples given below are completely regular.
• In an indiscrete space, any two points are topologically indistinguishable.
• In a pseudometric space, two points are topologically indistinguishable if and only if the distance between them is zero.
• In a seminormed vector space, x ≡ y if and only if ‖x − y‖ = 0.
• For example, let L2(R) be the space of all measurable functions from R to R which are square integrable (see Lp space). Then two functions f and g in L2(R) are topologically indistinguishable if and only if they are equal almost everywhere.
• In a topological group, x ≡ y if and only if x−1y ∈ cl{e} where cl{e} is the closure of the trivial subgroup. The equivalence classes are just the cosets of cl{e} (which is always a normal subgroup).
• Uniform spaces generalize both pseudometric spaces and topological groups. In a uniform space, x ≡ y if and only if the pair (x, y) belongs to every entourage. The intersection of all the entourages is an equivalence relation on X which is just that of topological indistinguishability.
• Let X have the initial topology with respect to a family of functions $\{f_{\alpha }:X\to Y_{\alpha }\}$. Then two points x and y in X will be topologically indistinguishable if the family $f_{\alpha }$ does not separate them (i.e. $f_{\alpha }(x)=f_{\alpha }(y)$ for all $\alpha $).
• Given any equivalence relation on a set X there is a topology on X for which the notion of topological indistinguishability agrees with the given equivalence relation. One can simply take the equivalence classes as a base for the topology. This is called the partition topology on X.
Specialization preorder
The topological indistinguishability relation on a space X can be recovered from a natural preorder on X called the specialization preorder. For points x and y in X this preorder is defined by
x ≤ y if and only if x ∈ cl{y}
where cl{y} denotes the closure of {y}. Equivalently, x ≤ y if the neighborhood system of x, denoted Nx, is contained in the neighborhood system of y:
x ≤ y if and only if Nx ⊂ Ny.
It is easy to see that this relation on X is reflexive and transitive and so defines a preorder. In general, however, this preorder will not be antisymmetric. Indeed, the equivalence relation determined by ≤ is precisely that of topological indistinguishability:
x ≡ y if and only if x ≤ y and y ≤ x.
A topological space is said to be symmetric (or R0) if the specialization preorder is symmetric (i.e. x ≤ y implies y ≤ x). In this case, the relations ≤ and ≡ are identical. Topological indistinguishability is better behaved in these spaces and easier to understand. Note that this class of spaces includes all regular and completely regular spaces.
Properties
Equivalent conditions
There are several equivalent ways of determining when two points are topologically indistinguishable. Let X be a topological space and let x and y be points of X. Denote the respective closures of x and y by cl{x} and cl{y}, and the respective neighborhood systems by Nx and Ny. Then the following statements are equivalent:
• x ≡ y
• for each open set U in X, U contains either both x and y or neither of them
• Nx = Ny
• x ∈ cl{y} and y ∈ cl{x}
• cl{x} = cl{y}
• x ∈ ∩Ny and y ∈ ∩Nx
• ∩Nx = ∩Ny
• x ∈ cl{y} and x ∈ ∩Ny
• x belongs to every open set and every closed set containing y
• a net or filter converges to x if and only if it converges to y
These conditions can be simplified in the case where X is symmetric space. For these spaces (in particular, for regular spaces), the following statements are equivalent:
• x ≡ y
• for each open set U, if x ∈ U then y ∈ U
• Nx ⊂ Ny
• x ∈ cl{y}
• x ∈ ∩Ny
• x belongs to every closed set containing y
• x belongs to every open set containing y
• every net or filter that converges to x converges to y
Equivalence classes
To discuss the equivalence class of x, it is convenient to first define the upper and lower sets of x. These are both defined with respect to the specialization preorder discussed above.
The lower set of x is just the closure of {x}:
$\mathop {\downarrow } x=\{y\in X:y\leq x\}={\textrm {cl}}\{x\}$
while the upper set of x is the intersection of the neighborhood system at x:
$\mathop {\uparrow } x=\{y\in X:x\leq y\}=\bigcap {\mathcal {N}}_{x}.$
The equivalence class of x is then given by the intersection
$[x]={\mathop {\downarrow } x}\cap {\mathop {\uparrow } x}.$
Since ↓x is the intersection of all the closed sets containing x and ↑x is the intersection of all the open sets containing x, the equivalence class [x] is the intersection of all the open sets and closed sets containing x.
Both cl{x} and ∩Nx will contain the equivalence class [x]. In general, both sets will contain additional points as well. In symmetric spaces (in particular, in regular spaces) however, the three sets coincide:
$[x]={\textrm {cl}}\{x\}=\bigcap {\mathcal {N}}_{x}.$
In general, the equivalence classes [x] will be closed if and only if the space is symmetric.
Continuous functions
Let f : X → Y be a continuous function. Then for any x and y in X
x ≡ y implies f(x) ≡ f(y).
The converse is generally false (There are quotients of T0 spaces which are trivial). The converse will hold if X has the initial topology induced by f. More generally, if X has the initial topology induced by a family of maps $f_{\alpha }:X\to Y_{\alpha }$ then
x ≡ y if and only if fα(x) ≡ fα(y) for all α.
It follows that two elements in a product space are topologically indistinguishable if and only if each of their components are topologically indistinguishable.
Kolmogorov quotient
Since topological indistinguishability is an equivalence relation on any topological space X, we can form the quotient space KX = X/≡. The space KX is called the Kolmogorov quotient or T0 identification of X. The space KX is, in fact, T0 (i.e. all points are topologically distinguishable). Moreover, by the characteristic property of the quotient map any continuous map f : X → Y from X to a T0 space factors through the quotient map q : X → KX.
Although the quotient map q is generally not a homeomorphism (since it is not generally injective), it does induce a bijection between the topology on X and the topology on KX. Intuitively, the Kolmogorov quotient does not alter the topology of a space. It just reduces the point set until points become topologically distinguishable.
See also
• Hausdorff space – Type of topological space
• Locally Hausdorff space
• Separation axiom – Axioms in topology defining notions of "separation"
• Specialization preorder – subject in topologyPages displaying wikidata descriptions as a fallback
• T0 space – Concept in topologyPages displaying short descriptions of redirect targets
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| Wikipedia |
Tame manifold
In geometry, a tame manifold is a manifold with a well-behaved compactification. More precisely, a manifold $M$ is called tame if it is homeomorphic to a compact manifold with a closed subset of the boundary removed.
Not to be confused with Tame topology.
The Whitehead manifold is an example of a contractible manifold that is not tame.
See also
• Closed manifold – compact manifold without boundaryPages displaying wikidata descriptions as a fallback
• Tameness theorem
References
• Gabai, David (2009), "Hyperbolic geometry and 3-manifold topology", in Mrowka, Tomasz S.; Ozsváth, Peter S. (eds.), Low dimensional topology, IAS/Park City Math. Ser., vol. 15, Providence, R.I.: Amer. Math. Soc., pp. 73–103, ISBN 978-0-8218-4766-4, MR 2503493
• Marden, Albert (2007), Outer circles, Cambridge University Press, doi:10.1017/CBO9780511618918, ISBN 978-0-521-83974-7, MR 2355387
• Tucker, Thomas W. (1974), "Non-compact 3-manifolds and the missing-boundary problem", Topology, 13 (3): 267–273, doi:10.1016/0040-9383(74)90019-6, ISSN 0040-9383, MR 0353317
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| Wikipedia |
Topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the function sin(1/x) on the half-open interval (0, 1], together with the origin, under the topology induced from the Euclidean plane:
$T=\left\{\left(x,\sin {\tfrac {1}{x}}\right):x\in (0,1]\right\}\cup \{(0,0)\}.$
Properties
The topologist's sine curve T is connected but neither locally connected nor path connected. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a path.
The space T is the continuous image of a locally compact space (namely, let V be the space {−1} ∪ (0, 1], and use the map f from V to T defined by f(−1) = (0,0) and f(x) = (x, sin(1/x)) for x > 0), but T is not locally compact itself.
The topological dimension of T is 1.
Variants
Two variants of the topologist's sine curve have other interesting properties.
The closed topologist's sine curve can be defined by taking the topologist's sine curve and adding its set of limit points, $\{(0,y)\mid y\in [-1,1]\}$; some texts define the topologist's sine curve itself as this closed version, as they prefer to use the term 'closed topologist's sine curve' to refer to another curve.[1] This space is closed and bounded and so compact by the Heine–Borel theorem, but has similar properties to the topologist's sine curve—it too is connected but neither locally connected nor path-connected.
The extended topologist's sine curve can be defined by taking the closed topologist's sine curve and adding to it the set $\{(x,1)\mid x\in [0,1]\}$. It is arc connected but not locally connected.
See also
• List of topologies
• Warsaw circle
References
1. Munkres, James R (1979). Topology; a First Course. Englewood Cliffs. p. 158. ISBN 9780139254956.
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Mineola, NY: Dover Publications, Inc., pp. 137–138, ISBN 978-0-486-68735-3, MR 1382863
• Weisstein, Eric W. "Topologist's Sine Curve". MathWorld.
| Wikipedia |
Topology optimization
Topology optimization (TO) is a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximizing the performance of the system. Topology optimization is different from shape optimization and sizing optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations.
The conventional topology optimization formulation uses a finite element method (FEM) to evaluate the design performance. The design is optimized using either gradient-based mathematical programming techniques such as the optimality criteria algorithm and the method of moving asymptotes or non gradient-based algorithms such as genetic algorithms.
Topology optimization has a wide range of applications in aerospace, mechanical, bio-chemical and civil engineering. Currently, engineers mostly use topology optimization at the concept level of a design process. Due to the free forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from topology optimization is often fine-tuned for manufacturability. Adding constraints to the formulation in order to increase the manufacturability is an active field of research. In some cases results from topology optimization can be directly manufactured using additive manufacturing; topology optimization is thus a key part of design for additive manufacturing.
Problem statement
A topology optimization problem can be written in the general form of an optimization problem as:
${\begin{aligned}&{\underset {\rho }{\operatorname {minimize} }}&&F=F(\mathbf {u(\rho ),\rho } )=\int _{\Omega }f(\mathbf {u(\rho ),\rho } )\mathrm {d} V\\&\operatorname {subject\;to} &&G_{0}(\rho )=\int _{\Omega }\rho \mathrm {d} V-V_{0}\leq 0\\&&&G_{j}(\mathbf {u} (\rho ),\rho )\leq 0{\text{ with }}j=1,...,m\end{aligned}}$
The problem statement includes the following:
• An objective function $F(\mathbf {u(\rho ),\rho } )$. This function represents the quantity that is being minimized for best performance. The most common objective function is compliance, where minimizing compliance leads to maximizing the stiffness of a structure.
• The material distribution as a problem variable. This is described by the density of the material at each location $\rho (\mathbf {x} )$. Material is either present, indicated by a 1, or absent, indicated by a 0. $\mathbf {u} =\mathbf {u} (\mathbf {\rho } )$ is a state field that satisfies a linear or nonlinear state equation depending on $\rho $.
• The design space $(\Omega )$. This indicates the allowable volume within which the design can exist. Assembly and packaging requirements, human and tool accessibility are some of the factors that need to be considered in identifying this space . With the definition of the design space, regions or components in the model that cannot be modified during the course of the optimization are considered as non-design regions.
• $\scriptstyle m$ constraints $G_{j}(\mathbf {u} (\rho ),\rho )\leq 0$ a characteristic that the solution must satisfy. Examples are the maximum amount of material to be distributed (volume constraint) or maximum stress values.
Evaluating $\mathbf {u(\rho )} $ often includes solving a differential equation. This is most commonly done using the finite element method since these equations do not have a known analytical solution.
Implementation methodologies
There are various implementation methodologies that have been used to solve topology optimization problems.
Discrete
Solving topology optimization problems in a discrete sense is done by discretizing the design domain into finite elements. The material densities inside these elements are then treated as the problem variables. In this case material density of one indicates the presence of material, while zero indicates an absence of material. Owing to the attainable topological complexity of the design being dependent on the number of elements, a large number is preferred. Large numbers of finite elements increases the attainable topological complexity, but come at a cost. Firstly, solving the FEM system becomes more expensive. Secondly, algorithms that can handle a large number (several thousands of elements is not uncommon) of discrete variables with multiple constraints are unavailable. Moreover, they are impractically sensitive to parameter variations.[1] In literature problems with up to 30000 variables have been reported.[2]
Solving the problem with continuous variables
The earlier stated complexities with solving topology optimization problems using binary variables has caused the community to search for other options. One is the modelling of the densities with continuous variables. The material densities can now also attain values between zero and one. Gradient based algorithms that handle large amounts of continuous variables and multiple constraints are available. But the material properties have to be modelled in a continuous setting. This is done through interpolation. One of the most implemented interpolation methodologies is the Solid Isotropic Material with Penalisation method (SIMP).[3][4] This interpolation is essentially a power law $E\;=\;E_{0}\,+\,\rho ^{p}(E_{1}-E_{0})$. It interpolates the Young's modulus of the material to the scalar selection field. The value of the penalisation parameter $p$ is generally taken between $[1,\,3]$. This has been shown to confirm the micro-structure of the materials.[5] In the SIMP method a lower bound on the Young's modulus is added, $E_{0}$, to make sure the derivatives of the objective function are non-zero when the density becomes zero. The higher the penalisation factor, the more SIMP penalises the algorithm in the use of non-binary densities. Unfortunately, the penalisation parameter also introduces non-convexities.[6]
Commercial software
There are several commercial topology optimization software on the market. Most of them use topology optimization as a hint how the optimal design should look like, and manual geometry re-construction is required. There are a few solutions which produce optimal designs ready for Additive Manufacturing.
Examples
Structural compliance
A stiff structure is one that has the least possible displacement when given certain set of boundary conditions. A global measure of the displacements is the strain energy (also called compliance) of the structure under the prescribed boundary conditions. The lower the strain energy the higher the stiffness of the structure. So, the objective function of the problem is to minimize the strain energy.
On a broad level, one can visualize that the more the material, the less the deflection as there will be more material to resist the loads. So, the optimization requires an opposing constraint, the volume constraint. This is in reality a cost factor, as we would not want to spend a lot of money on the material. To obtain the total material utilized, an integration of the selection field over the volume can be done.
Finally the elasticity governing differential equations are plugged in so as to get the final problem statement.
$\min _{\rho }\;\int _{\Omega }{\frac {1}{2}}\mathbf {\sigma } :\mathbf {\varepsilon } \,\mathrm {d} \Omega $ :\mathbf {\varepsilon } \,\mathrm {d} \Omega }
subject to:
• $\rho \,\in \,[0,\,1]$
• $\int _{\Omega }\rho \,\mathrm {d} \Omega \;\leq \;V^{*}$
• $\mathbf {\nabla } \cdot \mathbf {\sigma } \,+\,\mathbf {F} \;=\;{\mathbf {0} }$
• $\mathbf {\sigma } \;=\;{\mathsf {C}}:\mathbf {\varepsilon } $
But, a straightforward implementation in the finite element framework of such a problem is still infeasible owing to issues such as:
1. Mesh dependency—Mesh Dependency means that the design obtained on one mesh is not the one that will be obtained on another mesh. The features of the design become more intricate as the mesh gets refined.[7]
2. Numerical instabilities—The selection of region in the form of a chess board.[8]
Some techniques such as filtering based on image processing[9] are currently being used to alleviate some of these issues. Although it seemed like this was purely a heuristic approach for a long time, theoretical connections to nonlocal elasticity have been made to support the physical sense of these methods.[10]
Fluid-structure-interaction
Fluid-structure-interaction is a strongly coupled phenomenon and concerns the interaction between a stationary or moving fluid and an elastic structure. Many engineering applications and natural phenomena are subject to fluid-structure-interaction and to take such effects into consideration is therefore critical in the design of many engineering applications. Topology optimisation for fluid structure interaction problems has been studied in e.g. references[11][12][13] and.[14] Design solutions solved for different Reynolds numbers are shown below. The design solutions depend on the fluid flow with indicate that the coupling between the fluid and the structure is resolved in the design problems.
Design solution and velocity field for Re=1
Design solution and velocity field for Re=5
Design solution and pressure field for Re=10
Design solution and pressure field for Re=40
Design solutions for different Reynolds number for a wall inserted in a channel with a moving fluid.
Thermoelectric energy conversion
Thermoelectricity is a multi-physic problem which concerns the interaction and coupling between electric and thermal energy in semi conducting materials. Thermoelectric energy conversion can be described by two separately identified effects: The Seebeck effect and the Peltier effect. The Seebeck effect concerns the conversion of thermal energy into electric energy and the Peltier effect concerns the conversion of electric energy into thermal energy.[15] By spatially distributing two thermoelectric materials in a two dimensional design space with a topology optimisation methodology,[16] it is possible to exceed performance of the constitutive thermoelectric materials for thermoelectric coolers and thermoelectric generators.[17]
3F3D Form Follows Force 3D Printing
The current proliferation of 3D printer technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization combined with 3D printing can result in less weight, improved structural performance and shortened design-to-manufacturing cycle. As the designs, while efficient, might not be realisable with more traditional manufacturing techniques.
Design-dependent loads
The direction, magnitude, and location of a design-dependent load alter with topology optimization iterations. Therefore, dealing with such loads in a TO setting is a challenging task. One can find novel methods to deal with such loads (e.g. pressure load, self-weight, etc) in Refs.[18] [19] [20]
References
1. Sigmund, Ole; Maute, Kurt (2013). "Topology optimization approaches". Structural and Multidisciplinary Optimization. 48 (6): 1031–1055. doi:10.1007/s00158-013-0978-6. S2CID 124426387.
2. Beckers, M. (1999). "Topology optimization using a dual method with discrete variables" (PDF). Structural Optimization. 17: 14–24. doi:10.1007/BF01197709. S2CID 122845784.
3. Bendsøe, M. P. (1989). "Optimal shape design as a material distribution problem". Structural Optimization. 1 (4): 193–202. doi:10.1007/BF01650949. S2CID 18253872.
4. , a monograph of the subject.
5. Bendsøe, M. P.; Sigmund, O. (1999). "Material interpolation schemes in topology optimization" (PDF). Archive of Applied Mechanics. 69 (9–10): 635–654. Bibcode:1999AAM....69..635B. doi:10.1007/s004190050248. S2CID 11368603.
6. van Dijk, NP. Langelaar, M. van Keulen, F. Critical study of design parameterization in topology optimization; The influence of design parameterization on local minima.. 2nd International Conference on Engineering Optimization, 2010
7. Allaire, Grégoire; Henrot, Antoine (May 2001). "On some recent advances in shape optimization". Comptes Rendus de l'Académie des Sciences. Series IIB - Mechanics. Elsevier. 329 (5): 383–396. doi:10.1016/S1620-7742(01)01349-6. ISSN 1620-7742. Retrieved 2021-09-12.
8. Shukla, Avinash; Misra, Anadi; Kumar, Sunil (September 2013). "Checkerboard Problem in Finite Element Based Topology Optimization". International Journal of Advances in Engineering & Technology. CiteSeer. 6 (4): 1769–1774. CiteSeerX 10.1.1.670.6771. ISSN 2231-1963. Retrieved 2022-02-14.
9. Bourdin, Blaise (2001-03-30). "Filters in topology optimization". International Journal for Numerical Methods in Engineering. Wiley. 50 (9): 2143–2158. doi:10.1002/nme.116. ISSN 1097-0207. Retrieved 2020-08-02.
10. Sigmund, Ole; Maute, Kurt (October 2012). "Sensitivity filtering from a continuum mechanics perspective". Structural and Multidisciplinary Optimization. Springer. 46 (4): 471–475. doi:10.1007/s00158-012-0814-4. ISSN 1615-1488. Retrieved 2021-06-17.
11. Yoon, Gil Ho (2010). "Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation". International Journal for Numerical Methods in Engineering. 82 (5): 591–616. Bibcode:2010IJNME..82..591Y. doi:10.1002/nme.2777.
12. Picelli, R.; Vicente, W.M.; Pavanello, R. (2017). "Evolutionary topology optimization for structural compliance minimization considering design-dependent FSI loads". Finite Elements in Analysis and Design. 135: 44–55. doi:10.1016/j.finel.2017.07.005.
13. Jenkins, Nicholas; Maute, Kurt (2016). "An immersed boundary approach for shape and topology optimization of stationary fluid-structure interaction problems". Structural and Multidisciplinary Optimization. 54 (5): 1191–1208. doi:10.1007/s00158-016-1467-5. S2CID 124632210.
14. Lundgaard, Christian; Alexandersen, Joe; Zhou, Mingdong; Andreasen, Casper Schousboe; Sigmund, Ole (2018). "Revisiting density-based topology optimization for fluid-structure-interaction problems" (PDF). Structural and Multidisciplinary Optimization. 58 (3): 969–995. doi:10.1007/s00158-018-1940-4. S2CID 125798826.
15. Rowe, David Michael. Thermoelectrics handbook: macro to nano. CRC press, 2005.
16. Lundgaard, Christian; Sigmund, Ole (2018). "A density-based topology optimization methodology for thermoelectric energy conversion problems" (PDF). Structural and Multidisciplinary Optimization. 57 (4): 1427–1442. doi:10.1007/s00158-018-1919-1. S2CID 126031362.
17. Lundgaard, Christian; Sigmund, Ole; Bjørk, Rasmus (2018). "Topology Optimization of Segmented Thermoelectric Generators". Journal of Electronic Materials. 47 (12): 6959–6971. Bibcode:2018JEMat..47.6959L. doi:10.1007/s11664-018-6606-x. S2CID 105113187.
18. Kumar, P; Frouws, J S; Langelaar, M (2020). "Topology optimization of fluidic pressure-loaded structures and compliant mechanisms using the Darcy method". Structural and Multidisciplinary Optimization. 61 (4): 1637–1655. arXiv:1909.03292. doi:10.1007/s00158-019-02442-0.
19. Kumar, P; Langelaar, M (2021). "On topology optimization of design‐dependent pressure‐loaded three‐dimensional structures and compliant mechanisms". International Journal for Numerical Methods in Engineering. 122 (9): 2205–2220. arXiv:2009.05839. doi:10.1002/nme.6618.
20. Kumar, P (2022). "Topology optimization of stiff structures under self-weight for given volume using a smooth Heaviside function". Structural and Multidisciplinary Optimization. 65 (4): 1–17. arXiv:2111.13875. doi:10.1007/s00158-022-03232-x.
Further reading
• Recent Developments in the Commercial Implementation of Topology Optimization; Uwe Schramm, Ming Zhou; IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials: Status and Perspectives, 239–248; 2006 Springer.
• Industrial Implementation and Applications of Topology Optimization and Future Needs; Claus B.W. Pedersen; Peter Allinger; IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, 229-238; 2006 Springer.
• Topology optimization of 2D continua for minimum compliance using parallel computing Arash Mahdavi; Balaji Raghavan; Mary Frecker; Int Journal of Structural and Multidisciplinary Optimization, Volume 32, 121-132, 2006 Springer
• Modern Structural Optimization Concepts Applied to Topology Optimization Juan Pablo Leiva; Brian C. Watson and Iku Kosaka ; 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Material Conference. St. Louis, MO, pp. 1589–1596, 1999
External links
• Topology optimization animations
| Wikipedia |
Toral subalgebra
In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field).[1] Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algebra is abelian;[1][2] thus, its elements are simultaneously diagonalizable.
In semisimple and reductive Lie algebras
A subalgebra ${\mathfrak {h}}$ of a semisimple Lie algebra ${\mathfrak {g}}$ is called toral if the adjoint representation of ${\mathfrak {h}}$ on ${\mathfrak {g}}$, $\operatorname {ad} ({\mathfrak {h}})\subset {\mathfrak {gl}}({\mathfrak {g}})$ is a toral subalgebra. A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra, over an algebraically closed field of characteristic 0 is a Cartan subalgebra and vice versa.[3] In particular, a maximal toral Lie subalgebra in this setting is self-normalizing, coincides with its centralizer, and the Killing form of ${\mathfrak {g}}$ restricted to ${\mathfrak {h}}$ is nondegenerate.
For more general Lie algebras, a Cartan subalgebra may differ from a maximal toral subalgebra.
In a finite-dimensional semisimple Lie algebra ${\mathfrak {g}}$ over an algebraically closed field of a characteristic zero, a toral subalgebra exists.[1] In fact, if ${\mathfrak {g}}$ has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence, ${\mathfrak {g}}$ must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.
See also
• Maximal torus, in the theory of Lie groups
References
1. Humphreys 1972, Ch. II, § 8.1.
2. Proof (from Humphreys): Let $x\in {\mathfrak {h}}$. Since $\operatorname {ad} (x)$ is diagonalizable, it is enough to show the eigenvalues of $\operatorname {ad} _{\mathfrak {h}}(x)$ are all zero. Let $y\in {\mathfrak {h}}$ be an eigenvector of $\operatorname {ad} _{\mathfrak {h}}(x)$ with eigenvalue $\lambda $. Then $x$ is a sum of eigenvectors of $\operatorname {ad} _{\mathfrak {h}}(y)$ and then $-\lambda y=\operatorname {ad} _{\mathfrak {h}}(y)x$ is a linear combination of eigenvectors of $\operatorname {ad} _{\mathfrak {h}}(y)$ with nonzero eigenvalues. But, unless $\lambda =0$, we have that $-\lambda y$ is an eigenvector of $\operatorname {ad} _{\mathfrak {h}}(y)$ with eigenvalue zero, a contradiction. Thus, $\lambda =0$. $\square $
3. Humphreys 1972, Ch. IV, § 15.3. Corollary
• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
• Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7
| Wikipedia |
JSJ decomposition
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.
The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently.
The characteristic submanifold
An alternative version of the JSJ decomposition states:
A closed irreducible orientable 3-manifold M has a submanifold Σ that is a Seifert manifold (possibly disconnected and with boundary) whose complement is atoroidal (and possibly disconnected).
The submanifold Σ with the smallest number of boundary tori is called the characteristic submanifold of M; it is unique (up to isotopy). Cutting the manifold along the tori bounding the characteristic submanifold is also sometimes called a JSJ decomposition, though it may have more tori than the standard JSJ decomposition.
The boundary of the characteristic submanifold Σ is a union of tori that are almost the same as the tori appearing in the JSJ decomposition. However there is a subtle difference: if one of the tori in the JSJ decomposition is "non-separating", then the boundary of the characteristic submanifold has two parallel copies of it (and the region between them is a Seifert manifold isomorphic to the product of a torus and a unit interval). The set of tori bounding the characteristic submanifold can be characterised as the unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that closure of each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.
The JSJ decomposition is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. For example, the mapping torus of an Anosov map of a torus has a finite volume sol structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.
See also
• Geometrization conjecture
• Manifold decomposition
• Satellite knot
References
• Jaco, William H.; Shalen, Peter B (1979), "Seifert fibered spaces in 3-manifolds", Memoirs of the American Mathematical Society, 21 (220).
• Jaco, William; Shalen, Peter B. Seifert fibered spaces in 3-manifolds. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 91–99, Academic Press, New York-London, 1979.
• Jaco, William; Shalen, Peter B. A new decomposition theorem for irreducible sufficiently-large 3-manifolds. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 71–84, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.
• Johannson, Klaus, Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Mathematics, 761. Springer, Berlin, 1979. ISBN 3-540-09714-7
External links
• Allen Hatcher, Notes on Basic 3-Manifold Topology.
• William Jaco, JSJ Decomposition of 3-manifolds. This lecture gives a brief introduction to Seifert fibered 3-manifolds and provides the existence and uniqueness theorem of Jaco, Shalen, and Johannson for the JSJ decomposition of a 3-manifold.
• William Jaco, An Algorithm to Construct the JSJ Decomposition of a 3-manifold. An algorithm is given for constructing the JSJ-decomposition of a 3-manifold and deriving the Seifert invariants of the Characteristic submanifold.
| Wikipedia |
Tord Ganelius
Tord Hjalmar Ganelius (born 23 May 1925 in Stockholm, dead 14 March 2016 in Stockholm) was a Swedish mathematician and professor emeritus. He served as Permanent Secretary of the Royal Swedish Academy of Sciences and was a board member of the Nobel Foundation from 1981 to 1989.[1][2] His primary research interests were holomorphic functions and approximation theory.[3]
Education and career
Ganelius completed his Ph.D. in 1953 at Stockholms Högskola (known as Stockholm University since 1960) by presenting his dissertation Sequences of Analytic Functions and Their Zeros.[4]
He was an associate professor at Lund University in 1953–56 and in 1957 he was appointed Professor of Mathematics at Gothenburg University, where he served until 1981. During this time he was Dean of the Faculty of Science twice, in 1963–65 and 1977–80.[1] In 1972, he was elected a member of the Royal Swedish Academy of Sciences[5] and he was appointed the permanent secretary in 1981, a position he held until 1989.[2] He was a board member of the Nobel Foundation from 1981 to 1989, and every October he announced the Nobel Laureates in Physics, Chemistry and Economic Sciences.[1]
Tord Ganelius has also been a visiting professor at the University of Washington in 1962, Cornell University in 1967–68 and the University of California, San Diego in 1972–73.[1]
In 1966, Ganelius published the Swedish-language mathematics textbook ”Introduktion till matematiken”, which since 2006 has been available on-line.[6]
Family
Tord Ganelius is the son of Hjalmar and Ebba G. In 1951 he married Aggie Hemberg (born 1928).[1] They have four children: Per (1952), Truls (1955), Svante (1957) and Aggie Öhman (1963).[7]
References
1. Hans Uddling Katrin Paabo, ed. (1993). Vem är det, svensk biografisk handbok (in Swedish). Nordstedts. p. 368. ISBN 91-1-914072-X. Retrieved 17 October 2012.
2. Royal Swedish Academy of Sciences
3. Nationalencyklopedin (in Swedish). Höganäs: Bokförlaget Bra böcker. 1992. p. 329. ISBN 91-7024-621-1.
4. Ganelius, Tord Hjalmar (1953). Sequences of analytic functions and their zeros (in Swedish). Stockholm, London & Paris. libris 2660006.{{cite book}}: CS1 maint: location missing publisher (link)
5. "Memberpages Royal Swedish Academy of Sciences". Retrieved 17 October 2012.
6. "Introduktion till matematiken". Archived from the original on 9 October 2006. Retrieved 17 October 2012.
7. Paul Harnesk, ed. (1965). Vem är vem (in Swedish). Stockholm: Bokförlaget Vem är vem AB. p. 379. Retrieved 17 October 2012.
Authority control
International
• ISNI
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• Germany
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• MathSciNet
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People
• Deutsche Biographie
Other
• SNAC
• IdRef
| Wikipedia |
Mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Motivation
Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets K into open subsets U as K and U range throughout our original topological space, completed with their finite intersections (which must be open by definition of topology) and arbitrary unions (again which must be open). This gives a notion of continuity on the space of functions, so that we can consider continuous deformation of the homeomorphisms themselves: called homotopies. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the functional composition group structure already present on the space of homeomorphisms.
Definition
The term mapping class group has a flexible usage. Most often it is used in the context of a manifold M. The mapping class group of M is interpreted as the group of isotopy classes of automorphisms of M. So if M is a topological manifold, the mapping class group is the group of isotopy classes of homeomorphisms of M. If M is a smooth manifold, the mapping class group is the group of isotopy classes of diffeomorphisms of M. Whenever the group of automorphisms of an object X has a natural topology, the mapping class group of X is defined as $\operatorname {Aut} (X)/\operatorname {Aut} _{0}(X)$, where $\operatorname {Aut} _{0}(X)$ is the path-component of the identity in $\operatorname {Aut} (X)$. (Notice that in the compact-open topology, path components and isotopy classes coincide, i.e., two maps f and g are in the same path-component iff they are isotopic). For topological spaces, this is usually the compact-open topology. In the low-dimensional topology literature, the mapping class group of X is usually denoted MCG(X), although it is also frequently denoted $\pi _{0}(\operatorname {Aut} (X))$, where one substitutes for Aut the appropriate group for the category to which X belongs. Here $\pi _{0}$ denotes the 0-th homotopy group of a space.
So in general, there is a short exact sequence of groups:
$1\rightarrow \operatorname {Aut} _{0}(X)\rightarrow \operatorname {Aut} (X)\rightarrow \operatorname {MCG} (X)\rightarrow 1.$
Frequently this sequence is not split.[1]
If working in the homotopy category, the mapping class group of X is the group of homotopy classes of homotopy equivalences of X.
There are many subgroups of mapping class groups that are frequently studied. If M is an oriented manifold, $\operatorname {Aut} (M)$ would be the orientation-preserving automorphisms of M and so the mapping class group of M (as an oriented manifold) would be index two in the mapping class group of M (as an unoriented manifold) provided M admits an orientation-reversing automorphism. Similarly, the subgroup that acts as the identity on all the homology groups of M is called the Torelli group of M.
Examples
Sphere
In any category (smooth, PL, topological, homotopy)[2]
$\operatorname {MCG} (S^{2})\simeq \mathbb {Z} /2\mathbb {Z} ,$
corresponding to maps of degree ±1.
Torus
In the homotopy category
$\operatorname {MCG} (\mathbf {T} ^{n})\simeq \operatorname {GL} (n,\mathbb {Z} ).$
This is because the n-dimensional torus $\mathbf {T} ^{n}=(S^{1})^{n}$ is an Eilenberg–MacLane space.
For other categories if $n\geq 5$,[3] one has the following split-exact sequences:
In the category of topological spaces
$0\to \mathbb {Z} _{2}^{\infty }\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0$
In the PL-category
$0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0$
(⊕ representing direct sum). In the smooth category
$0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\oplus \sum _{i=0}^{n}{\binom {n}{i}}\Gamma _{i+1}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0$
where $\Gamma _{i}$ are the Kervaire–Milnor finite abelian groups of homotopy spheres and $\mathbb {Z} _{2}$ is the group of order 2.
Surfaces
Main article: Mapping class group of a surface
The mapping class groups of surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of $\operatorname {MCG} (\mathbf {T} ^{2})$ above), since they act on Teichmüller space and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to hyperbolic groups and to higher rank linear groups. They have many applications in Thurston's theory of geometric three-manifolds (for example, to surface bundles). The elements of this group have also been studied by themselves: an important result is the Nielsen–Thurston classification theorem, and a generating family for the group is given by Dehn twists which are in a sense the "simplest" mapping classes. Every finite group is a subgroup of the mapping class group of a closed, orientable surface,;[4] in fact one can realize any finite group as the group of isometries of some compact Riemann surface (which immediately implies that it injects in the mapping class group of the underlying topological surface).
Non-orientable surfaces
Some non-orientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the real projective plane $\mathbf {P} ^{2}(\mathbb {R} )$ is isotopic to the identity:
$\operatorname {MCG} (\mathbf {P} ^{2}(\mathbb {R} ))=1.$
The mapping class group of the Klein bottle K is:
$\operatorname {MCG} (K)=\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}.$
The four elements are the identity, a Dehn twist on a two-sided curve which does not bound a Möbius strip, the y-homeomorphism of Lickorish, and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity.
We also remark that the closed genus three non-orientable surface N3 (the connected sum of three projective planes) has:
$\operatorname {MCG} (N_{3})=\operatorname {GL} (2,\mathbb {Z} ).$
This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface $N\setminus C$ is a torus with a disk removed. As an unoriented surface, its mapping class group is $\operatorname {GL} (2,\mathbb {Z} )$. (Lemma 2.1[5]).
3-Manifolds
Mapping class groups of 3-manifolds have received considerable study as well, and are closely related to mapping class groups of 2-manifolds. For example, any finite group can be realized as the mapping class group (and also the isometry group) of a compact hyperbolic 3-manifold.[6]
Mapping class groups of pairs
Given a pair of spaces (X,A) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of (X,A) is defined as an automorphism of X that preserves A, i.e. f: X → X is invertible and f(A) = A.
Symmetry group of knot and links
If K ⊂ S3 is a knot or a link, the symmetry group of the knot (resp. link) is defined to be the mapping class group of the pair (S3, K). The symmetry group of a hyperbolic knot is known to be dihedral or cyclic, moreover every dihedral and cyclic group can be realized as symmetry groups of knots. The symmetry group of a torus knot is known to be of order two Z2.
Torelli group
Notice that there is an induced action of the mapping class group on the homology (and cohomology) of the space X. This is because (co)homology is functorial and Homeo0 acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy). The kernel of this action is the Torelli group, named after the Torelli theorem.
In the case of orientable surfaces, this is the action on first cohomology H1(Σ) ≅ Z2g. Orientation-preserving maps are precisely those that act trivially on top cohomology H2(Σ) ≅ Z. H1(Σ) has a symplectic structure, coming from the cup product; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are realized, yielding the short exact sequence:
$1\to \operatorname {Tor} (\Sigma )\to \operatorname {MCG} (\Sigma )\to \operatorname {Sp} (H^{1}(\Sigma ))\cong \operatorname {Sp} _{2g}(\mathbf {Z} )\to 1$
One can extend this to
$1\to \operatorname {Tor} (\Sigma )\to \operatorname {MCG} ^{*}(\Sigma )\to \operatorname {Sp} ^{\pm }(H^{1}(\Sigma ))\cong \operatorname {Sp} _{2g}^{\pm }(\mathbf {Z} )\to 1$
The symplectic group is well understood. Hence understanding the algebraic structure of the mapping class group often reduces to questions about the Torelli group.
Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the Torelli group vanishes.
Stable mapping class group
One can embed the surface $\Sigma _{g,1}$ of genus g and 1 boundary component into $\Sigma _{g+1,1}$ by attaching an additional hole on the end (i.e., gluing together $\Sigma _{g,1}$ and $\Sigma _{1,2}$), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the direct limit of these groups and inclusions yields the stable mapping class group, whose rational cohomology ring was conjectured by David Mumford (one of conjectures called the Mumford conjectures). The integral (not just rational) cohomology ring was computed in 2002 by Ib Madsen and Michael Weiss, proving Mumford's conjecture.
See also
• Braid groups, the mapping class groups of punctured discs
• Homotopy groups
• Homeotopy groups
• Lantern relation
References
1. Morita, Shigeyuki (1987). "Characteristic classes of surface bundles". Inventiones Mathematicae. 90 (3): 551–577. Bibcode:1987InMat..90..551M. doi:10.1007/bf01389178. MR 0914849.
2. Earle, Clifford J.; Eells, James (1967), "The diffeomorphism group of a compact Riemann surface", Bulletin of the American Mathematical Society, 73 (4): 557–559, doi:10.1090/S0002-9904-1967-11746-4, MR 0212840
3. Hatcher, A.E. (1978). "Concordance spaces, higher simple-homotopy theory, and applications". Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1. Proceedings of Symposia in Pure Mathematics. Vol. 32. pp. 3–21. doi:10.1090/pspum/032.1/520490. ISBN 978-0-8218-9320-3. MR 0520490.
4. Greenberg, Leon (1974). "Maximal groups and signatures". Discontinuous Groups and Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland. Annals of Mathematics Studies. Vol. 79. Princeton University Press. pp. 207–226. ISBN 978-1-4008-8164-2. MR 0379835.
5. Scharlemann, Martin (February 1982). "The complex of curves on nonorientable surfaces". Journal of the London Mathematical Society. s2-25 (1): 171–184. CiteSeerX 10.1.1.591.2588. doi:10.1112/jlms/s2-25.1.171.
6. Kojima, S. (August 1988). "Isometry transformations of hyperbolic 3-manifolds". Topology and Its Applications. 29 (3): 297–307. doi:10.1016/0166-8641(88)90027-2.
• Birman, Joan (1974). Braids, links and mapping class groups. Annals of Mathematical Studies. Vol. 82. Princeton, N.J.: Princeton University Press. ISBN 978-0691081496. MR 0375281.
• Casson, Andrew; Bleiler, Steve (2014) [1988]. Automorphisms of surfaces after Nielsen and Thurston. Cambridge University Press. ISBN 978-1-299-70610-1.
• Ivanov, Nikolai V. (2001). "9. Mapping class groups and arithmetic groups". Handbook of Geometric Topology. Elsevier. pp. 618–624. ISBN 978-0-08-053285-1.
• Farb, Benson; Margalit, Dan (2012). A Primer on Mapping Class Groups. Princeton University Press. ISBN 978-0-691-14794-9.
• Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I (PDF), IRMA Lectures in Mathematics and Theoretical Physics, vol. 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 978-3-03719-029-6, MR 2284826
• Lawton, Sean; Peterson, Elisha (2009), Papadopoulos, Athanase (ed.), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, vol. 13, European Mathematical Society (EMS), Zürich, arXiv:math/0511271, doi:10.4171/055, ISBN 978-3-03719-055-5, MR 2524085
• Papadopoulos, Athanase, ed. (2012), Handbook of Teichmüller theory. Vol. III, IRMA Lectures in Mathematics and Theoretical Physics, vol. 17, European Mathematical Society (EMS), Zürich, doi:10.4171/103, ISBN 978-3-03719-103-3, MR 2961353
• Papadopoulos, Athanase, ed. (2014), Handbook of Teichmüller theory. Vol. IV, IRMA Lectures in Mathematics and Theoretical Physics, vol. 19, European Mathematical Society (EMS), Zürich, doi:10.4171/117, ISBN 978-3-03719-117-0
Stable mapping class group
• Madsen, Ib; Weiss, Michael (2007). "The stable moduli space of Riemann surfaces: Mumford's conjecture". Annals of Mathematics. 165 (3): 843–941. arXiv:math/0212321. CiteSeerX 10.1.1.236.2025. doi:10.4007/annals.2007.165.843. JSTOR 20160047. S2CID 119721243.
External links
• Madsen-Weiss MCG Seminar; many references
| Wikipedia |
Torelli theorem
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field.[1] From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus $\geq 2$ are k-isomorphic for k any perfect field, so are the curves.[2]
This result has had many important extensions. It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is injective (on geometric points). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem. Secondly, to other period mappings. A case that has been investigated deeply is for K3 surfaces (by Viktor S. Kulikov, Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov)[3] and hyperkähler manifolds (by Misha Verbitsky, Eyal Markman and Daniel Huybrechts).[4]
Notes
1. James S. Milne, Jacobian Varieties, Theorem 12.1 in Cornell & Silverman (1986)
2. James S. Milne, Jacobian Varieties, Corollary 12.2 in Cornell & Silverman (1986)
3. Compact fibrations with hyperkähler fibers
4. Automorphisms of Hyperkähler manifolds
References
• Ruggiero Torelli (1913). "Sulle varietà di Jacobi". Rendiconti della Reale accademia nazionale dei Lincei. 22 (5): 98–103.
• André Weil (1957). "Zum Beweis des Torellischen Satzes". Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. IIa: 32–53.
• Cornell, Gary; Silverman, Joseph, eds. (1986), Arithmetic Geometry, New York: Springer-Verlag, ISBN 978-3-540-96311-0, MR 0861969
| Wikipedia |
Monomial ideal
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field.
A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal.
Definitions and Properties
Let $\mathbb {K} $ be a field and $R=\mathbb {K} [x]$ be the polynomial ring over $\mathbb {K} $ with n variables $x=x_{1},x_{2},\dotsc ,x_{n}$.
A monomial in $R$ is a product $x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{n}^{\alpha _{n}}$ for an n-tuple $\alpha =(\alpha _{1},\alpha _{2},\dotsc ,\alpha _{n})\in \mathbb {N} ^{n}$ of nonnegative integers.
The following three conditions are equivalent for an ideal $I\subseteq R$:
1. $I$ is generated by monomials,
2. If $ f=\sum _{\alpha \in \mathbb {N} ^{n}}c_{\alpha }x^{\alpha }\in I$, then $x^{\alpha }\in I$, provided that $c_{\alpha }$ is nonzero.
3. $I$ is torus fixed, i.e, given $(c_{1},c_{2},\dotsc ,c_{n})\in (\mathbb {K} ^{*})^{n}$, then $I$ is fixed under the action $f(x_{i})=c_{i}x_{i}$ for all $i$.
We say that $I\subseteq \mathbb {K} [x]$ is a monomial ideal if it satisfies any of these equivalent conditions.
Given a monomial ideal $I=(m_{1},m_{2},\dotsc ,m_{k})$, $f\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$ is in $I$ if and only if every monomial ideal term $f_{i}$ of $f$ is a multiple of one the $m_{j}$.[1]
Proof: Suppose $I=(m_{1},m_{2},\dotsc ,m_{k})$ and that $f\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$ is in $I$. Then $f=f_{1}m_{1}+f_{2}m_{2}+\dotsm +f_{k}m_{k}$, for some $f_{i}\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$.
For all $1\leqslant i\leqslant k$, we can express each $f_{i}$ as the sum of monomials, so that $f$ can be written as a sum of multiples of the $m_{i}$. Hence, $f$ will be a sum of multiples of monomial terms for at least one of the $m_{i}$.
Conversely, let $I=(m_{1},m_{2},\dotsc ,m_{k})$ and let each monomial term in $f\in \mathbb {K} [x_{1},x_{2},...,x_{n}]$ be a multiple of one of the $m_{i}$ in $I$. Then each monomial term in $I$ can be factored from each monomial in $f$. Hence $f$ is of the form $f=c_{1}m_{1}+c_{2}m_{2}+\dotsm +c_{k}m_{k}$ for some $c_{i}\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$, as a result $f\in I$.
The following illustrates an example of monomial and polynomial ideals.
Let $I=(xyz,y^{2})$ then the polynomial $x^{2}yz+3xy^{2}$ is in I, since each term is a multiple of an element in J, i.e., they can be rewritten as $x^{2}yz=x(xyz)$ and $3xy^{2}=3x(y^{2}),$ both in I. However, if $J=(xz^{2},y^{2})$, then this polynomial $x^{2}yz+3xy^{2}$ is not in J, since its terms are not multiples of elements in J.
Monomial Ideals and Young Diagrams
A monomial ideal can be interpreted as a Young diagram. Suppose $I\in \mathbb {R} [x,y]$, then $I$ can be interpreted in terms of the minimal monomials generators as $I=(x^{a_{1}}y^{b_{1}},x^{a_{2}}y^{b_{2}},\dotsc ,x^{a_{k}}y^{b_{k}})$, where $a_{1}>a_{2}>\dotsm >a_{k}\geq 0$ and $b_{k}>\dotsm >b_{2}>b_{1}\geq 0$. The minimal monomial generators of $I$ can be seen as the inner corners of the Young diagram. The minimal generators would determine where we would draw the staircase diagram.[2] The monomials not in $I$ lie inside the staircase, and these monomials form a vector space basis for the quotient ring $\mathbb {R} [x,y]/I$.
Consider the following example. Let $I=(x^{3},x^{2}y,y^{3})\subset \mathbb {R} [x,y]$ be a monomial ideal. Then the set of grid points $S={\{(3,0),(2,1),(0,3)}\}\subset \mathbb {N} ^{2}$ corresponds to the minimal monomial generators $x^{3}y^{0},x^{2}y^{1},x^{0}y^{3}$ in $I$. Then as the figure shows, the pink Young diagram consists of the monomials that are not in $I$. The points in the inner corners of the Young diagram, allow us to identify the minimal monomials $x^{0}y^{3},x^{2}y^{1},x^{3}y^{0}$ in $I$ as seen in the green boxes. Hence, $I=(y^{3},x^{2}y,x^{3})$.
In general, to any set of grid points, we can associate a Young diagram, so that the monomial ideal is constructed by determining the inner corners that make up the staircase diagram; likewise, given a monomial ideal, we can make up the Young diagram by looking at the $(a_{i},b_{j})$ and representing them as the inner corners of the Young diagram. The coordinates of the inner corners would represent the powers of the minimal monomials in $I$. Thus, monomial ideals can be described by Young diagrams of partitions.
Moreover, the $(\mathbb {C} ^{*})^{2}$-action on the set of $I\subset \mathbb {C} [x,y]$ such that $\dim _{\mathbb {C} }\mathbb {C} [x,y]/I=n$ as a vector space over $\mathbb {C} $ has fixed points corresponding to monomial ideals only, which correspond to partitions of size n, which are identified by Young diagrams with n boxes.
Monomial Ordering and Gröbner Basis
A monomial ordering is a well ordering $\geq $ on the set of monomials such that if $a,m_{1},m_{2}$ are monomials, then $am_{1}\geq am_{2}$.
By the monomial order, we can state the following definitions for a polynomial in $\mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$.
Definition[1]
1. Consider an ideal $I\subset \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$, and a fixed monomial ordering. The leading term of a nonzero polynomial $f\in \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$, denoted by $LT(f)$ is the monomial term of maximal order in $f$ and the leading term of $f=0$ is $0$.
2. The ideal of leading terms, denoted by $LT(I)$, is the ideal generated by the leading terms of every element in the ideal, that is, $LT(I)=(LT(f)\mid f\in I)$.
3. A Gröbner basis for an ideal $I\subset \mathbb {K} [x_{1},x_{2},\dotsc ,x_{n}]$ is a finite set of generators ${\{g_{1},g_{2},\dotsc ,g_{s}}\}$ for $I$ whose leading terms generate the ideal of all the leading terms in $I$, i.e., $I=(g_{1},g_{2},\dotsc ,g_{s})$ and $LT(I)=(LT(g_{1}),LT(g_{2}),\dotsc ,LT(g_{s}))$.
Note that $LT(I)$ in general depends on the ordering used; for example, if we choose the lexicographical order on $\mathbb {R} [x,y]$ subject to x > y, then $LT(2x^{3}y+9xy^{5}+19)=2x^{3}y$, but if we take y > x then $LT(2x^{3}y+9xy^{5}+19)=9xy^{5}$.
In addition, monomials are present on Gröbner basis and to define the division algorithm for polynomials with several variables.
Notice that for a monomial ideal $I=(g_{1},g_{2},\dotsc ,g_{s})\in \mathbb {F} [x_{1},x_{2},\dotsc ,x_{n}]$, the finite set of generators ${\{g_{1},g_{2},\dotsc ,g_{s}}\}$ is a Gröbner basis for $I$. To see this, note that any polynomial $f\in I$ can be expressed as $f=a_{1}g_{1}+a_{2}g_{2}+\dotsm +a_{s}g_{s}$ for $a_{i}\in \mathbb {F} [x_{1},x_{2},\dotsc ,x_{n}]$. Then the leading term of $f$ is a multiple for some $g_{i}$. As a result, $LT(I)$ is generated by the $g_{i}$ likewise.
See also
• Stanley–Reisner ring
• Hodge algebra
Footnotes
1. Dummit & Foote 2004
2. Miller & Sturmfels 2005
References
• Miller, Ezra; Sturmfels, Bernd (2005), Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, New York: Springer-Verlag, ISBN 0-387-22356-8
• Dummit, David S.; Foote, Richard M. (2004), Abstract Algebra (third ed.), New York: John Wiley & Sons, ISBN 978-0-471-43334-7
Further reading
• Cox, David. "Lectures on toric varieties" (PDF). Lecture 3. §4 and §5.
• Sturmfels, Bernd (1996). Gröbner Bases and Convex Polytopes. Providence, RI: American Mathematical Society.
• Taylor, Diana Kahn (1966). Ideals generated by monomials in an R-sequence (PhD thesis). University of Chicago. MR 2611561. ProQuest 302227382.
• Teissier, Bernard (2004). Monomial Ideals, Binomial Ideals, Polynomial Ideals (PDF).
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Toric manifold
In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an $n$-dimensional compact torus which is locally standard with the orbit space a simple convex polytope.[1][2]
The aim is to do combinatorics on the quotient polytope and obtain information on the manifold above. For example, the Euler characteristic and the cohomology ring of the manifold can be described in terms of the polytope.
The Atiyah and Guillemin-Sternberg theorem
This theorem states that the image of the moment map of a Hamiltonian toric action is the convex hull of the set of moments of the points fixed by the action. In particular, this image is a convex polytope.
References
1. Jeffrey, Lisa C. (1999), "Hamiltonian group actions and symplectic reduction", Symplectic geometry and topology (Park City, UT, 1997), IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, pp. 295–333, MR 1702947.
2. Masuda, Mikiya; Suh, Dong Youp (2008), "Classification problems of toric manifolds via topology", Toric topology, Contemp. Math., vol. 460, Amer. Math. Soc., Providence, RI, pp. 273–286, arXiv:0709.4579, doi:10.1090/conm/460/09024, MR 2428362.
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Toric section
A toric section is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone. Special cases have been known since antiquity, and the general case was studied by Jean Gaston Darboux.[1]
Mathematical formulae
In general, toric sections are fourth-order (quartic) plane curves[1] of the form
$\left(x^{2}+y^{2}\right)^{2}+ax^{2}+by^{2}+cx+dy+e=0.$
Spiric sections
A special case of a toric section is the spiric section, in which the intersecting plane is parallel to the rotational symmetry axis of the torus. They were discovered by the ancient Greek geometer Perseus in roughly 150 BC.[2] Well-known examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli.
Villarceau circles
Another special case is the Villarceau circles, in which the intersection is a circle despite the lack of any of the obvious sorts of symmetry that would entail a circular cross-section.[3]
General toric sections
More complicated figures such as an annulus can be created when the intersecting plane is perpendicular or oblique to the rotational symmetry axis.
References
1. Sym, Antoni (2009), "Darboux's greatest love", Journal of Physics A: Mathematical and Theoretical, 42 (40): 404001, doi:10.1088/1751-8113/42/40/404001.
2. Brieskorn, Egbert; Knörrer, Horst (1986), "Origin and generation of curves", Plane algebraic curves, Basel: Birkhäuser Verlag, pp. 2–65, doi:10.1007/978-3-0348-5097-1, ISBN 3-7643-1769-8, MR 0886476.
3. Schoenberg, I. J. (1985), "A direct approach to the Villarceau circles of a torus", Simon Stevin, 59 (4): 365–372, MR 0840858.
External links
• "The toric section: intersection of a torus with a plane" at "worlds of math and physics"
| Wikipedia |
Toric stack
In algebraic geometry, a toric stack is a stacky generalization of a toric variety. More precisely, a toric stack is obtained by replacing in the construction of a toric variety a step of taking GIT quotients with that of taking quotient stacks. Consequently, a toric variety is a coarse approximation of a toric stack. A toric orbifold is an example of a toric stack.
See also
• Stanley–Reisner ring
References
• Iwanari, Isamu (2009). "The category of toric stacks". Compositio Mathematica. 145 (3): 718–746. arXiv:math/0610548. doi:10.1112/S0010437X09003911. S2CID 13941792.
• Geraschenko, Anton; Satriano, Matthew (2015). "Toric stacks I: The theory of stacky fans". Transactions of the American Mathematical Society. 367 (2): 1033–1071. arXiv:1107.1906. doi:10.1090/S0002-9947-2014-06063-7. S2CID 5667546.
• Iwanari, I. (2009). "Integral Chow Rings of Toric Stacks". International Mathematics Research Notices. arXiv:0705.3524. doi:10.1093/imrn/rnp110. S2CID 12047977.
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Fermat point
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible[1] or, equivalently, the geometric median of the three vertices. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.
The Fermat point gives a solution to the geometric median and Steiner tree problems for three points.
Construction
The Fermat point of a triangle with largest angle at most 120° is simply its first isogonic center or X(13), which is constructed as follows:
1. Construct an equilateral triangle on each of two arbitrarily chosen sides of the given triangle.
2. Draw a line from each new vertex to the opposite vertex of the original triangle.
3. The two lines intersect at the Fermat point.
An alternative method is the following:
1. On each of two arbitrarily chosen sides, construct an isosceles triangle, with base the side in question, 30-degree angles at the base, and the third vertex of each isosceles triangle lying outside the original triangle.
2. For each isosceles triangle draw a circle, in each case with center on the new vertex of the isosceles triangle and with radius equal to each of the two new sides of that isosceles triangle.
3. The intersection inside the original triangle between the two circles is the Fermat point.
When a triangle has an angle greater than 120°, the Fermat point is sited at the obtuse-angled vertex.
In what follows "Case 1" means the triangle has an angle exceeding 120°. "Case 2" means no angle of the triangle exceeds 120°.
Location of X(13)
Fig. 2 shows the equilateral triangles △ARB, △AQC, △CPB attached to the sides of the arbitrary triangle △ABC. Here is a proof using properties of concyclic points to show that the three lines RC, BQ, AP in Fig 2 all intersect at the point F and cut one another at angles of 60°.
The triangles △RAC, △BAQ are congruent because the second is a 60° rotation of the first about A. Hence ∠ARF = ∠ABF and ∠AQF = ∠ACF. By the converse of the inscribed angle theorem applied to the segment AF, the points ARBF are concyclic (they lie on a circle). Similarly, the points AFCQ are concyclic.
∠ARB = 60°, so ∠AFB = 120°, using the inscribed angle theorem. Similarly, ∠AFC = 120°.
So ∠BFC = 120°. Therefore, ∠BFC + ∠BPC = 180°. Using the inscribed angle theorem, this implies that the points BPCF are concyclic. So, using the inscribed angle theorem applied to the segment BP, ∠BFP = ∠BCP = 60°. Because ∠BFP + ∠BFA = 180°, the point F lies on the line segment AP. So, the lines RC, BQ, AP are concurrent (they intersect at a single point). Q.E.D.
This proof applies only in Case 2, since if ∠BAC > 120°, point A lies inside the circumcircle of △BPC which switches the relative positions of A and F. However it is easily modified to cover Case 1. Then ∠AFB = ∠AFC = 60° hence ∠BFC = ∠AFB + ∠AFC = 120° which means BPCF is concyclic so ∠BFP = ∠BCP = 60° = ∠BFA. Therefore, A lies on FP.
The lines joining the centers of the circles in Fig. 2 are perpendicular to the line segments AP, BQ, CR. For example, the line joining the center of the circle containing △ARB and the center of the circle containing △AQC, is perpendicular to the segment AP. So, the lines joining the centers of the circles also intersect at 60° angles. Therefore, the centers of the circles form an equilateral triangle. This is known as Napoleon's Theorem.
Location of the Fermat point
Traditional geometry
Given any Euclidean triangle △ABC and an arbitrary point P let $d(P)=|PA|+|PB|+|PC|.$ The aim of this section is to identify a point P0 such that $d(P_{0})<d(P)$ for all $P\neq P_{0}.$ If such a point exists then it will be the Fermat point. In what follows Δ will denote the points inside the triangle and will be taken to include its boundary Ω.
A key result that will be used is the dogleg rule, which asserts that if a triangle and a polygon have one side in common and the rest of the triangle lies inside the polygon then the triangle has a shorter perimeter than the polygon:
If AB is the common side, extend AC to cut the polygon at the point X. Then the polygon's perimeter is, by the triangle inequality:
${\text{perimeter}}>|AB|+|AX|+|XB|=|AB|+|AC|+|CX|+|XB|\geq |AB|+|AC|+|BC|.$
Let P be any point outside Δ. Associate each vertex with its remote zone; that is, the half-plane beyond the (extended) opposite side. These 3 zones cover the entire plane except for Δ itself and P clearly lies in either one or two of them. If P is in two (say the B and C zones’ intersection) then setting $P'=A$ implies $d(P')=d(A)<d(P)$ by the dogleg rule. Alternatively if P is in only one zone, say the A-zone, then $d(P')<d(P)$ where P' is the intersection of AP and BC. So for every point P outside Δ there exists a point P' in Ω such that $d(P')<d(P).$
Case 1. The triangle has an angle ≥ 120°.
Without loss of generality, suppose that the angle at A is ≥ 120°. Construct the equilateral triangle △AFB and for any point P in Δ (except A itself) construct Q so that the triangle △AQP is equilateral and has the orientation shown. Then the triangle △ABP is a 60° rotation of the triangle △AFQ about A so these two triangles are congruent and it follows that $d(P)=|CP|+|PQ|+|QF|$ which is simply the length of the path CPQF. As P is constrained to lie within △ABC, by the dogleg rule the length of this path exceeds $|AC|+|AF|=d(A).$ Therefore, $d(A)<d(P)$ for all $P\in \Delta ,P\neq A.$ Now allow P to range outside Δ. From above a point $P'\in \Omega $ exists such that $d(P')<d(P)$ and as $d(A)\leq d(P')$ it follows that $d(A)<d(P)$ for all P outside Δ. Thus $d(A)<d(P)$ for all $P\neq A$ which means that A is the Fermat point of Δ. In other words, the Fermat point lies at the obtuse-angled vertex.
Case 2. The triangle has no angle ≥ 120°.
Construct the equilateral triangle △BCD, let P be any point inside Δ, and construct the equilateral triangle △CPQ. Then △CQD is a 60° rotation of △CPB about C so
$d(P)=|PA|+|PB|+|PC|=|AP|+|PQ|+|QD|$
which is simply the length of the path APQD. Let P0 be the point where AD and CF intersect. This point is commonly called the first isogonic center. Carry out the same exercise with P0 as you did with P, and find the point Q0. By the angular restriction P0 lies inside △ABC. Moreover, △BCF is a 60° rotation of △BDA about B, so Q0 must lie somewhere on AD. Since ∠CDB = 60° it follows that Q0 lies between P0 and D which means AP0Q0D is a straight line so $d(P_{0}=|AD|.$ Moreover, if $P\neq P_{0}$ then either P or Q won't lie on AD which means $d(P_{0})=|AD|<d(P).$ Now allow P to range outside Δ. From above a point $P'\in \Omega $ exists such that $d(P')<d(P)$ and as $d(P_{0})\leq d(P')$ it follows that $d(P_{0})<d(P)$ for all P outside Δ. That means P0 is the Fermat point of Δ. In other words, the Fermat point is coincident with the first isogonic center.
Vector analysis
Let O, A, B, C, X be any five points in a plane. Denote the vectors ${\overrightarrow {OA}},\ {\overrightarrow {OB}},\ {\overrightarrow {OC}},\ {\overrightarrow {OX}}$ by a, b, c, x respectively, and let i, j, k be the unit vectors from O along a, b, c.
${\begin{aligned}|\mathbf {a} |&=\mathbf {a\cdot i} =(\mathbf {a} -\mathbf {x} )\mathbf {\,\cdot \,i} +\mathbf {x\cdot i} \leq |\mathbf {a} -\mathbf {x} |+\mathbf {x\cdot i} ,\\|\mathbf {b} |&=\mathbf {b\cdot j} =(\mathbf {b} -\mathbf {x} )\mathbf {\,\cdot \,j} +\mathbf {x\cdot j} \leq |\mathbf {b} -\mathbf {x} |+\mathbf {x\cdot j} ,\\|\mathbf {c} |&=\mathbf {c\cdot k} =(\mathbf {c} -\mathbf {x} )\mathbf {\,\cdot \,k} +\mathbf {x\cdot k} \leq |\mathbf {c} -\mathbf {x} |+\mathbf {x\cdot k} .\end{aligned}}$
Adding a, b, c gives
$|\mathbf {a} |+|\mathbf {b} |+|\mathbf {c} |\leq |\mathbf {a} -\mathbf {x} |+|\mathbf {b} -\mathbf {x} |+|\mathbf {c} -\mathbf {x} |+x\cdot (\mathbf {i} +\mathbf {j} +\mathbf {k} ).$
If a, b, c meet at O at angles of 120° then i + j + k = 0, so
$|\mathbf {a} |+|\mathbf {b} |+|\mathbf {c} |\leq |\mathbf {a} -\mathbf {x} |+|\mathbf {b} -\mathbf {x} |+|\mathbf {c} -\mathbf {x} |$
for all x. In other words,
$|OA|+|OB|+|OC|\leq |XA|+|XB|+|XC|$
and hence O is the Fermat point of △ABC.
This argument fails when the triangle has an angle ∠C > 120° because there is no point O where a, b, c meet at angles of 120°. Nevertheless, it is easily fixed by redefining k = − (i + j) and placing O at C so that c = 0. Note that |k| ≤ 1 because the angle between the unit vectors i, j is ∠C which exceeds 120°. Since
$|\mathbf {0} |\leq |\mathbf {0} -\mathbf {x} |+\mathbf {x\cdot k} ,$
the third inequality still holds, the other two inequalities are unchanged. The proof now continues as above (adding the three inequalities and using i + j + k = 0) to reach the same conclusion that O (or in this case C) must be the Fermat point of △ABC.
Lagrange multipliers
Another approach to finding the point within a triangle, from which the sum of the distances to the vertices of the triangle is minimal, is to use one of the mathematical optimization methods; specifically, the method of Lagrange multipliers and the law of cosines.
We draw lines from the point within the triangle to its vertices and call them X, Y, Z. Also, let the lengths of these lines be x, y, z respectively. Let the angle between X and Y be α, Y and Z be β. Then the angle between X and Z is π − α − β. Using the method of Lagrange multipliers we have to find the minimum of the Lagrangian L, which is expressed as:
$L=x+y+z+\lambda _{1}(x^{2}+y^{2}-2xy\cos(\alpha )-a^{2})+\lambda _{2}(y^{2}+z^{2}-2yz\cos(\beta )-b^{2})+\lambda _{3}(z^{2}+x^{2}-2zx\cos(\alpha +\beta )-c^{2})$
where a, b, c are the lengths of the sides of the triangle.
Equating each of the five partial derivatives ${\tfrac {\partial L}{\partial x}},{\tfrac {\partial L}{\partial y}},{\tfrac {\partial L}{\partial z}},{\tfrac {\partial L}{\partial \alpha }},{\tfrac {\partial L}{\partial \beta }}$ to zero and eliminating λ1, λ2, λ3 eventually gives sin α = sin β and sin(α + β) = − sin β so α = β = 120°. However the elimination is a long and tedious business, and the end result covers only Case 2.
Properties
• When the largest angle of the triangle is not larger than 120°, X(13) is the Fermat point.
• The angles subtended by the sides of the triangle at X(13) are all equal to 120° (Case 2), or 60°, 60°, 120° (Case 1).
• The circumcircles of the three constructed equilateral triangles are concurrent at X(13).
• Trilinear coordinates for the first isogonic center, X(13):[2]
${\begin{aligned}&\csc \left(A+{\tfrac {\pi }{3}}\right):\csc \left(B+{\tfrac {\pi }{3}}\right):\csc \left(C+{\tfrac {\pi }{3}}\right)\\&=\sec \left(A-{\tfrac {\pi }{6}}\right):\sec \left(B-{\tfrac {\pi }{6}}\right):\sec \left(C-{\tfrac {\pi }{6}}\right).\end{aligned}}$
• Trilinear coordinates for the second isogonic center, X(14):[3]
${\begin{aligned}&\csc \left(A-{\tfrac {\pi }{3}}\right):\csc \left(B-{\tfrac {\pi }{3}}\right):\csc \left(C-{\tfrac {\pi }{3}}\right)\\&=\sec \left(A+{\tfrac {\pi }{6}}\right):\sec \left(B+{\tfrac {\pi }{6}}\right):\sec \left(C+{\tfrac {\pi }{6}}\right).\end{aligned}}$
• Trilinear coordinates for the Fermat point:
$1-u+uvw\sec \left(A-{\tfrac {\pi }{6}}\right):1-v+uvw\sec \left(B-{\tfrac {\pi }{6}}\right):1-w+uvw\sec \left(C-{\tfrac {\pi }{6}}\right)$
where u, v, w respectively denote the Boolean variables (A < 120°), (B < 120°), (C < 120°).
• The isogonal conjugate of X(13) is the first isodynamic point, X(15):[4]
$\sin \left(A+{\tfrac {\pi }{3}}\right):\sin \left(B+{\tfrac {\pi }{3}}\right):\sin \left(C+{\tfrac {\pi }{3}}\right).$
• The isogonal conjugate of X(14) is the second isodynamic point, X(16):[5]
$\sin \left(A-{\tfrac {\pi }{3}}\right):\sin \left(B-{\tfrac {\pi }{3}}\right):\sin \left(C-{\tfrac {\pi }{3}}\right).$
• The following triangles are equilateral:
• antipedal triangle of X(13)
• Antipedal triangle of X(14)
• Pedal triangle of X(15)
• Pedal triangle of X(16)
• Circumcevian triangle of X(15)
• Circumcevian triangle of X(16)
• The lines X(13)X(15) and X(14)X(16) are parallel to the Euler line. The three lines meet at the Euler infinity point, X(30).
• The points X(13), X(14), the circumcenter, and the nine-point center lie on a Lester circle.
• The line X(13)X(14) meets the Euler line at midpoint of X(2) and X(4).[6]
• The Fermat point lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.[7]
Aliases
The isogonic centers X(13) and X(14) are also known as the first Fermat point and the second Fermat point respectively. Alternatives are the positive Fermat point and the negative Fermat point. However these different names can be confusing and are perhaps best avoided. The problem is that much of the literature blurs the distinction between the Fermat point and the first Fermat point whereas it is only in Case 2 above that they are actually the same.
History
This question was proposed by Fermat, as a challenge to Evangelista Torricelli. He solved the problem in a similar way to Fermat's, albeit using the intersection of the circumcircles of the three regular triangles instead. His pupil, Viviani, published the solution in 1659.[8]
See also
• Geometric median or Fermat–Weber point, the point minimizing the sum of distances to more than three given points.
• Lester's theorem
• Triangle center
• Napoleon points
• Weber problem
References
1. Cut The Knot - The Fermat Point and Generalizations
2. Entry X(13) in the Encyclopedia of Triangle Centers Archived April 19, 2012, at the Wayback Machine
3. Entry X(14) in the Encyclopedia of Triangle Centers Archived April 19, 2012, at the Wayback Machine
4. Entry X(15) in the Encyclopedia of Triangle Centers Archived April 19, 2012, at the Wayback Machine
5. Entry X(16) in the Encyclopedia of Triangle Centers Archived April 19, 2012, at the Wayback Machine
6. Kimberling, Clark. "Encyclopedia of Triangle Centers".
7. Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers", Forum Geometricorum 6 (2006), 57--70. http://forumgeom.fau.edu/FG2006volume6/FG200607index.html
8. Weisstein, Eric W. "Fermat Points". MathWorld.
External links
• "Fermat-Torricelli problem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Fermat Point by Chris Boucher, The Wolfram Demonstrations Project.
• Fermat-Torricelli generalization at Dynamic Geometry Sketches Interactive sketch generalizes the Fermat-Torricelli point.
• A practical example of the Fermat point
• iOS Interactive sketch
Pierre de Fermat
Work
• Fermat's Last Theorem
• Fermat number
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Related
• List of things named after Pierre de Fermat
• Wiles's proof of Fermat's Last Theorem
• Fermat's Last Theorem in fiction
• Fermat Prize
• Fermat's Last Tango (2000 musical)
• Fermat's Last Theorem (popular science book)
| Wikipedia |
Torkel Franzén
Torkel Franzén (1 April 1950, Norrbotten County – 19 April 2006, Stockholm) was a Swedish academic.
Biography
Franzén worked at the Department of Computer Science and Electrical Engineering at Luleå University of Technology, Sweden, in the fields of mathematical logic and computer science. He was known for his work on Gödel's incompleteness theorems and for his contributions to Usenet.[1] He was active in the online science fiction fan community, and even issued his own electronic fanzine Frotz on his fiftieth birthday.[2] He died of bone cancer at age 56.[3]
Selected works
• Gödel's Theorem: An Incomplete Guide to its Use and Abuse. Wellesley, Massachusetts: A K Peters, Ltd., 2005. x + 172 pp. ISBN 1-56881-238-8.
• Inexhaustibility: A Non-Exhaustive Treatment. Wellesley, Massachusetts: A K Peters, Ltd., 2004. Lecture Notes in Logic, #16, Association for Symbolic Logic. ISBN 1-56881-174-8.
• The Popular Impact of Gödel's Incompleteness Theorem, Notices of the American Mathematical Society, 53, #4 (April 2006), pp. 440–443.
• Provability and Truth (Acta universitatis stockholmiensis, Stockholm Studies in Philosophy 9) (1987) ISBN 91-22-01158-7
See also
• Gödel's incompleteness theorems
References
1. In Memory Of, web page at the American Mathematical Society, accessed August 2, 2007.
2. "Frotz: An Electronic Oneshot". Archived from the original on October 1, 2007. Retrieved 2007-10-01., accessed online September 8, 2007.
3. Torkel Franzén is dead, 20 April 2006.
External links
• Home page
• Raatikainen, Panu. Review of Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Notices of the American Mathematical Society, Vol. 54, No. 3 (March 2007), pp. 380–3.
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Tornado diagram
Tornado diagrams, also called tornado plots, tornado charts or butterfly charts, are a special type of Bar chart, where the data categories are listed vertically instead of the standard horizontal presentation, and the categories are ordered so that the largest bar appears at the top of the chart, the second largest appears second from the top, and so on. They are so named because the final chart visually resembles either one half of or a complete tornado.
Purpose
Tornado diagrams are useful for deterministic sensitivity analysis – comparing the relative importance of variables. For each variable/uncertainty considered, one needs estimates for what the low, base, and high outcomes would be. The sensitive variable is modeled as having an uncertain value while all other variables are held at baseline values (stable).[1] This allows testing the sensitivity/risk associated with one uncertainty/variable. For example, if a decision maker needs to visually compare 100 budgetary items, and wishes to identify the ten items one should focus on, it would be nearly impossible to do using a standard bar graph. In a tornado diagram of the budget items, the top ten bars would represent the items that contribute the most to the variability of the outcome, and therefore what the decision maker should focus on.
References
1. PMBOK Guide Fifth Edition (2013) pg. 338 (4th Ed.,2008, pg. 298)
Further reading
• Technical note: constructing tornado diagrams with spreadsheets. Engineering Economist | June 22, 2006 | Eschenbach, Ted G
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| Wikipedia |
Toroid
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface.[1] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1-g).[2]
The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and should not be confused with toroids.
Toroidal structures occur in both natural and synthetic materials.[3]
Equations
A toroid is specified by the radius of revolution R measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference C and area A of the section):
Square toroid
The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.
$V=2\pi RA$
$S=2\pi RC$
Circular toroid
The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.
$V=2\pi ^{2}r^{2}R$
$S=4\pi ^{2}rR$
See also
• Toroidal inductors and transformers
• Toroidal propellers
• Annulus
• Solenoid
• Helix
Notes
1. Weisstein, Eric W. "Toroid". MathWorld.
2. Stewart, B.; "Adventures Among the Toroids:A Study of Orientable Polyhedra with Regular Faces", 2nd Edition, Stewart (1980).
3. Carroll, Gregory T.; Jongejan, Mahthild G. M.; Pijper, Dirk; Feringa, Ben L. (2010). "Spontaneous generation and patterning of chiral polymeric surface toroids". Chemical Science. 1 (4): 469. doi:10.1039/c0sc00159g. ISSN 2041-6520.
External links
• The dictionary definition of toroid at Wiktionary
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| Wikipedia |
Poloidal–toroidal decomposition
In vector calculus, a topic in pure and applied mathematics, a poloidal–toroidal decomposition is a restricted form of the Helmholtz decomposition. It is often used in the spherical coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids.[1]
Definition
Further information: Vector operator
For a three-dimensional vector field F with zero divergence
$\nabla \cdot \mathbf {F} =0,$
this F can be expressed as the sum of a toroidal field T and poloidal vector field P
$\mathbf {F} =\mathbf {T} +\mathbf {P} $
where r is a radial vector in spherical coordinates (r, θ, φ). The toroidal field is obtained from a scalar field, Ψ(r, θ, φ),[2] as the following curl,
$\mathbf {T} =\nabla \times (\mathbf {r} \Psi (\mathbf {r} ))$
and the poloidal field is derived from another scalar field Φ(r, θ, φ),[3] as a twice-iterated curl,
$\mathbf {P} =\nabla \times (\nabla \times (\mathbf {r} \Phi (\mathbf {r} )))\,.$
This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal, known as Chandrasekhar–Kendall function.[4]
Geometry
A toroidal vector field is tangential to spheres around the origin,[4]
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathbf{r} \cdot \mathbf{T} = 0 }
while the curl of a poloidal field is tangential to those spheres
$\mathbf {r} \cdot (\nabla \times \mathbf {P} )=0.$[5]
The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields Ψ and Φ vanishes on every sphere of radius r.[3]
Cartesian decomposition
A poloidal–toroidal decomposition also exists in Cartesian coordinates, but a mean-field flow has to be included in this case. For example, every solenoidal vector field can be written as
$\mathbf {F} (x,y,z)=\nabla \times g(x,y,z){\hat {\mathbf {z} }}+\nabla \times (\nabla \times h(x,y,z){\hat {\mathbf {z} }})+b_{x}(z){\hat {\mathbf {x} }}+b_{y}(z){\hat {\mathbf {y} }},$
where ${\hat {\mathbf {x} }},{\hat {\mathbf {y} }},{\hat {\mathbf {z} }}$ denote the unit vectors in the coordinate directions.[6]
See also
• Toroidal and poloidal
• Chandrasekhar–Kendall function
Notes
1. Subrahmanyan Chandrasekhar (1961). Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622.
2. Backus 1986, p. 87.
3. Backus 1986, p. 88.
4. Backus, Parker & Constable 1996, p. 178.
5. Backus, Parker & Constable 1996, p. 179.
6. Jones 2008, p. 17. sfn error: no target: CITEREFJones2008 (help)
References
• Hydrodynamic and hydromagnetic stability, Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.
• Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the boussinesq-equations, Schmitt, B. J. and von Wahl, W; in The Navier–Stokes Equations II — Theory and Numerical Methods, pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992.
• Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones, Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp. 247–264.
• Plane poloidal-toroidal decomposition of doubly periodic vector fields: Part 1. Fields with divergence and Part 2. Stokes equations. G. D. McBain. ANZIAM J. 47 (2005)
• Backus, George (1986), "Poloidal and toroidal fields in geomagnetic field modeling", Reviews of Geophysics, 24: 75–109, Bibcode:1986RvGeo..24...75B, doi:10.1029/RG024i001p00075.
• Backus, George; Parker, Robert; Constable, Catherine (1996), Foundations of Geomagnetism, Cambridge University Press, ISBN 0-521-41006-1.
• Jones, Chris, Dynamo Theory (PDF).
| Wikipedia |
Toroidal and poloidal coordinates
The terms toroidal and poloidal refer to directions relative to a torus of reference. They describe a three-dimensional coordinate system in which the poloidal direction follows a small circular ring around the surface, while the toroidal direction follows a large circular ring around the torus, encircling the central void.
For the vector fields, see Toroidal–poloidal decomposition.
The earliest use of these terms cited by the Oxford English Dictionary is by Walter M. Elsasser (1946) in the context of the generation of the Earth's magnetic field by currents in the core, with "toroidal" being parallel to lines of latitude and "poloidal" being in the direction of the magnetic field (i.e. towards the poles).
The OED also records the later usage of these terms in the context of toroidally confined plasmas, as encountered in magnetic confinement fusion. In the plasma context, the toroidal direction is the long way around the torus, the corresponding coordinate being denoted by z in the slab approximation or ζ or φ in magnetic coordinates; the poloidal direction is the short way around the torus, the corresponding coordinate being denoted by y in the slab approximation or θ in magnetic coordinates. (The third direction, normal to the magnetic surfaces, is often called the "radial direction", denoted by x in the slab approximation and variously ψ, χ, r, ρ, or s in magnetic coordinates.)
Example
As a simple example from the physics of magnetically confined plasmas, consider an axisymmetric system with circular, concentric magnetic flux surfaces of radius $r$ (a crude approximation to the magnetic field geometry in an early tokamak but topologically equivalent to any toroidal magnetic confinement system with nested flux surfaces) and denote the toroidal angle by $\zeta $ and the poloidal angle by $\theta $. Then the Toroidal/Poloidal coordinate system relates to standard Cartesian Coordinates by these transformation rules:
$x=(R_{0}+r\cos \theta )\cos \zeta $
$y=s_{\zeta }(R_{0}+r\cos \theta )\sin \zeta $
$z=s_{\theta }r\sin \theta .$
where $s_{\theta }=\pm 1,s_{\zeta }=\pm 1$.
The natural choice geometrically is to take $s_{\theta }=s_{\zeta }=+1$, giving the toroidal and poloidal directions shown by the arrows in the figure above, but this makes $r,\theta ,\zeta $ a left-handed curvilinear coordinate system. As it is usually assumed in setting up flux coordinates for describing magnetically confined plasmas that the set $r,\theta ,\zeta $ forms a right-handed coordinate system, $\nabla r\cdot \nabla \theta \times \nabla \zeta >0$, we must either reverse the poloidal direction by taking $s_{\theta }=-1,s_{\zeta }=+1$, or reverse the toroidal direction by taking $s_{\theta }=+1,s_{\zeta }=-1$. Both choices are used in the literature.
Kinematics
To study single particle motion in toroidally confined plasma devices, velocity and acceleration vectors must be known. Considering the natural choice $s_{\theta }=s_{\zeta }=+1$, the unit vectors of toroidal and poloidal coordinates system $\left(r,\theta ,\zeta \right)$ can be expressed as:
$\mathbf {e} _{r}={\begin{pmatrix}\cos \theta \cos \zeta \\\cos \theta \sin \zeta \\\sin \theta \end{pmatrix}}\quad \mathbf {e} _{\theta }={\begin{pmatrix}-\sin \theta \cos \zeta \\-\sin \theta \sin \zeta \\\cos \theta \end{pmatrix}}\quad \mathbf {e} _{\zeta }={\begin{pmatrix}-\sin \zeta \\\cos \zeta \\0\end{pmatrix}}$
according to Cartesian coordinates. The position vector is expressed as:
$\mathbf {r} =\left(r+R_{0}\cos \theta \right)\mathbf {e} _{r}-R_{0}\sin \theta \mathbf {e} _{\theta }$
The velocity vector is then given by:
$\mathbf {\dot {r}} ={\dot {r}}\mathbf {e} _{r}+r{\dot {\theta }}\mathbf {e} _{\theta }+{\dot {\zeta }}\left(R_{0}+r\cos \theta \right)\mathbf {e} _{\zeta }$
and the acceleration vector is:
${\begin{aligned}\mathbf {\ddot {r}} ={}&\left({\ddot {r}}-r{\dot {\theta }}^{2}-r{\dot {\zeta }}^{2}\cos ^{2}\theta -R_{0}{\dot {\zeta }}^{2}\cos \theta \right)\mathbf {e} _{r}\\[5pt]&{}+\left(2{\dot {r}}{\dot {\theta }}+r{\ddot {\theta }}+r{\dot {\zeta }}^{2}\cos \theta \sin \theta +R_{0}{\dot {\zeta }}^{2}\sin \theta \right)\mathbf {e} _{\theta }\\[5pt]&{}+\left(2{\dot {r}}{\dot {\zeta }}\cos \theta -2r{\dot {\theta }}{\dot {\zeta }}\sin \theta +{\ddot {\zeta }}\left(R_{0}+r\cos \theta \right)\right)\mathbf {e} _{\zeta }\end{aligned}}$
See also
Look up toroidal or poloidal in Wiktionary, the free dictionary.
• Toroidal coordinates
• Torus
• Zonal and poloidal
• Poloidal–toroidal decomposition
• Zonal flow (plasma)
References
• "Oxford English Dictionary Online". poloidal. Oxford University Press. Retrieved 2007-08-10.
• Elsasser, W. M. (1946). "Induction Effects in Terrestrial Magnetism, Part I. Theory". Phys. Rev. 69 (3–4): 106–116. doi:10.1103/PhysRev.69.106. Retrieved 2007-08-10.
| Wikipedia |
Toroidal embedding
In algebraic geometry, a toroidal embedding is an open embedding of algebraic varieties that locally looks like the embedding of the open torus into a toric variety. The notion was introduced by Mumford to prove the existence of semistable reductions of algebraic varieties over one-dimensional bases.
Definition
Let X be a normal variety over an algebraically closed field ${\bar {k}}$ and $U\subset X$ a smooth open subset. Then $U\hookrightarrow X$ is called a toroidal embedding if for every closed point x of X, there is an isomorphism of local ${\bar {k}}$-algebras:
${\widehat {\mathcal {O}}}_{X,x}\simeq {\widehat {\mathcal {O}}}_{X_{\sigma },t}$
for some affine toric variety $X_{\sigma }$ with a torus T and a point t such that the above isomorphism takes the ideal of $X-U$ to that of $X_{\sigma }-T$.
Let X be a normal variety over a field k. An open embedding $U\hookrightarrow X$ is said to a toroidal embedding if $U_{\bar {k}}\hookrightarrow X_{\bar {k}}$ is a toroidal embedding.
Examples
Tits' buildings
Main article: Tits' buildings
See also
• tropical compactification
References
• Kempf, G.; Knudsen, Finn Faye; Mumford, David; Saint-Donat, B. (1973), Toroidal embeddings. I, Lecture Notes in Mathematics, vol. 339, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070318, MR 0335518
• Abramovich, D., Denef, J. & Karu, K.: Weak toroidalization over non-closed fields. manuscripta math. (2013) 142: 257. doi:10.1007/s00229-013-0610-5
External links
• Toroidal embedding
| Wikipedia |
Toroidal coordinates
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci $F_{1}$ and $F_{2}$ in bipolar coordinates become a ring of radius $a$ in the $xy$ plane of the toroidal coordinate system; the $z$-axis is the axis of rotation. The focal ring is also known as the reference circle.
Definition
The most common definition of toroidal coordinates $(\tau ,\sigma ,\phi )$ is
$x=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\cos \phi $
$y=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\sin \phi $
$z=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}$
together with $\mathrm {sign} (\sigma )=\mathrm {sign} (z$). The $\sigma $ coordinate of a point $P$ equals the angle $F_{1}PF_{2}$ and the $\tau $ coordinate equals the natural logarithm of the ratio of the distances $d_{1}$ and $d_{2}$ to opposite sides of the focal ring
$\tau =\ln {\frac {d_{1}}{d_{2}}}.$
The coordinate ranges are $-\pi <\sigma \leq \pi $, $\tau \geq 0$ and $0\leq \phi <2\pi .$
Coordinate surfaces
Surfaces of constant $\sigma $ correspond to spheres of different radii
$\left(x^{2}+y^{2}\right)+\left(z-a\cot \sigma \right)^{2}={\frac {a^{2}}{\sin ^{2}\sigma }}$
that all pass through the focal ring but are not concentric. The surfaces of constant $\tau $ are non-intersecting tori of different radii
$z^{2}+\left({\sqrt {x^{2}+y^{2}}}-a\coth \tau \right)^{2}={\frac {a^{2}}{\sinh ^{2}\tau }}$
that surround the focal ring. The centers of the constant-$\sigma $ spheres lie along the $z$-axis, whereas the constant-$\tau $ tori are centered in the $xy$ plane.
Inverse transformation
The $(\sigma ,\tau ,\phi )$ coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle $\phi $ is given by the formula
$\tan \phi ={\frac {y}{x}}$
The cylindrical radius $\rho $ of the point P is given by
$\rho ^{2}=x^{2}+y^{2}=\left(a{\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\right)^{2}$
and its distances to the foci in the plane defined by $\phi $ is given by
$d_{1}^{2}=(\rho +a)^{2}+z^{2}$
$d_{2}^{2}=(\rho -a)^{2}+z^{2}$
The coordinate $\tau $ equals the natural logarithm of the focal distances
$\tau =\ln {\frac {d_{1}}{d_{2}}}$
whereas $|\sigma |$ equals the angle between the rays to the foci, which may be determined from the law of cosines
$\cos \sigma ={\frac {d_{1}^{2}+d_{2}^{2}-4a^{2}}{2d_{1}d_{2}}}.$
Or explicitly, including the sign,
$\sigma =\mathrm {sign} (z)\arccos {\frac {r^{2}-a^{2}}{\sqrt {(r^{2}-a^{2})^{2}+4a^{2}z^{2}}}}$
where $r={\sqrt {\rho ^{2}+z^{2}}}$.
The transformations between cylindrical and toroidal coordinates can be expressed in complex notation as
$z+i\rho \ =ia\coth {\frac {\tau +i\sigma }{2}},$
$\tau +i\sigma \ =\ln {\frac {z+i(\rho +a)}{z+i(\rho -a)}}.$
Scale factors
The scale factors for the toroidal coordinates $\sigma $ and $\tau $ are equal
$h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}$
whereas the azimuthal scale factor equals
$h_{\phi }={\frac {a\sinh \tau }{\cosh \tau -\cos \sigma }}$
Thus, the infinitesimal volume element equals
$dV={\frac {a^{3}\sinh \tau }{\left(\cosh \tau -\cos \sigma \right)^{3}}}\,d\sigma \,d\tau \,d\phi $
Differential Operators
The Laplacian is given by
${\begin{aligned}\nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sinh \tau }}&\left[\sinh \tau {\frac {\partial }{\partial \sigma }}\left({\frac {1}{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)\right.\\[8pt]&{}\quad +\left.{\frac {\partial }{\partial \tau }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sinh \tau \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\right]\end{aligned}}$
For a vector field
${\vec {n}}(\tau ,\sigma ,\phi )=n_{\tau }(\tau ,\sigma ,\phi ){\hat {e}}_{\tau }+n_{\sigma }(\tau ,\sigma ,\phi ){\hat {e}}_{\sigma }+n_{\phi }(\tau ,\sigma ,\phi ){\hat {e}}_{\phi },$
the Vector Laplacian is given by
${\begin{aligned}\Delta {\vec {n}}(\tau ,\sigma ,\phi )&=\nabla (\nabla \cdot {\vec {n}})-\nabla \times (\nabla \times {\vec {n}})\\&={\frac {1}{a^{2}}}{\vec {e}}_{\tau }\left\{n_{\tau }\left(-{\frac {\sinh ^{4}\tau +(\cosh \tau -\cos \sigma )^{2}}{\sinh ^{2}\tau }}\right)+n_{\sigma }(-\sinh \tau \sin \sigma )+{\frac {\partial n_{\tau }}{\partial \tau }}\left({\frac {(\cosh \tau -\cos \sigma )(1-\cosh \tau \cos \sigma )}{\sinh \tau }}\right)+\cdots \right.\\&\qquad +{\frac {\partial n_{\tau }}{\partial \sigma }}(-(\cosh \tau -\cos \sigma )\sin \sigma )+{\frac {\partial n_{\sigma }}{\partial \sigma }}(2(\cosh \tau -\cos \sigma )\sinh \tau )+{\frac {\partial n_{\sigma }}{\partial \tau }}(-2(\cosh \tau -\cos \sigma )\sin \sigma )+\cdots \\&\qquad +{\frac {\partial n_{\phi }}{\partial \phi }}\left({\frac {-2(\cosh \tau -\cos \sigma )(1-\cosh \tau \cos \sigma )}{\sinh ^{2}\tau }}\right)+{\frac {\partial ^{2}n_{\tau }}{{\partial \tau }^{2}}}(\cosh \tau -\cos \sigma )^{2}+{\frac {\partial ^{2}n_{\tau }}{{\partial \sigma }^{2}}}(-(\cosh \tau -\cos \sigma )^{2})+\cdots \\&\qquad \left.+{\frac {\partial ^{2}n_{\tau }}{{\partial \phi }^{2}}}{\frac {(\cosh \tau -\cos \sigma )^{2}}{\sinh ^{2}\tau }}\right\}\\&+{\frac {1}{a^{2}}}{\vec {e}}_{\sigma }\left\{n_{\tau }\left(-{\frac {(\cosh ^{2}\tau +1-2\cosh \tau \cos \sigma )\sin \sigma }{\sinh \tau }}\right)+n_{\sigma }\left(-\sinh ^{2}\tau -2\sin ^{2}\sigma \right)+\ldots \right.\\&\qquad \left.+{\frac {\partial n_{\tau }}{\partial \tau }}(2\sin \sigma (\cosh \tau -\cos \sigma ))+{\frac {\partial n_{\tau }}{\partial \sigma }}\left(-2\sinh \tau (\cosh \tau -\cos \sigma )\right)+\cdots \right.\\&\qquad \left.+{\frac {\partial n_{\sigma }}{\partial \tau }}\left({\frac {(\cosh \tau -\cos \sigma )(1-\cosh \tau \cos \sigma )}{\sinh \tau }}\right)+{\frac {\partial n_{\sigma }}{\partial \sigma }}(-(\cosh \tau -\cos \sigma )\sin \sigma )+\cdots \right.\\&\qquad \left.+{\frac {\partial n_{\phi }}{\partial \phi }}\left(2{\frac {(\cosh \tau -\cos \sigma )\sin \sigma }{\sinh \tau }}\right)+{\frac {\partial ^{2}n_{\sigma }}{{\partial \tau }^{2}}}(\cosh \tau -\cos \sigma )^{2}+{\frac {\partial ^{2}n_{\sigma }}{{\partial \sigma }^{2}}}(\cosh \tau -\cos \sigma )^{2}+\cdots \right.\\&\qquad \left.+{\frac {\partial ^{2}n_{\sigma }}{{\partial \phi }^{2}}}\left({\frac {(\cosh \tau -\cos \sigma )^{2}}{\sinh ^{2}\tau }}\right)\right\}\\&+{\frac {1}{a^{2}}}{\vec {e}}_{\phi }\left\{n_{\phi }\left(-{\frac {(\cosh \tau -\cos \sigma )^{2}}{\sinh ^{2}\tau }}\right)+{\frac {\partial n_{\tau }}{\partial \phi }}\left({\frac {2(\cosh \tau -\cos \sigma )(1-\cosh \tau \cos \sigma )}{\sinh ^{2}\tau }}\right)+\cdots \right.\\&\qquad \left.+{\frac {\partial n_{\sigma }}{\partial \phi }}\left(-{\frac {2(\cosh \tau -\cos \sigma )\sin \sigma }{\sinh \tau }}\right)+{\frac {\partial n_{\phi }}{\partial \tau }}\left({\frac {(\cosh \tau -\cos \sigma )(1-\cosh \tau \cos \sigma )}{\sinh \tau }}\right)+\cdots \right.\\&\qquad \left.+{\frac {\partial n_{\phi }}{\partial \sigma }}(-(\cosh \tau -\cos \sigma )\sin \sigma )+{\frac {\partial ^{2}n_{\phi }}{{\partial \tau }^{2}}}(\cosh \tau -\cos \sigma )^{2}+\cdots \right.\\&\qquad \left.+{\frac {\partial ^{2}n_{\phi }}{{\partial \sigma }^{2}}}(\cosh \tau -\cos \sigma )^{2}+{\frac {\partial ^{2}n_{\phi }}{{\partial \phi }^{2}}}\left({\frac {(\cosh \tau -\cos \sigma )^{2}}{\sinh ^{2}\tau }}\right)\right\}\end{aligned}}$
Other differential operators such as $\nabla \cdot \mathbf {F} $ and $\nabla \times \mathbf {F} $ can be expressed in the coordinates $(\sigma ,\tau ,\phi )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.
Toroidal harmonics
Standard separation
The 3-variable Laplace equation
$\nabla ^{2}\Phi =0$
admits solution via separation of variables in toroidal coordinates. Making the substitution
$\Phi =U{\sqrt {\cosh \tau -\cos \sigma }}$
A separable equation is then obtained. A particular solution obtained by separation of variables is:
$\Phi ={\sqrt {\cosh \tau -\cos \sigma }}\,\,S_{\nu }(\sigma )T_{\mu \nu }(\tau )V_{\mu }(\phi )$
where each function is a linear combination of:
$S_{\nu }(\sigma )=e^{i\nu \sigma }\,\,\,\,\mathrm {and} \,\,\,\,e^{-i\nu \sigma }$
$T_{\mu \nu }(\tau )=P_{\nu -1/2}^{\mu }(\cosh \tau )\,\,\,\,\mathrm {and} \,\,\,\,Q_{\nu -1/2}^{\mu }(\cosh \tau )$
$V_{\mu }(\phi )=e^{i\mu \phi }\,\,\,\,\mathrm {and} \,\,\,\,e^{-i\mu \phi }$
Where P and Q are associated Legendre functions of the first and second kind. These Legendre functions are often referred to as toroidal harmonics.
Toroidal harmonics have many interesting properties. If you make a variable substitution $z=\cosh \tau >1$ then, for instance, with vanishing order $\mu =0$ (the convention is to not write the order when it vanishes) and $\nu =0$
$Q_{-{\frac {1}{2}}}(z)={\sqrt {\frac {2}{1+z}}}K\left({\sqrt {\frac {2}{1+z}}}\right)$
and
$P_{-{\frac {1}{2}}}(z)={\frac {2}{\pi }}{\sqrt {\frac {2}{1+z}}}K\left({\sqrt {\frac {z-1}{z+1}}}\right)$
where $\,\!K$ and $\,\!E$ are the complete elliptic integrals of the first and second kind respectively. The rest of the toroidal harmonics can be obtained, for instance, in terms of the complete elliptic integrals, by using recurrence relations for associated Legendre functions.
The classic applications of toroidal coordinates are in solving partial differential equations, e.g., Laplace's equation for which toroidal coordinates allow a separation of variables or the Helmholtz equation, for which toroidal coordinates do not allow a separation of variables. Typical examples would be the electric potential and electric field of a conducting torus, or in the degenerate case, an electric current-ring (Hulme 1982).
An alternative separation
Alternatively, a different substitution may be made (Andrews 2006)
$\Phi ={\frac {U}{\sqrt {\rho }}}$
where
$\rho ={\sqrt {x^{2}+y^{2}}}={\frac {a\sinh \tau }{\cosh \tau -\cos \sigma }}.$
Again, a separable equation is obtained. A particular solution obtained by separation of variables is then:
$\Phi ={\frac {a}{\sqrt {\rho }}}\,\,S_{\nu }(\sigma )T_{\mu \nu }(\tau )V_{\mu }(\phi )$
where each function is a linear combination of:
$S_{\nu }(\sigma )=e^{i\nu \sigma }\,\,\,\,\mathrm {and} \,\,\,\,e^{-i\nu \sigma }$
$T_{\mu \nu }(\tau )=P_{\mu -1/2}^{\nu }(\coth \tau )\,\,\,\,\mathrm {and} \,\,\,\,Q_{\mu -1/2}^{\nu }(\coth \tau )$
$V_{\mu }(\phi )=e^{i\mu \phi }\,\,\,\,\mathrm {and} \,\,\,\,e^{-i\mu \phi }.$
Note that although the toroidal harmonics are used again for the T function, the argument is $\coth \tau $ rather than $\cosh \tau $ and the $\mu $ and $\nu $ indices are exchanged. This method is useful for situations in which the boundary conditions are independent of the spherical angle $\theta $, such as the charged ring, an infinite half plane, or two parallel planes. For identities relating the toroidal harmonics with argument hyperbolic cosine with those of argument hyperbolic cotangent, see the Whipple formulae.
References
• Byerly, W E. (1893) An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics Ginn & co. pp. 264–266
• Arfken G (1970). Mathematical Methods for Physicists (2nd ed.). Orlando, FL: Academic Press. pp. 112–115.
• Andrews, Mark (2006). "Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics". Journal of Electrostatics. 64 (10): 664–672. CiteSeerX 10.1.1.205.5658. doi:10.1016/j.elstat.2005.11.005.
• Hulme, A. (1982). "A note on the magnetic scalar potential of an electric current-ring". Mathematical Proceedings of the Cambridge Philosophical Society. 92 (1): 183–191. doi:10.1017/S0305004100059831.
Bibliography
• Morse P M, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw–Hill. p. 666.
• Korn G A, Korn T M (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 182. LCCN 59014456.
• Margenau H, Murphy G M (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 190–192. LCCN 55010911.
• Moon P H, Spencer D E (1988). "Toroidal Coordinates (η, θ, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (2nd ed., 3rd revised printing ed.). New York: Springer Verlag. pp. 112–115 (Section IV, E4Ry). ISBN 978-0-387-02732-6.
External links
• MathWorld description of toroidal coordinates
Orthogonal coordinate systems
Two dimensional
• Cartesian
• Polar (Log-polar)
• Parabolic
• Bipolar
• Elliptic
Three dimensional
• Cartesian
• Cylindrical
• Spherical
• Parabolic
• Paraboloidal
• Oblate spheroidal
• Prolate spheroidal
• Ellipsoidal
• Elliptic cylindrical
• Toroidal
• Bispherical
• Bipolar cylindrical
• Conical
• 6-sphere
| Wikipedia |
Toronto function
In mathematics, the Toronto function T(m,n,r) is a modification of the confluent hypergeometric function defined by Heatley (1943), Weisstein, as
$T(m,n,r)=r^{2n-m+1}e^{-r^{2}}{\frac {\Gamma ({\frac {1}{2}}m+{\frac {1}{2}})}{\Gamma (n+1)}}{}_{1}F_{1}( {\frac {1}{2}}}m+ {\frac {1}{2}}};n+1;r^{2}).$
Later, Heatley (1964) recomputed to 12 decimals the table of the M(R)-function, and gave some corrections of the original tables. The table was also extended from x = 4 to x = 16 (Heatley, 1965). An example of the Toronto function has appeared in a study on the theory of turbulence (Heatley, 1965).
References
• Heatley, A. H. (1943), "A short table of the Toronto function", Trans. Roy. Soc. Canada Sect. III., 37: 13–29, MR 0010055
• Heatley, A. H. (1964), "A short table of the Toronto function", Mathematics of Computation, 18, No.88: 361
• Heatley, A. H. (1965), "An extension of the table of the Toronto function", Mathematics of Computation, 19, No.89: 118-123
• Weisstein, E. W., "Toronto Function", From Math World - A Wolfram Web Resource
| Wikipedia |
Toronto space
In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality.
There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.[1]
The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete.[2]
References
1. Bonnet, Robert (1993), "On superatomic Boolean algebras", Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 411, Dordrecht: Kluwer Acad. Publ., pp. 31–62, MR 1261195.
2. van Mill, J.; Reed, George M. (1990), Open problems in topology, Volume 1, North-Holland, p. 15, ISBN 9780444887689.
| Wikipedia |
Torsion (algebra)
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element.
This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements.
This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules).
In the case of groups that are noncommutative, a torsion element is an element of finite order. Contrary to the commutative case, the torsion elements do not form a subgroup, in general.
Definition
An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., r m = 0. In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element, but this definition does not work well over more general rings.
A module M over a ring R is called a torsion module if all its elements are torsion elements, and torsion-free if zero is the only torsion element.[1] If the ring R is an integral domain then the set of all torsion elements forms a submodule of M, called the torsion submodule of M, sometimes denoted T(M). If R is not commutative, T(M) may or may not be a submodule. It is shown in (Lam 2007) that R is a right Ore ring if and only if T(M) is a submodule of M for all right R-modules. Since right Noetherian domains are Ore, this covers the case when R is a right Noetherian domain (which might not be commutative).
More generally, let M be a module over a ring R and S be a multiplicatively closed subset of R. An element m of M is called an S-torsion element if there exists an element s in S such that s annihilates m, i.e., s m = 0. In particular, one can take for S the set of regular elements of the ring R and recover the definition above.
An element g of a group G is called a torsion element of the group if it has finite order, i.e., if there is a positive integer m such that gm = e, where e denotes the identity element of the group, and gm denotes the product of m copies of g. A group is called a torsion (or periodic) group if all its elements are torsion elements, and a torsion-free group if its only torsion element is the identity element. Any abelian group may be viewed as a module over the ring Z of integers, and in this case the two notions of torsion coincide.
Examples
1. Let M be a free module over any ring R. Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero-divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as the module over K.
2. By contrast with example 1, any finite group (abelian or not) is periodic and finitely generated. Burnside's problem, conversely, asks whether any finitely generated periodic group must be finite? The answer is "no" in general, even if the period is fixed.
3. The torsion elements of the multiplicative group of a field are its roots of unity.
4. In the modular group, Γ obtained from the group SL(2, Z) of 2×2 integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element S or has order three and is conjugate to the element ST. In this case, torsion elements do not form a subgroup, for example, S · ST = T, which has infinite order.
5. The abelian group Q/Z, consisting of the rational numbers modulo 1, is periodic, i.e. every element has finite order. Analogously, the module K(t)/K[t] over the ring R = K[t] of polynomials in one variable is pure torsion. Both these examples can be generalized as follows: if R is an integral domain and Q is its field of fractions, then Q/R is a torsion R-module.
6. The torsion subgroup of (R/Z, +) is (Q/Z, +) while the groups (R, +) and (Z, +) are torsion-free. The quotient of a torsion-free abelian group by a subgroup is torsion-free exactly when the subgroup is a pure subgroup.
7. Consider a linear operator L acting on a finite-dimensional vector space V. If we view V as an F[L]-module in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the Cayley–Hamilton theorem), V is a torsion F[L]-module.
Case of a principal ideal domain
Suppose that R is a (commutative) principal ideal domain and M is a finitely generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that
$M\simeq F\oplus T(M),$
where F is a free R-module of finite rank (depending only on M) and T(M) is the torsion submodule of M. As a corollary, any finitely generated torsion-free module over R is free. This corollary does not hold for more general commutative domains, even for R = K[x,y], the ring of polynomials in two variables. For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a direct summand of it.
Torsion and localization
Assume that R is a commutative domain and M is an R-module. Let Q be the quotient field of the ring R. Then one can consider the Q-module
$M_{Q}=M\otimes _{R}Q,$
obtained from M by extension of scalars. Since Q is a field, a module over Q is a vector space, possibly infinite-dimensional. There is a canonical homomorphism of abelian groups from M to MQ, and the kernel of this homomorphism is precisely the torsion submodule T(M). More generally, if S is a multiplicatively closed subset of the ring R, then we may consider localization of the R-module M,
$M_{S}=M\otimes _{R}R_{S},$
which is a module over the localization RS. There is a canonical map from M to MS, whose kernel is precisely the S-torsion submodule of M. Thus the torsion submodule of M can be interpreted as the set of the elements that "vanish in the localization". The same interpretation continues to hold in the non-commutative setting for rings satisfying the Ore condition, or more generally for any right denominator set S and right R-module M.
Torsion in homological algebra
The concept of torsion plays an important role in homological algebra. If M and N are two modules over a commutative domain R (for example, two abelian groups, when R = Z), Tor functors yield a family of R-modules Tori (M,N). The S-torsion of an R-module M is canonically isomorphic to TorR1(M, RS/R) by the exact sequence of TorR*: The short exact sequence $0\to R\to R_{S}\to R_{S}/R\to 0$ of R-modules yields an exact sequence $0\to \operatorname {Tor} _{1}^{R}(M,R_{S}/R)\to M\to M_{S}$, hence $\operatorname {Tor} _{1}^{R}(M,R_{S}/R)$ is the kernel of the localisation map of M. The symbol Tor denoting the functors reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set S is a right denominator set.
Abelian varieties
The torsion elements of an abelian variety are torsion points or, in an older terminology, division points. On elliptic curves they may be computed in terms of division polynomials.
See also
• Analytic torsion
• Arithmetic dynamics
• Flat module
• Annihilator (ring theory)
• Localization of a module
• Rank of an abelian group
• Ray–Singer torsion
• Torsion-free abelian group
• Universal coefficient theorem
References
1. Roman 2008, p. 115, §4
Sources
• Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
• Irving Kaplansky, "Infinite abelian groups", University of Michigan, 1954.
• Michiel Hazewinkel (2001) [1994], "Torsion submodule", Encyclopedia of Mathematics, EMS Press
• Lam, Tsit Yuen (2007), Exercises in modules and rings, Problem Books in Mathematics, New York: Springer, pp. xviii+412, doi:10.1007/978-0-387-48899-8, ISBN 978-0-387-98850-4, MR 2278849
• Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, p. 446, ISBN 978-0-387-72828-5.
| Wikipedia |
Torsion-free module
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is torsion free if its torsion submodule is reduced to its zero element.
In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module.
Examples of torsion-free modules
Over a commutative ring R with total quotient ring K, a module M is torsion-free if and only if Tor1(K/R,M) vanishes. Therefore flat modules, and in particular free and projective modules, are torsion-free, but the converse need not be true. An example of a torsion-free module that is not flat is the ideal (x, y) of the polynomial ring k[x, y] over a field k, interpreted as a module over k[x, y].
Any torsionless module over a domain is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module which is not torsionless.
Structure of torsion-free modules
Over a Noetherian integral domain, torsion-free modules are the modules whose only associated prime is zero. More generally, over a Noetherian commutative ring the torsion-free modules are those modules all of whose associated primes are contained in the associated primes of the ring.
Over a Noetherian integrally closed domain, any finitely-generated torsion-free module has a free submodule such that the quotient by it is isomorphic to an ideal of the ring.
Over a Dedekind domain, a finitely-generated module is torsion-free if and only if it is projective, but is in general not free. Any such module is isomorphic to the sum of a finitely-generated free module and an ideal, and the class of the ideal is uniquely determined by the module.
Over a principal ideal domain, finitely-generated modules are torsion-free if and only if they are free.
Torsion-free covers
Over an integral domain, every module M has a torsion-free cover F → M from a torsion-free module F onto M, with the properties that any other torsion-free module mapping onto M factors through F, and any endomorphism of F over M is an automorphism of F. Such a torsion-free cover of M is unique up to isomorphism. Torsion-free covers are closely related to flat covers.
Torsion-free quasicoherent sheaves
A quasicoherent sheaf F over a scheme X is a sheaf of ${\mathcal {O}}_{X}$-modules such that for any open affine subscheme U = Spec(R) the restriction F|U is associated to some module M over R. The sheaf F is said to be torsion-free if all those modules M are torsion-free over their respective rings. Alternatively, F is torsion-free if and only if it has no local torsion sections.[1]
See also
• Torsion (algebra)
• torsion-free abelian group
• torsion-free abelian group of rank 1; the classification theory exists for this class.
References
1. Stacks Project, Tag 0AVQ.
• "Torsion-free_module", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Matlis, Eben (1972), Torsion-free modules, The University of Chicago Press, Chicago-London, MR 0344237
• The Stacks Project Authors, The Stacks Project
| Wikipedia |
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.[1] The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.
The term rank has a different meaning in the context of elementary abelian groups.
Definition
A subset {aα} of an abelian group A is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if
$\sum _{\alpha }n_{\alpha }a_{\alpha }=0,\quad n_{\alpha }\in \mathbb {Z} ,$
where all but finitely many coefficients nα are zero (so that the sum is, in effect, finite), then all coefficients are zero. Any two maximal linearly independent sets in A have the same cardinality, which is called the rank of A.
The rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group A is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted T(A). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group A/T(A) is the unique maximal torsion-free quotient of A and its rank coincides with the rank of A.
The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. For this, see finitely generated module#Generic rank.
Properties
• The rank of an abelian group A coincides with the dimension of the Q-vector space A ⊗ Q. If A is torsion-free then the canonical map A → A ⊗ Q is injective and the rank of A is the minimum dimension of Q-vector space containing A as an abelian subgroup. In particular, any intermediate group Zn < A < Qn has rank n.
• Abelian groups of rank 0 are exactly the periodic abelian groups.
• The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2.[2]
• Rank is additive over short exact sequences: if
$0\to A\to B\to C\to 0\;$
is a short exact sequence of abelian groups then rk B = rk A + rk C. This follows from the flatness of Q and the corresponding fact for vector spaces.
• Rank is additive over arbitrary direct sums:
$\operatorname {rank} \left(\bigoplus _{j\in J}A_{j}\right)=\sum _{j\in J}\operatorname {rank} (A_{j}),$
where the sum in the right hand side uses cardinal arithmetic.
Groups of higher rank
Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal d there exist torsion-free abelian groups of rank d that are indecomposable, i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well understood. Moreover, for every integer $n\geq 3$, there is a torsion-free abelian group of rank $2n-2$ that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups. Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.
Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers $n\geq k\geq 1$, there exists a torsion-free abelian group A of rank n such that for any partition $n=r_{1}+\cdots +r_{k}$ into k natural summands, the group A is the direct sum of k indecomposable subgroups of ranks $r_{1},r_{2},\ldots ,r_{k}$. Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of A.
Other surprising examples include torsion-free rank 2 groups An,m and Bn,m such that An is isomorphic to Bn if and only if n is divisible by m.
For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that
• K is indecomposable;
• K is generated by G and a single other element; and
• Every nonzero direct summand of G is decomposable.
Generalization
The notion of rank can be generalized for any module M over an integral domain R, as the dimension over R0, the quotient field, of the tensor product of the module with the field:
$\operatorname {rank} (M)=\dim _{R_{0}}M\otimes _{R}R_{0}$
It makes sense, since R0 is a field, and thus any module (or, to be more specific, vector space) over it is free.
It is a generalization, since every abelian group is a module over the integers. It easily follows that the dimension of the product over Q is the cardinality of maximal linearly independent subset, since for any torsion element x and any rational q,
$x\otimes _{\mathbf {Z} }q=0.$
See also
• Rank of a group
References
1. Page 46 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
2. Thomas, Simon; Schneider, Scott (2012), "Countable Borel equivalence relations", in Cummings, James; Schimmerling, Ernest (eds.), Appalachian Set Theory: 2006-2012, London Mathematical Society Lecture Note Series, vol. 406, Cambridge University Press, pp. 25–62, CiteSeerX 10.1.1.648.3113, doi:10.1017/CBO9781139208574.003, ISBN 9781107608504. On p. 46, Thomas and Schneider refer to "...this failure to classify even the rank 2 groups in a satisfactory way..."
| Wikipedia |
Torsion abelian group
In abstract algebra, a torsion abelian group is an abelian group in which every element has finite order.[1] For example, the torsion subgroup of an abelian group is a torsion abelian group.
See also
• Betti number
References
1. Dummit, David; Foote, Richard. Abstract Algebra, ISBN 978-0471433347, pp. 369
| Wikipedia |
Torsion-free abelian group
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order.
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While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case.[1]
Definitions
Main article: Abelian group
An abelian group $\langle G,+,0\rangle $ is said to be torsion-free if no element other than the identity $e$ is of finite order.[2][3][4] Explicitly, for any $n>0$, the only element $x\in G$ for which $nx=0$ is $x=0$.
A natural example of a torsion-free group is $\langle \mathbb {Z} ,+,0\rangle $, as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the free abelian group $\mathbb {Z} ^{r}$ is torsion-free for any $r\in \mathbb {N} $. An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a $\mathbb {Z} ^{r}$.
A non-finitely generated countable example is given by the additive group of the polynomial ring $\mathbb {Z} [X]$ (the free abelian group of countable rank).
More complicated examples are the additive group of the rational field $\mathbb {Q} $, or its subgroups such as $\mathbb {Z} [p^{-1}]$ (rational numbers whose denominator is a power of $p$). Yet more involved examples are given by groups of higher rank.
Groups of rank 1
Rank
Main article: Rank of an abelian group
The rank of an abelian group $A$ is the dimension of the $\mathbb {Q} $-vector space $\mathbb {Q} \otimes _{\mathbb {Z} }A$. Equivalently it is the maximal cardinality of a linearly independent (over $\mathbb {Z} $) subset of $A$.
If $A$ is torsion-free then it injects into $\mathbb {Q} \otimes _{\mathbb {Z} }A$. Thus, torsion-free abelian groups of rank 1 are exactly subgroups of the additive group $\mathbb {Q} $.
Classification
Torsion-free abelian groups of rank 1 have been completely classified. To do so one associates to a group $A$ a subset $\tau (A)$ of the prime numbers, as follows: pick any $x\in A\setminus \{0\}$, for a prime $p$ we say that $p\in \tau (A)$ if and only if $x\in p^{k}A$ for every $k\in \mathbb {N} $. This does not depend on the choice of $x$ since for another $y\in A\setminus \{0\}$ there exists $n,m\in \mathbb {Z} \setminus \{0\}$ such that $ny=mx$. Baer proved[5][6] that $\tau (A)$ is a complete isomorphism invariant for rank-1 torsion free abelian groups.
Classification problem in general
The hardness of a classification problem for a certain type of structures on a countable set can be quantified using model theory and descriptive set theory. In this sense it has been proved that the classification problem for countable torsion-free abelian groups is as hard as possible.[7]
Notes
1. See for instance the introduction to Thomas, Simon (2003), "The classification problem for torsion-free abelian groups of finite rank", J. Am. Math. Soc., 16 (1): 233–258, Zbl 1021.03043
2. Fraleigh (1976, p. 78)
3. Lang (2002, p. 42)
4. Hungerford (1974, p. 78)
5. Reinhold Baer (1937). "Abelian groups without elements of finite order". Duke Mathematical Journal. 3 (1): 68–122. doi:10.1215/S0012-7094-37-00308-9.
6. Phillip A. Griffith (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7. Chapter VII.
7. Paolini, Gianluca; Shelah, Saharon (2021). "Torsion-Free Abelian Groups are Borel Complete". arXiv:2102.12371 [math.LO].
References
• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
• Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016
• Hungerford, Thomas W. (1974), Algebra, New York: Springer-Verlag, ISBN 0-387-90518-9.
• Lang, Serge (2002), Algebra (Revised 3rd ed.), New York: Springer-Verlag, ISBN 0-387-95385-X.
• McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225
Groups
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E8
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| Wikipedia |
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.
In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950.[1] It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book Homological Algebra.[2]
Definition
Let R be a ring. Write R-Mod for the category of left R-modules and Mod-R for the category of right R-modules. (If R is commutative, the two categories can be identified.) For a fixed left R-module B, let $T(A)=A\otimes _{R}B$ for A in Mod-R. This is a right exact functor from Mod-R to the category of abelian groups Ab, and so it has left derived functors $L_{i}T$. The Tor groups are the abelian groups defined by
$\operatorname {Tor} _{i}^{R}(A,B)=(L_{i}T)(A),$
for an integer i. By definition, this means: take any projective resolution
$\cdots \to P_{2}\to P_{1}\to P_{0}\to A\to 0,$
and remove A, and form the chain complex:
$\cdots \to P_{2}\otimes _{R}B\to P_{1}\otimes _{R}B\to P_{0}\otimes _{R}B\to 0$
For each integer i, the group $\operatorname {Tor} _{i}^{R}(A,B)$ is the homology of this complex at position i. It is zero for i negative. Moreover, $\operatorname {Tor} _{0}^{R}(A,B)$ is the cokernel of the map $P_{1}\otimes _{R}B\to P_{0}\otimes _{R}B$, which is isomorphic to $A\otimes _{R}B$.
Alternatively, one can define Tor by fixing A and taking the left derived functors of the right exact functor G(B) = A ⊗R B. That is, tensor A with a projective resolution of B and take homology. Cartan and Eilenberg showed that these constructions are independent of the choice of projective resolution, and that both constructions yield the same Tor groups.[3] Moreover, for a fixed ring R, Tor is a functor in each variable (from R-modules to abelian groups).
For a commutative ring R and R-modules A and B, TorR
i
(A, B) is an R-module (using that A ⊗R B is an R-module in this case). For a non-commutative ring R, TorR
i
(A, B) is only an abelian group, in general. If R is an algebra over a ring S (which means in particular that S is commutative), then TorR
i
(A, B) is at least an S-module.
Properties
Here are some of the basic properties and computations of Tor groups.[4]
• TorR
0
(A, B) ≅ A ⊗R B for any right R-module A and left R-module B.
• TorR
i
(A, B) = 0 for all i > 0 if either A or B is flat (for example, free) as an R-module. In fact, one can compute Tor using a flat resolution of either A or B; this is more general than a projective (or free) resolution.[5]
• There are converses to the previous statement:
• If TorR
1
(A, B) = 0 for all B, then A is flat (and hence TorR
i
(A, B) = 0 for all i > 0).
• If TorR
1
(A, B) = 0 for all A, then B is flat (and hence TorR
i
(A, B) = 0 for all i > 0).
• By the general properties of derived functors, every short exact sequence 0 → K → L → M → 0 of right R-modules induces a long exact sequence of the form[6]
$\cdots \to \operatorname {Tor} _{2}^{R}(M,B)\to \operatorname {Tor} _{1}^{R}(K,B)\to \operatorname {Tor} _{1}^{R}(L,B)\to \operatorname {Tor} _{1}^{R}(M,B)\to K\otimes _{R}B\to L\otimes _{R}B\to M\otimes _{R}B\to 0,$
for any left R-module B. The analogous exact sequence also holds for Tor with respect to the second variable.
• Symmetry: for a commutative ring R, there is a natural isomorphism TorR
i
(A, B) ≅ TorR
i
(B, A).[7] (For R commutative, there is no need to distinguish between left and right R-modules.)
• If R is a commutative ring and u in R is not a zero divisor, then for any R-module B,
$\operatorname {Tor} _{i}^{R}(R/(u),B)\cong {\begin{cases}B/uB&i=0\\B[u]&i=1\\0&{\text{otherwise}}\end{cases}}$
where
$B[u]=\{x\in B:ux=0\}$
is the u-torsion subgroup of B. This is the explanation for the name Tor. Taking R to be the ring $\mathbb {Z} $ of integers, this calculation can be used to compute $\operatorname {Tor} _{1}^{\mathbb {Z} }(A,B)$ for any finitely generated abelian group A.
• Generalizing the previous example, one can compute Tor groups that involve the quotient of a commutative ring by any regular sequence, using the Koszul complex.[8] For example, if R is the polynomial ring k[x1, ..., xn] over a field k, then $\operatorname {Tor} _{*}^{R}(k,k)$ is the exterior algebra over k on n generators in Tor1.
• $\operatorname {Tor} _{i}^{\mathbb {Z} }(A,B)=0$ for all i ≥ 2. The reason: every abelian group A has a free resolution of length 1, since every subgroup of a free abelian group is free abelian.
• For any ring R, Tor preserves direct sums (possibly infinite) and filtered colimits in each variable.[9] For example, in the first variable, this says that
${\begin{aligned}\operatorname {Tor} _{i}^{R}\left(\bigoplus _{\alpha }M_{\alpha },N\right)&\cong \bigoplus _{\alpha }\operatorname {Tor} _{i}^{R}(M_{\alpha },N)\\\operatorname {Tor} _{i}^{R}\left(\varinjlim _{\alpha }M_{\alpha },N\right)&\cong \varinjlim _{\alpha }\operatorname {Tor} _{i}^{R}(M_{\alpha },N)\end{aligned}}$
• Flat base change: for a commutative flat R-algebra T, R-modules A and B, and an integer i,[10]
$\mathrm {Tor} _{i}^{R}(A,B)\otimes _{R}T\cong \mathrm {Tor} _{i}^{T}(A\otimes _{R}T,B\otimes _{R}T).$
It follows that Tor commutes with localization. That is, for a multiplicatively closed set S in R,
$S^{-1}\operatorname {Tor} _{i}^{R}(A,B)\cong \operatorname {Tor} _{i}^{S^{-1}R}\left(S^{-1}A,S^{-1}B\right).$
• For a commutative ring R and commutative R-algebras A and B, TorR
*
(A,B) has the structure of a graded-commutative algebra over R. Moreover, elements of odd degree in the Tor algebra have square zero, and there are divided power operations on the elements of positive even degree.[11]
Important special cases
• Group homology is defined by $H_{*}(G,M)=\operatorname {Tor} _{*}^{\mathbb {Z} [G]}(\mathbb {Z} ,M),$ where G is a group, M is a representation of G over the integers, and $\mathbb {Z} [G]$ is the group ring of G.
• For an algebra A over a field k and an A-bimodule M, Hochschild homology is defined by
$HH_{*}(A,M)=\operatorname {Tor} _{*}^{A\otimes _{k}A^{\text{op}}}(A,M).$
• Lie algebra homology is defined by $H_{*}({\mathfrak {g}},M)=\operatorname {Tor} _{*}^{U{\mathfrak {g}}}(R,M)$, where ${\mathfrak {g}}$ is a Lie algebra over a commutative ring R, M is a ${\mathfrak {g}}$-module, and $U{\mathfrak {g}}$ is the universal enveloping algebra.
• For a commutative ring R with a homomorphism onto a field k, $\operatorname {Tor} _{*}^{R}(k,k)$ is a graded-commutative Hopf algebra over k.[12] (If R is a Noetherian local ring with residue field k, then the dual Hopf algebra to $\operatorname {Tor} _{*}^{R}(k,k)$ is Ext*
R
(k,k).) As an algebra, $\operatorname {Tor} _{*}^{R}(k,k)$ is the free graded-commutative divided power algebra on a graded vector space π*(R).[13] When k has characteristic zero, π*(R) can be identified with the André-Quillen homology D*(k/R,k).[14]
See also
• Flat morphism
• Serre's intersection formula
• Derived tensor product
• Eilenberg–Moore spectral sequence
Notes
1. Weibel (1999).
2. Cartan & Eilenberg (1956), section VI.1.
3. Weibel (1994), section 2.4 and Theorem 2.7.2.
4. Weibel (1994), Chapters 2 and 3.
5. Weibel (1994), Lemma 3.2.8.
6. Weibel (1994), Definition 2.1.1.
7. Weibel (1994), Remark in section 3.1.
8. Weibel (1994), section 4.5.
9. Weibel (1994), Corollary 2.6.17.
10. Weibel (1994), Corollary 3.2.10.
11. Avramov & Halperin (1986), section 2.16; Stacks Project, Tag 09PQ.
12. Avramov & Halperin (1986), section 4.7.
13. Gulliksen & Levin (1969), Theorem 2.3.5; Sjödin (1980), Theorem 1.
14. Quillen (1970), section 7.
References
• Avramov, Luchezar; Halperin, Stephen (1986), "Through the looking glass: a dictionary between rational homotopy theory and local algebra", in J.-E. Roos (ed.), Algebra, algebraic topology, and their interactions (Stockholm, 1983), Lecture Notes in Mathematics, vol. 1183, Springer Nature, pp. 1–27, doi:10.1007/BFb0075446, ISBN 978-3-540-16453-1, MR 0846435
• Cartan, Henri; Eilenberg, Samuel (1999) [1956], Homological algebra, Princeton: Princeton University Press, ISBN 0-691-04991-2, MR 0077480
• Čech, Eduard (1935), "Les groupes de Betti d'un complexe infini" (PDF), Fundamenta Mathematicae, 25: 33–44, doi:10.4064/fm-25-1-33-44, JFM 61.0609.02
• Gulliksen, Tor; Levin, Gerson (1969), Homology of local rings, Queen's Papers in Pure and Applied Mathematics, vol. 20, Queen's University, MR 0262227
• Quillen, Daniel (1970), "On the (co-)homology of commutative rings", Applications of categorical algebra, Proc. Symp. Pure Mat., vol. 17, American Mathematical Society, pp. 65–87, MR 0257068
• Sjödin, Gunnar (1980), "Hopf algebras and derivations", Journal of Algebra, 64: 218–229, doi:10.1016/0021-8693(80)90143-X, MR 0575792
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
• Weibel, Charles (1999), "History of homological algebra", History of topology (PDF), Amsterdam: North-Holland, pp. 797–836, MR 1721123
External links
• The Stacks Project Authors, The Stacks Project
| Wikipedia |
Torsion of a curve
In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.
Definition
Let r be a space curve parametrized by arc length s and with the unit tangent vector T. If the curvature κ of r at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors
$\mathbf {N} ={\frac {\mathbf {T} '}{\kappa }},\quad \mathbf {B} =\mathbf {T} \times \mathbf {N} $
respectively, where the prime denotes the derivative of the vector with respect to the parameter s. The torsion τ measures the speed of rotation of the binormal vector at the given point. It is found from the equation
$\mathbf {B} '=-\tau \mathbf {N} .$
which means
$\tau =-\mathbf {N} \cdot \mathbf {B} '.$
As $\mathbf {N} \cdot \mathbf {B} =0$, this is equivalent to $\tau =\mathbf {N} '\cdot \mathbf {B} $.
Remark: The derivative of the binormal vector is perpendicular to both the binormal and the tangent, hence it has to be proportional to the principal normal vector. The negative sign is simply a matter of convention: it is a byproduct of the historical development of the subject.
Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
Properties
• A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane.
• The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and non-zero is a helix. The torsion is positive for a right-handed[1] helix and is negative for a left-handed one.
Alternative description
Let r = r(t) be the parametric equation of a space curve. Assume that this is a regular parametrization and that the curvature of the curve does not vanish. Analytically, r(t) is a three times differentiable function of t with values in R3 and the vectors
$\mathbf {r'} (t),\mathbf {r''} (t)$
are linearly independent.
Then the torsion can be computed from the following formula:
$\tau ={\frac {\det \left({\mathbf {r} ',\mathbf {r} '',\mathbf {r} '''}\right)}{\left\|{\mathbf {r} '\times \mathbf {r} ''}\right\|^{2}}}={\frac {\left({\mathbf {r} '\times \mathbf {r} ''}\right)\cdot \mathbf {r} '''}{\left\|{\mathbf {r} '\times \mathbf {r} ''}\right\|^{2}}}.$
Here the primes denote the derivatives with respect to t and the cross denotes the cross product. For r = (x, y, z), the formula in components is
$\tau ={\frac {x'''\left(y'z''-y''z'\right)+y'''\left(x''z'-x'z''\right)+z'''\left(x'y''-x''y'\right)}{\left(y'z''-y''z'\right)^{2}+\left(x''z'-x'z''\right)^{2}+\left(x'y''-x''y'\right)^{2}}}.$
Notes
1. Weisstein, Eric W. "Torsion". mathworld.wolfram.com.
References
• Pressley, Andrew (2001), Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, ISBN 1-85233-152-6
Wikimedia Commons has media related to Graphical illustrations of the torsion of space curves.
Various notions of curvature defined in differential geometry
Differential geometry
of curves
• Curvature
• Torsion of a curve
• Frenet–Serret formulas
• Radius of curvature (applications)
• Affine curvature
• Total curvature
• Total absolute curvature
Differential geometry
of surfaces
• Principal curvatures
• Gaussian curvature
• Mean curvature
• Darboux frame
• Gauss–Codazzi equations
• First fundamental form
• Second fundamental form
• Third fundamental form
Riemannian geometry
• Curvature of Riemannian manifolds
• Riemann curvature tensor
• Ricci curvature
• Scalar curvature
• Sectional curvature
Curvature of connections
• Curvature form
• Torsion tensor
• Cocurvature
• Holonomy
| Wikipedia |
Torsion sheaf
In mathematics, a torsion sheaf is a sheaf of abelian groups ${\mathcal {F}}$ on a site for which, for every object U, the space of sections $\Gamma (U,{\mathcal {F}})$ is a torsion abelian group. Similarly, for a prime number p, we say a sheaf ${\mathcal {F}}$ is p-torsion if every section over any object is killed by a power of p.
A torsion sheaf on an étale site is the union of its constructible subsheaves.[1]
See also
• Twisted sheaf
Notes
1. Milne 2012, Remark 17.6
References
• Milne, James S. (2012). "Lectures on Étale Cohomology" (PDF).
• J. S. Milne, Étale Cohomology
• Fu, Lei (2015). Etale Cohomology Theory. Nankai Tracts in Mathematics. Vol. 14. doi:10.1142/9569. ISBN 978-981-4675-08-6.
| Wikipedia |
Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet–Serret frame about the tangent vector). In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting".
More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described concretely as a tensor, or as a vector-valued 2-form on the manifold. If ∇ is an affine connection on a differential manifold, then the torsion tensor is defined, in terms of vector fields X and Y, by
$T(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$
where [X,Y] is the Lie bracket of vector fields.
Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which absorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the contorsion tensor. Absorption of torsion also plays a fundamental role in the study of G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory.
The torsion tensor
Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇. The torsion tensor (sometimes called the Cartan (torsion) tensor) of ∇ is the vector-valued 2-form defined on vector fields X and Y by
$T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$
where [X, Y] is the Lie bracket of two vector fields. By the Leibniz rule, T(fX, Y) = T(X, fY) = fT(X, Y) for any smooth function f. So T is tensorial, despite being defined in terms of the connection which is a first order differential operator: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields.
Components of the torsion tensor
The components of the torsion tensor $T^{c}{}_{ab}$ in terms of a local basis (e1, ..., en) of sections of the tangent bundle can be derived by setting X = ei, Y = ej and by introducing the commutator coefficients γkijek := [ei, ej]. The components of the torsion are then
$T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.$
Here ${\Gamma ^{k}}_{ij}$ are the connection coefficients defining the connection. If the basis is holonomic then the Lie brackets vanish, $\gamma ^{k}{}_{ij}=0$. So $T^{k}{}_{ij}=2\Gamma ^{k}{}_{[ij]}$. In particular (see below), while the geodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.
The torsion form
The torsion form, an alternative characterization of torsion, applies to the frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω, a gl(n)-valued one-form which maps vertical vectors to the generators of the right action in gl(n) and equivariantly intertwines the right action of GL(n) on the tangent bundle of FM with the adjoint representation on gl(n). The frame bundle also carries a canonical one-form θ, with values in Rn, defined at a frame u ∈ FxM (regarded as a linear function u : Rn → TxM) by
$\theta (X)=u^{-1}(\pi _{*}(X))$
where π : FM → M is the projection mapping for the principal bundle and π∗ is its push-forward. The torsion form is then
$\Theta =d\theta +\omega \wedge \theta .$
Equivalently, Θ = Dθ, where D is the exterior covariant derivative determined by the connection.
The torsion form is a (horizontal) tensorial form with values in Rn, meaning that under the right action of g ∈ GL(n) it transforms equivariantly:
$R_{g}^{*}\Theta =g^{-1}\cdot \Theta $
where g acts on the right-hand side through its adjoint representation on Rn.
Torsion form in a frame
See also: connection form
The torsion form may be expressed in terms of a connection form on the base manifold M, written in a particular frame of the tangent bundle (e1, ..., en). The connection form expresses the exterior covariant derivative of these basic sections:
$D\mathbf {e} _{i}=\mathbf {e} _{j}{\omega ^{j}}_{i}.$
The solder form for the tangent bundle (relative to this frame) is the dual basis θi ∈ T∗M of the ei, so that θi(ej) = δij (the Kronecker delta). Then the torsion 2-form has components
$\Theta ^{k}=d\theta ^{k}+{\omega ^{k}}_{j}\wedge \theta ^{j}={T^{k}}_{ij}\theta ^{i}\wedge \theta ^{j}.$
In the rightmost expression,
${T^{k}}_{ij}=\theta ^{k}\left(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-\left[\mathbf {e} _{i},\mathbf {e} _{j}\right]\right)$
are the frame-components of the torsion tensor, as given in the previous definition.
It can be easily shown that Θi transforms tensorially in the sense that if a different frame
${\tilde {\mathbf {e} }}_{i}=\mathbf {e} _{j}{g^{j}}_{i}$
for some invertible matrix-valued function (gji), then
${\tilde {\Theta }}^{i}={\left(g^{-1}\right)^{i}}_{j}\Theta ^{j}.$
In other terms, Θ is a tensor of type (1, 2) (carrying one contravariant and two covariant indices).
Alternatively, the solder form can be characterized in a frame-independent fashion as the TM-valued one-form θ on M corresponding to the identity endomorphism of the tangent bundle under the duality isomorphism End(TM) ≈ TM ⊗ T∗M. Then the torsion 2-form is a section
$\Theta \in {\text{Hom}}\left( \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right)$
given by
$\Theta =D\theta ,$
where D is the exterior covariant derivative. (See connection form for further details.)
Irreducible decomposition
The torsion tensor can be decomposed into two irreducible parts: a trace-free part and another part which contains the trace terms. Using the index notation, the trace of T is given by
$a_{i}=T^{k}{}_{ik},$
and the trace-free part is
$B^{i}{}_{jk}=T^{i}{}_{jk}+{\frac {1}{n-1}}\delta ^{i}{}_{j}a_{k}-{\frac {1}{n-1}}\delta ^{i}{}_{k}a_{j},$
where δij is the Kronecker delta.
Intrinsically, one has
$T\in \operatorname {Hom} \left( \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right).$
The trace of T, tr T, is an element of T∗M defined as follows. For each vector fixed X ∈ TM, T defines an element T(X) of Hom(TM, TM) via
$T(X):Y\mapsto T(X\wedge Y).$
Then (tr T)(X) is defined as the trace of this endomorphism. That is,
$(\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X)).$
The trace-free part of T is then
$T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T),$
where ι denotes the interior product.
Curvature and the Bianchi identities
The curvature tensor of ∇ is a mapping TM × TM → End(TM) defined on vector fields X, Y, and Z by
$R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z.$
For vectors at a point, this definition is independent of how the vectors are extended to vector fields away from the point (thus it defines a tensor, much like the torsion).
The Bianchi identities relate the curvature and torsion as follows.[1] Let ${\mathfrak {S}}$ denote the cyclic sum over X, Y, and Z. For instance,
${\mathfrak {S}}\left(R\left(X,Y\right)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.$
Then the following identities hold
1. Bianchi's first identity:
${\mathfrak {S}}\left(R\left(X,Y\right)Z\right)={\mathfrak {S}}\left(T\left(T(X,Y),Z\right)+\left(\nabla _{X}T\right)\left(Y,Z\right)\right)$
2. Bianchi's second identity:
${\mathfrak {S}}\left(\left(\nabla _{X}R\right)\left(Y,Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0$
The curvature form and Bianchi identities
The curvature form is the gl(n)-valued 2-form
$\Omega =D\omega =d\omega +\omega \wedge \omega $
where, again, D denotes the exterior covariant derivative. In terms of the curvature form and torsion form, the corresponding Bianchi identities are[2]
1. $D\Theta =\Omega \wedge \theta $
2. $D\Omega =0.$
Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At a point u of FxM, one has[3]
${\begin{aligned}R(X,Y)Z&=u\left(2\Omega \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right)\left(u^{-1}(Z)\right),\\T(X,Y)&=u\left(2\Theta \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right),\end{aligned}}$
where again u : Rn → TxM is the function specifying the frame in the fibre, and the choice of lift of the vectors via π−1 is irrelevant since the curvature and torsion forms are horizontal (they vanish on the ambiguous vertical vectors).
Characterizations and interpretations
Throughout this section, M is assumed to be a differentiable manifold, and ∇ a covariant derivative on the tangent bundle of M unless otherwise noted.
Twisting of reference frames
In the classical differential geometry of curves, the Frenet-Serret formulas describe how a particular moving frame (the Frenet-Serret frame) twists along a curve. In physical terms, the torsion corresponds to the angular momentum of an idealized top pointing along the tangent of the curve.
The case of a manifold with a (metric) connection admits an analogous interpretation. Suppose that an observer is moving along a geodesic for the connection. Such an observer is ordinarily thought of as inertial since they experience no acceleration. Suppose that in addition the observer carries with themselves a system of rigid straight measuring rods (a coordinate system). Each rod is a straight segment; a geodesic. Assume that each rod is parallel transported along the trajectory. The fact that these rods are physically carried along the trajectory means that they are Lie-dragged, or propagated so that the Lie derivative of each rod along the tangent vanishes. They may, however, experience torque (or torsional forces) analogous to the torque felt by the top in the Frenet-Serret frame. This force is measured by the torsion.
More precisely, suppose that the observer moves along a geodesic path γ(t) and carries a measuring rod along it. The rod sweeps out a surface as the observer travels along the path. There are natural coordinates (t, x) along this surface, where t is the parameter time taken by the observer, and x is the position along the measuring rod. The condition that the tangent of the rod should be parallel translated along the curve is
$\left.\nabla _{\frac {\partial }{\partial t}}{\frac {\partial }{\partial x}}\right|_{x=0}=0.$
Consequently, the torsion is given by
$\left.T\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial t}}\right)\right|_{x=0}=\left.\nabla _{\frac {\partial }{\partial x}}{\frac {\partial }{\partial t}}\right|_{x=0}.$
If this is not zero, then the marked points on the rod (the x = constant curves) will trace out helices instead of geodesics. They will tend to rotate around the observer. Note that for this argument it was not essential that $\gamma (t)$ is a geodesic. Any curve would work.
This interpretation of torsion plays a role in the theory of teleparallelism, also known as Einstein–Cartan theory, an alternative formulation of relativity theory.
The torsion of a filament
In materials science, and especially elasticity theory, ideas of torsion also play an important role. One problem models the growth of vines, focusing on the question of how vines manage to twist around objects.[4] The vine itself is modeled as a pair of elastic filaments twisted around one another. In its energy-minimizing state, the vine naturally grows in the shape of a helix. But the vine may also be stretched out to maximize its extent (or length). In this case, the torsion of the vine is related to the torsion of the pair of filaments (or equivalently the surface torsion of the ribbon connecting the filaments), and it reflects the difference between the length-maximizing (geodesic) configuration of the vine and its energy-minimizing configuration.
Torsion and vorticity
In fluid dynamics, torsion is naturally associated to vortex lines.
Geodesics and the absorption of torsion
Suppose that γ(t) is a curve on M. Then γ is an affinely parametrized geodesic provided that
$\nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0$
for all time t in the domain of γ. (Here the dot denotes differentiation with respect to t, which associates with γ the tangent vector pointing along it.) Each geodesic is uniquely determined by its initial tangent vector at time t = 0, ${\dot {\gamma }}(0)$.
One application of the torsion of a connection involves the geodesic spray of the connection: roughly the family of all affinely parametrized geodesics. Torsion is the ambiguity of classifying connections in terms of their geodesic sprays:
• Two connections ∇ and ∇′ which have the same affinely parametrized geodesics (i.e., the same geodesic spray) differ only by torsion.[5]
More precisely, if X and Y are a pair of tangent vectors at p ∈ M, then let
$\Delta (X,Y)=\nabla _{X}{\tilde {Y}}-\nabla '_{X}{\tilde {Y}}$
be the difference of the two connections, calculated in terms of arbitrary extensions of X and Y away from p. By the Leibniz product rule, one sees that Δ does not actually depend on how X and Y{{′}} are extended (so it defines a tensor on M). Let S and A be the symmetric and alternating parts of Δ:
$S(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)+\Delta (Y,X)\right)$
$A(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)$
Then
• $A(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)$ is the difference of the torsion tensors.
• ∇ and ∇′ define the same families of affinely parametrized geodesics if and only if S(X, Y) = 0.
In other words, the symmetric part of the difference of two connections determines whether they have the same parametrized geodesics, whereas the skew part of the difference is determined by the relative torsions of the two connections. Another consequence is:
• Given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics. The difference between these two connections is in fact a tensor, the contorsion tensor.
This is a generalization of the fundamental theorem of Riemannian geometry to general affine (possibly non-metric) connections. Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is known as absorption of torsion, and it is one of the stages of Cartan's equivalence method.
See also
• Contorsion tensor
• Curtright field
• Curvature tensor
• Levi-Civita connection
• Torsion coefficient
• Torsion of curves
Notes
1. Kobayashi & Nomizu 1963, Volume 1, Proposition III.5.2.
2. Kobayashi & Nomizu 1963, Volume 1, III.2.
3. Kobayashi & Nomizu 1963, Volume 1, III.5.
4. Goriely et al. 2006.
5. See Spivak (1999) Volume II, Addendum 1 to Chapter 6. See also Bishop and Goldberg (1980), section 5.10.
References
• Bishop, R.L.; Goldberg, S.I. (1980), Tensor analysis on manifolds, Dover Publications
• Cartan, É. (1923), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)", Annales Scientifiques de l'École Normale Supérieure, 40: 325–412, doi:10.24033/asens.751
• Cartan, É. (1924), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)", Annales Scientifiques de l'École Normale Supérieure, 41: 1–25, doi:10.24033/asens.753
• Elzanowski, M.; Epstein, M. (1985), "Geometric characterization of hyperelastic uniformity", Archive for Rational Mechanics and Analysis, 88 (4): 347–357, Bibcode:1985ArRMA..88..347E, doi:10.1007/BF00250871, S2CID 120127682
• Goriely, A.; Robertson-Tessi, M.; Tabor, M.; Vandiver, R. (2006), "Elastic growth models" (PDF), BIOMAT-2006, Springer-Verlag, archived from the original (PDF) on 2006-12-29
• Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976), "General relativity with spin and torsion: Foundations and prospects", Rev. Mod. Phys., 48 (3): 393–416, Bibcode:1976RvMP...48..393H, doi:10.1103/revmodphys.48.393, 393.
• Kibble, T.W.B. (1961), "Lorentz invariance and the gravitational field", J. Math. Phys., 2 (2): 212–221, Bibcode:1961JMP.....2..212K, doi:10.1063/1.1703702, 212.
• Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, vol. 1 & 2 (New ed.), Wiley-Interscience (published 1996), ISBN 0-471-15733-3
• Poplawski, N.J. (2009), Spacetime and fields, arXiv:0911.0334, Bibcode:2009arXiv0911.0334P
• Schouten, J.A. (1954), Ricci Calculus, Springer-Verlag
• Schrödinger, E. (1950), Space-Time Structure, Cambridge University Press
• Sciama, D.W. (1964), "The physical structure of general relativity", Rev. Mod. Phys., 36 (1): 463, Bibcode:1964RvMP...36..463S, doi:10.1103/RevModPhys.36.463
• Spivak, M. (1999), A comprehensive introduction to differential geometry, Volume II, Houston, Texas: Publish or Perish, ISBN 0-914098-71-3
Various notions of curvature defined in differential geometry
Differential geometry
of curves
• Curvature
• Torsion of a curve
• Frenet–Serret formulas
• Radius of curvature (applications)
• Affine curvature
• Total curvature
• Total absolute curvature
Differential geometry
of surfaces
• Principal curvatures
• Gaussian curvature
• Mean curvature
• Darboux frame
• Gauss–Codazzi equations
• First fundamental form
• Second fundamental form
• Third fundamental form
Riemannian geometry
• Curvature of Riemannian manifolds
• Riemann curvature tensor
• Ricci curvature
• Scalar curvature
• Sectional curvature
Curvature of connections
• Curvature form
• Torsion tensor
• Cocurvature
• Holonomy
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
• Euclidean geometry
• Exterior calculus
• Multilinear algebra
• Tensor algebra
• Tensor calculus
• Physics
• Engineering
• Computer vision
• Continuum mechanics
• Electromagnetism
• General relativity
• Transport phenomena
Notation
• Abstract index notation
• Einstein notation
• Index notation
• Multi-index notation
• Penrose graphical notation
• Ricci calculus
• Tetrad (index notation)
• Van der Waerden notation
• Voigt notation
Tensor
definitions
• Tensor (intrinsic definition)
• Tensor field
• Tensor density
• Tensors in curvilinear coordinates
• Mixed tensor
• Antisymmetric tensor
• Symmetric tensor
• Tensor operator
• Tensor bundle
• Two-point tensor
Operations
• Covariant derivative
• Exterior covariant derivative
• Exterior derivative
• Exterior product
• Hodge star operator
• Lie derivative
• Raising and lowering indices
• Symmetrization
• Tensor contraction
• Tensor product
• Transpose (2nd-order tensors)
Related
abstractions
• Affine connection
• Basis
• Cartan formalism (physics)
• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
• Manifold
• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
• Nonmetricity tensor
• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
• Angular momentum tensor
• Spin tensor
• Cauchy stress tensor
• stress–energy tensor
• Einstein tensor
• EM tensor
• Gluon field strength tensor
• Metric tensor (GR)
Mathematicians
• Élie Cartan
• Augustin-Louis Cauchy
• Elwin Bruno Christoffel
• Albert Einstein
• Leonhard Euler
• Carl Friedrich Gauss
• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Jan Arnoldus Schouten
• Woldemar Voigt
• Hermann Weyl
| Wikipedia |
Dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimensions, a dihedral angle represents the angle between two hyperplanes. The planes of a flying machine are said to be at positive dihedral angle when both starboard and port main planes (commonly called "wings") are upwardly inclined to the lateral axis; when downwardly inclined they are said to be at a negative dihedral angle.
Types of angles
2D angles
Right
Interior
Exterior
2D angle pairs
Adjacent
Vertical
Complementary
Supplementary
Transversal
3D angles
Dihedral
Mathematical background
When the two intersecting planes are described in terms of Cartesian coordinates by the two equations
$a_{1}x+b_{1}y+c_{1}z+d_{1}=0$
$a_{2}x+b_{2}y+c_{2}z+d_{2}=0$
the dihedral angle, $\varphi $ between them is given by:
$\cos \varphi ={\frac {\left\vert a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}\right\vert }{{\sqrt {a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}{\sqrt {a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}}$
and satisfies $0\leq \varphi \leq \pi /2.$
Alternatively, if nA and nB are normal vector to the planes, one has
$\cos \varphi ={\frac {\left\vert \mathbf {n} _{\mathrm {A} }\cdot \mathbf {n} _{\mathrm {B} }\right\vert }{|\mathbf {n} _{\mathrm {A} }||\mathbf {n} _{\mathrm {B} }|}}$
where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths.[1]
The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.
However the absolute values can be and should be avoided when considering the dihedral angle of two half planes whose boundaries are the same line. In this case, the half planes can be described by a point P of their intersection, and three vectors b0, b1 and b2 such that P + b0, P + b1 and P + b2 belong respectively to the intersection line, the first half plane, and the second half plane. The dihedral angle of these two half planes is defined by
$\cos \varphi ={\frac {(\mathbf {b} _{0}\times \mathbf {b} _{1})\cdot (\mathbf {b} _{0}\times \mathbf {b} _{2})}{|\mathbf {b} _{0}\times \mathbf {b} _{1}||\mathbf {b} _{0}\times \mathbf {b} _{2}|}}$,
and satisfies $0\leq \varphi <\pi .$ In this case, switching the two half-planes gives the same result, and so does replacing $\mathbf {b} _{0}$ with $-\mathbf {b} _{0}.$ In chemistry (see below), we define a dihedral angle such that replacing $\mathbf {b} _{0}$ with $-\mathbf {b} _{0}$ changes the sign of the angle, which can be between −π and π.
In polymer physics
In some scientific areas such as polymer physics, one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positions r1, r2, r3, etc. then bond vectors are defined by u1=r2−r1, u2=r3−r2, and ui=ri+1−ri, more generally.[2] This is the case for kinematic chains or amino acids in a protein structure. In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If u1, u2 and u3 are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval (−π, π]. This dihedral angle is defined by[3]
${\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3}))}{|\mathbf {u} _{2}|\,|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}$
or, using the function atan2,
$\varphi =\operatorname {atan2} (\mathbf {u} _{2}\cdot ((\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})),|\mathbf {u} _{2}|\,(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).$
This dihedral angle does not depend on the orientation of the chain (order in which the point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3. Both operations do not change the cosine, but change the sign of the sine. Thus, together, they do not change the angle.
A simpler formula for the same dihedral angle is the following (the proof is given below)
${\begin{aligned}\cos \varphi &={\frac {(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}}\\\sin \varphi &={\frac {|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})}{|\mathbf {u} _{1}\times \mathbf {u} _{2}|\,|\mathbf {u} _{2}\times \mathbf {u} _{3}|}},\end{aligned}}$
or equivalently,
$\varphi =\operatorname {atan2} (|\mathbf {u} _{2}|\,\mathbf {u} _{1}\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3}),(\mathbf {u} _{1}\times \mathbf {u} _{2})\cdot (\mathbf {u} _{2}\times \mathbf {u} _{3})).$
This can be deduced from previous formulas by using the vector quadruple product formula, and the fact that a scalar triple product is zero if it contains twice the same vector:
$(\mathbf {u} _{1}\times \mathbf {u} _{2})\times (\mathbf {u} _{2}\times \mathbf {u} _{3})=[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{1}]\mathbf {u} _{2}-[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{2}]\mathbf {u} _{1}=[(\mathbf {u} _{2}\times \mathbf {u} _{3})\cdot \mathbf {u} _{1}]\mathbf {u} _{2}$
Given the definition of the cross product, this means that $\varphi $ is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say the usual cases) are $\varphi =\pi $, $\varphi =+\pi /3$ and $\varphi =-\pi /3$, which are called the trans, gauche+, and gauche− conformations.
In stereochemistry
Configuration names
according to dihedral angle
syn n-Butane in the
gauche− conformation (−60°)
Newman projection
syn n-Butane
sawhorse projection
In stereochemistry, a torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a chemical bond.[4][5] Every set of three non-colinear atoms of a molecule defines a half-plane. As explained above, when two such half-planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation.[6] Stereochemical arrangements corresponding to angles between 0° and ±90° are called syn (s), those corresponding to angles between ±90° and 180° anti (a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal (c) and those between 0° and ±30° or ±150° and 180° are called periplanar (p).
The two types of terms can be combined so as to define four ranges of angle; 0° to ±30° synperiplanar (sp); 30° to 90° and −30° to −90° synclinal (sc); 90° to 150° and −90° to −150° anticlinal (ac); ±150° to 180° antiperiplanar (ap). The synperiplanar conformation is also known as the syn- or cis-conformation; antiperiplanar as anti or trans; and synclinal as gauche or skew.
For example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn-conformation shown above, with a dihedral angle of 60° is less stable than the anti-conformation with a dihedral angle of 180°.
For macromolecular usage the symbols T, C, G+, G−, A+ and A− are recommended (ap, sp, +sc, −sc, +ac and −ac respectively).
Proteins
A Ramachandran plot (also known as a Ramachandran diagram or a [φ,ψ] plot), originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan,[7] is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure. In a protein chain three dihedral angles are defined:
• ω (omega) is the angle in the chain Cα − C' − N − Cα,
• φ (phi) is the angle in the chain C' − N − Cα − C'
• ψ (psi) is the angle in the chain N − Cα − C' − N (called φ′ by Ramachandran)
The figure at right illustrates the location of each of these angles (but it does not show correctly the way they are defined).[8]
The planarity of the peptide bond usually restricts ω to be 180° (the typical trans case) or 0° (the rare cis case). The distance between the Cα atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, respectively. The vast majority of the peptide bonds in proteins are trans, though the peptide bond to the nitrogen of proline has an increased prevalence of cis compared to other amino-acid pairs.[9]
The side chain dihedral angles are designated with χn (chi-n).[10] They tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche−, and gauche+ conformations. The stability of certain sidechain dihedral angles is affected by the values φ and ψ.[11] For instance, there are direct steric interactions between the Cγ of the side chain in the gauche+ rotamer and the backbone nitrogen of the next residue when ψ is near -60°.[12] This is evident from statistical distributions in backbone-dependent rotamer libraries.
Converting from dihedral angles to Cartesian coordinates in chains
It is common to represent polymers backbones, notably proteins, in internal coordinates; that is, a list of consecutive dihedral angles and bond lengths. However, some types of computational chemistry instead use cartesian coordinates. In computational structure optimization, some programs need to flip back and forth between these representations during their iterations. This task can dominate the calculation time. For processes with many iterations or with long chains, it can also introduce cumulative numerical inaccuracy. While all conversion algorithms produce mathematically identical results, they differ in speed and numerical accuracy.[13]
Geometry
Every polyhedron has a dihedral angle at every edge describing the relationship of the two faces that share that edge. This dihedral angle, also called the face angle, is measured as the internal angle with respect to the polyhedron. An angle of 0° means the face normal vectors are antiparallel and the faces overlap each other, which implies that it is part of a degenerate polyhedron. An angle of 180° means the faces are parallel, as in a tiling. An angle greater than 180° exists on concave portions of a polyhedron.
Every dihedral angle in an edge-transitive polyhedron has the same value. This includes the 5 Platonic solids, the 13 Catalan solids, the 4 Kepler–Poinsot polyhedra, the two quasiregular solids, and two quasiregular dual solids.
Law of cosines for dihedral angle
Given 3 faces of a polyhedron which meet at a common vertex P and have edges AP, BP and CP, the cosine of the dihedral angle between the faces containing APC and BPC is:[14]
$\cos \varphi ={\frac {\cos(\angle \mathrm {APB} )-\cos(\angle \mathrm {APC} )\cos(\angle \mathrm {BPC} )}{\sin(\angle \mathrm {APC} )\sin(\angle \mathrm {BPC} )}}$
This can be deduced from Spherical law of cosines
See also
• Atropisomer
References
1. "Angle Between Two Planes". TutorVista.com. Retrieved 2018-07-06.
2. Kröger, Martin (2005). Models for polymeric and anisotropic liquids. Springer. ISBN 3540262105.
3. Blondel, Arnaud; Karplus, Martin (7 Dec 1998). "New formulation for derivatives of torsion angles and improper torsion angles in molecular mechanics: Elimination of singularities". Journal of Computational Chemistry. 17 (9): 1132–1141. doi:10.1002/(SICI)1096-987X(19960715)17:9<1132::AID-JCC5>3.0.CO;2-T.
4. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Torsion angle". doi:10.1351/goldbook.T06406
5. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Dihedral angle". doi:10.1351/goldbook.D01730
6. Anslyn, Eric; Dennis Dougherty (2006). Modern Physical Organic Chemistry. University Science. p. 95. ISBN 978-1891389313.
7. Ramachandran, G. N.; Ramakrishnan, C.; Sasisekharan, V. (1963). "Stereochemistry of polypeptide chain configurations". Journal of Molecular Biology. 7: 95–9. doi:10.1016/S0022-2836(63)80023-6. PMID 13990617.
8. Richardson, J. S. (1981). "The Anatomy and Taxonomy of Protein Structure". Anatomy and Taxonomy of Protein Structures. Advances in Protein Chemistry. Vol. 34. pp. 167–339. doi:10.1016/S0065-3233(08)60520-3. ISBN 9780120342341. PMID 7020376.
9. Singh J, Hanson J, Heffernan R, Paliwal K, Yang Y, Zhou Y (August 2018). "Detecting Proline and Non-Proline Cis Isomers in Protein Structures from Sequences Using Deep Residual Ensemble Learning". Journal of Chemical Information and Modeling. 58 (9): 2033–2042. doi:10.1021/acs.jcim.8b00442. PMID 30118602. S2CID 52031431.
10. "Side Chain Conformation".
11. Dunbrack, RL Jr.; Karplus, M (20 March 1993). "Backbone-dependent rotamer library for proteins. Application to side-chain prediction". Journal of Molecular Biology. 230 (2): 543–74. doi:10.1006/jmbi.1993.1170. PMID 8464064.
12. Dunbrack, RL Jr; Karplus, M (May 1994). "Conformational analysis of the backbone-dependent rotamer preferences of protein sidechains". Nature Structural Biology. 1 (5): 334–40. doi:10.1038/nsb0594-334. PMID 7664040. S2CID 9157373.
13. Parsons, J.; Holmes, J. B.; Rojas, J. M.; Tsai, J.; Strauss, C. E. (2005), "Practical conversion from torsion space to cartesian space for in silico protein synthesis", Journal of Computational Chemistry, 26 (10): 1063–1068, doi:10.1002/jcc.20237, PMID 15898109, S2CID 2279574
14. "dihedral angle calculator polyhedron". www.had2know.com. Archived from the original on 25 November 2015. Retrieved 25 October 2015.
External links
• The Dihedral Angle in Woodworking at Tips.FM
• Analysis of the 5 Regular Polyhedra gives a step-by-step derivation of these exact values.
| Wikipedia |
Torsor (algebraic geometry)
In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topologies. The notion also generalizes a Galois extension in abstract algebra.
The category of torsors over a fixed base forms a stack. Conversely, a prestack can be stackified by taking the category of torsors (over the prestack).
Definition
Given a smooth algebraic group G, a G-torsor (or a principal G-bundle) P over a scheme X is a scheme (or even algebraic space) with an action of G that is locally trivial in the given Grothendieck topology in the sense that the base change $Y\times _{X}P$ along some covering map $Y\to X$ is isomorphic to the trivial torsor $Y\times G\to Y$ (G acts only on the second factor).[1] Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme $G_{X}=X\times G$ (i.e., $G_{X}$ acts simply transitively on $P$.)
The definition may be formulated in the sheaf-theoretic language: a sheaf P on the category of X-schemes with some Grothendieck topology is a G-torsor if there is a covering $\{U_{i}\to X\}$ in the topology, called the local trivialization, such that the restriction of P to each $U_{i}$ is a trivial $G_{U_{i}}$-torsor.
A line bundle is nothing but a $\mathbb {G} _{m}$-bundle, and, like a line bundle, the two points of views of torsors, geometric and sheaf-theoretic, are used interchangeably (by permitting P to be a stack like an algebraic space if necessary[2]).
It is common to consider a torsor for not just a group scheme but more generally for a group sheaf (e.g., fppf group sheaf).
Examples and basic properties
Examples
• A $\operatorname {GL} _{n}$-torsor on X is a principal $\operatorname {GL} _{n}$-bundle on X.
• If $L/K$ is a finite Galois extension, then $\operatorname {Spec} L\to \operatorname {Spec} K$ is a $\operatorname {Gal} (L/K)$-torsor (roughly because the Galois group acts simply transitively on the roots.) This fact is a basis for Galois descent. See integral extension for a generalization.
Remark: A G-torsor P over X is isomorphic to a trivial torsor if and only if $P(X)=\operatorname {Mor} (X,P)$ is nonempty. (Proof: if there is an $s:X\to P$, then $X\times G\to P,(x,g)\mapsto s(x)g$ is an isomorphism.)
Let P be a G-torsor with a local trivialization $\{U_{i}\to X\}$ in étale topology. A trivial torsor admits a section: thus, there are elements $s_{i}\in P(U_{i})$. Fixing such sections $s_{i}$, we can write uniquely $s_{i}g_{ij}=s_{j}$ on $U_{ij}$ with $g_{ij}\in G(U_{ij})$. Different choices of $s_{i}$ amount to 1-coboundaries in cohomology; that is, the $g_{ij}$ define a cohomology class in the sheaf cohomology (more precisely Čech cohomology with sheaf coefficient) group $H^{1}(X,G)$.[3] A trivial torsor corresponds to the identity element. Conversely, it is easy to see any class in $H^{1}(X,G)$ defines a G-torsor on X, unique up to an isomorphism.
If G is a connected algebraic group over a finite field $\mathbf {F} _{q}$, then any G-bundle over $\operatorname {Spec} \mathbf {F} _{q}$ is trivial. (Lang's theorem.)
Reduction of a structure group
Most of constructions and terminology regarding principal bundles in algebraic topology carry over in verbatim to G-bundles. For example, if $P\to X$ is a G-bundle and G acts from the left on a scheme F, then one can form the associated bundle $P\times ^{G}F\to X$ with fiber F. In particular, if H is a closed subgroup of G, then for any H-bundle P, $P\times ^{H}G$ is a G-bundle called the induced bundle.
If P is a G-bundle that is isomorphic to the induced bundle $P'\times ^{H}G$ for some H-bundle P', then P is said to admit a reduction of structure group from G to H.
Let X be a smooth projective curve over an algebraically closed field k, G a semisimple algebraic group and P a G-bundle on a relative curve $X_{R}=X\times _{\operatorname {Spec} k}\operatorname {Spec} R$, R a finitely generated k-algebra. Then a theorem of Drinfeld and Simpson states that, if G is simply connected and split, there is an étale morphism $R\to R'$ such that $P\times _{X_{R}}X_{R'}$ admits a reduction of structure group to a Borel subgroup of G.[4][5]
Invariants
If P is a parabolic subgroup of a smooth affine group scheme G with connected fibers, then its degree of instability, denoted by $\deg _{i}(P)$, is the degree of its Lie algebra $\operatorname {Lie} (P)$ as a vector bundle on X. The degree of instability of G is then $\deg _{i}(G)=\max\{\deg _{i}(P)\mid P\subset G{\text{ parabolic subgroups}}\}$. If G is an algebraic group and E is a G-torsor, then the degree of instability of E is the degree of the inner form ${}^{E}G=\operatorname {Aut} _{G}(E)$ of G induced by E (which is a group scheme over X); i.e., $\deg _{i}(E)=\deg _{i}({}^{E}G)$. E is said to be semi-stable if $\deg _{i}(E)\leq 0$ and is stable if $\deg _{i}(E)<0$.
Examples of torsors in applied mathematics
According to John Baez, energy, voltage, position and the phase of a quantum-mechanical wavefunction are all examples of torsors in everyday physics; in each case, only relative comparisons can be measured, but a reference point must be chosen arbitrarily to make absolute values meaningful. However, the comparative values of relative energy, voltage difference, displacements and phase differences are not torsors, but can be represented by simpler structures such as real numbers, vectors or angles.[6]
In basic calculus, he cites indefinite integrals as being examples of torsors.[6]
See also
• Beauville–Laszlo theorem
• Moduli stack of principal bundles
• Fundamental group scheme
Notes
1. Algebraic stacks, Example 2.3. harvnb error: no target: CITEREFAlgebraic_stacks (help)
2. Behrend 1993, Lemma 4.3.1 harvnb error: no target: CITEREFBehrend1993 (help)
3. Milne 1980, The discussion preceding Proposition 4.6.
4. http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Oct27(Higgs).pdf
5. http://www.math.harvard.edu/~lurie/282ynotes/LectureXIV-Borel.pdf
6. Baez, John (December 27, 2009). "Torsors Made Easy". math.ucr.edu. Retrieved 2022-11-22.
References
• Behrend, K. The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles. PhD dissertation.
• Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), Algebraic stacks, archived from the original on 2008-05-05
• Milne, James S. (1980), Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, ISBN 978-0-691-08238-7, MR 0559531
Further reading
• Brian Conrad, Finiteness theorems for algebraic groups over function �fields
| Wikipedia |
Torsten Carleman
Torsten Carleman (8 July 1892, Visseltofta, Osby Municipality – 11 January 1949, Stockholm), born Tage Gillis Torsten Carleman, was a Swedish mathematician, known for his results in classical analysis and its applications. As the director of the Mittag-Leffler Institute for more than two decades, Carleman was the most influential mathematician in Sweden.
Torsten Carleman
Born(1892-07-08)8 July 1892
Visseltofta
Died11 January 1949(1949-01-11) (aged 56)
Stockholm
NationalitySwedish
Alma materUppsala University
Known forCarleman's condition
Carleman's inequality
Denjoy–Carleman theorem
mean ergodic theorem
Carleman kernel
Carleman formulae
AwardsBjörkénska priset (1941)
Scientific career
FieldsMathematics
InstitutionsLund University
Stockholm University
Mittag-Leffler Institute
Doctoral advisorErik Albert Holmgren
Doctoral studentsÅke Pleijel
Hans Rådström
Work
The dissertation of Carleman under Erik Albert Holmgren, as well as his work in the early 1920s, was devoted to singular integral equations. He developed the spectral theory of integral operators with Carleman kernels, that is, kernels K(x, y) such that K(y, x) = K(x, y) for almost every (x, y), and
$\int |K(x,y)|^{2}dy<\infty $
for almost every x.[1][2]
In the mid-1920s, Carleman developed the theory of quasi-analytic functions. He proved the necessary and sufficient condition for quasi-analyticity, now called the Denjoy–Carleman theorem.[3] As a corollary, he obtained a sufficient condition for the determinacy of the moment problem.[4] As one of the steps in the proof of the Denjoy–Carleman theorem in Carleman (1926), he introduced the Carleman inequality
$\sum _{n=1}^{\infty }\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}\leq e\sum _{n=1}^{\infty }a_{n},$
valid for any sequence of non-negative real numbers ak.[5]
At about the same time, he established the Carleman formulae in complex analysis, which reconstruct an analytic function in a domain from its values on a subset of the boundary. He also proved a generalisation of Jensen's formula, now called the Jensen–Carleman formula.[6]
In the 1930s, independently of John von Neumann, he discovered the mean ergodic theorem.[7] Later, he worked in the theory of partial differential equations, where he introduced the Carleman estimates,[8] and found a way to study the spectral asymptotics of Schrödinger operators.[9]
In 1932, following the work of Henri Poincaré, Erik Ivar Fredholm, and Bernard Koopman, he devised the Carleman embedding (also called Carleman linearization), a way to embed a finite-dimensional system of nonlinear differential equations du⁄dt = P(u) for u: Rk → R, where the components of P are polynomials in u, into an infinite-dimensional system of linear differential equations.[10][11]
In 1933 Carleman published a short proof of what is now called the Denjoy–Carleman–Ahlfors theorem.[12] This theorem states that the number of asymptotic values attained by an entire function of order ρ along curves in the complex plane going outwards toward infinite absolute value is less than or equal to 2ρ.
In 1935, Torsten Carleman introduced a generalisation of Fourier transform, which foreshadowed the work of Mikio Sato on hyperfunctions;[13] his notes were published in Carleman (1944). He considered the functions f of at most polynomial growth, and showed that every such function can be decomposed as f = f+ + f−, where f+ and f− are analytic in the upper and lower half planes, respectively, and that this representation is essentially unique. Then he defined the Fourier transform of (f+, f−) as another such pair (g+, g−). Though conceptually different, the definition coincides with the one given later by Laurent Schwartz for tempered distributions.[13] Carleman's definition gave rise to numerous extensions.[13][14]
Returning to mathematical physics in the 1930s, Carleman gave the first proof of global existence for Boltzmann's equation in the kinetic theory of gases (his result applies to the space-homogeneous case).[15] The results were published posthumously in Carleman (1957).
Carleman supervised the Ph.D. theses of Ulf Hellsten, Karl Persson (Dagerholm), Åke Pleijel and (jointly with Fritz Carlson) of Hans Rådström.
Life
Carleman was born in Visseltofta to Alma Linnéa Jungbeck and Karl Johan Carleman, a school teacher.[6] He studied at Växjö Cathedral School, graduating in 1910.
He continued his studies at Uppsala University, being one of the active members of the Uppsala Mathematical Society. Kjellberg recalls:
He was a genius! My older friends in Uppsala used to tell me about the wonderful years they had had when Carleman was there. He was the most active speaker in the Uppsala Mathematical Society and a well-trained gymnast. When people left the seminar crossing the Fyris River, he walked on his hands on the railing of the bridge.[16]
From 1917 he was docent at Uppsala University, and from 1923 — a full professor at Lund University. In 1924 he was appointed professor at Stockholm University. He was elected a member of the Royal Swedish Academy of Sciences in 1926. From 1927, he was director of the Mittag-Leffler Institute and editor of Acta Mathematica.[6]
From 1929 to 1946 Carleman was married to Anna-Lisa Lemming (1885–1954),[17] the half-sister[18] of the athlete Eric Lemming who won four golden medals and three bronze at the Olympic Games.[19] During this period he was also known as a recognized fascist, anti-semite and xenophobe. His interaction with William Feller before the former departure to the United States was not particularly pleasant, at some point being reported due to his opinion that "Jews and foreigners should be executed".[20]
Carlson remembers Carleman as: "secluded and taciturn, who looked at life and people with a bitter humour. In his heart, he was inclined to kindliness towards those around him, and strove to assist them swiftly."[6] Towards the end of his life, he remarked to his students that "professors ought to be shot at the age of fifty."[21]
During the last decades of his life, Carleman abused alcohol, according to Norbert Wiener[22][23] and William Feller.[24] His final years were plagued by neuralgia. At the end of 1948, he developed the liver disease jaundice; he died from complications of the disease.[6][23]
Selected publications
• Carleman, T. (1926). Les fonctions quasi analytiques (in French). Paris: Gauthier-Villars. JFM 52.0255.02.
• Carleman, T. (1944). L'Intégrale de Fourier et Questions que s'y Rattachent (in French). Uppsala: Publications Scientifiques de l'Institut Mittag-Leffler. MR 0014165.
• Carleman, T. (1957). Problèmes mathématiques dans la théorie cinétique des gaz (in French). Uppsala: Publ. Sci. Inst. Mittag-Leffler. MR 0098477.
• Carleman, Torsten (1960), Pleijel, Ake; Lithner, Lars; Odhnoff, Jan (eds.), Edition Complete Des Articles De Torsten Carleman, Litos reprotryk and l'Institut mathematique Mittag-Leffler
See also
• Carleman's condition
• Carleman's inequality
• Carleman's equation
• Carleman matrix
• Denjoy-Carleman theorem
Notes
1. Dieudonné, Jean (1981). History of functional analysis. North-Holland Mathematics Studies. Vol. 49. Amsterdam–New York: North-Holland Publishing Co. pp. 168–171. ISBN 0-444-86148-3. MR 0605488.
2. Ahiezer, N. I. (1947). "Integral operators with Carleman kernels". Uspekhi Mat. Nauk (in Russian). 2 (5(21)): 93–132. MR 0028526.
3. Mandelbrojt, S. (1942). "Analytic functions and classes of infinitely differentiable functions". Rice Inst. Pamphlet. 29 (1). MR 0006354.
4. Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd. MR 0184042.
5. Pečarić, Josip; Stolarsky, Kenneth B. (2001). "Carleman's inequality: history and new generalizations". Aequationes Mathematicae. 61 (1–2): 49–62. doi:10.1007/s000100050160. MR 1820809. S2CID 121175099.
6. Carlson, F. (1950). "Torsten Carleman". Acta Math. (in French). 82 (1): i–vi. doi:10.1007/BF02398273. MR 1555457.
7. Wiener, N. (1939). "The ergodic theorem". Duke Math. J. 5 (1): 1–18. doi:10.1215/S0012-7094-39-00501-6. MR 1546100. Zbl 0021.23501.
8. Kenig, Carlos E. (1987). "Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems". Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986). Providence, RI: Amer. Math. Soc. pp. 948–960. MR 0934297.
9. Clark, Colin (1967). "The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems". SIAM Rev. 9 (4): 627–646. doi:10.1137/1009105. MR 0510064.
10. Kowalski, Krzysztof; Steeb, Willi-Hans (1991). Nonlinear dynamical systems and Carleman linearization. River Edge, NJ: World Scientific Publishing Co., Inc. p. 7. ISBN 981-02-0587-2. MR 1178493.
11. Kowalski, K (1994). Methods of Hilbert spaces in the theory of nonlinear dynamical systems. River Edge, NJ: World Scientific Publishing Co., Inc. ISBN 981-02-1753-6. MR 1296251.
12. Torsten Carleman (April 3, 1933). "Sur une inégalité différentielle dans la théorie des fonctions analytiques". Comptes Rendus de l'Académie des Sciences. 196: 995–7.
13. Kiselman, Christer O. (2002). "Generalized Fourier transformations: The work of Bochner and Carleman viewed in the light of the theories of Schwartz and Sato". Microlocal analysis and complex Fourier analysis (PDF). River Edge, NJ: World Sci. Publ. pp. 166–185. MR 2068535.
14. Singh, U. N. (1992). "The Carleman-Fourier transform and its applications". Functional analysis and operator theory. Lecture Notes in Math. Vol. 1511. Berlin: Springer. pp. 181–214. MR 1180762.
15. Cercignani, C. (2008), 134 years of Boltzmann equation. Boltzmann's legacy, ESI Lect. Math. Phys., Zürich: Eur. Math. Soc., pp. 107–127, doi:10.4171/057-1/8, MR 2509759
16. Kjellberg, B. (1995). "Mathematicians in Uppsala — some recollections". In A. Vretblad (ed.). Festschrift in honour of Lennart Carleson and Yngve Domar. Proc. Conf. at Dept. of Math. (in Swedish). Uppsala: Uppsala Univ. pp. 87–95.
17. Swedish Death Index, which is a Windows based digital data base, shows different dates (1940 and 1946) of their divorce; Maligranda (2003) lists the year of divorce as 1940. Her original name was Anna Lovisa Lemming, born July 20, 1885.
18. Thus according to the Swedish Church birth records. Note that several sources, including Maligranda (2003), state that she was the daughter of Eric Lemming.
19. Webpage of the Swedish Olympic Committee Archived 2012-05-23 at the Wayback Machine
20. Siegmund-Schultze, Reinhard (2009). Mathematicians fleeing from Nazi Germany: Individual fates and global impact. Princeton, New Jersey: Princeton University Press. p. 135. ISBN 978-0-691-14041-4. MR 2522825.
21. Gårding, Lars (1998). Mathematics and mathematicians. Mathematics in Sweden before 1950. History of Mathematics. Vol. 13. Providence, RI: American Mathematical Society. p. 206. ISBN 0-8218-0612-2. MR 1488153.
22. "He died of drink.... During meetings he was often a bit drunk, and afterwards in Paris I saw him come to Mandelbrojt's apartment for an advance on the travel money due him, red-eyed, with a three-day beard." Wiener, Norbert (1956). I am a mathematician: The later life of a prodigy (later republished by MIT Press ed.). Garden City, N. Y.: Doubleday and Co. pp. 317–318. ISBN 9780026273008. MR 0077455.
The mathematician Szolem Mandelbrojt was an uncle of Benoit Mandelbrot.
23. Maligranda, Lech (2003), "Torsten Carleman", The MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland, retrieved 13 December 2011
24. Siegmund-Schultze, R. (2009). "Alternative (non-American) host countries". Mathematicians fleeing from Nazi Germany: Individual fates and global impact. Princeton, New Jersey: Princeton University Press. p. 135. ISBN 978-1400831401. MR 0252285.
External links
• Torsten Carleman at the Mathematics Genealogy Project
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| Wikipedia |
Torus action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold).
A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties).
Linear action of a torus
A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus T is acting on a finite-dimensional vector space V, then there is a direct sum decomposition:
$V=\bigoplus _{\chi }V_{\chi }$
where
• $\chi :T\to \mathbb {G} _{m}$ is a group homomorphism, a character of T.
• $V_{\chi }=\{v\in V|t\cdot v=\chi (t)v\}$, T-invariant subspace called the weight subspace of weight $\chi $.
The decomposition exists because the linear action determines (and is determined by) a linear representation $\pi :T\to \operatorname {GL} (V)$ and then $\pi (T)$ consists of commuting diagonalizable linear transformations, upon extending the base field.
If V does not have finite dimension, the existence of such a decomposition is tricky but one easy case when decomposition is possible is when V is a union of finite-dimensional representations ($\pi $ is called rational; see below for an example). Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum.
Example: Let $S=k[x_{0},\dots ,x_{n}]$ be a polynomial ring over an infinite field k. Let $T=\mathbb {G} _{m}^{r}$ act on it as algebra automorphisms by: for $t=(t_{1},\dots ,t_{r})\in T$
$t\cdot x_{i}=\chi _{i}(t)x_{i}$
where
$\chi _{i}(t)=t_{1}^{\alpha _{i,1}}\dots t_{r}^{\alpha _{i,r}},$ $\alpha _{i,j}$ = integers.
Then each $x_{i}$ is a T-weight vector and so a monomial $x_{0}^{m_{0}}\dots x_{r}^{m_{r}}$ is a T-weight vector of weight $\sum m_{i}\chi _{i}$. Hence,
$S=\bigoplus _{m_{0},\dots m_{n}\geq 0}S_{m_{0}\chi _{0}+\dots +m_{n}\chi _{n}}.$
Note if $\chi _{i}(t)=t$ for all i, then this is the usual decomposition of the polynomial ring into homogeneous components.
Białynicki-Birula decomposition
The Białynicki-Birula decomposition says that a smooth algebraic T-variety admits a T-stable cellular decomposition.
It is often described as algebraic Morse theory.[1]
See also
• Sumihiro's theorem
• GKM variety
• Equivariant cohomology
• monomial ideal
References
1. "Konrad Voelkel » Białynicki-Birula and Motivic Decompositions «".
• Altmann, Klaus; Ilten, Nathan Owen; Petersen, Lars; Süß, Hendrik; Vollmert, Robert (2012-08-15). The Geometry of T-Varieties. arXiv:1102.5760. doi:10.4171/114. ISBN 978-3-03719-114-9.
• A. Bialynicki-Birula, "Some Theorems on Actions of Algebraic Groups," Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480–497
• M. Brion, C. Procesi, Action d'un tore dans une variété projective, in Operator algebras, unitary representations, and invariant theory (Paris 1989), Prog. in Math. 92 (1990), 509–539.
| Wikipedia |
Torus bundle
A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three-manifolds.
Construction
To obtain a torus bundle: let $f$ be an orientation-preserving homeomorphism of the two-dimensional torus $T$ to itself. Then the three-manifold $M(f)$ is obtained by
• taking the Cartesian product of $T$ and the unit interval and
• gluing one component of the boundary of the resulting manifold to the other boundary component via the map $f$.
Then $M(f)$ is the torus bundle with monodromy $f$.
Examples
For example, if $f$ is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle $M(f)$ is the three-torus: the Cartesian product of three circles.
Seeing the possible kinds of torus bundles in more detail requires an understanding of William Thurston's geometrization program. Briefly, if $f$ is finite order, then the manifold $M(f)$ has Euclidean geometry. If $f$ is a power of a Dehn twist then $M(f)$ has Nil geometry. Finally, if $f$ is an Anosov map then the resulting three-manifold has Sol geometry.
These three cases exactly correspond to the three possibilities for the absolute value of the trace of the action of $f$ on the homology of the torus: either less than two, equal to two, or greater than two.
References
• Jeffrey R. Weeks (2002). The Shape of Space (Second ed.). Marcel Dekker, Inc. ISBN 978-0824707095.
| Wikipedia |
Toroidal graph
In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
Examples
Any graph that can be embedded in a plane can also be embedded in a torus. A toroidal graph of genus 1 can be embedded in a torus but not in a plane. The Heawood graph, the complete graph K7 (and hence K5 and K6), the Petersen graph (and hence the complete bipartite graph K3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks,[1] and all Möbius ladders are toroidal. More generally, any graph with crossing number 1 is toroidal. Some graphs with greater crossing numbers are also toroidal: the Möbius–Kantor graph, for example, has crossing number 4 and is toroidal.[2]
Properties
Any toroidal graph has chromatic number at most 7.[3] The complete graph K7 provides an example of a toroidal graph with chromatic number 7.[4]
Any triangle-free toroidal graph has chromatic number at most 4.[5]
By a result analogous to Fáry's theorem, any toroidal graph may be drawn with straight edges in a rectangle with periodic boundary conditions.[6] Furthermore, the analogue of Tutte's spring theorem applies in this case.[7] Toroidal graphs also have book embeddings with at most 7 pages.[8]
Obstructions
By the Robertson–Seymour theorem, there exists a finite set H of minimal non-toroidal graphs, such that a graph is toroidal if and only if it has no graph minor in H. That is, H forms the set of forbidden minors for the toroidal graphs. The complete set H is not known, but it has at least 17,523 graphs. Alternatively, there are at least 250,815 non-toroidal graphs that are minimal in the topological minor ordering. A graph is toroidal if and only if it has none of these graphs as a topological minor.[9]
Gallery
• Two isomorphic Cayley graphs of the quaternion group.
• Cayley graph of the quaternion group embedded in the torus.
• Video of Cayley graph of the quaternion group embedded in the torus.
• The Heawood graph and associated map embedded in the torus.
• The Pappus graph and associated map embedded in the torus.
See also
• Planar graph
• Topological graph theory
• Császár polyhedron
Notes
1. Orbanić et al. (2004).
2. Marušič & Pisanski (2000).
3. Heawood (1890).
4. Chartrand & Zhang (2008).
5. Kronk & White (1972).
6. Kocay, Neilson & Szypowski (2001).
7. Gortler, Gotsman & Thurston (2006).
8. Endo (1997).
9. Myrvold & Woodcock (2018).
References
• Chartrand, Gary; Zhang, Ping (2008), Chromatic graph theory, CRC Press, ISBN 978-1-58488-800-0.
• Endo, Toshiki (1997), "The pagenumber of toroidal graphs is at most seven", Discrete Mathematics, 175 (1–3): 87–96, doi:10.1016/S0012-365X(96)00144-6, MR 1475841.
• Gortler, Steven J.; Gotsman, Craig; Thurston, Dylan (2006), "Discrete one-forms on meshes and applications to 3D mesh parameterization" (PDF), Computer Aided Geometric Design, 23 (2): 83–112, doi:10.1016/j.cagd.2005.05.002, MR 2189438, S2CID 135438.
• Heawood, P. J. (1890), "Map colouring theorems", Quarterly Journal of Mathematics, First Series, 24: 322–339.
• Kocay, W.; Neilson, D.; Szypowski, R. (2001), "Drawing graphs on the torus" (PDF), Ars Combinatoria, 59: 259–277, MR 1832459, archived from the original (PDF) on 2004-12-24, retrieved 2018-09-06.
• Kronk, Hudson V.; White, Arthur T. (1972), "A 4-color theorem for toroidal graphs", Proceedings of the American Mathematical Society, American Mathematical Society, 34 (1): 83–86, doi:10.2307/2037902, JSTOR 2037902, MR 0291019.
• Marušič, Dragan; Pisanski, Tomaž (2000), "The remarkable generalized Petersen graph G(8,3)", Math. Slovaca, 50: 117–121, CiteSeerX 10.1.1.28.7183, hdl:10338.dmlcz/133137, MR 1763113, Zbl 0984.05044.
• Myrvold, Wendy; Woodcock, Jennifer (2018), "A large set of torus obstructions and how they were discovered", Electronic Journal of Combinatorics, 25 (1): P1.16, doi:10.37236/3797
• Neufeld, Eugene; Myrvold, Wendy (1997), "Practical toroidality testing", Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 574–580, ISBN 978-0-89871-390-9.
• Orbanić, Alen; Pisanski, Tomaž; Randić, Milan; Servatius, Brigitte (2004), "Blanuša double" (PDF), Math. Commun., 9 (1): 91–103, CiteSeerX 10.1.1.361.2772.
| Wikipedia |
Toshmuhammad Qori-Niyoziy
Toshmuhammad Qori-Niyoziy (Uzbek Cyrillic: Тошмуҳаммад Ниёзович Қори-Ниёзий, Russian: Ташмухамед Ниязович Кары-Ниязов, Tashmukhamed Niyazovich Kary-Niyazov; 2 September [O.S. 21 August] 1897 — 17 March 1970) was an Uzbek mathematician and historian who served as the first president of the Academy of Sciences of the Uzbek SSR.
Toshmuhammad Qori-Niyoziy
Born2 September [O.S. 21 August] 1897
Khujand, Samarkand Oblast, Russian Empire
Died17 March 1970(1970-03-17) (aged 72)
Tashkent, Uzbek SSR, Soviet Union
EducationDoctor of Sciences
Alma materTashkent University
SpouseOishakhon Urazaeva
AwardsHero of Socialist Labor
Early life
Born in Khujand on 2 September [O.S. 21 August] 1897 to a shoemaker, he initially received schooling in a maktab, but attended for less than a year due to abuse from the teacher. His family went on to move to Skobelev (now Fergana), where he eventually attended a Russian school and graduated with excellent marks in the mid-1910s.[lower-alpha 1] In 1917, he became a teacher at a school he founded in Kokand, which quickly became a regional school. Initially having volunteered to serve as head of schools for the Skobelev district, he went on to serve as director of the Uzbek Pedagogical College in Kokand from 1920 to 1925. Several years later he graduated from the Faculty of Physics and Mathematics at Central Asian State University in Tashkent; he defended his thesis in Uzbek.[1][2][3]
His wife Oishakhon, who he married in 1920 in a Muslim ceremony, was one of the first women teachers in the Uzbek SSR. She frequently advised him on his philology work, including the first Uzbek dictionaries that they worked on together.[2]
Career
Whilst a university student, he was tasked with teaching advanced math classes such as analytic geometry in the Uzbek language. After graduating he continued to teach university-level mathematics in the Uzbek language, becoming the first Uzbek to receive the title of professor in 1931. That year, he became a member of the Communist Party.[4] From then to 1933 he served as a rector at the university, although he did not receive his doctorate of physics and mathematics until 1939. He then became the Deputy Chairman of the Committee of the Uzbek SSR for Science, Culture and Art, and worked on the transition of the Uzbek alphabet to a Cyrillic script. He also devoted a considerable amount of time to researching the history of Uzbekistan and historic academic works, with a special focus on astronomy and archaeology. As part of his research about early astronomy in what is present-day Uzbekistan, he had to read through numerous Arabic manuscripts. In addition to his academic work, he held various political offices, serving as a deputy in the Supreme Soviet of the USSR for the 1st and 2nd convocations. He also authored numerous textbooks and academic papers, including the first Uzbek-language math textbooks and papers about Uzbek culture and society.[5][6][7][8]
World War II
In June 1941, he led alongside Mikhail Gerasimov a scientific expedition to examine the tomb of Timur in Samarkand. An inscription on the tomb threatened to bring about a catastrophe to whoever opened it, and shortly after it was opened, Nazi Germany began invading the Soviet Union. After the remains were reburied with Muslim rites in 1942, some in Uzbekistan credited the Soviet victory in the Battle of Stalingrad to the reburial.[9][1][10]
After the German invasion of the Soviet Union, his only son Shavkat applied to go to the frontlines with the Red Army. Being skilled in mathematics like his father, he was chosen for artillery school. After surviving the war, Shavkat went on to graduate from the F.E. Dzerzhinky Military Academy and follow in his father's footsteps with a career in mathematics, but specialized in ballistics and rocket technology.[2]
When the Academy of Sciences of the Uzbek SSR was established in 1943, Qori-Niyazov was made its first president and held the post until 1947.[11]
Postwar
In 1946 Qori-Niyoziy became a professor at the Tashkent Institute of Engineers and Agricultural Mechanization. For his paper "Ulugbek's Astronomical School" he was awarded the Stalin Prize. In 1954 he became a member of the International Astronomical Union, in 1967 he became a corresponding member of the International Academy of the History of Science, and that same year on 1 September he was awarded the title Hero of Socialist Labour for his work promoting academics in the Uzbek SSR. His work included serving as editor-in-chief of the Uzbek science magazine Fan va turmush and deputy chairman of the board for preserving historic and cultural monuments of Uzbekistan. During the course of his work, he travelled to various foreign countries including Afghanistan, Bulgaria, India, Italy, and Japan. He died on 17 March 1970 and was buried in the Chigatoy Cemetery.[1][6][7]
Awards and honors
• Honoured Scientist of the Uzbek SSR (1939)[1]
• Three Orders of Lenin (4 November 1944, 1 March 1965, 1 September 1967)
• Three Orders of the Red Banner of Labour (23 November 1946, 16 January 1950, 27 October 1953)
• Stalin Prize (1952)
• Hero of Socialist Labour (1 September 1967)
• Beruniy State Prize (1970)
• Order of Outstanding Merit (2002, posthumously)
Notes
1. Sources differ if he graduated in 1915 or 1916
References
1. Karimov, Timur. "Кары-Ниязов Ташмухамед Ниязович". warheroes.ru. Retrieved 1 June 2022.
2. ""Родоначальник современной науки Узбекистана…". Ко дню рождения Ташмухаммада Ниязовича Кары-Ниязова – Kultura.uz". www.kultura.uz. Retrieved 11 November 2022.
3. Вопросы истории естествознания и техники (in Russian). Nauka. 1971.
4. Prominent Personalities in the USSR. Scarecrow Press. 1968.
5. "Речь депутата Т.Н. Кары-Ниязов". Pravda Vostoka (in Russian). No. 112 (6724). 9 June 1945. p. 2.
6. "Тошмуҳаммад Ниёзович Қори-Ниёзий". Sovet Uzbekistoni. 18 March 1970. p. 3.
7. ҚОРИ-НИЁЗИЙ OʻzME. Q-harfi Birinchi jild. Toshkent, 2000-2005-yillar
8. "Юбилей Узбекского ученого". Ogonyok (21): 9. 1949.
9. "Gur Emir Mausoleum in Samarkand". pagetour.org. Retrieved 11 November 2022.
10. McChesney, R. D. (25 May 2021). Four Central Asian Shrines: A Socio-Political History of Architecture. BRILL. ISBN 978-90-04-45959-5.
11. Ўзбекистонда ижтимоий фанлар (in Russian). Izd-vo "Fan". 1978. pp. 28–30.
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| Wikipedia |
Toshmuhammad Sarimsoqov
Toshmuhammad Sarimsoqov (Uzbek Cyrillic: Тошмуҳаммад Алиевич Саримсоқов, Russian: Ташмухамед Алиевич Сарымсаков, Tashmukhamed Alievich Sarymsakov; 4 September [O.S. 22 August] 1915 – 17 December 1995) was an Uzbek mathematician who served as president of the Academy of Sciences of the Uzbek SSR from 1947 to 1952.
Toshmuhammad Sarimsoqov
Тошмуҳаммад Саримсоқов
Born4 September [O.S. 22 August] 1915
Shahrixon, Russian Empire
Died17 December 1995 (aged 80)
Tashkent, Uzbekistan
EducationDoctor of Physical and Mathematical Sciences
Alma materCentral Asian State University
AwardsHero of Socialist Labour
Early life and education
Born in 1915 in Shahrixon to an Uzbek family, in 1931 he graduated from a Russian secondary school in Kokand; he subsequently enrolled in the Central Asia State University. There, he was one of the first students of Vsevolod Romanovsky. After graduating from the Faculty of Physics and Mathematics of the university in 1936, he remained at the university, where he attended graduate school. At the same time, he worked as assistant and associate professor. After briefly serving in the Red Army he returned to the university in 1942 to defend his doctoral dissertation. That year he received his Doctor of Sciences degree.[1][2][3]
He became a member of the Communist party in 1944 and served as a deputy in the third convocation of the Supreme Soviet of the USSR.[4][5]
Career
In 1943 he became the rector of his university, and held that post until June 1944. When the Academy of Sciences of the Uzbek SSR was founded in 1943, he became its vice president. In 1947 he became its president, and held the post until 1952. He then returned to being the rector of Central Asia State University, where he remained until 1958. From 1959 to 1971 he served as the minister of higher education of the Uzbek SSR before again returning to being the rector of the university, which had been renamed to Tashkent State University in 1960. In 1983 he returned to working at the Academy of Sciences of the Uzbek SSR, and in 1988 he became advisor to the Presidium of the Academy of Sciences of the Uzbek SSR. His main areas of study were probability, statistics, and functional analysis. During his career he authored over 170 academic papers. His work on the theory of non-homogeneous Markov chains is cited in modern academic papers.[6] For his work, he was awarded the title Hero of Socialist Labour on 3 April 1990. After Uzbekistan became independent he worked as an advisor to the President of the Academy of Sciences of Uzbekistan. He died in Tashkent on 18 December 1995.[1][7][8]
Awards
• Hero of Socialist Labour (3 April 1990)
• Four Order of Lenin (23 January 1946, 16 January 1950, 15 September 1961, 3 April 1990)
• Three Order of the Red Banner of Labour (4 November 1944, 11 January 1957, 9 September 1971)
• Order of the Badge of Honour (1 March 1965)
• Order of the October Revolution (3 October 1975)
• Medal "For Labour Valour" (6 November 1951)
• Order of Outstanding Merit (2002)
• Stalin Prize (1948)
• Biruni State Prize (1967)
• Honoured Worker of Science and Technology of the Uzbek SSR (1960)[5]
References
1. Аюпов, Ш. А. (ed.). "Математическая жизнь - Ташмухамед Алиевич Сарымсаков" [Mathematical life - Toshmuhamad Aliyevich Sarimsoqov] (PDF). Узбекский Математический Журнал (in Russian). Ташкент: Издательство «Фан» Академии Наук Республики Узбекистан. 2005 (3): 3–10. ISSN 2010-7269.
2. Боголюбов, А. Н.; Матвиевская, Г. П. (1997). Всеволод Иванович Романовский, 1879–1954 [Vsevolod Ivanovich Romanovsky, 1879–1954]. Научно-биографическая серия (in Russian). Москва: Наука. ISBN 978-5-02003-644-4.
3. Kostyrya, Vyacheslav (1977). "Могучая крона знаний" [Mighty Crown of Knowledge]. Ogonyok (in Russian) (26): 16–17.
4. Академия Наук Узбекской ССР [Academy of Sciences of the Uzbek SSR] (in Russian). Издательство «Фан» Узбекской ССР. 1983.
5. Каримов, Тимур. "Сарымсаков Ташмухамед Алиевич" [Sarymsakov Tashmukhamed Alievich]. Международный патриотический интернет-проект «Герои страны» (in Russian). Retrieved 13 November 2022.
6. Iosifescu, Marius (2007) [Originally published 1980]. Finite Markov Processes and Their Applications. Mineola, New York: Dover Publications. p. 286. ISBN 978-0-48645-869-4.
7. Салахитдинов, М. С. (1993). Академии наук Республики Узбекистан – 50 лет [Academy of Sciences of the Republic of Uzbekistan – 50 years] (in Russian). Фан.
8. Ўзбекистон миллий энциклопедияси (PDF) (in Uzbek). Vol. 7. Tashkent: O‘zbekiston milliy ensiklopediyasi Davlat ilmiy nashriyoti. 2004. Retrieved 9 February 2023.
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| Wikipedia |
Law of total probability
In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name.
Part of a series on statistics
Probability theory
• Probability
• Axioms
• Determinism
• System
• Indeterminism
• Randomness
• Probability space
• Sample space
• Event
• Collectively exhaustive events
• Elementary event
• Mutual exclusivity
• Outcome
• Singleton
• Experiment
• Bernoulli trial
• Probability distribution
• Bernoulli distribution
• Binomial distribution
• Normal distribution
• Probability measure
• Random variable
• Bernoulli process
• Continuous or discrete
• Expected value
• Markov chain
• Observed value
• Random walk
• Stochastic process
• Complementary event
• Joint probability
• Marginal probability
• Conditional probability
• Independence
• Conditional independence
• Law of total probability
• Law of large numbers
• Bayes' theorem
• Boole's inequality
• Venn diagram
• Tree diagram
Statement
The law of total probability is[1] a theorem that states, in its discrete case, if $\left\{{B_{n}:n=1,2,3,\ldots }\right\}$ is a finite or countably infinite partition of a sample space (in other words, a set of pairwise disjoint events whose union is the entire sample space) and each event $B_{n}$ is measurable, then for any event $A$ of the same sample space:
$P(A)=\sum _{n}P(A\cap B_{n})$
or, alternatively,[1]
$P(A)=\sum _{n}P(A\mid B_{n})P(B_{n}),$
where, for any $n$ for which $P(B_{n})=0$ these terms are simply omitted from the summation, because $P(A\mid B_{n})$ is finite.
The summation can be interpreted as a weighted average, and consequently the marginal probability, $P(A)$, is sometimes called "average probability";[2] "overall probability" is sometimes used in less formal writings.[3]
The law of total probability can also be stated for conditional probabilities:
$P({A|C})={\frac {P({A,C})}{P(C)}}={\frac {\sum \limits _{n}{P({A,{B_{n}},C})}}{P(C)}}={\frac {\sum \limits _{n}P({A\mid {B_{n}},C})P({{B_{n}}\mid C})P(C)}{P(C)}}=\sum \limits _{n}P({A\mid {B_{n}},C})P({{B_{n}}\mid C})$
Taking the $B_{n}$ as above, and assuming $C$ is an event independent of any of the $B_{n}$:
$P(A\mid C)=\sum _{n}P(A\mid C\cap B_{n})P(B_{n})$
Continuous case
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let $(\Omega ,{\mathcal {F}},P)$ be a probability space. Suppose $X$ is a random variable with distribution function $F_{X}$, and $A$ an event on $(\Omega ,{\mathcal {F}},P)$. Then the law of total probability states
$P(A)=\int _{-\infty }^{\infty }P(A|X=x)dF_{X}(x).$
If $X$ admits a density function $f_{X}$, then the result is
$P(A)=\int _{-\infty }^{\infty }P(A|X=x)f_{X}(x)dx.$
Moreover, for the specific case where $A=\{Y\in B\}$, where $B$ is a Borel set, then this yields
$P(Y\in B)=\int _{-\infty }^{\infty }P(Y\in B|X=x)f_{X}(x)dx.$
Example
Suppose that two factories supply light bulbs to the market. Factory X's bulbs work for over 5000 hours in 99% of cases, whereas factory Y's bulbs work for over 5000 hours in 95% of cases. It is known that factory X supplies 60% of the total bulbs available and Y supplies 40% of the total bulbs available. What is the chance that a purchased bulb will work for longer than 5000 hours?
Applying the law of total probability, we have:
${\begin{aligned}P(A)&=P(A\mid B_{X})\cdot P(B_{X})+P(A\mid B_{Y})\cdot P(B_{Y})\\[4pt]&={99 \over 100}\cdot {6 \over 10}+{95 \over 100}\cdot {4 \over 10}={{594+380} \over 1000}={974 \over 1000}\end{aligned}}$
where
• $P(B_{X})={6 \over 10}$ is the probability that the purchased bulb was manufactured by factory X;
• $P(B_{Y})={4 \over 10}$ is the probability that the purchased bulb was manufactured by factory Y;
• $P(A\mid B_{X})={99 \over 100}$ is the probability that a bulb manufactured by X will work for over 5000 hours;
• $P(A\mid B_{Y})={95 \over 100}$ is the probability that a bulb manufactured by Y will work for over 5000 hours.
Thus each purchased light bulb has a 97.4% chance to work for more than 5000 hours.
Other names
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. One author uses the terminology of the "Rule of Average Conditional Probabilities",[4] while another refers to it as the "continuous law of alternatives" in the continuous case.[5] This result is given by Grimmett and Welsh[6] as the partition theorem, a name that they also give to the related law of total expectation.
See also
• Law of total expectation
• Law of total variance
• Law of total covariance
• Law of total cumulance
• Marginal distribution
Notes
1. Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. ISBN 1-58488-059-7 page 31.
2. Paul E. Pfeiffer (1978). Concepts of probability theory. Courier Dover Publications. pp. 47–48. ISBN 978-0-486-63677-1.
3. Deborah Rumsey (2006). Probability for dummies. For Dummies. p. 58. ISBN 978-0-471-75141-0.
4. Jim Pitman (1993). Probability. Springer. p. 41. ISBN 0-387-97974-3.
5. Kenneth Baclawski (2008). Introduction to probability with R. CRC Press. p. 179. ISBN 978-1-4200-6521-3.
6. Probability: An Introduction, by Geoffrey Grimmett and Dominic Welsh, Oxford Science Publications, 1986, Theorem 1B.
References
• Introduction to Probability and Statistics by Robert J. Beaver, Barbara M. Beaver, Thomson Brooks/Cole, 2005, page 159.
• Theory of Statistics, by Mark J. Schervish, Springer, 1995.
• Schaum's Outline of Probability, Second Edition, by John J. Schiller, Seymour Lipschutz, McGraw–Hill Professional, 2010, page 89.
• A First Course in Stochastic Models, by H. C. Tijms, John Wiley and Sons, 2003, pages 431–432.
• An Intermediate Course in Probability, by Alan Gut, Springer, 1995, pages 5–6.
| Wikipedia |
Total absolute curvature
In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve.[1]
If the curve is parameterized by its arc length, the total absolute curvature can be expressed by the formula
$\int |\kappa (s)|ds,$
where s is the arc length parameter and κ is the curvature. This is almost the same as the formula for the total curvature, but differs in using the absolute value instead of the signed curvature.[2]
Because the total curvature of a simple closed curve in the Euclidean plane is always exactly 2π, the total absolute curvature of a simple closed curve is also always at least 2π. It is exactly 2π for a convex curve, and greater than 2π whenever the curve has any non-convexities.[2] When a smooth simple closed curve undergoes the curve-shortening flow, its total absolute curvature decreases monotonically until the curve becomes convex, after which its total absolute curvature remains fixed at 2π until the curve collapses to a point.[3][4]
The total absolute curvature may also be defined for curves in three-dimensional Euclidean space. Again, it is at least 2π (this is Fenchel's theorem), but may be larger. If a space curve is surrounded by a sphere, the total absolute curvature of the sphere equals the expected value of the central projection of the curve onto a plane tangent to a random point of the sphere.[5] According to the Fáry–Milnor theorem, every nontrivial smooth knot must have total absolute curvature greater than 4π.[2]
References
1. Brook, Alexander; Bruckstein, Alfred M.; Kimmel, Ron (2005), "On similarity-invariant fairness measures", in Kimmel, Ron; Sochen, Nir A.; Weickert, Joachim (eds.), Scale Space and PDE Methods in Computer Vision: 5th International Conference, Scale-Space 2005, Hofgeismar, Germany, April 7-9, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3459, Springer-Verlag, pp. 456–467, doi:10.1007/11408031_39.
2. Chen, Bang-Yen (2000), "Riemannian submanifolds", Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, pp. 187–418, doi:10.1016/S1874-5741(00)80006-0, MR 1736854. See in particular section 21.1, "Rotation index and total curvature of a curve", pp. 359–360.
3. Brakke, Kenneth A. (1978), The motion of a surface by its mean curvature (PDF), Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., Appendix B, Proposition 2, p. 230, ISBN 0-691-08204-9, MR 0485012.
4. Chou, Kai-Seng; Zhu, Xi-Ping (2001), The Curve Shortening Problem, Boca Raton, Florida: Chapman & Hall/CRC, Lemma 5.5, p. 130, and Section 6.1, pp. 144–147, doi:10.1201/9781420035704, ISBN 1-58488-213-1, MR 1888641.
5. Banchoff, Thomas F. (1970), "Total central curvature of curves", Duke Mathematical Journal, 37 (2): 281–289, doi:10.1215/S0012-7094-70-03736-1, MR 0259815.
Various notions of curvature defined in differential geometry
Differential geometry
of curves
• Curvature
• Torsion of a curve
• Frenet–Serret formulas
• Radius of curvature (applications)
• Affine curvature
• Total curvature
• Total absolute curvature
Differential geometry
of surfaces
• Principal curvatures
• Gaussian curvature
• Mean curvature
• Darboux frame
• Gauss–Codazzi equations
• First fundamental form
• Second fundamental form
• Third fundamental form
Riemannian geometry
• Curvature of Riemannian manifolds
• Riemann curvature tensor
• Ricci curvature
• Scalar curvature
• Sectional curvature
Curvature of connections
• Curvature form
• Torsion tensor
• Cocurvature
• Holonomy
| Wikipedia |
Total algebra
In abstract algebra, the total algebra of a monoid is a generalization of the monoid ring that allows for infinite sums of elements of a ring. Suppose that S is a monoid with the property that, for all $s\in S$, there exist only finitely many ordered pairs $(t,u)\in S\times S$ for which $tu=s$. Let R be a ring. Then the total algebra of S over R is the set $R^{S}$ of all functions $\alpha :S\to R$ with the addition law given by the (pointwise) operation:
$(\alpha +\beta )(s)=\alpha (s)+\beta (s)$
and with the multiplication law given by:
$(\alpha \cdot \beta )(s)=\sum _{tu=s}\alpha (t)\beta (u).$
The sum on the right-hand side has finite support, and so is well-defined in R.
These operations turn $R^{S}$ into a ring. There is an embedding of R into $R^{S}$, given by the constant functions, which turns $R^{S}$ into an R-algebra.
An example is the ring of formal power series, where the monoid S is the natural numbers. The product is then the Cauchy product.
References
• Nicolas Bourbaki (1989), Algebra, Springer: §III.2
| Wikipedia |
Total curvature
In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:
$\int _{a}^{b}k(s)\,ds=2\pi N.$
The total curvature of a closed curve is always an integer multiple of 2π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces.
Comparison to surfaces
Further information: Gaussian_curvature § Total_curvature
This relationship between a local geometric invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher-dimensional Riemannian geometry such as the Gauss–Bonnet theorem.
Invariance
According to the Whitney–Graustein theorem, the total curvature is invariant under a regular homotopy of a curve: it is the degree of the Gauss map. However, it is not invariant under homotopy: passing through a kink (cusp) changes the turning number by 1.
By contrast, winding number about a point is invariant under homotopies that do not pass through the point, and changes by 1 if one passes through the point.
Generalizations
A finite generalization is that the exterior angles of a triangle, or more generally any simple polygon, add up to 360° = 2π radians, corresponding to a turning number of 1. More generally, polygonal chains that do not go back on themselves (no 180° angles) have well-defined total curvature, interpreting the curvature as point masses at the angles.
The total absolute curvature of a curve is defined in almost the same way as the total curvature, but using the absolute value of the curvature instead of the signed curvature. It is 2π for convex curves in the plane, and larger for non-convex curves.[1] It can also be generalized to curves in higher dimensional spaces by flattening out the tangent developable to γ into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in n-dimensional space is
$\int _{a}^{b}\left|\gamma ''(s)\right|\operatorname {sgn} \kappa _{n-1}(s)\,ds$
where κn−1 is last Frenet curvature (the torsion of the curve) and sgn is the signum function.
The minimum total absolute curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2π for the unknot, but by the Fáry–Milnor theorem it is at least 4π for any other knot.[2]
References
1. Chen, Bang-Yen (2000), "Riemannian submanifolds", Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, pp. 187–418, doi:10.1016/S1874-5741(00)80006-0, MR 1736854. See in particular section 21.1, "Rotation index and total curvature of a curve", pp. 359–360.
2. Milnor, John W. (1950), "On the Total Curvature of Knots", Annals of Mathematics, Second Series, 52 (2): 248–257, doi:10.2307/1969467, JSTOR 1969467
Further reading
• Kuhnel, Wolfgang (2005), Differential Geometry: Curves - Surfaces - Manifolds (2nd ed.), American Mathematical Society, ISBN 978-0-8218-3988-1 (translated by Bruce Hunt)
• Sullivan, John M. (2008), "Curves of finite total curvature", Discrete differential geometry, Oberwolfach Semin., vol. 38, Birkhäuser, Basel, pp. 137–161, arXiv:math/0606007, doi:10.1007/978-3-7643-8621-4_7, MR 2405664, S2CID 117955587
| Wikipedia |
Total derivative
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function.[1]: 198–203
Not to be confused with Total differential or Total derivative (fluid mechanics).
Part of a series of articles about
Calculus
• Fundamental theorem
• Limits
• Continuity
• Rolle's theorem
• Mean value theorem
• Inverse function theorem
Differential
Definitions
• Derivative (generalizations)
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• infinitesimal
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The total derivative as a linear map
Let $U\subseteq \mathbb {R} ^{n}$ be an open subset. Then a function $f:U\to \mathbb {R} ^{m}$ is said to be (totally) differentiable at a point $a\in U$ if there exists a linear transformation $df_{a}:\mathbb {R} ^{n}\to \mathbb {R} ^{m}$ such that
$\lim _{x\to a}{\frac {\|f(x)-f(a)-df_{a}(x-a)\|}{\|x-a\|}}=0.$
The linear map $df_{a}$ is called the (total) derivative or (total) differential of $f$ at $a$. Other notations for the total derivative include $D_{a}f$ and $Df(a)$. A function is (totally) differentiable if its total derivative exists at every point in its domain.
Conceptually, the definition of the total derivative expresses the idea that $df_{a}$ is the best linear approximation to $f$ at the point $a$. This can be made precise by quantifying the error in the linear approximation determined by $df_{a}$. To do so, write
$f(a+h)=f(a)+df_{a}(h)+\varepsilon (h),$
where $\varepsilon (h)$ equals the error in the approximation. To say that the derivative of $f$ at $a$ is $df_{a}$ is equivalent to the statement
$\varepsilon (h)=o(\lVert h\rVert ),$
where $o$ is little-o notation and indicates that $\varepsilon (h)$ is much smaller than $\lVert h\rVert $ as $h\to 0$. The total derivative $df_{a}$ is the unique linear transformation for which the error term is this small, and this is the sense in which it is the best linear approximation to $f$.
The function $f$ is differentiable if and only if each of its components $f_{i}\colon U\to \mathbb {R} $ is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain. It is true that if $f$ is differentiable at $a$, then each partial derivative $\partial f/\partial x_{i}$ exists at $a$. The converse does not hold: it can happen that all of the partial derivatives of $f$ at $a$ exist, but $f$ is not differentiable at $a$. This means that the function is very "rough" at $a$, to such an extreme that its behavior cannot be adequately described by its behavior in the coordinate directions. When $f$ is not so rough, this cannot happen. More precisely, if all the partial derivatives of $f$ at $a$ exist and are continuous in a neighborhood of $a$, then $f$ is differentiable at $a$. When this happens, then in addition, the total derivative of $f$ is the linear transformation corresponding to the Jacobian matrix of partial derivatives at that point.[2]
The total derivative as a differential form
When the function under consideration is real-valued, the total derivative can be recast using differential forms. For example, suppose that $f\colon \mathbb {R} ^{n}\to \mathbb {R} $ is a differentiable function of variables $x_{1},\ldots ,x_{n}$. The total derivative of $f$ at $a$ may be written in terms of its Jacobian matrix, which in this instance is a row matrix:
$Df_{a}={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(a)&\cdots &{\frac {\partial f}{\partial x_{n}}}(a)\end{bmatrix}}.$
The linear approximation property of the total derivative implies that if
$\Delta x={\begin{bmatrix}\Delta x_{1}&\cdots &\Delta x_{n}\end{bmatrix}}^{\mathsf {T}}$
is a small vector (where the ${\mathsf {T}}$ denotes transpose, so that this vector is a column vector), then
$f(a+\Delta x)-f(a)\approx Df_{a}\cdot \Delta x=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(a)\cdot \Delta x_{i}.$
Heuristically, this suggests that if $dx_{1},\ldots ,dx_{n}$ are infinitesimal increments in the coordinate directions, then
$df_{a}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(a)\cdot dx_{i}.$
In fact, the notion of the infinitesimal, which is merely symbolic here, can be equipped with extensive mathematical structure. Techniques, such as the theory of differential forms, effectively give analytical and algebraic descriptions of objects like infinitesimal increments, $dx_{i}$. For instance, $dx_{i}$ may be inscribed as a linear functional on the vector space $\mathbb {R} ^{n}$. Evaluating $dx_{i}$ at a vector $h$ in $\mathbb {R} ^{n}$ measures how much $h$ points in the $i$th coordinate direction. The total derivative $df_{a}$ is a linear combination of linear functionals and hence is itself a linear functional. The evaluation $df_{a}(h)$ measures how much $h$ points in the direction determined by $f$ at $a$, and this direction is the gradient. This point of view makes the total derivative an instance of the exterior derivative.
Suppose now that $f$ is a vector-valued function, that is, $f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}$. In this case, the components $f_{i}$ of $f$ are real-valued functions, so they have associated differential forms $df_{i}$. The total derivative $df$ amalgamates these forms into a single object and is therefore an instance of a vector-valued differential form.
The chain rule for total derivatives
Main article: Chain rule
The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions $f$ and $g$, the total derivative of the composite function $g\circ f$ at $a$ satisfies
$d(g\circ f)_{a}=dg_{f(a)}\cdot df_{a}.$
If the total derivatives of $f$ and $g$ are identified with their Jacobian matrices, then the composite on the right-hand side is simply matrix multiplication. This is enormously useful in applications, as it makes it possible to account for essentially arbitrary dependencies among the arguments of a composite function.
Example: Differentiation with direct dependencies
Suppose that f is a function of two variables, x and y. If these two variables are independent, so that the domain of f is $\mathbb {R} ^{2}$, then the behavior of f may be understood in terms of its partial derivatives in the x and y directions. However, in some situations, x and y may be dependent. For example, it might happen that f is constrained to a curve $y=y(x)$. In this case, we are actually interested in the behavior of the composite function $f(x,y(x))$. The partial derivative of f with respect to x does not give the true rate of change of f with respect to changing x because changing x necessarily changes y. However, the chain rule for the total derivative takes such dependencies into account. Write $\gamma (x)=(x,y(x))$. Then, the chain rule says
$d(f\circ \gamma )_{x_{0}}=df_{(x_{0},y(x_{0}))}\cdot d\gamma _{x_{0}}.$
By expressing the total derivative using Jacobian matrices, this becomes:
${\frac {df(x,y(x))}{dx}}(x_{0})={\frac {\partial f}{\partial x}}(x_{0},y(x_{0}))\cdot {\frac {\partial x}{\partial x}}(x_{0})+{\frac {\partial f}{\partial y}}(x_{0},y(x_{0}))\cdot {\frac {\partial y}{\partial x}}(x_{0}).$
Suppressing the evaluation at $x_{0}$ for legibility, we may also write this as
${\frac {df(x,y(x))}{dx}}={\frac {\partial f}{\partial x}}{\frac {\partial x}{\partial x}}+{\frac {\partial f}{\partial y}}{\frac {\partial y}{\partial x}}.$
This gives a straightforward formula for the derivative of $f(x,y(x))$ in terms of the partial derivatives of $f$ and the derivative of $y(x)$.
For example, suppose
$f(x,y)=xy.$
The rate of change of f with respect to x is usually the partial derivative of f with respect to x; in this case,
${\frac {\partial f}{\partial x}}=y.$
However, if y depends on x, the partial derivative does not give the true rate of change of f as x changes because the partial derivative assumes that y is fixed. Suppose we are constrained to the line
$y=x.$
Then
$f(x,y)=f(x,x)=x^{2},$
and the total derivative of f with respect to x is
${\frac {df}{dx}}=2x,$
which we see is not equal to the partial derivative $\partial f/\partial x$. Instead of immediately substituting for y in terms of x, however, we can also use the chain rule as above:
${\frac {df}{dx}}={\frac {\partial f}{\partial x}}+{\frac {\partial f}{\partial y}}{\frac {dy}{dx}}=y+x\cdot 1=x+y=2x.$
Example: Differentiation with indirect dependencies
While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose $L(t,x_{1},\dots ,x_{n})$ is a function of time $t$ and $n$ variables $x_{i}$ which themselves depend on time. Then, the time derivative of $L$ is
${\frac {dL}{dt}}={\frac {d}{dt}}L{\bigl (}t,x_{1}(t),\ldots ,x_{n}(t){\bigr )}.$
The chain rule expresses this derivative in terms of the partial derivatives of $L$ and the time derivatives of the functions $x_{i}$:
${\frac {dL}{dt}}={\frac {\partial L}{\partial t}}+\sum _{i=1}^{n}{\frac {\partial L}{\partial x_{i}}}{\frac {dx_{i}}{dt}}={\biggl (}{\frac {\partial }{\partial t}}+\sum _{i=1}^{n}{\frac {dx_{i}}{dt}}{\frac {\partial }{\partial x_{i}}}{\biggr )}(L).$
This expression is often used in physics for a gauge transformation of the Lagrangian, as two Lagrangians that differ only by the total time derivative of a function of time and the $n$ generalized coordinates lead to the same equations of motion. An interesting example concerns the resolution of causality concerning the Wheeler–Feynman time-symmetric theory. The operator in brackets (in the final expression above) is also called the total derivative operator (with respect to $t$).
For example, the total derivative of $f(x(t),y(t))$ is
${\frac {df}{dt}}={\partial f \over \partial x}{dx \over dt}+{\partial f \over \partial y}{dy \over dt}.$
Here there is no $\partial f/\partial t$ term since $f$ itself does not depend on the independent variable $t$ directly.
Total differential equation
Main article: Total differential equation
A total differential equation is a differential equation expressed in terms of total derivatives. Since the exterior derivative is coordinate-free, in a sense that can be given a technical meaning, such equations are intrinsic and geometric.
Application to equation systems
In economics, it is common for the total derivative to arise in the context of a system of equations.[1]: pp. 217–220 For example, a simple supply-demand system might specify the quantity q of a product demanded as a function D of its price p and consumers' income I, the latter being an exogenous variable, and might specify the quantity supplied by producers as a function S of its price and two exogenous resource cost variables r and w. The resulting system of equations
$q=D(p,I),$
$q=S(p,r,w),$
determines the market equilibrium values of the variables p and q. The total derivative $dp/dr$ of p with respect to r, for example, gives the sign and magnitude of the reaction of the market price to the exogenous variable r. In the indicated system, there are a total of six possible total derivatives, also known in this context as comparative static derivatives: dp / dr, dp / dw, dp / dI, dq / dr, dq / dw, and dq / dI. The total derivatives are found by totally differentiating the system of equations, dividing through by, say dr, treating dq / dr and dp / dr as the unknowns, setting dI = dw = 0, and solving the two totally differentiated equations simultaneously, typically by using Cramer's rule.
See also
• Directional derivative – Instantaneous rate of change of the function
• Fréchet derivative – Derivative defined on normed spaces - generalization of the total derivative
• Gateaux derivative – Generalization of the concept of directional derivative
• Generalizations of the derivative – Fundamental construction of differential calculus
• Gradient#Total derivative – Multivariate derivative (mathematics)
References
1. Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). McGraw-Hill. ISBN 0-07-010813-7.
2. Abraham, Ralph; Marsden, J. E.; Ratiu, Tudor (2012). Manifolds, Tensor Analysis, and Applications. Springer Science & Business Media. p. 78. ISBN 9781461210290.
• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
• From thesaurus.maths.org total derivative
External links
• Weisstein, Eric W. "Total Derivative". MathWorld.
• Ronald D. Kriz (2007) Envisioning total derivatives of scalar functions of two dimensions using raised surfaces and tangent planes from Virginia Tech
Analysis in topological vector spaces
Basic concepts
• Abstract Wiener space
• Classical Wiener space
• Bochner space
• Convex series
• Cylinder set measure
• Infinite-dimensional vector function
• Matrix calculus
• Vector calculus
Derivatives
• Differentiable vector–valued functions from Euclidean space
• Differentiation in Fréchet spaces
• Fréchet derivative
• Total
• Functional derivative
• Gateaux derivative
• Directional
• Generalizations of the derivative
• Hadamard derivative
• Holomorphic
• Quasi-derivative
Measurability
• Besov measure
• Cylinder set measure
• Canonical Gaussian
• Classical Wiener measure
• Measure like set functions
• infinite-dimensional Gaussian measure
• Projection-valued
• Vector
• Bochner / Weakly / Strongly measurable function
• Radonifying function
Integrals
• Bochner
• Direct integral
• Dunford
• Gelfand–Pettis/Weak
• Regulated
• Paley–Wiener
Results
• Cameron–Martin theorem
• Inverse function theorem
• Nash–Moser theorem
• Feldman–Hájek theorem
• No infinite-dimensional Lebesgue measure
• Sazonov's theorem
• Structure theorem for Gaussian measures
Related
• Crinkled arc
• Covariance operator
Functional calculus
• Borel functional calculus
• Continuous functional calculus
• Holomorphic functional calculus
Applications
• Banach manifold (bundle)
• Convenient vector space
• Choquet theory
• Fréchet manifold
• Hilbert manifold
| Wikipedia |
Total coloring
In graph theory, total coloring is a type of graph coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be proper in the sense that no adjacent edges, no adjacent vertices and no edge and either endvertex are assigned the same color. The total chromatic number χ″(G) of a graph G is the fewest colors needed in any total coloring of G.
The total graph T = T(G) of a graph G is a graph such that (i) the vertex set of T corresponds to the vertices and edges of G and (ii) two vertices are adjacent in T if and only if their corresponding elements are either adjacent or incident in G. Then total coloring of G becomes a (proper) vertex coloring of T(G). A total coloring is a partitioning of the vertices and edges of the graph into total independent sets.
Some inequalities for χ″(G):
1. χ″(G) ≥ Δ(G) + 1.
2. χ″(G) ≤ Δ(G) + 1026. (Molloy, Reed 1998)
3. χ″(G) ≤ Δ(G) + 8 ln8(Δ(G)) for sufficiently large Δ(G). (Hind, Molloy, Reed 1998)
4. χ″(G) ≤ ch′(G) + 2.
Here Δ(G) is the maximum degree; and ch′(G), the edge choosability.
Total coloring arises naturally since it is simply a mixture of vertex and edge colorings. The next step is to look for any Brooks-typed or Vizing-typed upper bound on the total chromatic number in terms of maximum degree.
The total coloring version of maximum degree upper bound is a difficult problem that has eluded mathematicians for 50 years. A trivial lower bound for χ″(G) is Δ(G) + 1. Some graphs such as cycles of length $n\not \equiv 0{\bmod {3}}$ and complete bipartite graphs of the form $K_{n,n}$ need Δ(G) + 2 colors but no graph has been found that requires more colors. This leads to the speculation that every graph needs either Δ(G) + 1 or Δ(G) + 2 colors, but never more:
Total coloring conjecture (Behzad, Vizing). $\chi ''(G)\leq \Delta (G)+2.$
Apparently, the term "total coloring" and the statement of total coloring conjecture were independently introduced by Behzad and Vizing in numerous occasions between 1964 and 1968 (see Jensen & Toft). The conjecture is known to hold for a few important classes of graphs, such as all bipartite graphs and most planar graphs except those with maximum degree 6. The planar case can be completed if Vizing's planar graph conjecture is true. Also, if the list coloring conjecture is true, then $\chi ''(G)\leq \Delta (G)+3.$
Results related to total coloring have been obtained. For example, Kilakos and Reed (1993) proved that the fractional chromatic number of the total graph of a graph G is at most Δ(G) + 2.
References
• Hind, Hugh; Molloy, Michael; Reed, Bruce (1998). "Total coloring with Δ + poly(log Δ) colors". SIAM Journal on Computing. 28 (3): 816–821. doi:10.1137/S0097539795294578.
• Jensen, Tommy R.; Toft, Bjarne (1995). Graph coloring problems. New York: Wiley-Interscience. ISBN 0-471-02865-7.
• Kilakos, Kyriakos; Reed, Bruce (1993). "Fractionally colouring total graphs". Combinatorica. 13 (4): 435–440. doi:10.1007/BF01303515. S2CID 31163141.
• Molloy, Michael; Reed, Bruce (1998). "A bound on the total chromatic number". Combinatorica. 18 (2): 241–280. doi:10.1007/PL00009820. hdl:1807/9465. S2CID 9600550.
| Wikipedia |
Total operating characteristic
The total operating characteristic (TOC) is a statistical method to compare a Boolean variable versus a rank variable. TOC can measure the ability of an index variable to diagnose either presence or absence of a characteristic. The diagnosis of presence or absence depends on whether the value of the index is above a threshold. TOC considers multiple possible thresholds. Each threshold generates a two-by-two contingency table, which contains four entries: hits, misses, false alarms, and correct rejections.[1]
The receiver operating characteristic (ROC) also characterizes diagnostic ability, although ROC reveals less information than the TOC. For each threshold, ROC reveals two ratios, hits/(hits + misses) and false alarms/(false alarms + correct rejections), while TOC shows the total information in the contingency table for each threshold.[2] The TOC method reveals all of the information that the ROC method provides, plus additional important information that ROC does not reveal, i.e. the size of every entry in the contingency table for each threshold. TOC also provides the popular area under the curve (AUC) of the ROC.
TOC is applicable to measure diagnostic ability in many fields including but not limited to: land change science, medical imaging, weather forecasting, remote sensing, and materials testing.
Basic concept
The procedure to construct the TOC curve compares the Boolean variable to the index variable by diagnosing each observation as either presence or absence, depending on how the index relates to various thresholds. If an observation's index is greater than or equal to a threshold, then the observation is diagnosed as presence, otherwise the observation is diagnosed as absence. The contingency table that results from the comparison between the Boolean variable and the diagnosis for a single threshold has four central entries. The four central entries are hits (H), misses (M), false alarms (F), and correct rejections (C). The total number of observations is P + Q. The terms “true positives”, “false negatives”, “false positives” and “true negatives” are equivalent to hits, misses, false alarms and correct rejections, respectively. The entries can be formulated in a two-by-two contingency table or confusion matrix, as follows:
Diagnosis
Boolean
Presence Absence Boolean total
Presence Hits (H) Misses (M) H + M = P
Absence False alarms (F) Correct rejections (C) F + C = Q
Diagnosis total H + F M + C P + Q
Four bits of information determine all the entries in the contingency table, including its marginal totals. For example, if we know H, M, F, and C, then we can compute all the marginal totals for any threshold. Alternatively, if we know H/P, F/Q, P, and Q, then we can compute all the entries in the table.[1] Two bits of information are not sufficient to complete the contingency table. For example, if we know only H/P and F/Q, which is what ROC shows, then it is impossible to know all the entries in the table.[1]
History
Robert Gilmore Pontius Jr, professor of Geography at Clark University, and Kangping Si in 2014 first developed the TOC for application in land change science.
TOC space
The TOC curve with four boxes indicates how a point on the TOC curve reveals the hits, misses, false alarms, and correct rejections. The TOC curve is an effective way to show the total information in the contingency table for all thresholds. The data used to create this TOC curve is available for download here. This dataset has 30 observations, each of which consists of values for a Boolean variable and an index variable. The observations are ranked from the greatest to the least value of the index. There are 31 thresholds, consisting of the 30 values of the index and one additional threshold that is greater than all the index values, which creates the point at the origin (0,0). Each point is labeled to indicate the value of each threshold. The horizontal axes ranges from 0 to 30 which is the number of observations in the dataset (P + Q). The vertical axis ranges from 0 to 10, which is the Boolean variable's number of presence observations P (i.e. hits + misses). TOC curves also show the threshold at which the diagnosed amount of presence matches the Boolean amount of presence, which is the threshold point that lies directly under the point where the maximum line meets the hits + misses line, as the TOC curve on the left illustrates. For a more detailed explanation of the construction of the TOC curve, please see Pontius Jr, Robert Gilmore; Si, Kangping (2014). "The total operating characteristic to measure diagnostic ability for multiple thresholds." International Journal of Geographical Information Science 28 (3): 570–583.”[1]
The following four pieces of information are the central entries in the contingency table for each threshold:
1. The number of hits at each threshold is the distance between the threshold's point and the horizontal axis.
2. The number of misses at each threshold is the distance between the threshold's point and the hits + misses horizontal line across the top of the graph.
3. The number of false alarms at each threshold is the distance between threshold's point and the blue dashed Maximum line that bounds the left side of the TOC space.
4. The number of correct rejections at each threshold is the distance between the threshold's point and the purple dashed Minimum line that bounds the right side of the TOC space.
TOC vs. ROC curves
These figures are the TOC and ROC curves using the same data and thresholds. Consider the point that corresponds to a threshold of 74. The TOC curve shows the number of hits, which is 3, and hence the number of misses, which is 7. Additionally, the TOC curve shows that the number of false alarms is 4 and the number of correct rejections is 16. At any given point in the ROC curve, it is possible to glean values for the ratios of false alarms/(false alarms+correct rejections) and hits/(hits+misses). For example, at threshold 74, it is evident that the x coordinate is 0.2 and the y coordinate is 0.3. However, these two values are insufficient to construct all entries of the underlying two-by-two contingency table.
Interpreting TOC curves
It is common to report the area under the curve (AUC) to summarize a TOC or ROC curve. However, condensing diagnostic ability into a single number fails to appreciate the shape of the curve. The following three TOC curves are TOC curves that have an AUC of 0.75 but have different shapes.
This TOC curve on the left exemplifies an instance in which the index variable has a high diagnostic ability at high thresholds near the origin, but random diagnostic ability at low thresholds near the upper right of the curve. The curve shows accurate diagnosis of presence until the curve reaches a threshold of 86. The curve then levels off and predicts around the random line.
This TOC curve exemplifies an instance in which the index variable has a medium diagnostic ability at all thresholds. The curve is consistently above the random line.
This TOC curve exemplifies an instance in which the index variable has random diagnostic ability at high thresholds and high diagnostic ability at low thresholds. The curve follows the random line at the highest thresholds near the origin, then the index variable diagnoses absence correctly as thresholds decrease near the upper right corner.
Area under the curve
When measuring diagnostic ability, a commonly reported measure is the area under the curve (AUC). The AUC is calculable from the TOC and the ROC. The value of the AUC is consistent for the same data whether you are calculating the area under the curve for a TOC curve or a ROC curve. The AUC indicates the probability that the diagnosis ranks a randomly chosen observation of Boolean presence higher than a randomly chosen observation of Boolean absence.[3] The AUC is appealing to many researchers because AUC summarizes diagnostic ability in a single number, however, the AUC has come under critique as a potentially misleading measure, especially for spatially explicit analyses.[3][4] Some features of the AUC that draw criticism include the fact that 1) AUC ignores the thresholds; 2) AUC summarizes the test performance over regions of the TOC or ROC space in which one would rarely operate; 3) AUC weighs omission and commission errors equally; 4) AUC does not give information about the spatial distribution of model errors; and, 5) the selection of spatial extent highly influences the rate of accurately diagnosed absences and the AUC scores.[5] However, most of those criticisms apply to many other metrics.
When using normalized units, the area under the curve (often referred to as simply the AUC) is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one (assuming 'positive' ranks higher than 'negative').[6] This can be seen as follows: the area under the curve is given by (the integral boundaries are reversed as large T has a lower value on the x-axis)
$TPR(T):T\rightarrow y(x)$
$FPR(T):T\rightarrow x$
$A=\int _{x=0}^{1}{\mbox{TPR}}({\mbox{FPR}}^{-1}(x))\,dx=\int _{\infty }^{-\infty }{\mbox{TPR}}(T){\mbox{FPR}}'(T)\,dT=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }I(T'>T)f_{1}(T')f_{0}(T)\,dT'\,dT=P(X_{1}>X_{0})$
where $X_{1}$ is the score for a positive instance and $X_{0}$ is the score for a negative instance, and $f_{0}$ and $f_{1}$ are probability densities as defined in previous section.
It can further be shown that the AUC is closely related to the Mann–Whitney U,[7][8] which tests whether positives are ranked higher than negatives. It is also equivalent to the Wilcoxon test of ranks.[8] The AUC is related to the Gini coefficient ($G_{1}$) by the formula $G_{1}=2{\mbox{AUC}}-1$, where:
$G_{1}=1-\sum _{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})$[9]
In this way, it is possible to calculate the AUC by using an average of a number of trapezoidal approximations.
It is also common to calculate the area under the TOC convex hull (ROC AUCH = ROCH AUC) as any point on the line segment between two prediction results can be achieved by randomly using one or the other system with probabilities proportional to the relative length of the opposite component of the segment.[10] It is also possible to invert concavities – just as in the figure the worse solution can be reflected to become a better solution; concavities can be reflected in any line segment, but this more extreme form of fusion is much more likely to overfit the data.[11]
Another problem with TOC AUC is that reducing the TOC Curve to a single number ignores the fact that it is about the tradeoffs between the different systems or performance points plotted and not the performance of an individual system, as well as ignoring the possibility of concavity repair, so that related alternative measures such as Informedness or DeltaP are recommended.[12][13] These measures are essentially equivalent to the Gini for a single prediction point with DeltaP' = informedness = 2AUC-1, whilst DeltaP = markedness represents the dual (viz. predicting the prediction from the real class) and their geometric mean is the Matthews correlation coefficient.
Whereas TOC AUC varies between 0 and 1 — with an uninformative classifier yielding 0.5 — the alternative measures known as informedness, Certainty [12] and Gini coefficient (in the single parameterization or single system case) all have the advantage that 0 represents chance performance whilst 1 represents perfect performance, and −1 represents the "perverse" case of full informedness always giving the wrong response.[14] Bringing chance performance to 0 allows these alternative scales to be interpreted as Kappa statistics. Informedness has been shown to have desirable characteristics for machine learning versus other common definitions of Kappa such as Cohen kappa and Fleiss kappa.[15]
Sometimes it can be more useful to look at a specific region of the TOC curve rather than at the whole curve. It is possible to compute partial AUC.[16] For example, one could focus on the region of the curve with low false positive rate, which is often of prime interest for population screening tests.[17] Another common approach for classification problems in which P ≪ N (common in bioinformatics applications) is to use a logarithmic scale for the x-axis.[18]
References
1. Pontius, Robert Gilmore; Si, Kangping (2014). "The total operating characteristic to measure diagnostic ability for multiple thresholds". International Journal of Geographical Information Science. 28 (3): 570–583. doi:10.1080/13658816.2013.862623. S2CID 29204880.
2. Pontius, Robert Gilmore; Parmentier, Benoit (2014). "Recommendations for using the Relative Operating Characteristic (ROC)". Landscape Ecology. 29 (3): 367–382. doi:10.1007/s10980-013-9984-8. S2CID 254740981.
3. Halligan, Steve; Altman, Douglas G.; Mallett, Susan (2015). "Disadvantages of using the area under the receiver operating characteristic curve to assess imaging tests: A discussion and proposal for an alternative approach". European Radiology. 25 (4): 932–939. doi:10.1007/s00330-014-3487-0. PMC 4356897. PMID 25599932.
4. Powers, David Martin Ward (2012). "The problem of Area Under the Curve". 2012 IEEE International Conference on Information Science and Technology. pp. 567–573. doi:10.1109/ICIST.2012.6221710. ISBN 978-1-4577-0345-4. S2CID 11072457.
5. Lobo, Jorge M.; Jiménez-Valverde, Alberto; Real, Raimundo (2008). "AUC: a misleading measure of the performance of predictive distribution models". Global Ecology and Biogeography. 17 (2): 145–151. doi:10.1111/j.1466-8238.2007.00358.x.
6. Fawcett, Tom (2006); An introduction to ROC analysis, Pattern Recognition Letters, 27, 861–874.
7. Hanley, James A.; McNeil, Barbara J. (1982). "The Meaning and Use of the Area under a Receiver Operating Characteristic (ROC) Curve". Radiology. 143 (1): 29–36. doi:10.1148/radiology.143.1.7063747. PMID 7063747. S2CID 10511727.
8. Mason, Simon J.; Graham, Nicholas E. (2002). "Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation" (PDF). Quarterly Journal of the Royal Meteorological Society. 128 (584): 2145–2166. Bibcode:2002QJRMS.128.2145M. CiteSeerX 10.1.1.458.8392. doi:10.1256/003590002320603584. S2CID 121841664. Archived from the original (PDF) on 2008-11-20.
9. Hand, David J.; and Till, Robert J. (2001); A simple generalization of the area under the ROC curve for multiple class classification problems, Machine Learning, 45, 171–186.
10. Provost, F.; Fawcett, T. (2001). "Robust classification for imprecise environments". Machine Learning. 42 (3): 203–231. arXiv:cs/0009007. doi:10.1023/a:1007601015854. S2CID 5415722.
11. Flach, P.A.; Wu, S. (2005). "Repairing concavities in ROC curves." (PDF). 19th International Joint Conference on Artificial Intelligence (IJCAI'05). pp. 702–707.
12. Powers, David MW (2012). "ROC-ConCert: ROC-Based Measurement of Consistency and Certainty" (PDF). Spring Congress on Engineering and Technology (SCET). Vol. 2. IEEE. pp. 238–241.
13. Powers, David M.W. (2012). "The Problem of Area Under the Curve". International Conference on Information Science and Technology.
14. Powers, David M. W. (2003). "Recall and Precision versus the Bookmaker" (PDF). Proceedings of the International Conference on Cognitive Science (ICSC-2003), Sydney Australia, 2003, pp. 529–534.
15. Powers, David M. W. (2012). "The Problem with Kappa" (PDF). Conference of the European Chapter of the Association for Computational Linguistics (EACL2012) Joint ROBUS-UNSUP Workshop. Archived from the original (PDF) on 2016-05-18. Retrieved 2012-07-20.
16. McClish, Donna Katzman (1989-08-01). "Analyzing a Portion of the ROC Curve". Medical Decision Making. 9 (3): 190–195. doi:10.1177/0272989X8900900307. PMID 2668680. S2CID 24442201.
17. Dodd, Lori E.; Pepe, Margaret S. (2003). "Partial AUC Estimation and Regression". Biometrics. 59 (3): 614–623. doi:10.1111/1541-0420.00071. PMID 14601762. S2CID 23054670.
18. Karplus, Kevin (2011); Better than Chance: the importance of null models, University of California, Santa Cruz, in Proceedings of the First International Workshop on Pattern Recognition in Proteomics, Structural Biology and Bioinformatics (PR PS BB 2011)
Further reading
• Pontius Jr, Robert Gilmore; Si, Kangping (2014). "The total operating characteristic to measure diagnostic ability for multiple thresholds". International Journal of Geographical Information Science. 28 (3): 570–583. doi:10.1080/13658816.2013.862623. S2CID 29204880.
• Pontius Jr, Robert Gilmore; Parmentier, Benoit (2014). "Recommendations for using the Relative Operating Characteristic (ROC)". Landscape Ecology. 29 (3): 367–382. doi:10.1007/s10980-013-9984-8. S2CID 254740981.
• Mas, Jean-François; Filho, Britaldo Soares; Pontius Jr, Robert Gilmore; Gutiérrez, Michelle Farfán; Rodrigues, Hermann (2013). "A suite of tools for ROC analysis of spatial models". ISPRS International Journal of Geo-Information. 2 (3): 869–887. Bibcode:2013IJGI....2..869M. doi:10.3390/ijgi2030869.
• Pontius Jr, Robert Gilmore; Pacheco, Pablo (2004). "Calibration and validation of a model of forest disturbance in the Western Ghats, India 1920–1990". GeoJournal. 61 (4): 325–334. doi:10.1007/s10708-004-5049-5. S2CID 155073463.
• Pontius Jr, Robert Gilmore; Batchu, Kiran (2003). "Using the relative operating characteristic to quantify certainty in prediction of location of land cover change in India". Transactions in GIS. 7 (4): 467–484. doi:10.1111/1467-9671.00159. S2CID 14452746.
• Pontius Jr, Robert Gilmore; Schneider, Laura (2001). "Land-use change model validation by a ROC method for the Ipswich watershed, Massachusetts, USA". Agriculture, Ecosystems & Environment. 85 (1–3): 239–248. doi:10.1016/s0167-8809(01)00187-6.
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| Wikipedia |
Total order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation $\leq $ on some set $X$, which satisfies the following for all $a,b$ and $c$ in $X$:
1. $a\leq a$ (reflexive).
2. If $a\leq b$ and $b\leq c$ then $a\leq c$ (transitive).
3. If $a\leq b$ and $b\leq a$ then $a=b$ (antisymmetric).
4. $a\leq b$ or $b\leq a$ (strongly connected, formerly called total).
Reflexivity (1.) already follows from connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.[1] Total orders are sometimes also called simple,[2] connex,[3] or full orders.[4]
A set equipped with a total order is a totally ordered set;[5] the terms simply ordered set,[2] linearly ordered set,[3][5] and loset[6][7] are also used. The term chain is sometimes defined as a synonym of totally ordered set,[5] but refers generally to some sort of totally ordered subsets of a given partially ordered set.
An extension of a given partial order to a total order is called a linear extension of that partial order.
Strict and non-strict total orders
A strict total order on a set $X$ is a strict partial order on $X$ in which any two distinct elements are comparable. That is, a strict total order is a binary relation $<$ on some set $X$, which satisfies the following for all $a,b$ and $c$ in $X$:
1. Not $a<a$ (irreflexive).
2. If $a<b$ then not $b<a$ (asymmetric).
3. If $a<b$ and $b<c$ then $a<c$ (transitive).
4. If $a\neq b$, then $a<b$ or $b<a$ (connected).
Asymmetry follows from transitivity and irreflexivity;[8] moreover, irreflexivity follows from asymmetry.[9]
For delimitation purposes, a total order as defined in the lead is sometimes called non-strict order. For each (non-strict) total order $\leq $ there is an associated relation $<$, called the strict total order associated with $\leq $ that can be defined in two equivalent ways:
• $a<b$ if $a\leq b$ and $a\neq b$ (reflexive reduction).
• $a<b$ if not $b\leq a$ (i.e., $<$ is the complement of the converse of $\leq $).
Conversely, the reflexive closure of a strict total order $<$ is a (non-strict) total order.
Examples
• Any subset of a totally ordered set X is totally ordered for the restriction of the order on X.
• The unique order on the empty set, ∅, is a total order.
• Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).
• If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on X by setting x1 ≤ x2 if and only if f(x1) ≤ f(x2).
• The lexicographical order on the Cartesian product of a family of totally ordered sets, indexed by a well ordered set, is itself a total order.
• The set of real numbers ordered by the usual "less than or equal to" (≤) or "greater than or equal to" (≥) relations is totally ordered. Hence each subset of the real numbers is totally ordered, such as the natural numbers, integers, and rational numbers. Each of these can be shown to be the unique (up to an order isomorphism) "initial example" of a totally ordered set with a certain property, (here, a total order A is initial for a property, if, whenever B has the property, there is an order isomorphism from A to a subset of B):[10]
• The natural numbers form an initial non-empty totally ordered set with no upper bound.
• The integers form an initial non-empty totally ordered set with neither an upper nor a lower bound.
• The rational numbers form an initial totally ordered set which is dense in the real numbers. Moreover, the reflexive reduction < is a dense order on the rational numbers.
• The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined below).
• Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic to the real numbers.
• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc., is a strict total order.
Chains
The term chain is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to a subset of a partially ordered set that is totally ordered for the induced order.[1][11] Typically, the partially ordered set is a set of subsets of a given set that is ordered by inclusion, and the term is used for stating properties of the set of the chains. This high number of nested levels of sets explains the usefulness of the term.
A common example of the use of chain for referring to totally ordered subsets is Zorn's lemma which asserts that, if every chain in a partially ordered set X has an upper bound in X, then X contains at least one maximal element.[12] Zorn's lemma is commonly used with X being a set of subsets; in this case, the upperbound is obtained by proving that the union of the elements of a chain in X is in X. This is the way that is generally used to prove that a vector space has Hamel bases and that a ring has maximal ideals.
In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its opposite order. In this case, a chain can be identified with a monotone sequence, and is called an ascending chain or a descending chain, depending whether the sequence is increasing or decreasing.[13]
A partially ordered set has the descending chain condition if every descending chain eventually stabilizes.[14] For example, an order is well founded if it has the descending chain condition. Similarly, the ascending chain condition means that every ascending chain eventually stabilizes. For example, a Noetherian ring is a ring whose ideals satisfy the ascending chain condition.
In other contexts, only chains that are finite sets are considered. In this case, one talks of a finite chain, often shortened as a chain. In this case, the length of a chain is the number of inequalities (or set inclusions) between consecutive elements of the chain; that is, the number minus one of elements in the chain.[15] Thus a singleton set is a chain of length zero, and an ordered pair is a chain of length one. The dimension of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the dimension of a vector space is the maximal length of chains of linear subspaces, and the Krull dimension of a commutative ring is the maximal length of chains of prime ideals.
"Chain" may also be used for some totally ordered subsets of structures that are not partially ordered sets. An example is given by regular chains of polynomials. Another example is the use of "chain" as a synonym for a walk in a graph.
Further concepts
Lattice theory
One may define a totally ordered set as a particular kind of lattice, namely one in which we have
$\{a\vee b,a\wedge b\}=\{a,b\}$ for all a, b.
We then write a ≤ b if and only if $a=a\wedge b$. Hence a totally ordered set is a distributive lattice.
Finite total orders
A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words, a total order on a set with k elements induces a bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with order type ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).
Category theory
Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being maps which respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).
A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.
Order topology
For any totally ordered set X we can define the open intervals
• (a, b) = {x | a < x and x < b},
• (−∞, b) = {x | x < b},
• (a, ∞) = {x | a < x}, and
• (−∞, ∞) = X.
We can use these open intervals to define a topology on any ordered set, the order topology.
When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in general).
The order topology induced by a total order may be shown to be hereditarily normal.
Completeness
A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upper bound. For example, the set of real numbers R is complete but the set of rational numbers Q is not. In other words, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers.
There are a number of results relating properties of the order topology to the completeness of X:
• If the order topology on X is connected, X is complete.
• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is two points a and b in X with a < b such that no c satisfies a < c < b.)
• X is complete if and only if every bounded set that is closed in the order topology is compact.
A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line). There are order-preserving homeomorphisms between these examples.
Sums of orders
For any two disjoint total orders $(A_{1},\leq _{1})$ and $(A_{2},\leq _{2})$, there is a natural order $\leq _{+}$ on the set $A_{1}\cup A_{2}$, which is called the sum of the two orders or sometimes just $A_{1}+A_{2}$:
For $x,y\in A_{1}\cup A_{2}$, $x\leq _{+}y$ holds if and only if one of the following holds:
1. $x,y\in A_{1}$ and $x\leq _{1}y$
2. $x,y\in A_{2}$ and $x\leq _{2}y$
3. $x\in A_{1}$ and $y\in A_{2}$
Intuitively, this means that the elements of the second set are added on top of the elements of the first set.
More generally, if $(I,\leq )$ is a totally ordered index set, and for each $i\in I$ the structure $(A_{i},\leq _{i})$ is a linear order, where the sets $A_{i}$ are pairwise disjoint, then the natural total order on $\bigcup _{i}A_{i}$ is defined by
For $x,y\in \bigcup _{i\in I}A_{i}$, $x\leq y$ holds if:
1. Either there is some $i\in I$ with $x\leq _{i}y$
2. or there are some $i<j$ in $I$ with $x\in A_{i}$, $y\in A_{j}$
Decidability
The first-order theory of total orders is decidable, i.e. there is an algorithm for deciding which first-order statements hold for all total orders. Using interpretability in S2S, the monadic second-order theory of countable total orders is also decidable.[16]
Orders on the Cartesian product of totally ordered sets
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product of two totally ordered sets are:
• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.
• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.
• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of the corresponding strict total orders). This is also a partial order.
All three can similarly be defined for the Cartesian product of more than two sets.
Applied to the vector space Rn, each of these make it an ordered vector space.
See also examples of partially ordered sets.
A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding total preorder on that subset.
Related structures
Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗
Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗
Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗
Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗
Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗
Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗
Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗
Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗
Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all $a,b$ and $S\neq \varnothing :$ :} ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$
Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table.
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.
A group with a compatible total order is a totally ordered group.
There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both data results in a separation relation.[17]
See also
• Artinian ring – ring that satisfies the descending chain condition on idealsPages displaying wikidata descriptions as a fallback
• Countryman line
• Order theory – Branch of mathematics
• Permutation – Mathematical version of an order change
• Prefix order – generalization of the notion of prefix of a string, and of the notion of a treePages displaying wikidata descriptions as a fallback – a downward total partial order
• Suslin's problem – the proposition, independent of ZFC, that a nonempty unbounded complete dense total order satisfying the countable chain condition is isomorphic to the realsPages displaying wikidata descriptions as a fallback
• Well-order – Class of mathematical orderings
Notes
1. Halmos 1968, Ch.14.
2. Birkhoff 1967, p. 2.
3. Schmidt & Ströhlein 1993, p. 32.
4. Fuchs 1963, p. 2.
5. Davey & Priestley 1990, p. 3.
6. Strohmeier, Alfred; Genillard, Christian; Weber, Mats (1 August 1990). "Ordering of characters and strings". ACM SIGAda Ada Letters (7): 84. doi:10.1145/101120.101136. S2CID 38115497.
7. Ganapathy, Jayanthi (1992). "Maximal Elements and Upper Bounds in Posets". Pi Mu Epsilon Journal. 9 (7): 462–464. ISSN 0031-952X. JSTOR 24340068.
8. Let $a<b$, assume for contradiction that also $b<a$. Then $a<a$ by transitivity, which contradicts irreflexivity.
9. If $a<a$, the not $a<a$ by asymmetry.
10. This definition resembles that of an initial object of a category, but is weaker.
11. Roland Fraïssé (December 2000). Theory of Relations. Studies in Logic and the Foundations of Mathematics. Vol. 145 (1st ed.). Elsevier. ISBN 978-0-444-50542-2. Here: p. 35
12. Brian A. Davey and Hilary Ann Priestley (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press. ISBN 0-521-36766-2. LCCN 89009753. Here: p. 100
13. Yiannis N. Moschovakis (2006) Notes on set theory, Undergraduate Texts in Mathematics (Birkhäuser) ISBN 0-387-28723-X, p. 116
14. that is, beyond some index, all further sequence members are equal
15. Davey and Priestly 1990, Def.2.24, p. 37
16. Weyer, Mark (2002). "Decidability of S1S and S2S". Automata, Logics, and Infinite Games. Lecture Notes in Computer Science. Vol. 2500. Springer. pp. 207–230. doi:10.1007/3-540-36387-4_12. ISBN 978-3-540-00388-5.
17. Macpherson, H. Dugald (2011), "A survey of homogeneous structures", Discrete Mathematics, 311 (15): 1599–1634, doi:10.1016/j.disc.2011.01.024
References
• Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications. Vol. 25. Providence: Am. Math. Soc.
• Davey, Brian A.; Priestley, Hilary Ann (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press. ISBN 0-521-36766-2. LCCN 89009753.
• Fuchs, L (1963). Partially Ordered Algebraic Systems. Pergamon Press.
• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN 0-7167-0442-0
• Halmos, Paul R. (1968). Naive Set Theory. Princeton: Nostrand.
• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4
• Rosenstein, Joseph G. (1982). Linear orderings. New York: Academic Press.
• Schmidt, Gunther; Ströhlein, Thomas (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Berlin: Springer-Verlag. ISBN 978-3-642-77970-1.
External links
• "Totally ordered set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Order theory
• Topics
• Glossary
• Category
Key concepts
• Binary relation
• Boolean algebra
• Cyclic order
• Lattice
• Partial order
• Preorder
• Total order
• Weak ordering
Results
• Boolean prime ideal theorem
• Cantor–Bernstein theorem
• Cantor's isomorphism theorem
• Dilworth's theorem
• Dushnik–Miller theorem
• Hausdorff maximal principle
• Knaster–Tarski theorem
• Kruskal's tree theorem
• Laver's theorem
• Mirsky's theorem
• Szpilrajn extension theorem
• Zorn's lemma
Properties & Types (list)
• Antisymmetric
• Asymmetric
• Boolean algebra
• topics
• Completeness
• Connected
• Covering
• Dense
• Directed
• (Partial) Equivalence
• Foundational
• Heyting algebra
• Homogeneous
• Idempotent
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• Bounded
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• Join and meet
• Reflexive
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• Chain-complete
• Graded
• Eulerian
• Strict
• Prefix order
• Preorder
• Total
• Semilattice
• Semiorder
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• Total
• Tolerance
• Transitive
• Well-founded
• Well-quasi-ordering (Better)
• (Pre) Well-order
Constructions
• Composition
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• Lexicographic order
• Linear extension
• Product order
• Reflexive closure
• Series-parallel partial order
• Star product
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Topology & Orders
• Alexandrov topology & Specialization preorder
• Ordered topological vector space
• Normal cone
• Order topology
• Order topology
• Topological vector lattice
• Banach
• Fréchet
• Locally convex
• Normed
Related
• Antichain
• Cofinal
• Cofinality
• Comparability
• Graph
• Duality
• Filter
• Hasse diagram
• Ideal
• Net
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• Order morphism
• Embedding
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• Riesz space
• Upper set
• Young's lattice
| Wikipedia |
Weak ordering
In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders.[1]
Not to be confused with Weak order of permutations.
Transitive binary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗
Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗
Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗
Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗
Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗
Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗
Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗
Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗
Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗
Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗
Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y
Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric
Definitions, for all $a,b$ and $S\neq \varnothing :$ :} ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$
Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table.
There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic (interconvertable with no loss of information): they may be axiomatized as strict weak orderings (strictly partially ordered sets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of the elements into disjoint subsets, together with a total order on the subsets). In many cases another representation called a preferential arrangement based on a utility function is also possible.
Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library.[2]
Examples
In horse racing, the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context) dead heats, so the outcome of a horse race may be modeled by a weak ordering.[3] In an example from the Maryland Hunt Cup steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied for second place, with the remaining horses farther back; three horses did not finish.[4] In the weak ordering describing this outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before all the other horses that finished, and the three horses that did not finish would be placed last in the order but tied with each other.
The points of the Euclidean plane may be ordered by their distance from the origin, giving another example of a weak ordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to a common circle centered at the origin), and infinitely many points within these subsets. Although this ordering has a smallest element (the origin itself), it does not have any second-smallest elements, nor any largest element.
Opinion polling in political elections provides an example of a type of ordering that resembles weak orderings, but is better modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another, or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they are within the margin of error of each other. However, if candidate $x$ is statistically tied with $y,$ and $y$ is statistically tied with $z,$ it might still be possible for $x$ to be clearly better than $z,$ so being tied is not in this case a transitive relation. Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings.[5]
Axiomatizations
Suppose throughout that $\,<\,$ is a homogeneous binary relation on a set $S$ (that is, $\,<\,$ is a subset of $S\times S$) and as usual, write $x<y$ and say that $x<y$ holds or is true if and only if $(x,y)\in \,<.\,$
Strict weak orderings
Preliminaries on incomparability and transitivity of incomparability
Two elements $x$ and $y$ of $S$ are said to be incomparable with respect to $\,<\,$ if neither $x<y$ nor $y<x$ is true.[1] Incomparability with respect to $\,<\,$ is itself a homogeneous symmetric relation on $S$ that is reflexive if and only if $\,<\,$ is irreflexive (meaning that $x<x$ is always false), which may be assumed so that transitivity is the only property this "incomparability relation" needs in order to be an equivalence relation. Define also an induced homogeneous relation $\,\lesssim \,$ on $S$ by declaring that
$x\lesssim y{\text{ is true }}\quad {\text{ if and only if }}\quad y<x{\text{ is false}}$
where importantly, this definition is not necessarily the same as: $x\lesssim y$ if and only if $x<y{\text{ or }}x=y.$ Two elements $x,y\in S$ are incomparable with respect to $\,<\,$ if and only if $x{\text{ and }}y$ are equivalent with respect to $\,\lesssim \,$ (or less verbosely, $\,\lesssim $-equivalent), which by definition means that both $x\lesssim y{\text{ and }}y\lesssim x$ are true. The relation "are incomparable with respect to $\,<$" is thus identical to (that is, equal to) the relation "are $\,\lesssim $-equivalent" (so in particular, the former is transitive if and only if the latter is). When $\,<\,$ is irreflexive then the property known as "transitivity of incomparability" (defined below) is exactly the condition necessary and sufficient to guarantee that the relation "are $\,\lesssim $-equivalent" does indeed form an equivalence relation on $S.$ When this is the case, it allows any two elements $x,y\in S$ satisfying $x\lesssim y{\text{ and }}y\lesssim x$ to be identified as a single object (specifically, they are identified together in their common equivalence class).
Definition
A strict weak ordering on a set $S$ is a strict partial order $\,<\,$ on $S$ for which the incomparability relation induced on $S$ by $\,<\,$ is a transitive relation.[1] Explicitly, a strict weak order on $S$ is a homogeneous relation $\,<\,$ on $S$ that has all four of the following properties:
1. Irreflexivity: For all $x\in S,$ it is not true that $x<x.$
• This condition holds if and only if the induced relation $\,\lesssim \,$ on $S$ is reflexive, where $\,\lesssim \,$ is defined by declaring that $x\lesssim y$ is true if and only if $y<x$ is false.
2. Transitivity: For all $x,y,z\in S,$ if $x<y{\text{ and }}y<z$ then $x<z.$
3. Asymmetry: For all $x,y\in S,$ if $x<y$ is true then $y<x$ is false.
4. Transitivity of incomparability: For all $x,y,z\in S,$ if $x$ is incomparable with $y$ (meaning that neither $x<y$ nor $y<x$ is true) and if $y$ is incomparable with $z,$ then $x$ is incomparable with $z.$
• Two elements $x{\text{ and }}y$ are incomparable with respect to $\,<\,$ if and only if they are equivalent with respect to the induced relation $\,\lesssim \,$ (which by definition means that $x\lesssim y{\text{ and }}y\lesssim x$ are both true), where as before, $x\lesssim y$ is declared to be true if and only if $y<x$ is false. Thus this condition holds if and only if the symmetric relation on $S$ defined by "$x{\text{ and }}y$ are equivalent with respect to $\,\lesssim \,$" is a transitive relation, meaning that whenever $x{\text{ and }}y$ are $\,\lesssim $-equivalent and also $y{\text{ and }}z$ are $\,\lesssim $-equivalent then necessarily $x{\text{ and }}z$ are $\,\lesssim $-equivalent. This can also be restated as: whenever $x\lesssim y{\text{ and }}y\lesssim x$ and also $y\lesssim z{\text{ and }}z\lesssim y$ then necessarily $x\lesssim z{\text{ and }}z\lesssim x.$
Properties (1), (2), and (3) are the defining properties of a strict partial order, although there is some redundancy in this list as asymmetry (3) implies irreflexivity (1), and also because irreflexivity (1) and transitivity (2) together imply asymmetry (3).[6] The incomparability relation is always symmetric and it will be reflexive if and only if $\,<\,$ is an irreflexive relation (which is assumed by the above definition). Consequently, a strict partial order $\,<\,$ is a strict weak order if and only if its induced incomparability relation is an equivalence relation. In this case, its equivalence classes partition $S$ and moreover, the set ${\mathcal {P}}$ of these equivalence classes can be strictly totally ordered by a binary relation, also denoted by $\,<,$ that is defined for all $A,B\in {\mathcal {P}}$ by:
$A<B{\text{ if and only if }}a<b$ for some (or equivalently, for all) representatives $a\in A{\text{ and }}b\in B.$
Conversely, any strict total order on a partition ${\mathcal {P}}$ of $S$ gives rise to a strict weak ordering $\,<\,$ on $S$ defined by $a<b$ if and only if there exists sets $A,B\in {\mathcal {P}}$ in this partition such that $a\in A,b\in B,{\text{ and }}A<B.$
Not every partial order obeys the transitive law for incomparability. For instance, consider the partial order in the set $\{a,b,c\}$ defined by the relationship $b<c.$ The pairs $a,b{\text{ and }}a,c$ are incomparable but $b$ and $c$ are related, so incomparability does not form an equivalence relation and this example is not a strict weak ordering.
For transitivity of incomparability, each of the following conditions is necessary, and for strict partial orders also sufficient:
• If $x<y$ then for all $z,$ either $x<z{\text{ or }}z<y$ or both.
• If $x$ is incomparable with $y$ then for all $z$, either ($x<z{\text{ and }}y<z$) or ($z<x{\text{ and }}z<y$) or ($z$ is incomparable with $x$ and $z$ is incomparable with $y$).
Total preorders
Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a preorder in which any two elements are comparable.[7] A total preorder $\,\lesssim \,$ satisfies the following properties:
• Transitivity: For all $x,y,{\text{ and }}z,$ if $x\lesssim y{\text{ and }}y\lesssim z$ then $x\lesssim z.$
• Strong connectedness: For all $x{\text{ and }}y,$ $x\lesssim y{\text{ or }}y\lesssim x.$
• Which implies reflexivity: for all $x,$ $x\lesssim x.$
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preorders are sometimes also called preference relations.
The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we take the converse of the complement: for a strict weak ordering $\,<,$ define a total preorder $\,\lesssim \,$ by setting $x\lesssim y$ whenever it is not the case that $y<x.$ In the other direction, to define a strict weak ordering < from a total preorder $\,\lesssim ,$ set $x<y$ whenever it is not the case that $y\lesssim x.$[8]
In any preorder there is a corresponding equivalence relation where two elements $x$ and $y$ are defined as equivalent if $x\lesssim y{\text{ and }}y\lesssim x.$ In the case of a total preorder the corresponding partial order on the set of equivalence classes is a total order. Two elements are equivalent in a total preorder if and only if they are incomparable in the corresponding strict weak ordering.
Ordered partitions
A partition of a set $S$ is a family of non-empty disjoint subsets of $S$ that have $S$ as their union. A partition, together with a total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition[9] and by Theodore Motzkin a list of sets.[10] An ordered partition of a finite set may be written as a finite sequence of the sets in the partition: for instance, the three ordered partitions of the set $\{a,b\}$ are
$\{a\},\{b\},$
$\{b\},\{a\},\;{\text{ and }}$
$\{a,b\}.$
In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partition gives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in the partition, and otherwise inherit the order of the sets that contain them.
Representation by functions
For sets of sufficiently small cardinality, a third axiomatization is possible, based on real-valued functions. If $X$ is any set then a real-valued function $f:X\to \mathbb {R} $ on $X$ induces a strict weak order on $X$ by setting
$a<b{\text{ if and only if }}f(a)<f(b).$
The associated total preorder is given by setting $a{}\lesssim {}b{\text{ if and only if }}f(a)\leq f(b)$ and the associated equivalence by setting $a{}\sim {}b{\text{ if and only if }}f(a)=f(b).$
The relations do not change when $f$ is replaced by $g\circ f$ (composite function), where $g$ is a strictly increasing real-valued function defined on at least the range of $f.$ Thus for example, a utility function defines a preference relation. In this context, weak orderings are also known as preferential arrangements.[11]
If $X$ is finite or countable, every weak order on $X$ can be represented by a function in this way.[12] However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the lexicographic order on $\mathbb {R} ^{n}.$ Thus, while in most preference relation models the relation defines a utility function up to order-preserving transformations, there is no such function for lexicographic preferences.
More generally, if $X$ is a set, $Y$ is a set with a strict weak ordering $\,<,\,$ and $f:X\to Y$ is a function, then $f$ induces a strict weak ordering on $X$ by setting
$a<b{\text{ if and only if }}f(a)<f(b).$
As before, the associated total preorder is given by setting $a{}\lesssim {}b{\text{ if and only if }}f(a){}\lesssim {}f(b),$ and the associated equivalence by setting $a{}\sim {}b{\text{ if and only if }}f(a){}\sim {}f(b).$ It is not assumed here that $f$ is an injective function, so a class of two equivalent elements on $Y$ may induce a larger class of equivalent elements on $X.$ Also, $f$ is not assumed to be a surjective function, so a class of equivalent elements on $Y$ may induce a smaller or empty class on $X.$ However, the function $f$ induces an injective function that maps the partition on $X$ to that on $Y.$ Thus, in the case of finite partitions, the number of classes in $X$ is less than or equal to the number of classes on $Y.$
Related types of ordering
Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.[13] A strict weak order that is trichotomous is called a strict total order.[14] The total preorder which is the inverse of its complement is in this case a total order.
For a strict weak order $\,<\,$ another associated reflexive relation is its reflexive closure, a (non-strict) partial order $\,\leq .$ The two associated reflexive relations differ with regard to different $a$ and $b$ for which neither $a<b$ nor $b<a$: in the total preorder corresponding to a strict weak order we get $a\lesssim b$ and $b\lesssim a,$ while in the partial order given by the reflexive closure we get neither $a\leq b$ nor $b\leq a.$ For strict total orders these two associated reflexive relations are the same: the corresponding (non-strict) total order.[14] The reflexive closure of a strict weak ordering is a type of series-parallel partial order.
All weak orders on a finite set
Combinatorial enumeration
Main article: ordered Bell number
The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an $n$-element set is given by the following sequence (sequence A000670 in the OEIS):
Number of n-element binary relations of different types
Elements Any Transitive Reflexive Symmetric Preorder Partial order Total preorder Total order Equivalence relation
0111111111
1221211111
216134843322
3512171646429191365
465,5363,9944,0961,024355219752415
n 2n2 2n2−n 2n(n+1)/2 $ \sum _{k=0}^{n}k!S(n,k)$ n! $ \sum _{k=0}^{n}S(n,k)$
OEIS A002416 A006905 A053763 A006125 A000798 A001035 A000670 A000142 A000110
Note that S(n, k) refers to Stirling numbers of the second kind.
These numbers are also called the Fubini numbers or ordered Bell numbers.
For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are three ways of partitioning the items into one singleton set and one group of two tied items, and each of these partitions gives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering is reversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons, which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.
Adjacency structure
Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves that add or remove a single order relation to or from a given ordering. For instance, for three elements, the ordering in which all three elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strict weak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weak orderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes, and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in the ordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as a Dedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies. For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of moves that add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph that has the weak orderings as its vertices, and these moves as its edges, forms a partial cube.[15]
Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and the dichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderings on the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedron itself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in the corresponding weak ordering.[16] In this geometric representation the partial cube of moves on weak orderings is the graph describing the covering relation of the face lattice of the permutohedron.
For instance, for $n=3,$ the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon (again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon, six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie, and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.
Applications
As mentioned above, weak orders have applications in utility theory.[12] In linear programming and other types of combinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, determined by a real-valued objective function; the phenomenon of ties in these orderings is called "degeneracy", and several types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to prevent problems caused by degeneracy.[17]
Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographic breadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of a graph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a total order on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that is the output of the algorithm.[18]
In the Standard Library for the C++ programming language, the set and multiset data types sort their input by a comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict weak ordering.[2]
See also
• Comparability – Property of elements related by inequalities
• Preorder – Reflexive and transitive binary relation
• Weak component – Partition of vertices of a directed graph − the equivalent subsets in the finest weak ordering consistent with a given relation
References
1. Roberts, Fred; Tesman, Barry (2011), Applied Combinatorics (2nd ed.), CRC Press, Section 4.2.4 Weak Orders, pp. 254–256, ISBN 9781420099836.
2. Josuttis, Nicolai M. (2012), The C++ Standard Library: A Tutorial and Reference, Addison-Wesley, p. 469, ISBN 9780132977739.
3. de Koninck, J. M. (2009), Those Fascinating Numbers, American Mathematical Society, p. 4, ISBN 9780821886311.
4. Baker, Kent (April 29, 2007), "The Bruce hangs on for Hunt Cup victory: Bug River, Lear Charm finish in dead heat for second", The Baltimore Sun, archived from the original on March 29, 2015.
5. Regenwetter, Michel (2006), Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications, Cambridge University Press, pp. 113ff, ISBN 9780521536660.
6. Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007), Transitive Closures of Binary Relations I (PDF), Prague: School of Mathematics - Physics Charles University, p. 1, S2CID 47676001, archived from the original (PDF) on 2018-04-06, Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".
7. Such a relation is also called strongly connected.
8. Ehrgott, Matthias (2005), Multicriteria Optimization, Springer, Proposition 1.9, p. 10, ISBN 9783540276593.
9. Stanley, Richard P. (1997), Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, p. 297.
10. Motzkin, Theodore S. (1971), "Sorting numbers for cylinders and other classification numbers", Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Providence, R.I.: Amer. Math. Soc., pp. 167–176, MR 0332508.
11. Gross, O. A. (1962), "Preferential arrangements", The American Mathematical Monthly, 69 (1): 4–8, doi:10.2307/2312725, JSTOR 2312725, MR 0130837.
12. Roberts, Fred S. (1979), Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences, Encyclopedia of Mathematics and its Applications, vol. 7, Addison-Wesley, Theorem 3.1, ISBN 978-0-201-13506-0.
13. Luce, R. Duncan (1956), "Semiorders and a theory of utility discrimination" (PDF), Econometrica, 24 (2): 178–191, doi:10.2307/1905751, JSTOR 1905751, MR 0078632.
14. Velleman, Daniel J. (2006), How to Prove It: A Structured Approach, Cambridge University Press, p. 204, ISBN 9780521675994.
15. Eppstein, David; Falmagne, Jean-Claude; Ovchinnikov, Sergei (2008), Media Theory: Interdisciplinary Applied Mathematics, Springer, Section 9.4, Weak Orders and Cubical Complexes, pp. 188–196.
16. Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, p. 18.
17. Chvátal, Vašek (1983), Linear Programming, Macmillan, pp. 29–38, ISBN 9780716715870.
18. Habib, Michel; Paul, Christophe; Viennot, Laurent (1999), "Partition refinement techniques: an interesting algorithmic tool kit", International Journal of Foundations of Computer Science, 10 (2): 147–170, doi:10.1142/S0129054199000125, MR 1759929.
Order theory
• Topics
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Key concepts
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| Wikipedia |
Total ring of fractions
In abstract algebra, the total quotient ring[1] or total ring of fractions[2] is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring. If the homomorphism from R to the new ring is to be injective, no further elements can be given an inverse.
Algebraic structure → Ring theory
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Definition
Let $R$ be a commutative ring and let $S$ be the set of elements which are not zero divisors in $R$; then $S$ is a multiplicatively closed set. Hence we may localize the ring $R$ at the set $S$ to obtain the total quotient ring $S^{-1}R=Q(R)$.
If $R$ is a domain, then $S=R-\{0\}$ and the total quotient ring is the same as the field of fractions. This justifies the notation $Q(R)$, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.
Since $S$ in the construction contains no zero divisors, the natural map $R\to Q(R)$ is injective, so the total quotient ring is an extension of $R$.
Examples
• For a product ring A × B, the total quotient ring Q(A × B) is the product of total quotient rings Q(A) × Q(B). In particular, if A and B are integral domains, it is the product of quotient fields.
• For the ring of holomorphic functions on an open set D of complex numbers, the total quotient ring is the ring of meromorphic functions on D, even if D is not connected.
• In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero-divisors is the group of units of the ring, $R^{\times }$, and so $Q(R)=(R^{\times })^{-1}R$. But since all these elements already have inverses, $Q(R)=R$.
• In a commutative von Neumann regular ring R, the same thing happens. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a = axa for some x in R, giving the equation a(xa − 1) = 0. Since a is not a zero divisor, xa = 1, showing a is a unit. Here again, $Q(R)=R$.
• In algebraic geometry one considers a sheaf of total quotient rings on a scheme, and this may be used to give the definition of a Cartier divisor.
The total ring of fractions of a reduced ring
Proposition — Let A be a reduced ring that has only finitely many minimal prime ideals, ${\mathfrak {p}}_{1},\dots ,{\mathfrak {p}}_{r}$ (e.g., a Noetherian reduced ring). Then
$Q(A)\simeq \prod _{i=1}^{r}Q(A/{\mathfrak {p}}_{i}).$
Geometrically, $\operatorname {Spec} (Q(A))$ is the Artinian scheme consisting (as a finite set) of the generic points of the irreducible components of $\operatorname {Spec} (A)$.
Proof: Every element of Q(A) is either a unit or a zero divisor. Thus, any proper ideal I of Q(A) is contained in the set of zero divisors of Q(A); that set equals the union of the minimal prime ideals ${\mathfrak {p}}_{i}Q(A)$ since Q(A) is reduced. By prime avoidance, I must be contained in some ${\mathfrak {p}}_{i}Q(A)$. Hence, the ideals ${\mathfrak {p}}_{i}Q(A)$ are maximal ideals of Q(A). Also, their intersection is zero. Thus, by the Chinese remainder theorem applied to Q(A),
$Q(A)\simeq \prod _{i}Q(A)/{\mathfrak {p}}_{i}Q(A)$.
Let S be the multiplicatively closed set of non-zero-divisors of A. By exactness of localization,
$Q(A)/{\mathfrak {p}}_{i}Q(A)=A[S^{-1}]/{\mathfrak {p}}_{i}A[S^{-1}]=(A/{\mathfrak {p}}_{i})[S^{-1}]$,
which is already a field and so must be $Q(A/{\mathfrak {p}}_{i})$. $\square $
Generalization
If $R$ is a commutative ring and $S$ is any multiplicatively closed set in $R$, the localization $S^{-1}R$ can still be constructed, but the ring homomorphism from $R$ to $S^{-1}R$ might fail to be injective. For example, if $0\in S$, then $S^{-1}R$ is the trivial ring.
Citations
1. Matsumura 1980, p. 12.
2. Matsumura 1989, p. 21.
References
• Matsumura, Hideyuki (1980), Commutative algebra (2nd ed.), Benjamin/Cummings, ISBN 978-0-8053-7026-3, OCLC 988482880
• Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge University Press, ISBN 978-0-521-36764-6, OCLC 23133540
| Wikipedia |
Total set
In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals $T$ with the property that if a vector $x\in X$ satisfies $f(x)=0$ for all $f\in T,$ then $x=0$ is the zero vector.[1]
In a more general setting, a subset $T$ of a topological vector space $X$ is a total set or fundamental set if the linear span of $T$ is dense in $X.$[2]
See also
• Kadec norm – All infinite-dimensional, separable Banach spaces are homeomorphicPages displaying short descriptions of redirect targets
• Degenerate bilinear form – Possible x & y for x-E conjugatesPages displaying wikidata descriptions as a fallback
• Dual system
• Topologies on spaces of linear maps
References
1. Klauder, John R. (2010). A Modern Approach to Functional Integration. Springer Science & Business Media. p. 91. ISBN 9780817647902.
2. Lomonosov, L. I. "Total set". Encyclopedia of Mathematics. Springer. Retrieved 14 September 2014.
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| Wikipedia |
Total subset
In mathematics, more specifically in functional analysis, a subset $T$ of a topological vector space $X$ is said to be a total subset of $X$ if the linear span of $T$ is a dense subset of $X.$[1] This condition arises frequently in many theorems of functional analysis.
Examples
Unbounded self-adjoint operators on Hilbert spaces are defined on total subsets.
See also
• Dense subset – Subset whose closure is the whole spacePages displaying short descriptions of redirect targets
• Positive linear operator
• Topological vector spaces – Vector space with a notion of nearnessPages displaying short descriptions of redirect targets
References
1. Schaefer & Wolff 1999, p. 80.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Functional analysis (topics – glossary)
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• Sobolev
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| Wikipedia |
Total variation
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.
Not to be confused with Total variation distance of probability measures.
Historical note
The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper (Jordan 1881).[1] He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons.
Definitions
Total variation for functions of one real variable
Definition 1.1. The total variation of a real-valued (or more generally complex-valued) function $f$, defined on an interval $[a,b]\subset \mathbb {R} $ is the quantity
$V_{a}^{b}(f)=\sup _{\mathcal {P}}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,$
where the supremum runs over the set of all partitions ${\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}[a,b]\right\}$ of the given interval.
Total variation for functions of n > 1 real variables
Definition 1.2. Let Ω be an open subset of Rn. Given a function f belonging to L1(Ω), the total variation of f in Ω is defined as
$V(f,\Omega ):=\sup \left\{\int _{\Omega }f(x)\operatorname {div} \phi (x)\,\mathrm {d} x\colon \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \phi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},$
where
• $C_{c}^{1}(\Omega ,\mathbb {R} ^{n})$ is the set of continuously differentiable vector functions of compact support contained in $\Omega $,
• $\Vert \;\Vert _{L^{\infty }(\Omega )}$ is the essential supremum norm, and
• $\operatorname {div} $ is the divergence operator.
This definition does not require that the domain $\Omega \subseteq \mathbb {R} ^{n}$ of the given function be a bounded set.
Classical total variation definition
Following Saks (1937, p. 10), consider a signed measure $\mu $ on a measurable space $(X,\Sigma )$: then it is possible to define two set functions ${\overline {\mathrm {W} }}(\mu ,\cdot )$ and ${\underline {\mathrm {W} }}(\mu ,\cdot )$, respectively called upper variation and lower variation, as follows
${\overline {\mathrm {W} }}(\mu ,E)=\sup \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma $
${\underline {\mathrm {W} }}(\mu ,E)=\inf \left\{\mu (A)\mid A\in \Sigma {\text{ and }}A\subset E\right\}\qquad \forall E\in \Sigma $
clearly
${\overline {\mathrm {W} }}(\mu ,E)\geq 0\geq {\underline {\mathrm {W} }}(\mu ,E)\qquad \forall E\in \Sigma $
Definition 1.3. The variation (also called absolute variation) of the signed measure $\mu $ is the set function
$|\mu |(E)={\overline {\mathrm {W} }}(\mu ,E)+\left|{\underline {\mathrm {W} }}(\mu ,E)\right|\qquad \forall E\in \Sigma $
and its total variation is defined as the value of this measure on the whole space of definition, i.e.
$\|\mu \|=|\mu |(X)$
Modern definition of total variation norm
Saks (1937, p. 11) uses upper and lower variations to prove the Hahn–Jordan decomposition: according to his version of this theorem, the upper and lower variation are respectively a non-negative and a non-positive measure. Using a more modern notation, define
$\mu ^{+}(\cdot )={\overline {\mathrm {W} }}(\mu ,\cdot )\,,$
$\mu ^{-}(\cdot )=-{\underline {\mathrm {W} }}(\mu ,\cdot )\,,$
Then $\mu ^{+}$ and $\mu ^{-}$ are two non-negative measures such that
$\mu =\mu ^{+}-\mu ^{-}$
$|\mu |=\mu ^{+}+\mu ^{-}$
The last measure is sometimes called, by abuse of notation, total variation measure.
Total variation norm of complex measures
If the measure $\mu $ is complex-valued i.e. is a complex measure, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow Rudin (1966, pp. 137–139) and define the total variation of the complex-valued measure $\mu $ as follows
Definition 1.4. The variation of the complex-valued measure $\mu $ is the set function
$|\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma $
where the supremum is taken over all partitions $\pi $ of a measurable set $E$ into a countable number of disjoint measurable subsets.
This definition coincides with the above definition $|\mu |=\mu ^{+}+\mu ^{-}$ for the case of real-valued signed measures.
Total variation norm of vector-valued measures
The variation so defined is a positive measure (see Rudin (1966, p. 139)) and coincides with the one defined by 1.3 when $\mu $ is a signed measure: its total variation is defined as above. This definition works also if $\mu $ is a vector measure: the variation is then defined by the following formula
$|\mu |(E)=\sup _{\pi }\sum _{A\in \pi }\|\mu (A)\|\qquad \forall E\in \Sigma $
where the supremum is as above. This definition is slightly more general than the one given by Rudin (1966, p. 138) since it requires only to consider finite partitions of the space $X$: this implies that it can be used also to define the total variation on finite-additive measures.
Total variation of probability measures
Main article: Total variation distance of probability measures
The total variation of any probability measure is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are probability measures, the total variation distance of probability measures can be defined as $\|\mu -\nu \|$ where the norm is the total variation norm of signed measures. Using the property that $(\mu -\nu )(X)=0$, we eventually arrive at the equivalent definition
$\|\mu -\nu \|=|\mu -\nu |(X)=2\sup \left\{\,\left|\mu (A)-\nu (A)\right|:A\in \Sigma \,\right\}$
and its values are non-trivial. The factor $2$ above is usually dropped (as is the convention in the article total variation distance of probability measures). Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event. For a categorical distribution it is possible to write the total variation distance as follows
$\delta (\mu ,\nu )=\sum _{x}\left|\mu (x)-\nu (x)\right|\;.$
It may also be normalized to values in $[0,1]$ by halving the previous definition as follows
$\delta (\mu ,\nu )={\frac {1}{2}}\sum _{x}\left|\mu (x)-\nu (x)\right|$[2]
Basic properties
Total variation of differentiable functions
The total variation of a $C^{1}({\overline {\Omega }})$ function $f$ can be expressed as an integral involving the given function instead of as the supremum of the functionals of definitions 1.1 and 1.2.
The form of the total variation of a differentiable function of one variable
Theorem 1. The total variation of a differentiable function $f$, defined on an interval $[a,b]\subset \mathbb {R} $, has the following expression if $f'$ is Riemann integrable
$V_{a}^{b}(f)=\int _{a}^{b}|f'(x)|\mathrm {d} x$
If $f$ is differentiable and monotonic, then the above simplifies to
$V_{a}^{b}(f)=|f(a)-f(b)|$
For any differentiable function $f$, we can decompose the domain interval $[a,b]$, into subintervals $[a,a_{1}],[a_{1},a_{2}],\dots ,[a_{N},b]$ (with $a<a_{1}<a_{2}<\cdots <a_{N}<b$) in which $f$ is locally monotonic, then the total variation of $f$ over $[a,b]$ can be written as the sum of local variations on those subintervals:
${\begin{aligned}V_{a}^{b}(f)&=V_{a}^{a_{1}}(f)+V_{a_{1}}^{a_{2}}(f)+\,\cdots \,+V_{a_{N}}^{b}(f)\\[0.3em]&=|f(a)-f(a_{1})|+|f(a_{1})-f(a_{2})|+\,\cdots \,+|f(a_{N})-f(b)|\end{aligned}}$
The form of the total variation of a differentiable function of several variables
Theorem 2. Given a $C^{1}({\overline {\Omega }})$ function $f$ defined on a bounded open set $\Omega \subseteq \mathbb {R} ^{n}$, with $\partial \Omega $ of class $C^{1}$, the total variation of $f$ has the following expression
$V(f,\Omega )=\int _{\Omega }\left|\nabla f(x)\right|\mathrm {d} x$ .
Proof
The first step in the proof is to first prove an equality which follows from the Gauss–Ostrogradsky theorem.
Lemma
Under the conditions of the theorem, the following equality holds:
$\int _{\Omega }f\operatorname {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi $
Proof of the lemma
From the Gauss–Ostrogradsky theorem:
$\int _{\Omega }\operatorname {div} \mathbf {R} =\int _{\partial \Omega }\mathbf {R} \cdot \mathbf {n} $
by substituting $\mathbf {R} :=f\mathbf {\varphi } $ :=f\mathbf {\varphi } } , we have:
$\int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=\int _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} $
where $\mathbf {\varphi } $ is zero on the border of $\Omega $ by definition:
$\int _{\Omega }\operatorname {div} \left(f\mathbf {\varphi } \right)=0$
$\int _{\Omega }\partial _{x_{i}}\left(f\mathbf {\varphi } _{i}\right)=0$
$\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f+f\partial _{x_{i}}\mathbf {\varphi } _{i}=0$
$\int _{\Omega }f\partial _{x_{i}}\mathbf {\varphi } _{i}=-\int _{\Omega }\mathbf {\varphi } _{i}\partial _{x_{i}}f$
$\int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f$
Proof of the equality
Under the conditions of the theorem, from the lemma we have:
$\int _{\Omega }f\operatorname {div} \mathbf {\varphi } =-\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\leq \left|\int _{\Omega }\mathbf {\varphi } \cdot \nabla f\right|\leq \int _{\Omega }\left|\mathbf {\varphi } \right|\cdot \left|\nabla f\right|\leq \int _{\Omega }\left|\nabla f\right|$
in the last part $\mathbf {\varphi } $ could be omitted, because by definition its essential supremum is at most one.
On the other hand, we consider $\theta _{N}:=-\mathbb {I} _{\left[-N,N\right]}\mathbb {I} _{\{\nabla f\neq 0\}}{\frac {\nabla f}{\left|\nabla f\right|}}$ and $\theta _{N}^{*}$ which is the up to $\varepsilon $ approximation of $\theta _{N}$ in $C_{c}^{1}$ with the same integral. We can do this since $C_{c}^{1}$ is dense in $L^{1}$. Now again substituting into the lemma:
${\begin{aligned}&\lim _{N\to \infty }\int _{\Omega }f\operatorname {div} \theta _{N}^{*}\\[4pt]&=\lim _{N\to \infty }\int _{\{\nabla f\neq 0\}}\mathbb {I} _{\left[-N,N\right]}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\lim _{N\to \infty }\int _{\left[-N,N\right]\cap {\{\nabla f\neq 0\}}}\nabla f\cdot {\frac {\nabla f}{\left|\nabla f\right|}}\\[4pt]&=\int _{\Omega }\left|\nabla f\right|\end{aligned}}$
This means we have a convergent sequence of $ \int _{\Omega }f\operatorname {div} \mathbf {\varphi } $ that tends to $ \int _{\Omega }\left|\nabla f\right|$ as well as we know that $ \int _{\Omega }f\operatorname {div} \mathbf {\varphi } \leq \int _{\Omega }\left|\nabla f\right|$. Q.E.D.
It can be seen from the proof that the supremum is attained when
$\varphi \to {\frac {-\nabla f}{\left|\nabla f\right|}}.$
The function $f$ is said to be of bounded variation precisely if its total variation is finite.
Total variation of a measure
The total variation is a norm defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a Banach space, called the ca space, relative to this norm. It is contained in the larger Banach space, called the ba space, consisting of finitely additive (as opposed to countably additive) measures, also with the same norm. The distance function associated to the norm gives rise to the total variation distance between two measures μ and ν.
For finite measures on R, the link between the total variation of a measure μ and the total variation of a function, as described above, goes as follows. Given μ, define a function $\varphi \colon \mathbb {R} \to \mathbb {R} $ by
$\varphi (t)=\mu ((-\infty ,t])~.$
Then, the total variation of the signed measure μ is equal to the total variation, in the above sense, of the function $\varphi $. In general, the total variation of a signed measure can be defined using Jordan's decomposition theorem by
$\|\mu \|_{TV}=\mu _{+}(X)+\mu _{-}(X)~,$
for any signed measure μ on a measurable space $(X,\Sigma )$.
Applications
Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like optimal control, numerical analysis, and calculus of variations, where the solution to a certain problem has to minimize its value. As an example, use of the total variation functional is common in the following two kind of problems
• Numerical analysis of differential equations: it is the science of finding approximate solutions to differential equations. Applications of total variation to these problems are detailed in the article "total variation diminishing"
• Image denoising: in image processing, denoising is a collection of methods used to reduce the noise in an image reconstructed from data obtained by electronic means, for example data transmission or sensing. "Total variation denoising" is the name for the application of total variation to image noise reduction; further details can be found in the papers of (Rudin, Osher & Fatemi 1992) and (Caselles, Chambolle & Novaga 2007). A sensible extension of this model to colour images, called Colour TV, can be found in (Blomgren & Chan 1998).
See also
• Bounded variation
• p-variation
• Total variation diminishing
• Total variation denoising
• Quadratic variation
• Total variation distance of probability measures
• Kolmogorov–Smirnov test
• Anisotropic diffusion
Notes
1. According to Golubov & Vitushkin (2001).
2. Gibbs, Alison; Francis Edward Su (2002). "On Choosing and Bounding Probability Metrics" (PDF). p. 7. Retrieved 8 April 2017.
Historical references
• Arzelà, Cesare (7 May 1905), "Sulle funzioni di due variabili a variazione limitata (On functions of two variables of bounded variation)", Rendiconto delle Sessioni della Reale Accademia delle Scienze dell'Istituto di Bologna, Nuova serie (in Italian), IX (4): 100–107, JFM 36.0491.02, archived from the original on 2007-08-07.
• Golubov, Boris I. (2001) [1994], "Arzelà variation", Encyclopedia of Mathematics, EMS Press.
• Golubov, Boris I. (2001) [1994], "Fréchet variation", Encyclopedia of Mathematics, EMS Press.
• Golubov, Boris I. (2001) [1994], "Hardy variation", Encyclopedia of Mathematics, EMS Press.
• Golubov, Boris I. (2001) [1994], "Pierpont variation", Encyclopedia of Mathematics, EMS Press.
• Golubov, Boris I. (2001) [1994], "Vitali variation", Encyclopedia of Mathematics, EMS Press.
• Golubov, Boris I. (2001) [1994], "Tonelli plane variation", Encyclopedia of Mathematics, EMS Press.
• Golubov, Boris I.; Vitushkin, Anatoli G. (2001) [1994], "Variation of a function", Encyclopedia of Mathematics, EMS Press
• Jordan, Camille (1881), "Sur la série de Fourier", Comptes rendus hebdomadaires des séances de l'Académie des sciences (in French), 92: 228–230, JFM 13.0184.01 (available at Gallica). This is, according to Boris Golubov, the first paper on functions of bounded variation.
• Hahn, Hans (1921), Theorie der reellen Funktionen (in German), Berlin: Springer Verlag, pp. VII+600, JFM 48.0261.09.
• Vitali, Giuseppe (1908) [17 dicembre 1907], "Sui gruppi di punti e sulle funzioni di variabili reali (On groups of points and functions of real variables)", Atti dell'Accademia delle Scienze di Torino (in Italian), 43: 75–92, JFM 39.0101.05, archived from the original on 2009-03-31. The paper containing the first proof of Vitali covering theorem.
References
• Adams, C. Raymond; Clarkson, James A. (1933), "On definitions of bounded variation for functions of two variables", Transactions of the American Mathematical Society, 35 (4): 824–854, doi:10.1090/S0002-9947-1933-1501718-2, JFM 59.0285.01, MR 1501718, Zbl 0008.00602.
• Cesari, Lamberto (1936), "Sulle funzioni a variazione limitata (On the functions of bounded variation)", Annali della Scuola Normale Superiore, II (in Italian), 5 (3–4): 299–313, JFM 62.0247.03, MR 1556778, Zbl 0014.29605. Available at Numdam.
• Leoni, Giovanni (2017), A First Course in Sobolev Spaces: Second Edition, Graduate Studies in Mathematics, American Mathematical Society, pp. xxii+734, ISBN 978-1-4704-2921-8.
• Saks, Stanisław (1937). Theory of the Integral. Monografie Matematyczne. Vol. 7 (2nd ed.). Warszawa–Lwów: G.E. Stechert & Co. pp. VI+347. JFM 63.0183.05. Zbl 0017.30004.. (available at the Polish Virtual Library of Science). English translation from the original French by Laurence Chisholm Young, with two additional notes by Stefan Banach.
• Rudin, Walter (1966), Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics (1st ed.), New York: McGraw-Hill, pp. xi+412, MR 0210528, Zbl 0142.01701.
External links
One variable
• "Total variation" on PlanetMath.
One and more variables
• Function of bounded variation at Encyclopedia of Mathematics
Measure theory
• Rowland, Todd. "Total Variation". MathWorld..
• Jordan decomposition at PlanetMath..
• Jordan decomposition at Encyclopedia of Mathematics
Applications
• Caselles, Vicent; Chambolle, Antonin; Novaga, Matteo (2007), The discontinuity set of solutions of the TV denoising problem and some extensions, SIAM, Multiscale Modeling and Simulation, vol. 6 n. 3, archived from the original on 2011-09-27 (a work dealing with total variation application in denoising problems for image processing).
• Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad (1992), "Nonlinear total variation based noise removal algorithms", Physica D: Nonlinear Phenomena, Physica D: Nonlinear Phenomena 60.1: 259-268, 60 (1–4): 259–268, Bibcode:1992PhyD...60..259R, doi:10.1016/0167-2789(92)90242-F.
• Blomgren, Peter; Chan, Tony F. (1998), "Color TV: total variation methods for restoration of vector-valued images", IEEE Transactions on Image Processing, Image Processing, IEEE Transactions on, vol. 7, no. 3: 304-309, 7 (3): 304, Bibcode:1998ITIP....7..304B, doi:10.1109/83.661180, PMID 18276250.
• Tony F. Chan and Jackie (Jianhong) Shen (2005), Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, SIAM, ISBN 0-89871-589-X (with in-depth coverage and extensive applications of Total Variations in modern image processing, as started by Rudin, Osher, and Fatemi).
| Wikipedia |
Total variation distance of probability measures
In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes called the statistical distance, statistical difference or variational distance.
Not to be confused with Total variation.
Definition
Consider a measurable space $(\Omega ,{\mathcal {F}})$ and probability measures $P$ and $Q$ defined on $(\Omega ,{\mathcal {F}})$. The total variation distance between $P$ and $Q$ is defined as:[1]
$\delta (P,Q)=\sup _{A\in {\mathcal {F}}}\left|P(A)-Q(A)\right|.$
Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.
Properties
Relation to other distances
The total variation distance is related to the Kullback–Leibler divergence by Pinsker’s inequality:
$\delta (P,Q)\leq {\sqrt {{\frac {1}{2}}D_{\mathrm {KL} }(P\parallel Q)}}.$
One also has the following inequality, due to Bretagnolle and Huber[2] (see, also, Tsybakov[3]), which has the advantage of providing a non-vacuous bound even when $D_{\mathrm {KL} }(P\parallel Q)>2$:
$\delta (P,Q)\leq {\sqrt {1-e^{-D_{\mathrm {KL} }(P\parallel Q)}}}.$
When $\Omega $ is countable, the total variation distance is related to the L1 norm by the identity:[4]
$\delta (P,Q)={\frac {1}{2}}\|P-Q\|_{1}={\frac {1}{2}}\sum _{\omega \in \Omega }|P(\{\omega \})-Q(\{\omega \})|$
The total variation distance is related to the Hellinger distance $H(P,Q)$ as follows:[5]
$H^{2}(P,Q)\leq \delta (P,Q)\leq {\sqrt {2}}H(P,Q).$
These inequalities follow immediately from the inequalities between the 1-norm and the 2-norm.
Connection to transportation theory
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is $c(x,y)={\mathbf {1} }_{x\neq y}$, that is,
${\frac {1}{2}}\|P-Q\|_{1}=\delta (P,Q)=\inf\{\mathbb {P} (X\neq Y):{\text{Law}}(X)=P,{\text{Law}}(Y)=Q\}=\inf _{\pi }\operatorname {E} _{\pi }[{\mathbf {1} }_{x\neq y}],$
where the expectation is taken with respect to the probability measure $\pi $ on the space where $(x,y)$ lives, and the infimum is taken over all such $\pi $ with marginals $P$ and $Q$, respectively.[6]
See also
• Total variation
• Kolmogorov–Smirnov test
• Wasserstein metric
References
1. Chatterjee, Sourav. "Distances between probability measures" (PDF). UC Berkeley. Archived from the original (PDF) on July 8, 2008. Retrieved 21 June 2013.
2. Bretagnolle, J.; Huber, C, Estimation des densités: risque minimax, Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977), pp. 342–363, Lecture Notes in Math., 649, Springer, Berlin, 1978, Lemma 2.1 (French).
3. Tsybakov, Alexandre B., Introduction to nonparametric estimation, Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. Springer Series in Statistics. Springer, New York, 2009. xii+214 pp. ISBN 978-0-387-79051-0, Equation 2.25.
4. David A. Levin, Yuval Peres, Elizabeth L. Wilmer, Markov Chains and Mixing Times, 2nd. rev. ed. (AMS, 2017), Proposition 4.2, p. 48.
5. Harsha, Prahladh (September 23, 2011). "Lecture notes on communication complexity" (PDF).
6. Villani, Cédric (2009). Optimal Transport, Old and New. Grundlehren der mathematischen Wissenschaften. Vol. 338. Springer-Verlag Berlin Heidelberg. p. 10. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3.
| Wikipedia |
Totally disconnected space
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the only connected subsets.
Not to be confused with extremally disconnected space.
An important example of a totally disconnected space is the Cantor set, which is homeomorphic to the set of p-adic integers. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers.
Definition
A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets.[1][2] Analogously, a topological space $X$ is totally path-disconnected if all path-components in $X$ are the one-point sets.
Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. That is, a topological space $X$ is totally separated space if and only if for every $x\in X$, the intersection of all clopen neighborhoods of $x$ is the singleton $\{x\}$. Equivalently, for each pair of distinct points $x,y\in X$, there is a pair of disjoint open neighborhoods $U,V$ of $x,y$ such that $X=U\sqcup V$.
Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take $X$ to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then $X$ is totally disconnected but its quasicomponents are not singletons. For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.
Unfortunately in the literature (for instance [3]), totally disconnected spaces are sometimes called hereditarily disconnected, while the terminology totally disconnected is used for totally separated spaces.
Examples
The following are examples of totally disconnected spaces:
• Discrete spaces
• The rational numbers
• The irrational numbers
• The p-adic numbers; more generally, all profinite groups are totally disconnected.
• The Cantor set and the Cantor space
• The Baire space
• The Sorgenfrey line
• Every Hausdorff space of small inductive dimension 0 is totally disconnected
• The Erdős space ℓ2$\,\cap \,\mathbb {Q} ^{\omega }$ is a totally disconnected Hausdorff space that does not have small inductive dimension 0.
• Extremally disconnected Hausdorff spaces
• Stone spaces
• The Knaster–Kuratowski fan provides an example of a connected space, such that the removal of a single point produces a totally disconnected space.
Properties
• Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
• Totally disconnected spaces are T1 spaces, since singletons are closed.
• Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every compact metric space is a continuous image of the Cantor set.
• A locally compact Hausdorff space has small inductive dimension 0 if and only if it is totally disconnected.
• Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.
• It is in general not true that every open set in a totally disconnected space is also closed.
• It is in general not true that the closure of every open set in a totally disconnected space is open, i.e. not every totally disconnected Hausdorff space is extremally disconnected.
Constructing a totally disconnected quotient space of any given space
Let $X$ be an arbitrary topological space. Let $x\sim y$ if and only if $y\in \mathrm {conn} (x)$ (where $\mathrm {conn} (x)$ denotes the largest connected subset containing $x$). This is obviously an equivalence relation whose equivalence classes are the connected components of $X$. Endow $X/{\sim }$ with the quotient topology, i.e. the finest topology making the map $m:x\mapsto \mathrm {conn} (x)$ continuous. With a little bit of effort we can see that $X/{\sim }$ is totally disconnected.
In fact this space is not only some totally disconnected quotient but in a certain sense the biggest: The following universal property holds: For any totally disconnected space $Y$ and any continuous map $f:X\rightarrow Y$, there exists a unique continuous map ${\breve {f}}:(X/\sim )\rightarrow Y$ with $f={\breve {f}}\circ m$.
See also
• Extremally disconnected space
• Totally disconnected group
Citations
1. Rudin 1991, p. 395 Appendix A7.
2. Munkres 2000, pp. 152.
3. Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Sigma Series in Pure Mathematics. ISBN 3-88538-006-4.
References
• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Willard, Stephen (2004), General topology, Dover Publications, ISBN 978-0-486-43479-7, MR 2048350 (reprint of the 1970 original, MR0264581)
| Wikipedia |
Partially ordered ring
In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order $\,\leq \,$ on the underlying set A that is compatible with the ring operations in the sense that it satisfies:
$x\leq y{\text{ implies }}x+z\leq y+z$
and
$0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y$
for all $x,y,z\in A$.[1] Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring $(A,\leq )$ where $A$'s partially ordered additive group is Archimedean.[2]
An ordered ring, also called a totally ordered ring, is a partially ordered ring $(A,\leq )$ where $\,\leq \,$ is additionally a total order.[1][2]
An l-ring, or lattice-ordered ring, is a partially ordered ring $(A,\leq )$ where $\,\leq \,$ is additionally a lattice order.
Properties
The additive group of a partially ordered ring is always a partially ordered group.
The set of non-negative elements of a partially ordered ring (the set of elements $x$ for which $0\leq x,$ also called the positive cone of the ring) is closed under addition and multiplication, that is, if $P$ is the set of non-negative elements of a partially ordered ring, then $P+P\subseteq P$ and $P\cdot P\subseteq P.$ Furthermore, $P\cap (-P)=\{0\}.$
The mapping of the compatible partial order on a ring $A$ to the set of its non-negative elements is one-to-one;[1] that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists.
If $S\subseteq A$ is a subset of a ring $A,$ and:
1. $0\in S$
2. $S\cap (-S)=\{0\}$
3. $S+S\subseteq S$
4. $S\cdot S\subseteq S$
then the relation $\,\leq \,$ where $x\leq y$ if and only if $y-x\in S$ defines a compatible partial order on $A$ (that is, $(A,\leq )$ is a partially ordered ring).[2]
In any l-ring, the absolute value $|x|$ of an element $x$ can be defined to be $x\vee (-x),$ where $x\vee y$ denotes the maximal element. For any $x$ and $y,$
$|x\cdot y|\leq |x|\cdot |y|$
holds.[3]
f-rings
An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring $(A,\leq )$ in which $x\wedge y=0$[4] and $0\leq z$ imply that $zx\wedge y=xz\wedge y=0$ for all $x,y,z\in A.$ They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled "Lattice-ordered rings", in an attempt to restrict the class of l-rings so as to eliminate a number of pathological examples. For example, Birkhoff and Pierce demonstrated an l-ring with 1 in which 1 is not positive, even though it is a square.[2] The additional hypothesis required of f-rings eliminates this possibility.
Example
Let $X$ be a Hausdorff space, and ${\mathcal {C}}(X)$ be the space of all continuous, real-valued functions on $X.$ ${\mathcal {C}}(X)$ is an Archimedean f-ring with 1 under the following pointwise operations:
$[f+g](x)=f(x)+g(x)$
$[fg](x)=f(x)\cdot g(x)$
$[f\wedge g](x)=f(x)\wedge g(x).$
[2]
From an algebraic point of view the rings ${\mathcal {C}}(X)$ are fairly rigid. For example, localisations, residue rings or limits of rings of the form ${\mathcal {C}}(X)$ are not of this form in general. A much more flexible class of f-rings containing all rings of continuous functions and resembling many of the properties of these rings is the class of real closed rings.
Properties
• A direct product of f-rings is an f-ring, an l-subring of an f-ring is an f-ring, and an l-homomorphic image of an f-ring is an f-ring.[3]
• $|xy|=|x||y|$ in an f-ring.[3]
• The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.[5]
• Every ordered ring is an f-ring, so every sub-direct union of ordered rings is also an f-ring. Assuming the axiom of choice, a theorem of Birkhoff shows the converse, and that an l-ring is an f-ring if and only if it is l-isomorphic to a sub-direct union of ordered rings.[2] Some mathematicians take this to be the definition of an f-ring.[3]
Formally verified results for commutative ordered rings
IsarMathLib, a library for the Isabelle theorem prover, has formal verifications of a few fundamental results on commutative ordered rings. The results are proved in the ring1 context.[6]
Suppose $(A,\leq )$ is a commutative ordered ring, and $x,y,z\in A.$ Then:
by
The additive group of $A$ is an ordered group OrdRing_ZF_1_L4
$x\leq y{\text{ if and only if }}x-y\leq 0$ OrdRing_ZF_1_L7
$x\leq y$ and $0\leq z$ imply
$xz\leq yz$ and $zx\leq zy$
OrdRing_ZF_1_L9
$0\leq 1$ ordring_one_is_nonneg
$|xy|=|x||y|$ OrdRing_ZF_2_L5
$|x+y|\leq |x|+|y|$ ord_ring_triangle_ineq
$x$ is either in the positive set, equal to 0 or in minus the positive set. OrdRing_ZF_3_L2
The set of positive elements of $(A,\leq )$ is closed under multiplication if and only if $A$ has no zero divisors. OrdRing_ZF_3_L3
If $A$ is non-trivial ($0\neq 1$), then it is infinite. ord_ring_infinite
See also
• Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
• Ordered field – Algebraic object with an ordered structure
• Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Ordered topological vector space
• Ordered vector space – Vector space with a partial order
• Partially ordered space – Partially ordered topological space
• Riesz space – Partially ordered vector space, ordered as a lattice
References
1. Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7.
2. Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389.
3. Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez (ed.). Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8.
4. $\wedge $ denotes infimum.
5. Hager, Anthony W.; Jorge Martinez (2002). "Functorial rings of quotients—III: The maximum in Archimedean f-rings". Journal of Pure and Applied Algebra. 169: 51–69. doi:10.1016/S0022-4049(01)00060-3.
6. "IsarMathLib" (PDF). Retrieved 2009-03-31.
Further reading
• Birkhoff, G.; R. Pierce (1956). "Lattice-ordered rings". Anais da Academia Brasileira de Ciências. 28: 41–69.
• Gillman, Leonard; Jerison, Meyer Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp
External links
• "Ordered ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Partially Ordered Ring at PlanetMath.
| Wikipedia |
Totally bounded space
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space).
The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.
In metric spaces
A metric space $(M,d)$ is totally bounded if and only if for every real number $\varepsilon >0$, there exists a finite collection of open balls of radius $\varepsilon $ whose centers lie in M and whose union contains M. Equivalently, the metric space M is totally bounded if and only if for every $\varepsilon >0$, there exists a finite cover such that the radius of each element of the cover is at most $\varepsilon $. This is equivalent to the existence of a finite ε-net.[1] A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded.[2]
Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded). The reverse is true for subsets of Euclidean space (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded:[3] every discrete ball of radius $\varepsilon =1/2$ or less is a singleton, and no finite union of singletons can cover an infinite set.
Uniform (topological) spaces
A metric appears in the definition of total boundedness only to ensure that each element of the finite cover is of comparable size, and can be weakened to that of a uniform structure. A subset S of a uniform space X is totally bounded if and only if, for any entourage E, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, E replaces the "size" ε, and a subset is of size E if its Cartesian square is a subset of E.)[4]
The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchy completion: a space is totally bounded if and only if its (Cauchy) completion is compact.
Examples and elementary properties
• Every compact set is totally bounded, whenever the concept is defined.
• Every totally bounded set is bounded.
• A subset of the real line, or more generally of finite-dimensional Euclidean space, is totally bounded if and only if it is bounded.[5][3]
• The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded (in the norm topology) if and only if the space has finite dimension.
• Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem.
• A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.[3]
• The closure of a totally bounded subset is again totally bounded.[6]
Comparison with compact sets
In metric spaces, a set is compact if and only if it is complete and totally bounded;[5] without the axiom of choice only the forward direction holds. Precompact sets share a number of properties with compact sets.
• Like compact sets, a finite union of totally bounded sets is totally bounded.
• Unlike compact sets, every subset of a totally bounded set is again totally bounded.
• The continuous image of a compact set is compact. The uniformly continuous image of a precompact set is precompact.
In topological groups
Although the notion of total boundedness is closely tied to metric spaces, the greater algebraic structure of topological groups allows one to trade away some separation properties. For example, in metric spaces, a set is compact if and only if complete and totally bounded. Under the definition below, the same holds for any topological vector space (not necessarily Hausdorff nor complete).[6][7][8]
The general logical form of the definition is: a subset $S$ of a space $X$ is totally bounded if and only if, given any size $E,$ there exists a finite cover ${\mathcal {O}}$ of $S$ such that each element of ${\mathcal {O}}$ has size at most $E.$ $X$ is then totally bounded if and only if it is totally bounded when considered as a subset of itself.
We adopt the convention that, for any neighborhood $U\subseteq X$ of the identity, a subset $S\subseteq X$ is called (left) $U$-small if and only if $(-S)+S\subseteq U.$[6] A subset $S$ of a topological group $X$ is (left) totally bounded if it satisfies any of the following equivalent conditions:
1. Definition: For any neighborhood $U$ of the identity $0,$ there exist finitely many $x_{1},\ldots ,x_{n}\in X$ such that $ S\subseteq \bigcup _{j=1}^{n}\left(x_{j}+U\right):=\left(x_{1}+U\right)+\cdots +\left(x_{n}+U\right).$
2. For any neighborhood $U$ of $0,$ there exists a finite subset $F\subseteq X$ such that $S\subseteq F+U$ (where the right hand side is the Minkowski sum $F+U:=\{f+u:f\in F,u\in U\}$).
3. For any neighborhood $U$ of $0,$ there exist finitely many subsets $B_{1},\ldots ,B_{n}$ of $X$ such that $S\subseteq B_{1}\cup \cdots \cup B_{n}$ and each $B_{j}$ is $U$-small.[6]
4. For any given filter subbase ${\mathcal {B}}$ of the identity element's neighborhood filter ${\mathcal {N}}$ (which consists of all neighborhoods of $0$ in $X$) and for every $B\in {\mathcal {B}},$ there exists a cover of $S$ by finitely many $B$-small subsets of $X.$[6]
5. $S$ is Cauchy bounded: for every neighborhood $U$ of the identity and every countably infinite subset $I$ of $S,$ there exist distinct $x,y\in I$ such that $x-y\in U.$[6] (If $S$ is finite then this condition is satisfied vacuously).
6. Any of the following three sets satisfies (any of the above definitions of) being (left) totally bounded:
1. The closure ${\overline {S}}=\operatorname {cl} _{X}S$ of $S$ in $X.$[6]
• This set being in the list means that the following characterization holds: $S$ is (left) totally bounded if and only if $\operatorname {cl} _{X}S$ is (left) totally bounded (according to any of the defining conditions mentioned above). The same characterization holds for the other sets listed below.
2. The image of $S$ under the canonical quotient $X\to X/{\overline {\{0\}}},$ which is defined by $x\mapsto x+{\overline {\{0\}}}$ (where $0$ is the identity element).
3. The sum $S+\operatorname {cl} _{X}\{0\}.$[9]
The term pre-compact usually appears in the context of Hausdorff topological vector spaces.[10][11] In that case, the following conditions are also all equivalent to $S$ being (left) totally bounded:
1. In the completion ${\widehat {X}}$ of $X,$ the closure $\operatorname {cl} _{\widehat {X}}S$ of $S$ is compact.[10][12]
2. Every ultrafilter on $S$ is a Cauchy filter.
The definition of right totally bounded is analogous: simply swap the order of the products.
Condition 4 implies any subset of $\operatorname {cl} _{X}\{0\}$ is totally bounded (in fact, compact; see § Comparison with compact sets above). If $X$ is not Hausdorff then, for example, $\{0\}$ is a compact complete set that is not closed.[6]
Topological vector spaces
See also: Topological vector spaces § Properties
Any topological vector space is an abelian topological group under addition, so the above conditions apply. Historically, definition 1(b) was the first reformulation of total boundedness for topological vector spaces; it dates to a 1935 paper of John von Neumann.[13]
This definition has the appealing property that, in a locally convex space endowed with the weak topology, the precompact sets are exactly the bounded sets.
For separable Banach spaces, there is a nice characterization of the precompact sets (in the norm topology) in terms of weakly convergent sequences of functionals: if $X$ is a separable Banach space, then $S\subseteq X$ is precompact if and only if every weakly convergent sequence of functionals converges uniformly on $S.$[14]
Interaction with convexity
• The balanced hull of a totally bounded subset of a topological vector space is again totally bounded.[6][15]
• The Minkowski sum of two compact (totally bounded) sets is compact (resp. totally bounded).
• In a locally convex (Hausdorff) space, the convex hull and the disked hull of a totally bounded set $K$ is totally bounded if and only if $K$ is complete.[16]
See also
• Compact space
• Locally compact space
• Measure of non-compactness
• Orthocompact space
• Paracompact space
• Relatively compact subspace
References
1. Sutherland 1975, p. 139.
2. "Cauchy sequences, completeness, and a third formulation of compactness" (PDF). Harvard Mathematics Department.
3. Willard 2004, p. 182.
4. Willard, Stephen (1970). Loomis, Lynn H. (ed.). General topology. Reading, Mass.: Addison-Wesley. p. 262. C.f. definition 39.7 and lemma 39.8.
5. Kolmogorov, A. N.; Fomin, S. V. (1957) [1954]. Elements of the theory of functions and functional analysis,. Vol. 1. Translated by Boron, Leo F. Rochester, N.Y.: Graylock Press. pp. 51–3.
6. Narici & Beckenstein 2011, pp. 47–66.
7. Narici & Beckenstein 2011, pp. 55–56.
8. Narici & Beckenstein 2011, pp. 55–66.
9. Schaefer & Wolff 1999, pp. 12–35.
10. Schaefer & Wolff 1999, p. 25.
11. Trèves 2006, p. 53.
12. Jarchow 1981, pp. 56–73.
13. von Neumann, John (1935). "On Complete Topological Spaces". Transactions of the American Mathematical Society. 37 (1): 1–20. doi:10.2307/1989693. ISSN 0002-9947.
14. Phillips, R. S. (1940). "On Linear Transformations". Annals of Mathematics: 525.
15. Narici & Beckenstein 2011, pp. 156–175.
16. Narici & Beckenstein 2011, pp. 67–113.
Bibliography
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Sutherland, W. A. (1975). Introduction to metric and topological spaces. Oxford University Press. ISBN 0-19-853161-3. Zbl 0304.54002.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
| Wikipedia |
Strong orientation
In graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that makes it into a strongly connected graph.
Strong orientations have been applied to the design of one-way road networks. According to Robbins' theorem, the graphs with strong orientations are exactly the bridgeless graphs. Eulerian orientations and well-balanced orientations provide important special cases of strong orientations; in turn, strong orientations may be generalized to totally cyclic orientations of disconnected graphs. The set of strong orientations of a graph forms a partial cube, with adjacent orientations in this structure differing in the orientation of a single edge. It is possible to find a single orientation in linear time, but it is #P-complete to count the number of strong orientations of a given graph.
Application to traffic control
Robbins (1939) introduces the problem of strong orientation with a story about a town, whose streets and intersections are represented by the given graph G. According to Robbins' story, the people of the town want to be able to repair any segment of road during the weekdays, while still allowing any part of the town to be reached from any other part using the remaining roads as two-way streets. On the weekends, all roads are open, but because of heavy traffic volume, they wish to convert all roads to one-way streets and again allow any part of town to be reached from any other part. Robbins' theorem states that a system of roads is suitable for weekday repairs if and only if it is suitable for conversion to a one-way system on weekends. For this reason, his result is sometimes known as the one-way street theorem.[1]
Subsequently to the work of Robbins, a series of papers by Roberts and Xu modeled more carefully the problem of turning a grid of two-way city streets into one-way streets, and examined the effect of this conversion on the distances between pairs of points within the grid. As they showed, the traditional one-way layout in which parallel streets alternate in direction is not optimal in keeping the pairwise distances as small as possible. However, the improved orientations that they found include points where the traffic from two one-way blocks meets itself head-on, which may be viewed as a flaw in their solutions.
Related types of orientation
If an undirected graph has an Euler tour, an Eulerian orientation of the graph (an orientation for which every vertex has indegree equal to its outdegree) may be found by orienting the edges consistently around the tour.[2] These orientations are automatically strong orientations.
A theorem of Nash-Williams (1960, 1969) states that every undirected graph G has a well-balanced orientation. This is an orientation with the property that, for every pair of vertices u and v in G, the number of pairwise edge-disjoint directed paths from u to v in the resulting directed graph is at least $\left\lfloor {\frac {k}{2}}\right\rfloor $, where k is the maximum number of paths in a set of edge-disjoint undirected paths from u to v. Nash-Williams' orientations also have the property that they are as close as possible to being Eulerian orientations: at each vertex, the indegree and the outdegree are within one of each other. The existence of well-balanced orientations, together with Menger's theorem, immediately implies Robbins' theorem: by Menger's theorem, a 2-edge-connected graph has at least two edge-disjoint paths between every pair of vertices, from which it follows that any well-balanced orientation must be strongly connected. More generally this result implies that every 2k-edge-connected undirected graph can be oriented to form a k-edge-connected directed graph.
A totally cyclic orientation of a graph G is an orientation in which each edge belongs to a directed cycle. For connected graphs, this is the same thing as a strong orientation, but totally cyclic orientations may also be defined for disconnected graphs, and are the orientations in which each connected component of G becomes strongly connected. Robbins' theorem can be restated as saying that a graph has a totally cyclic orientation if and only if it does not have a bridge. Totally cyclic orientations are dual to acyclic orientations (orientations that turn G into a directed acyclic graph) in the sense that, if G is a planar graph, and orientations of G are transferred to orientations of the planar dual graph of G by turning each edge 90 degrees clockwise, then a totally cyclic orientation of G corresponds in this way to an acyclic orientation of the dual graph and vice versa.[3][4] The number of different totally cyclic orientations of any graph G is TG(0,2) where TG is the Tutte polynomial of the graph, and dually the number of acyclic orientations is TG(2,0).[5] As a consequence, Robbins' theorem implies that the Tutte polynomial has a root at the point (0,2) if and only if the graph G has a bridge.
If a strong orientation has the property that all directed cycles pass through a single edge st (equivalently, if flipping the orientation of an edge produces an acyclic orientation) then the acyclic orientation formed by reversing st is a bipolar orientation. Every bipolar orientation is related to a strong orientation in this way.[6]
Flip graphs
If G is a 3-edge-connected graph, and X and Y are any two different strong orientations of G, then it is possible to transform X into Y by changing the orientation of a single edge at a time, at each step preserving the property that the orientation is strong.[7] Therefore, the flip graph whose vertices correspond to the strong orientations of G, and whose edges correspond to pairs of strong orientations that differ in the direction of a single edge, forms a partial cube.
Algorithms and complexity
A strong orientation of a given bridgeless undirected graph may be found in linear time by performing a depth-first search of the graph, orienting all edges in the depth-first search tree away from the tree root, and orienting all the remaining edges (which must necessarily connect an ancestor and a descendant in the depth-first search tree) from the descendant to the ancestor.[8] If an undirected graph G with bridges is given, together with a list of ordered pairs of vertices that must be connected by directed paths, it is possible in polynomial time to find an orientation of G that connects all the given pairs, if such an orientation exists. However, the same problem is NP-complete when the input may be a mixed graph.[9]
It is #P-complete to count the number of strong orientations of a given graph G, even when G is planar and bipartite.[3][10] However, for dense graphs (more specifically, graphs in which each vertex has a linear number of neighbors), the number of strong orientations may be estimated by a fully polynomial-time randomized approximation scheme.[3][11] The problem of counting strong orientations may also be solved exactly, in polynomial time, for graphs of bounded treewidth.[3]
Notes
1. Koh & Tay (2002).
2. Schrijver (1983).
3. Welsh (1997).
4. Noy (2001).
5. Las Vergnas (1980).
6. de Fraysseix, Ossona de Mendez & Rosenstiehl (1995).
7. Fukuda, Prodon & Sakuma (2001).
8. See e.g. Atallah (1984) and Roberts (1978).
9. Arkin & Hassin (2002).
10. Vertigan & Welsh (1992).
11. Alon, Frieze & Welsh (1995).
References
• Alon, Noga; Frieze, Alan; Welsh, Dominic (1995), "Polynomial time randomized approximation schemes for Tutte-Gröthendieck invariants: the dense case", Random Structures & Algorithms, 6 (4): 459–478, doi:10.1002/rsa.3240060409, MR 1368847
• Arkin, Esther M.; Hassin, Refael (2002), "A note on orientations of mixed graphs" (PDF), Discrete Applied Mathematics, 116 (3): 271–278, doi:10.1016/S0166-218X(01)00228-1, MR 1878572.
• Atallah, Mikhail J. (1984), "Parallel strong orientation of an undirected graph", Information Processing Letters, 18 (1): 37–39, doi:10.1016/0020-0190(84)90072-3, MR 0742079.
• de Fraysseix, Hubert; Ossona de Mendez, Patrice; Rosenstiehl, Pierre (1995), "Bipolar orientations revisited", Discrete Applied Mathematics, 56 (2–3): 157–179, doi:10.1016/0166-218X(94)00085-R, MR 1318743.
• Fukuda, Komei; Prodon, Alain; Sakuma, Tadashi (2001), "Notes on acyclic orientations and the shelling lemma", Theoretical Computer Science, 263 (1–2): 9–16, doi:10.1016/S0304-3975(00)00226-7, MR 1846912.
• Koh, K. M.; Tay, E. G. (2002), "Optimal orientations of graphs and digraphs: a survey", Graphs and Combinatorics, 18 (4): 745–756, doi:10.1007/s003730200060, MR 1964792, S2CID 34821155.
• Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10.1016/0095-8956(80)90082-9, MR 0586435.
• Nash-Williams, C. St. J. A. (1960), "On orientations, connectivity and odd-vertex-pairings in finite graphs.", Canadian Journal of Mathematics, 12: 555–567, doi:10.4153/cjm-1960-049-6, MR 0118684.
• Nash-Williams, C. St. J. A. (1969), "Well-balanced orientations of finite graphs and unobtrusive odd-vertex-pairings", Recent Progress in Combinatorics (Proc. Third Waterloo Conf. on Combinatorics, 1968), New York: Academic Press, pp. 133–149, MR 0253933.
• Noy, Marc (2001), "Acyclic and totally cyclic orientations in planar graphs", The American Mathematical Monthly, 108 (1): 66–68, doi:10.2307/2695680, JSTOR 2695680, MR 1857074.
• Robbins, H. E. (1939), "A theorem on graphs, with an application to a problem on traffic control", American Mathematical Monthly, 46 (5): 281–283, doi:10.2307/2303897, JSTOR 2303897.
• Roberts, Fred S. (1978), "Chapter 2. The One-Way Street Problem", Graph Theory and its Applications to Problems of Society, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 29, Philadelphia, Pa.: Society for Industrial and Applied Mathematics (SIAM), pp. 7–14, ISBN 9780898710267, MR 0508050.
• Roberts, Fred S.; Xu, Yonghua (1988), "On the optimal strongly connected orientations of city street graphs. I. Large grids", SIAM Journal on Discrete Mathematics, 1 (2): 199–222, doi:10.1137/0401022, MR 0941351.
• Roberts, Fred S.; Xu, Yonghua (1989), "On the optimal strongly connected orientations of city street graphs. II. Two east-west avenues or north-south streets", Networks, 19 (2): 221–233, doi:10.1002/net.3230190204, MR 0984567.
• Roberts, Fred S.; Xu, Yonghua (1992), "On the optimal strongly connected orientations of city street graphs. III. Three east-west avenues or north-south streets", Networks, 22 (2): 109–143, doi:10.1002/net.3230220202, MR 1148018.
• Roberts, Fred S.; Xu, Yong Hua (1994), "On the optimal strongly connected orientations of city street graphs. IV. Four east-west avenues or north-south streets", Discrete Applied Mathematics, 49 (1–3): 331–356, doi:10.1016/0166-218X(94)90217-8, MR 1272496.
• Schrijver, A. (1983), "Bounds on the number of Eulerian orientations" (PDF), Combinatorica, 3 (3–4): 375–380, doi:10.1007/BF02579193, MR 0729790, S2CID 13708977.
• Vertigan, D. L.; Welsh, D. J. A. (1992), "The computational complexity of the Tutte plane: the bipartite case", Combinatorics, Probability and Computing, 1 (2): 181–187, doi:10.1017/S0963548300000195, MR 1179248.
• Welsh, Dominic (1997), "Approximate counting", Surveys in combinatorics, 1997 (London), London Math. Soc. Lecture Note Ser., vol. 241, Cambridge: Cambridge Univ. Press, pp. 287–323, doi:10.1017/CBO9780511662119.010, MR 1477750.
| Wikipedia |
Totally imaginary number field
In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic fields, and, more generally, CM fields.
Any number field that is Galois over the rationals must be either totally real or totally imaginary.
References
• Section 13.1 of Alaca, Şaban; Williams, Kenneth S. (2004), Introductory algebraic number theory, Cambridge University Press, ISBN 978-0-521-54011-7
| Wikipedia |
Indescribable cardinal
In set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by Hanf & Scott (1961).
A cardinal number $\kappa $ is called $\Pi _{m}^{n}$-indescribable if for every $\Pi _{m}$ proposition $\phi $, and set $A\subseteq V_{\kappa }$ with $(V_{\kappa +n},\in ,A)\vDash \phi $ there exists an $\alpha <\kappa $ with $(V_{\alpha +n},\in ,A\cap V_{\alpha })\vDash \phi $.[1] Following Lévy's hierarchy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. $\Sigma _{m}^{n}$-indescribable cardinals are defined in a similar way, but with an outermost existential quantifier. Prior to defining the structure $(V_{\kappa +n},\in ,A)$, one new predicate symbol is added to the language of set theory, which is interpreted as $A$.[2]p.295 The idea is that $\kappa $ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties.
The cardinal number $\kappa $ is called totally indescribable if it is $\Pi _{m}^{n}$-indescribable for all positive integers m and n.
If $\alpha $ is an ordinal, the cardinal number $\kappa $ is called $\alpha $-indescribable if for every formula $\phi $ and every subset $U$ of $V_{\kappa }$ such that $\phi (U)$ holds in $V_{\kappa +\alpha }$ there is a some $\lambda <\kappa $ such that $\phi (U\cap V_{\lambda })$ holds in $V_{\lambda +\alpha }$. If $\alpha $ is infinite then $\alpha $-indescribable ordinals are totally indescribable, and if $\alpha $ is finite they are the same as $\Pi _{\omega }^{\alpha }$-indescribable ordinals. There is no $\alpha $ that is $\alpha $-indescribable, nor does $\alpha $-indescribability necessarily imply $\beta $-indescribability for any $\beta <\alpha $, but there is an alternative notion of shrewd cardinals that makes sense when $\alpha \geq \kappa $: there is $\lambda <\kappa $ and $\beta $ such that $\phi (U\cap V_{\lambda })$ holds in $V_{\lambda +\beta }$.[3]
Historical note
Originally, a cardinal κ was called Q-indescribable if for every Q-formula $\phi $ and relation $A$, if $(\kappa ,<,A)\vDash \phi $ then there exists an $\alpha <\kappa $ such that $(\alpha ,\in ,A\upharpoonright \alpha )\vDash \phi $.[4][5] Using this definition, $\kappa $ is $\Pi _{0}^{1}$-indescribable iff $\kappa $ is regular and greater than $\aleph _{0}$.[5]p.207 The cardinals $\kappa $ satisfying the above version based on the cumulative hierarchy were called strongly Q-indescribable.[6]
Equivalent conditions
A cardinal is $\Sigma _{n+1}^{1}$-indescribable iff it is $\Pi _{n}^{1}$-indescribable.[7] A cardinal is inaccessible if and only if it is $\Pi _{n}^{0}$-indescribable for all positive integers $n$, equivalently iff it is $\Pi _{2}^{0}$-indescribable, equivalently if it is $\Sigma _{1}^{1}$-indescribable.
$\Pi _{1}^{1}$-indescribable cardinals are the same as weakly compact cardinals.
The indescribability condition is equivalent to $V_{\kappa }$ satisfying the reflection principle (which is provable in ZFC), but extended by allowing higher-order formulae with a second-order free variable. [8]
If V=L, then for a natural number n>0, an uncountable cardinal is Π1
n
-indescribable iff it's (n+1)-stationary.[9]
Enforceable classes
For a class $X$ of ordinals and a $\Gamma $-indescribable cardinal $\kappa $, $X$ is said to be enforced at $\alpha $ (by some formula $\phi $ of $\Gamma $) if there is a $\Gamma $-formula $\phi $ and an $A\subseteq V_{\kappa }$ such that $(V_{\kappa },\in ,A)\vDash \phi $, but for no $\beta <\alpha $ with $\beta \notin X$ does $(V_{\beta },\in ,A\cap V_{\beta })\vDash \phi $ hold.[1]p.277 This gives a tool to show necessary properties of indescribable cardinals.
Properties
The property of $\kappa $ being $\Pi _{n}^{1}$-indescribable is $\Pi _{n+1}^{1}$ over $V_{\kappa }$, i.e. there is a $\Pi _{n+1}^{1}$ sentence that $V_{\kappa }$ satisfies iff $\kappa $ is $\Pi _{n}^{1}$-indescribable.[10] For $m>1$, the property of being $\Pi _{n}^{m}$-indescribable is $\Sigma _{n}^{m}$ and the property of being $\Sigma _{n}^{m}$-indescribable is $\Pi _{n}^{m}$.[10] Thus, for $m>1$, every cardinal that is either $\Pi _{n+1}^{m}$-indescribable or $\Sigma _{n+1}^{m}$-indescribable is both $\Pi _{n}^{m}$-indescribable and $\Sigma _{n}^{m}$-indescribable and the set of such cardinals below it is stationary. The consistency strength of $\Sigma _{n}^{m}$-indescribable cardinals is below that of $\Pi _{n}^{m}$-indescribable, but for $m>1$ it is consistent with ZFC that the least $\Sigma _{n}^{m}$-indescribable exists and is above the least $\Pi _{n}^{m}$-indescribable cardinal (this is proved from consistency of ZFC with $\Pi _{n}^{m}$-indescribable cardinal and a $\Sigma _{n}^{m}$-indescribable cardinal above it).
Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for $\Pi _{n}^{m}$- and $\Sigma _{n}^{m}$-indescribability.
Measurable cardinals are $\Pi _{1}^{2}$-indescribable, but the smallest measurable cardinal is not $\Sigma _{1}^{2}$-indescribable. However, assuming choice, there are many totally indescribable cardinals below any measurable cardinal.
For $n\geq 1$, ZFC+"there is a $\Sigma _{n}^{1}$-indescribable cardinal" is equiconsistent with ZFC+"there is a $\Sigma _{n}^{1}$-indescribable cardinal $\kappa $ such that $2^{\kappa }>\kappa ^{+}$", i.e. "GCH fails at a $\Sigma _{n}^{1}$-indescribable cardinal".[7]
References
1. Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
2. T. Jech, 'Set Theory: The Third Millennium Edition, revised and expanded'. Springer Monographs in Mathematics (2006). ISBN 3-540-44085-2.
3. M. Rathjen, "The Higher Infinite in Proof Theory" (1995), p.20. Accessed 28 July 2023.
4. K. Kunen, "Indescribability and the Continuum" (1971). Appearing in Axiomatic Set Theory: Proceedings of Symposia in Pure Mathematics, vol. 13 part 1, pp.199--203
5. Azriel Lévy, "The Sizes of the Indescribable Cardinals" (1971). Appearing in Axiomatic Set Theory: Proceedings of Symposia in Pure Mathematics, vol. 13 part 1, pp.205--218
6. W. Richter, P. Aczel, "Inductive Definitions and Reflecting Properties of Admissible Ordinals" (1974)
7. K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of Symbolic Logic vol. 56, iss. 2 (1991), pp.439--457.
8. K. Hauser, "Indescribable Cardinals and Elementary Embeddings". Journal of Symbolic Logic vol. 56, no. 2 (1991), pp.439--457
9. Bagaria, Magidor, Mancilla. "On the Consistency Strength of Hyperstationarity", p.3. (2019)
10. Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. p. 64. doi:10.1007/978-3-540-88867-3_2. ISBN 3-540-00384-3.
• Hanf, W. P.; Scott, D. S. (1961), "Classifying inaccessible cardinals", Notices of the American Mathematical Society, 8: 445, ISSN 0002-9920
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. doi:10.1007/978-3-540-88867-3_2. ISBN 3-540-00384-3.
| Wikipedia |
Linearly ordered group
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. This may have different meanings. We say that (G, ≤) is a:
• left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G,
• right-ordered group if ≤ is right-invariant, that is a ≤ b implies ac ≤ bc for all a, b, c in G,
• bi-ordered group if ≤ is bi-invariant, that is it is both left- and right-invariant.
A group G is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on G. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be left- or right-orderable; however there exist left-orderable groups which are not bi-orderable.
Further definitions
In this section $\leq $ is a left-invariant order on a group $G$ with identity element $e$. All that is said applies to right-invariant orders with the obvious modifications. Note that $\leq $ being left-invariant is equivalent to the order $\leq '$ defined by $g\leq 'h$ if and only if $h^{-1}\leq g^{-1}$ being right-invariant. In particular a group being left-orderable is the same as it being right-orderable.
In analogy with ordinary numbers we call an element $g\not =e$ of an ordered group positive if $e\leq g$. The set of positive elements in an ordered group is called the positive cone, it is often denoted with $G_{+}$; the slightly different notation $G^{+}$ is used for the positive cone together with the identity element.[1]
The positive cone $G_{+}$ characterises the order $\leq $; indeed, by left-invariance we see that $g\leq h$ if and only if $g^{-1}h\in G_{+}$. In fact a left-ordered group can be defined as a group $G$ together with a subset $P$ satisfying the two conditions that:
1. for $g,h\in P$ we have also $gh\in P$;
2. let $P^{-1}=\{g^{-1},g\in P\}$, then $G$ is the disjoint union of $P,P^{-1}$ and $\{e\}$.
The order $\leq _{P}$ associated with $P$ is defined by $g\leq _{P}h\Leftrightarrow g^{-1}h\in P$; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of $\leq _{P}$ is $P$.
The left-invariant order $\leq $ is bi-invariant if and only if it is conjugacy invariant, that is if $g\leq h$ then for any $x\in G$ we have $xgx^{-1}\leq xhx^{-1}$ as well. This is equivalent to the positive cone being stable under inner automorphisms.
If $a\in G$, then the absolute value of $a$, denoted by $|a|$, is defined to be:
$|a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\-a,&{\text{otherwise}}.\end{cases}}$
If in addition the group $G$ is abelian, then for any $a,b\in G$ a triangle inequality is satisfied: $|a+b|\leq |a|+|b|$.
Examples
Any left- or right-orderable group is torsion-free, that is it contains no elements of finite order besides the identity. Conversely, F. W. Levi showed that a torsion-free abelian group is bi-orderable;[2] this is still true for nilpotent groups[3] but there exist torsion-free, finitely presented groups which are not left-orderable.
Archimedean ordered groups
Otto Hölder showed that every Archimedean group (a bi-ordered group satisfying an Archimedean property) is isomorphic to a subgroup of the additive group of real numbers, (Fuchs & Salce 2001, p. 61). If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, ${\widehat {G}}$ of the closure of a l.o. group under $n$th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each $g\in {\widehat {G}}$ the exponential maps $g^{\cdot }:(\mathbb {R} ,+)\to ({\widehat {G}},\cdot ):\lim _{i}q_{i}\in \mathbb {Q} \mapsto \lim _{i}g^{q_{i}}$ are well defined order preserving/reversing, topological group isomorphisms. Completing a l.o. group can be difficult in the non-Archimedean case. In these cases, one may classify a group by its rank: which is related to the order type of the largest sequence of convex subgroups.
Other examples
Free groups are left-orderable. More generally this is also the case for right-angled Artin groups.[4] Braid groups are also left-orderable.[5]
The group given by the presentation $\langle a,b|a^{2}ba^{2}b^{-1},b^{2}ab^{2}a^{-1}\rangle $ is torsion-free but not left-orderable;[6] note that it is a 3-dimensional crystallographic group (it can be realised as the group generated by two glided half-turns with orthogonal axes and the same translation length), and it is the same group that was proven to be a counterexample to the unit conjecture. More generally the topic of orderability of 3--manifold groups is interesting for its relation with various topological invariants.[7] There exists a 3-manifold group which is left-orderable but not bi-orderable[8] (in fact it does not satisfy the weaker property of being locally indicable).
Left-orderable groups have also attracted interest from the perspective of dynamical systems cas it is known that a countable group is left-orderable if and only if it acts on the real line by homeomorphisms.[9] Non-examples related to this paradigm are lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in $\mathrm {SL} _{n}(\mathbb {Z} )$ are not left-orderable;[10] a wide generalisation of this has been recently announced.[11]
See also
• Cyclically ordered group
• Hahn embedding theorem
• Partially ordered group
Notes
1. Deroin, Navas & Rivas 2014, 1.1.1.
2. Levi 1942.
3. Deroin, Navas & Rivas 2014, 1.2.1.
4. Duchamp, Gérard; Thibon, Jean-Yves (1992). "Simple orderings for free partially commutative groups". International Journal of Algebra and Computation. 2 (3): 351–355. doi:10.1142/S0218196792000219. Zbl 0772.20017.
5. Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert (2002). Why are braids orderable?. Paris: Société Mathématique de France. p. xiii + 190. ISBN 2-85629-135-X.
6. Deroin, Navas & Rivas 2014, 1.4.1.
7. Boyer, Steven; Rolfsen, Dale; Wiest, Bert (2005). "Orderable 3-manifold groups". Annales de l'Institut Fourier. 55 (1): 243–288. doi:10.5802/aif.2098. Zbl 1068.57001.
8. Bergman, George (1991). "Right orderable groups that are not locally indicable". Pacific Journal of Mathematics. 147 (2): 243–248. doi:10.2140/pjm.1991.147.243. Zbl 0677.06007.
9. Deroin, Navas & Rivas 2014, Proposition 1.1.8.
10. Witte, Dave (1994). "Arithmetic groups of higher \(\mathbb{Q}\)-rank cannot act on \(1\)-manifolds". Proceedings of the American Mathematical Society. 122 (2): 333–340. doi:10.2307/2161021. JSTOR 2161021. Zbl 0818.22006.
11. Deroin, Bertrand; Hurtado, Sebastian (2020). "Non left-orderability of lattices in higher rank semi-simple Lie groups". arXiv:2008.10687 [math.GT].
References
• Deroin, Bertrand; Navas, Andrés; Rivas, Cristóbal (2014). "Groups, orders and dynamics". arXiv:1408.5805 [math.GT].
• Levi, F.W. (1942), "Ordered groups.", Proc. Indian Acad. Sci., A16 (4): 256–263, doi:10.1007/BF03174799, S2CID 198139979
• Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715
• Ghys, É. (2001), "Groups acting on the circle.", L'Enseignement Mathématique, 47: 329–407
| Wikipedia |
Ordered ring
In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R:[1]
• if a ≤ b then a + c ≤ b + c.
• if 0 ≤ a and 0 ≤ b then 0 ≤ ab.
Examples
Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers.[2] (The rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and i.
Positive elements
In analogy with the real numbers, we call an element c of an ordered ring R positive if 0 < c, and negative if c < 0. 0 is considered to be neither positive nor negative.
The set of positive elements of an ordered ring R is often denoted by R+. An alternative notation, favored in some disciplines, is to use R+ for the set of nonnegative elements, and R++ for the set of positive elements.
Absolute value
If $a$ is an element of an ordered ring R, then the absolute value of $a$, denoted $|a|$, is defined thus:
$|a|:={\begin{cases}a,&{\mbox{if }}0\leq a,\\-a,&{\mbox{otherwise}},\end{cases}}$
where $-a$ is the additive inverse of $a$ and 0 is the additive identity element.
Discrete ordered rings
A discrete ordered ring or discretely ordered ring is an ordered ring in which there is no element between 0 and 1. The integers are a discrete ordered ring, but the rational numbers are not.
Basic properties
For all a, b and c in R:
• If a ≤ b and 0 ≤ c, then ac ≤ bc.[3] This property is sometimes used to define ordered rings instead of the second property in the definition above.
• |ab| = |a| |b|.[4]
• An ordered ring that is not trivial is infinite.[5]
• Exactly one of the following is true: a is positive, -a is positive, or a = 0.[6] This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
• In an ordered ring, no negative element is a square:[7] Firstly, 0 is square. Now if a ≠ 0 and a = b2 then b ≠ 0 and a = (-b)2; as either b or -b is positive, a must be nonnegative.
See also
• Ordered field – Algebraic object with an ordered structure
• Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Ordered topological vector space
• Ordered vector space – Vector space with a partial order
• Partially ordered ring – Ring with a compatible partial order
• Partially ordered space – Partially ordered topological space
• Riesz space – Partially ordered vector space, ordered as a lattice, also called vector lattice
• Ordered semirings
Notes
The list below includes references to theorems formally verified by the IsarMathLib project.
1. Lam, T. Y. (1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, American Mathematical Society, ISBN 0-8218-0702-1, Zbl 0516.12001
• Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2nd ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439, Zbl 0980.16001
2. OrdRing_ZF_1_L9
3. OrdRing_ZF_2_L5
4. ord_ring_infinite
5. OrdRing_ZF_3_L2, see also OrdGroup_decomp
6. OrdRing_ZF_1_L12
| Wikipedia |
Totally positive matrix
In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number.[1] A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero). Some authors use "totally positive" to include all totally non-negative matrices.
Not to be confused with Positive matrix and Positive-definite matrix.
Definition
Let $\mathbf {A} =(A_{ij})_{ij}$ be an n × n matrix. Consider any $p\in \{1,2,\ldots ,n\}$ and any p × p submatrix of the form $\mathbf {B} =(A_{i_{k}j_{\ell }})_{k\ell }$ where:
$1\leq i_{1}<\ldots <i_{p}\leq n,\qquad 1\leq j_{1}<\ldots <j_{p}\leq n.$
Then A is a totally positive matrix if:[2]
$\det(\mathbf {B} )>0$
for all submatrices $\mathbf {B} $ that can be formed this way.
History
Topics which historically led to the development of the theory of total positivity include the study of:[2]
• the spectral properties of kernels and matrices which are totally positive,
• ordinary differential equations whose Green's function is totally positive (by M. G. Krein and some colleagues in the mid-1930s),
• the variation diminishing properties (started by I. J. Schoenberg in 1930),
• Pólya frequency functions (by I. J. Schoenberg in the late 1940s and early 1950s).
Examples
For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.
See also
• Compound matrix
References
1. George M. Phillips (2003), "Total Positivity", Interpolation and Approximation by Polynomials, Springer, p. 274, ISBN 9780387002156
2. Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus
Further reading
• Allan Pinkus (2009), Totally Positive Matrices, Cambridge University Press, ISBN 9780521194082
External links
• Spectral Properties of Totally Positive Kernels and Matrices, Allan Pinkus
• Parametrizations of Canonical Bases and Totally Positive Matrices, Arkady Berenstein
• Tensor Product Multiplicities, Canonical Bases And Totally Positive Varieties (2001), A. Berenstein , A. Zelevinsky
Matrix classes
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With specific applications
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| Wikipedia |
Totally real number field
In number theory, a number field F is called totally real if for each embedding of F into the complex numbers the image lies inside the real numbers. Equivalent conditions are that F is generated over Q by one root of an integer polynomial P, all of the roots of P being real; or that the tensor product algebra of F with the real field, over Q, is isomorphic to a tensor power of R.
For example, quadratic fields F of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial P irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will not be totally real, although it is a field of real numbers.
The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two.
Any number field that is Galois over the rationals must be either totally real or totally imaginary.
See also
• Totally imaginary number field
• CM-field, a totally imaginary quadratic extension of a totally real field
References
• Hida, Haruzo (1993), Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts, vol. 26, Cambridge University Press, ISBN 978-0-521-43569-7
| Wikipedia |
Cubic field
In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
Definition
If K is a field extension of the rational numbers Q of degree [K:Q] = 3, then K is called a cubic field. Any such field is isomorphic to a field of the form
$\mathbf {Q} [x]/(f(x))$
where f is an irreducible cubic polynomial with coefficients in Q. If f has three real roots, then K is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, f has a non-real root, then K is called a complex cubic field.
A cubic field K is called a cyclic cubic field if it contains all three roots of its generating polynomial f. Equivalently, K is a cyclic cubic field if it is a Galois extension of Q, in which case its Galois group over Q is cyclic of order three. This can only happen if K is totally real. It is a rare occurrence in the sense that if the set of cubic fields is ordered by discriminant, then the proportion of cubic fields which are cyclic approaches zero as the bound on the discriminant approaches infinity.[1]
A cubic field is called a pure cubic field if it can be obtained by adjoining the real cube root ${\sqrt[{3}]{n}}$ of a cube-free positive integer n to the rational number field Q. Such fields are always complex cubic fields since each positive number has two complex non-real cube roots.
Examples
• Adjoining the real cube root of 2 to the rational numbers gives the cubic field $\mathbf {Q} ({\sqrt[{3}]{2}})$. This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in absolute value), namely −108.[2]
• The complex cubic field obtained by adjoining to Q a root of x3 + x2 − 1 is not pure. It has the smallest discriminant (in absolute value) of all cubic fields, namely −23.[3]
• Adjoining a root of x3 + x2 − 2x − 1 to Q yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49.[4]
• The field obtained by adjoining to Q a root of x3 + x2 − 3x − 1 is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field.[5]
• No cyclotomic fields are cubic because the degree of a cyclotomic field is equal to φ(n), where φ is Euler's totient function, which only takes on even values except for φ(1) = φ(2) = 1.
Galois closure
A cyclic cubic field K is its own Galois closure with Galois group Gal(K/Q) isomorphic to the cyclic group of order three. However, any other cubic field K is a non-Galois extension of Q and has a field extension N of degree two as its Galois closure. The Galois group Gal(N/Q) is isomorphic to the symmetric group S3 on three letters.
Associated quadratic field
The discriminant of a cubic field K can be written uniquely as df2 where d is a fundamental discriminant. Then, K is cyclic if and only if d = 1, in which case the only subfield of K is Q itself. If d ≠ 1 then the Galois closure N of K contains a unique quadratic field k whose discriminant is d (in the case d = 1, the subfield Q is sometimes considered as the "degenerate" quadratic field of discriminant 1). The conductor of N over k is f, and f2 is the relative discriminant of N over K. The discriminant of N is d3f4.[6][7]
The field K is a pure cubic field if and only if d = −3. This is the case for which the quadratic field contained in the Galois closure of K is the cyclotomic field of cube roots of unity.[7]
Discriminant
Since the sign of the discriminant of a number field K is (−1)r2, where r2 is the number of conjugate pairs of complex embeddings of K into C, the discriminant of a cubic field will be positive precisely when the field is totally real, and negative if it is a complex cubic field.
Given some real number N > 0 there are only finitely many cubic fields K whose discriminant DK satisfies |DK| ≤ N.[9] Formulae are known which calculate the prime decomposition of DK, and so it can be explicitly calculated.[10]
Unlike quadratic fields, several non-isomorphic cubic fields K1, ..., Km may share the same discriminant D. The number m of these fields is called the multiplicity[11] of the discriminant D. Some small examples are m = 2 for D = −1836, 3969, m = 3 for D = −1228, 22356, m = 4 for D = −3299, 32009, and m = 6 for D = −70956, 3054132.
Any cubic field K will be of the form K = Q(θ) for some number θ that is a root of an irreducible polynomial
$f(X)=X^{3}-aX+b$
where a and b are integers. The discriminant of f is Δ = 4a3 − 27b2. Denoting the discriminant of K by D, the index i(θ) of θ is then defined by Δ = i(θ)2D.
In the case of a non-cyclic cubic field K this index formula can be combined with the conductor formula D = f2d to obtain a decomposition of the polynomial discriminant Δ = i(θ)2f2d into the square of the product i(θ)f and the discriminant d of the quadratic field k associated with the cubic field K, where d is squarefree up to a possible factor 22 or 23. Georgy Voronoy gave a method for separating i(θ) and f in the square part of Δ.[12]
The study of the number of cubic fields whose discriminant is less than a given bound is a current area of research. Let N+(X) (respectively N−(X)) denote the number of totally real (respectively complex) cubic fields whose discriminant is bounded by X in absolute value. In the early 1970s, Harold Davenport and Hans Heilbronn determined the first term of the asymptotic behaviour of N±(X) (i.e. as X goes to infinity).[13][14] By means of an analysis of the residue of the Shintani zeta function, combined with a study of the tables of cubic fields compiled by Karim Belabas (Belabas 1997) and some heuristics, David P. Roberts conjectured a more precise asymptotic formula:[15]
$N^{\pm }(X)\sim {\frac {A_{\pm }}{12\zeta (3)}}X+{\frac {4\zeta ({\frac {1}{3}})B_{\pm }}{5\Gamma ({\frac {2}{3}})^{3}\zeta ({\frac {5}{3}})}}X^{\frac {5}{6}}$
where A± = 1 or 3, B± = 1 or ${\sqrt {3}}$, according to the totally real or complex case, ζ(s) is the Riemann zeta function, and Γ(s) is the Gamma function. Proofs of this formula have been published by Bhargava, Shankar & Tsimerman (2013) using methods based on Bhargava's earlier work, as well as by Taniguchi & Thorne (2013) based on the Shintani zeta function.
Unit group
According to Dirichlet's unit theorem, the torsion-free unit rank r of an algebraic number field K with r1 real embeddings and r2 pairs of conjugate complex embeddings is determined by the formula r = r1 + r2 − 1. Hence a totally real cubic field K with r1 = 3, r2 = 0 has two independent units ε1, ε2 and a complex cubic field K with r1 = r2 = 1 has a single fundamental unit ε1. These fundamental systems of units can be calculated by means of generalized continued fraction algorithms by Voronoi,[16] which have been interpreted geometrically by Delone and Faddeev.[17]
Notes
1. Harvey Cohn computed an asymptotic for the number of cyclic cubic fields (Cohn 1954), while Harold Davenport and Hans Heilbronn computed the asymptotic for all cubic fields (Davenport & Heilbronn 1971).
2. Cohen 1993, §B.3 contains a table of complex cubic fields
3. Cohen 1993, §B.3
4. Cohen 1993, §B.4 contains a table of totally real cubic fields and indicates which are cyclic
5. Cohen 1993, §B.4
6. Hasse 1930
7. Cohen 1993, §6.4.5
8. The exact counts were computed by Michel Olivier and are available at . The first-order asymptotic is due to Harold Davenport and Hans Heilbronn (Davenport & Heilbronn 1971). The second-order term was conjectured by David P. Roberts (Roberts 2001) and a proof has been published by Manjul Bhargava, Arul Shankar, and Jacob Tsimerman (Bhargava, Shankar & Tsimerman 2013).
9. H. Minkowski, Diophantische Approximationen, chapter 4, §5.
10. Llorente, P.; Nart, E. (1983). "Effective determination of the decomposition of the rational primes in a cubic field". Proceedings of the American Mathematical Society. 87 (4): 579–585. doi:10.1090/S0002-9939-1983-0687621-6.
11. Mayer, D. C. (1992). "Multiplicities of dihedral discriminants". Math. Comp. 58 (198): 831–847 and S55–S58. Bibcode:1992MaCom..58..831M. doi:10.1090/S0025-5718-1992-1122071-3.
12. G. F. Voronoi, Concerning algebraic integers derivable from a root of an equation of the third degree, Master's Thesis, St. Petersburg, 1894 (Russian).
13. Davenport & Heilbronn 1971
14. Their work can also be interpreted as a computation of the average size of the 3-torsion part of the class group of a quadratic field, and thus constitutes one of the few proven cases of the Cohen–Lenstra conjectures: see, e.g. Bhargava, Manjul; Varma, Ila (2014), The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders, arXiv:1401.5875, Bibcode:2014arXiv1401.5875B, This theorem [of Davenport and Heilbronn] yields the only two proven cases of the Cohen-Lenstra heuristics for class groups of quadratic fields.
15. Roberts 2001, Conjecture 3.1
16. Voronoi, G. F. (1896). On a generalization of the algorithm of continued fractions (in Russian). Warsaw: Doctoral Dissertation.
17. Delone, B. N.; Faddeev, D. K. (1964). The theory of irrationalities of the third degree. Translations of Mathematical Monographs. Vol. 10. Providence, Rhode Island: American Mathematical Society.
References
• Şaban Alaca, Kenneth S. Williams, Introductory algebraic number theory, Cambridge University Press, 2004.
• Belabas, Karim (1997), "A fast algorithm to compute cubic fields", Mathematics of Computation, 66 (219): 1213–1237, doi:10.1090/s0025-5718-97-00846-6, MR 1415795
• Bhargava, Manjul; Shankar, Arul; Tsimerman, Jacob (2013), "On the Davenport–Heilbronn theorem and second order terms", Inventiones Mathematicae, 193 (2): 439–499, arXiv:1005.0672, Bibcode:2013InMat.193..439B, doi:10.1007/s00222-012-0433-0, MR 3090184, S2CID 253738365
• Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Berlin, New York: Springer-Verlag, ISBN 978-3-540-55640-4, MR 1228206
• Cohn, Harvey (1954), "The density of abelian cubic fields", Proceedings of the American Mathematical Society, 5 (3): 476–477, doi:10.2307/2031963, JSTOR 2031963, MR 0064076
• Davenport, Harold; Heilbronn, Hans (1971), "On the density of discriminants of cubic fields. II", Proceedings of the Royal Society A, 322 (1551): 405–420, Bibcode:1971RSPSA.322..405D, doi:10.1098/rspa.1971.0075, MR 0491593, S2CID 122814162
• Hasse, Helmut (1930), "Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage", Mathematische Zeitschrift (in German), 31 (1): 565–582, doi:10.1007/BF01246435, S2CID 121649559
• Roberts, David P. (2001), "Density of cubic field discriminants", Mathematics of Computation, 70 (236): 1699–1705, arXiv:math/9904190, doi:10.1090/s0025-5718-00-01291-6, MR 1836927, S2CID 7524750
• Taniguchi, Takashi; Thorne, Frank (2013), "Secondary terms in counting functions for cubic fields", Duke Mathematical Journal, 162 (13): 2451–2508, arXiv:1102.2914, doi:10.1215/00127094-2371752, MR 3127806, S2CID 16463250
External links
• Media related to Cubic field at Wikimedia Commons
| Wikipedia |
Symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n one has P(Xσ(1), Xσ(2), …, Xσ(n)) = P(X1, X2, …, Xn).
This article is about individual symmetric polynomials. For the ring of symmetric polynomials, see ring of symmetric functions.
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.
Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play important roles alongside the elementary ones. The resulting structures, and in particular the ring of symmetric functions, are of great importance in combinatorics and in representation theory.
Examples
The following polynomials in two variables X1 and X2 are symmetric:
$X_{1}^{3}+X_{2}^{3}-7$
$4X_{1}^{2}X_{2}^{2}+X_{1}^{3}X_{2}+X_{1}X_{2}^{3}+(X_{1}+X_{2})^{4}$
as is the following polynomial in three variables X1, X2, X3:
$X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3}$
There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is
$\prod _{1\leq i<j\leq n}(X_{i}-X_{j})^{2}$
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant).
On the other hand, the polynomial in two variables
$X_{1}-X_{2}$
is not symmetric, since if one exchanges $X_{1}$ and $X_{2}$ one gets a different polynomial, $X_{2}-X_{1}$. Similarly in three variables
$X_{1}^{4}X_{2}^{2}X_{3}+X_{1}X_{2}^{4}X_{3}^{2}+X_{1}^{2}X_{2}X_{3}^{4}$
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric:
$X_{1}^{4}X_{2}^{2}X_{3}+X_{1}X_{2}^{4}X_{3}^{2}+X_{1}^{2}X_{2}X_{3}^{4}+X_{1}^{4}X_{2}X_{3}^{2}+X_{1}X_{2}^{2}X_{3}^{4}+X_{1}^{2}X_{2}^{4}X_{3}$
Applications
Galois theory
Main article: Galois theory
One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field. These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.
This yields the approach to solving polynomial equations by inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomials in the roots), how can one recover the roots? This leads to studying solutions of polynomials using the permutation group of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory.
Relation with the roots of a monic univariate polynomial
Consider a monic polynomial in t of degree n
$P=t^{n}+a_{n-1}t^{n-1}+\cdots +a_{2}t^{2}+a_{1}t+a_{0}$
with coefficients ai in some field K. There exist n roots x1,…,xn of P in some possibly larger field (for instance if K is the field of real numbers, the roots will exist in the field of complex numbers); some of the roots might be equal, but the fact that one has all roots is expressed by the relation
$P=t^{n}+a_{n-1}t^{n-1}+\cdots +a_{2}t^{2}+a_{1}t+a_{0}=(t-x_{1})(t-x_{2})\cdots (t-x_{n}).$
By comparing coefficients one finds that
${\begin{aligned}a_{n-1}&=-x_{1}-x_{2}-\cdots -x_{n}\\a_{n-2}&=x_{1}x_{2}+x_{1}x_{3}+\cdots +x_{2}x_{3}+\cdots +x_{n-1}x_{n}=\textstyle \sum _{1\leq i<j\leq n}x_{i}x_{j}\\&{}\ \,\vdots \\a_{1}&=(-1)^{n-1}(x_{2}x_{3}\cdots x_{n}+x_{1}x_{3}x_{4}\cdots x_{n}+\cdots +x_{1}x_{2}\cdots x_{n-2}x_{n}+x_{1}x_{2}\cdots x_{n-1})=\textstyle (-1)^{n-1}\sum _{i=1}^{n}\prod _{j\neq i}x_{j}\\a_{0}&=(-1)^{n}x_{1}x_{2}\cdots x_{n}.\end{aligned}}$
These are in fact just instances of Vieta's formulas. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression: although for a given polynomial P there may be qualitative differences between the roots (like lying in the base field K or not, being simple or multiple roots), none of this affects the way the roots occur in these expressions.
Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing P, and considering them as indeterminates rather than as constants in an appropriate field; the coefficients ai then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign $(-1)^{n-i}$, are known as the elementary symmetric polynomials in x1, …, xn. A basic fact, known as the fundamental theorem of symmetric polynomials, states that any symmetric polynomial in n variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the coefficients of the polynomial, and in particular that its value lies in the base field K that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than K in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identities, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.
Special kinds of symmetric polynomials
There are a few types of symmetric polynomials in the variables X1, X2, …, Xn that are fundamental.
Elementary symmetric polynomials
Main article: Elementary symmetric polynomial
For each nonnegative integer k, the elementary symmetric polynomial ek(X1, …, Xn) is the sum of all distinct products of k distinct variables. (Some authors denote it by σk instead.) For k = 0 there is only the empty product so e0(X1, …, Xn) = 1, while for k > n, no products at all can be formed, so ek(X1, X2, …, Xn) = 0 in these cases. The remaining n elementary symmetric polynomials are building blocks for all symmetric polynomials in these variables: as mentioned above, any symmetric polynomial in the variables considered can be obtained from these elementary symmetric polynomials using multiplications and additions only. In fact one has the following more detailed facts:
• any symmetric polynomial P in X1, …, Xn can be written as a polynomial expression in the polynomials ek(X1, …, Xn) with 1 ≤ k ≤ n;
• this expression is unique up to equivalence of polynomial expressions;
• if P has integral coefficients, then the polynomial expression also has integral coefficients.
For example, for n = 2, the relevant elementary symmetric polynomials are e1(X1, X2) = X1 + X2, and e2(X1, X2) = X1X2. The first polynomial in the list of examples above can then be written as
$X_{1}^{3}+X_{2}^{3}-7=e_{1}(X_{1},X_{2})^{3}-3e_{2}(X_{1},X_{2})e_{1}(X_{1},X_{2})-7$
(for a proof that this is always possible see the fundamental theorem of symmetric polynomials).
Monomial symmetric polynomials
Powers and products of elementary symmetric polynomials work out to rather complicated expressions. If one seeks basic additive building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial in X1, …, Xn can be written as X1α1…Xnαn where the exponents αi are natural numbers (possibly zero); writing α = (α1,…,αn) this can be abbreviated to X α. The monomial symmetric polynomial mα(X1, …, Xn) is defined as the sum of all monomials xβ where β ranges over all distinct permutations of (α1,…,αn). For instance one has
$m_{(3,1,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}X_{3}+X_{1}X_{2}^{3}X_{3}+X_{1}X_{2}X_{3}^{3}$,
$m_{(3,2,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}^{2}X_{3}+X_{1}^{3}X_{2}X_{3}^{2}+X_{1}^{2}X_{2}^{3}X_{3}+X_{1}^{2}X_{2}X_{3}^{3}+X_{1}X_{2}^{3}X_{3}^{2}+X_{1}X_{2}^{2}X_{3}^{3}.$
Clearly mα = mβ when β is a permutation of α, so one usually considers only those mα for which α1 ≥ α2 ≥ … ≥ αn, in other words for which α is a partition of an integer. These monomial symmetric polynomials form a vector space basis: every symmetric polynomial P can be written as a linear combination of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in P. In particular if P has integer coefficients, then so will the linear combination.
The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤ k ≤ n one has
$e_{k}(X_{1},\ldots ,X_{n})=m_{\alpha }(X_{1},\ldots ,X_{n})$ where α is the partition of k into k parts 1 (followed by n − k zeros).
Power-sum symmetric polynomials
Main article: Power sum symmetric polynomial
For each integer k ≥ 1, the monomial symmetric polynomial m(k,0,…,0)(X1, …, Xn) is of special interest. It is the power sum symmetric polynomial, defined as
$p_{k}(X_{1},\ldots ,X_{n})=X_{1}^{k}+X_{2}^{k}+\cdots +X_{n}^{k}.$
All symmetric polynomials can be obtained from the first n power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely,
Any symmetric polynomial in X1, …, Xn can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials p1(X1, …, Xn), …, pn(X1, …, Xn).
In particular, the remaining power sum polynomials pk(X1, …, Xn) for k > n can be so expressed in the first n power sum polynomials; for example
$p_{3}(X_{1},X_{2})=\textstyle {\frac {3}{2}}p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2})-{\frac {1}{2}}p_{1}(X_{1},X_{2})^{3}.$
In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in n variables with integral coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials. For an example, for n = 2, the symmetric polynomial
$m_{(2,1)}(X_{1},X_{2})=X_{1}^{2}X_{2}+X_{1}X_{2}^{2}$
has the expression
$m_{(2,1)}(X_{1},X_{2})=\textstyle {\frac {1}{2}}p_{1}(X_{1},X_{2})^{3}-{\frac {1}{2}}p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2}).$
Using three variables one gets a different expression
${\begin{aligned}m_{(2,1)}(X_{1},X_{2},X_{3})&=X_{1}^{2}X_{2}+X_{1}X_{2}^{2}+X_{1}^{2}X_{3}+X_{1}X_{3}^{2}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2}\\&=p_{1}(X_{1},X_{2},X_{3})p_{2}(X_{1},X_{2},X_{3})-p_{3}(X_{1},X_{2},X_{3}).\end{aligned}}$
The corresponding expression was valid for two variables as well (it suffices to set X3 to zero), but since it involves p3, it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n. But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e1 which coincides with the first power sum) in terms of power sum polynomials. The Newton identities provide an explicit method to do this; it involves division by integers up to n, which explains the rational coefficients. Because of these divisions, the mentioned statement fails in general when coefficients are taken in a field of finite characteristic; however, it is valid with coefficients in any ring containing the rational numbers.
Complete homogeneous symmetric polynomials
Main article: Complete homogeneous symmetric polynomial
For each nonnegative integer k, the complete homogeneous symmetric polynomial hk(X1, …, Xn) is the sum of all distinct monomials of degree k in the variables X1, …, Xn. For instance
$h_{3}(X_{1},X_{2},X_{3})=X_{1}^{3}+X_{1}^{2}X_{2}+X_{1}^{2}X_{3}+X_{1}X_{2}^{2}+X_{1}X_{2}X_{3}+X_{1}X_{3}^{2}+X_{2}^{3}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2}+X_{3}^{3}.$
The polynomial hk(X1, …, Xn) is also the sum of all distinct monomial symmetric polynomials of degree k in X1, …, Xn, for instance for the given example
${\begin{aligned}h_{3}(X_{1},X_{2},X_{3})&=m_{(3)}(X_{1},X_{2},X_{3})+m_{(2,1)}(X_{1},X_{2},X_{3})+m_{(1,1,1)}(X_{1},X_{2},X_{3})\\&=(X_{1}^{3}+X_{2}^{3}+X_{3}^{3})+(X_{1}^{2}X_{2}+X_{1}^{2}X_{3}+X_{1}X_{2}^{2}+X_{1}X_{3}^{2}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2})+(X_{1}X_{2}X_{3}).\\\end{aligned}}$
All symmetric polynomials in these variables can be built up from complete homogeneous ones: any symmetric polynomial in X1, …, Xn can be obtained from the complete homogeneous symmetric polynomials h1(X1, …, Xn), …, hn(X1, …, Xn) via multiplications and additions. More precisely:
Any symmetric polynomial P in X1, …, Xn can be written as a polynomial expression in the polynomials hk(X1, …, Xn) with 1 ≤ k ≤ n.
If P has integral coefficients, then the polynomial expression also has integral coefficients.
For example, for n = 2, the relevant complete homogeneous symmetric polynomials are h1(X1, X2) = X1 + X2 and h2(X1, X2) = X12 + X1X2 + X22. The first polynomial in the list of examples above can then be written as
$X_{1}^{3}+X_{2}^{3}-7=-2h_{1}(X_{1},X_{2})^{3}+3h_{1}(X_{1},X_{2})h_{2}(X_{1},X_{2})-7.$
As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond hn(X1, …, Xn), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased.
An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities
$\sum _{i=0}^{k}(-1)^{i}e_{i}(X_{1},\ldots ,X_{n})h_{k-i}(X_{1},\ldots ,X_{n})=0$, for all k > 0, and any number of variables n.
Since e0(X1, …, Xn) and h0(X1, …, Xn) are both equal to 1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the hk(X1, …, Xn) with 1 ≤ k ≤ n: one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones.
Schur polynomials
Main article: Schur polynomial
Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to representation theory. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.
Symmetric polynomials in algebra
Symmetric polynomials are important to linear algebra, representation theory, and Galois theory. They are also important in combinatorics, where they are mostly studied through the ring of symmetric functions, which avoids having to carry around a fixed number of variables all the time.
Alternating polynomials
Main article: Alternating polynomials
Analogous to symmetric polynomials are alternating polynomials: polynomials that, rather than being invariant under permutation of the entries, change according to the sign of the permutation.
These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant.
See also
• Symmetric function
• Newton's identities
• Stanley symmetric function
• Muirhead's inequality
References
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
• Macdonald, I.G. (1979), Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford: Clarendon Press.
• I.G. Macdonald (1995), Symmetric Functions and Hall Polynomials, second ed. Oxford: Clarendon Press. ISBN 0-19-850450-0 (paperback, 1998).
• Richard P. Stanley (1999), Enumerative Combinatorics, Vol. 2. Cambridge: Cambridge University Press. ISBN 0-521-56069-1
Authority control: National
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Perfect totient number
In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.
Examples
For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and 9 = 6 + 2 + 1, so 9 is a perfect totient number.
The first few perfect totient numbers are
3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, ... (sequence A082897 in the OEIS).
Notation
In symbols, one writes
$\varphi ^{i}(n)={\begin{cases}\varphi (n),&{\text{ if }}i=1\\\varphi (\varphi ^{i-1}(n)),&{\text{ if }}i\geq 2\end{cases}}$
for the iterated totient function. Then if c is the integer such that
$\displaystyle \varphi ^{c}(n)=2,$
one has that n is a perfect totient number if
$n=\sum _{i=1}^{c+1}\varphi ^{i}(n).$
Multiples and powers of three
It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that
$\displaystyle \varphi (3^{k})=\varphi (2\times 3^{k})=2\times 3^{k-1}.$
Venkataraman (1975) found another family of perfect totient numbers: if p = 4 × 3k + 1 is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are
0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... (sequence A005537 in the OEIS).
More generally if p is a prime number greater than 3, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3kp where p is prime and k > 3.
References
• Pérez-Cacho Villaverde, Laureano (1939). "Sobre la suma de indicadores de ordenes sucesivos". Revista Matematica Hispano-Americana. 5 (3): 45–50.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. New York: Springer-Verlag. p. §B41. ISBN 0-387-20860-7.
• Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003). "On perfect totient numbers" (PDF). Journal of Integer Sequences. 6 (4): 03.4.5. Bibcode:2003JIntS...6...45I. MR 2051959. Archived from the original (PDF) on 2017-08-12. Retrieved 2007-02-07.
• Luca, Florian (2006). "On the distribution of perfect totients" (PDF). Journal of Integer Sequences. 9 (4): 06.4.4. Bibcode:2006JIntS...9...44L. MR 2247943. Retrieved 2007-02-07.
• Mohan, A. L.; Suryanarayana, D. (1982). "Perfect totient numbers". Number theory (Mysore, 1981). Lecture Notes in Mathematics, vol. 938, Springer-Verlag. pp. 101–105. MR 0665442.
• Venkataraman, T. (1975). "Perfect totient number". The Mathematics Student. 43: 178. MR 0447089.
• Hyvärinen, Tuukka (2015). "Täydelliset totienttiluvut". Tampere: Tampereen yliopisto.
This article incorporates material from Perfect Totient Number on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Totient summatory function
In number theory, the totient summatory function $\Phi (n)$ is a summatory function of Euler's totient function defined by:
$\Phi (n):=\sum _{k=1}^{n}\varphi (k),\quad n\in \mathbf {N} $
It is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.
Properties
Using Möbius inversion to the totient function, we obtain
$\Phi (n)=\sum _{k=1}^{n}k\sum _{d\mid k}{\frac {\mu (d)}{d}}={\frac {1}{2}}\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor \left(1+\left\lfloor {\frac {n}{k}}\right\rfloor \right)$
Φ(n) has the asymptotic expansion
$\Phi (n)\sim {\frac {1}{2\zeta (2)}}n^{2}+O\left(n\log n\right),$
where ζ(2) is the Riemann zeta function for the value 2.
Φ(n) is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.
The summatory of reciprocal totient function
The summatory of reciprocal totient function is defined as
$S(n):=\sum _{k=1}^{n}{\frac {1}{\varphi (k)}}$
Edmund Landau showed in 1900 that this function has the asymptotic behavior
$S(n)\sim A(\gamma +\log n)+B+O\left({\frac {\log n}{n}}\right)$
where γ is the Euler–Mascheroni constant,
$A=\sum _{k=1}^{\infty }{\frac {\mu (k)^{2}}{k\varphi (k)}}={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}=\prod _{p}\left(1+{\frac {1}{p(p-1)}}\right)$
and
$B=\sum _{k=1}^{\infty }{\frac {\mu (k)^{2}\log k}{k\,\varphi (k)}}=A\,\prod _{p}\left({\frac {\log p}{p^{2}-p+1}}\right).$
The constant A = 1.943596... is sometimes known as Landau's totient constant. The sum $\textstyle \sum _{k=1}^{\infty }{\frac {1}{k\varphi (k)}}$ is convergent and equal to:
$\sum _{k=1}^{\infty }{\frac {1}{k\varphi (k)}}=\zeta (2)\prod _{p}\left(1+{\frac {1}{p^{2}(p-1)}}\right)=2.20386\ldots $
In this case, the product over the primes in the right side is a constant known as totient summatory constant,[1] and its value is:
$\prod _{p}\left(1+{\frac {1}{p^{2}(p-1)}}\right)=1.339784\ldots $
See also
• Arithmetic function
References
1. OEIS: A065483
• Weisstein, Eric W. "Totient Summatory Function". MathWorld.
External links
• Totient summatory function
• Decimal expansion of totient constant product(1 + 1/(p^2*(p-1))), p prime >= 2)
| Wikipedia |
Totative
In number theory, a totative of a given positive integer n is an integer k such that 0 < k ≤ n and k is coprime to n. Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n.
Distribution
The distribution of totatives has been a subject of study. Paul Erdős conjectured that, writing the totatives of n as
$0<a_{1}<a_{2}\cdots <a_{\phi (n)}<n,$
the mean square gap satisfies
$\sum _{i=1}^{\phi (n)-1}(a_{i+1}-a_{i})^{2}<Cn^{2}/\phi (n)$
for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.[1]
See also
• Reduced residue system
References
1. Montgomery, H.L.; Vaughan, R.C. (1986). "On the distribution of reduced residues". Ann. Math. 2. 123: 311–333. doi:10.2307/1971274. Zbl 0591.10042.
• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B40. ISBN 978-0-387-20860-2. Zbl 1058.11001.
Further reading
• Sándor, Jozsef; Crstici, Borislav (2004), Handbook of number theory II, Dordrecht: Kluwer Academic, pp. 242–250, ISBN 1-4020-2546-7, Zbl 1079.11001
External links
• Weisstein, Eric W. "Totative". MathWorld.
• totative at PlanetMath.
| Wikipedia |
Touchard polynomials
The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by
$T_{n}(x)=\sum _{k=0}^{n}S(n,k)x^{k}=\sum _{k=0}^{n}\left\{{n \atop k}\right\}x^{k},$
Not to be confused with Bell polynomials.
where $S(n,k)=\left\{{n \atop k}\right\}$is a Stirling number of the second kind, i.e., the number of partitions of a set of size n into k disjoint non-empty subsets.[1][2][3][4]
The first few Touchard polynomials are
$T_{1}(x)=x,$
$T_{2}(x)=x^{2}+x,$
$T_{3}(x)=x^{3}+3x+x,$
$T_{4}(x)=x^{4}+6x^{3}+7x^{2}+x,$
$T_{5}(x)=x^{5}+10x^{4}+25x^{3}+15x^{2}+x.$
Properties
Basic properties
The value at 1 of the nth Touchard polynomial is the nth Bell number, i.e., the number of partitions of a set of size n:
$T_{n}(1)=B_{n}.$
If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(Xn) = Tn(λ), leading to the definition:
$T_{n}(x)=e^{-x}\sum _{k=0}^{\infty }{\frac {x^{k}k^{n}}{k!}}.$
Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities:
$T_{n}(\lambda +\mu )=\sum _{k=0}^{n}{n \choose k}T_{k}(\lambda )T_{n-k}(\mu ).$
The Touchard polynomials constitute the only polynomial sequence of binomial type with the coefficient of x equal 1 in every polynomial.
The Touchard polynomials satisfy the Rodrigues-like formula:
$T_{n}\left(e^{x}\right)=e^{-e^{x}}{\frac {d^{n}}{dx^{n}}}\;e^{e^{x}}.$
The Touchard polynomials satisfy the recurrence relation
$T_{n+1}(x)=x\left(1+{\frac {d}{dx}}\right)T_{n}(x)$
and
$T_{n+1}(x)=x\sum _{k=0}^{n}{n \choose k}T_{k}(x).$
In the case x = 1, this reduces to the recurrence formula for the Bell numbers.
A generalization of both this formula and the definition, is a generalization of Spivey's formula[5]
$T_{n+m}(x)=\sum _{k=0}^{n}\left\{{n \atop k}\right\}x^{k}\sum _{j=0}^{m}{\binom {m}{j}}k^{m-j}T_{j}(x)$
Using the umbral notation Tn(x)=Tn(x), these formulas become:
$T_{n}(\lambda +\mu )=\left(T(\lambda )+T(\mu )\right)^{n},$
$T_{n+1}(x)=x\left(1+T(x)\right)^{n}.$
The generating function of the Touchard polynomials is
$\sum _{n=0}^{\infty }{T_{n}(x) \over n!}t^{n}=e^{x\left(e^{t}-1\right)},$
which corresponds to the generating function of Stirling numbers of the second kind.
Touchard polynomials have contour integral representation:
$T_{n}(x)={\frac {n!}{2\pi i}}\oint {\frac {e^{x({e^{t}}-1)}}{t^{n+1}}}\,dt.$
Zeroes
All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967.[6]
The absolute value of the leftmost zero is bounded from above by[7]
${\frac {1}{n}}{\binom {n}{2}}+{\frac {n-1}{n}}{\sqrt {{\binom {n}{2}}^{2}-{\frac {2n}{n-1}}\left({\binom {n}{3}}+3{\binom {n}{4}}\right)}},$
although it is conjectured that the leftmost zero grows linearly with the index n.
The Mahler measure $M(T_{n})$of the Touchard polynomials can be estimated as follows:[8]
${\frac {\lbrace \textstyle {n \atop \Omega _{n}}\rbrace }{\binom {n}{\Omega _{n}}}}\leq M(T_{n})\leq {\sqrt {n+1}}\left\{{n \atop K_{n}}\right\},$
where $\Omega _{n}$ and $K_{n}$ are the smallest of the maximum two k indices such that $\lbrace \textstyle {n \atop k}\rbrace /{\binom {n}{k}}$ and $\lbrace \textstyle {n \atop k}\rbrace $ are maximal, respectively.
Generalizations
• Complete Bell polynomial $B_{n}(x_{1},x_{2},\dots ,x_{n})$ may be viewed as a multivariate generalization of Touchard polynomial $T_{n}(x)$, since $T_{n}(x)=B_{n}(x,x,\dots ,x).$
• The Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order:
$T_{n}(x)={\frac {n!}{\pi }}\int _{0}^{\pi }e^{x{\bigl (}e^{\cos(\theta )}\cos(\sin(\theta ))-1{\bigr )}}\cos {\bigl (}xe^{\cos(\theta )}\sin(\sin(\theta ))-n\theta {\bigr )}\,d\theta \,.$
See also
• Bell polynomials
References
1. Roman, Steven (1984). The Umbral Calculus. Dover. ISBN 0-486-44139-3.
2. Boyadzhiev, Khristo N. "Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals". Abstract and Applied Analysis. 2009: 1–18. arXiv:0909.0979. Bibcode:2009AbApA2009....1B. doi:10.1155/2009/168672.
3. Brendt, Bruce C. "RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU" (PDF). Retrieved 23 November 2013.
4. Weisstein, Eric W. "Bell Polynomial". MathWorld.
5. "Implications of Spivey's Bell Number Formula". cs.uwaterloo.ca. Retrieved 2023-05-28.
6. Harper, L. H. (1967). "Stirling behavior is asymptotically normal". The Annals of Mathematical Statistics. 38 (2): 410–414. doi:10.1214/aoms/1177698956.
7. Mező, István; Corcino, Roberto B. (2015). "The estimation of the zeros of the Bell and r-Bell polynomials". Applied Mathematics and Computation. 250: 727–732. doi:10.1016/j.amc.2014.10.058.
8. István, Mező. "On the Mahler measure of the Bell polynomials". Retrieved 7 November 2017.
• Touchard, Jacques (1939), "Sur les cycles des substitutions", Acta Mathematica, 70 (1): 243–297, doi:10.1007/BF02547349, ISSN 0001-5962, MR 1555449
| Wikipedia |
Toufik Mansour
Toufik Mansour is an Israeli mathematician working in algebraic combinatorics. He is a member of the Druze community and is the first Israeli Druze to become a professional mathematician.[1]
Toufik Mansour
Born(1968-01-17)17 January 1968
Scientific career
FieldsMathematician
Mansour obtained his Ph.D. in mathematics from the University of Haifa in 2001 under Alek Vainshtein.[2] As of 2007, he is a professor of mathematics at the University of Haifa.[3] He served as chair of the department from 2015 to 2017. He has previously been a faculty member of the Center for Combinatorics at Nankai University from 2004 to 2007, and at The John Knopfmacher Center for Applicable Analysis and Number Theory at the University of the Witwatersrand.
Mansour is an expert on Discrete Mathematics and its applications. In particular, he is interested in permutation patterns, colored permutations, set partitions, combinatorics on words, and compositions. He has written more than 260 research papers, which means that he publishes a paper roughly every 20 days, or that he produces one publication page roughly every day.
Books
• Heubach, Silvia; Mansour, Toufik (2010), Combinatorics of Compositions and Words, Discrete Mathematics and its Applications, Boca Raton, Florida: CRC Press, ISBN 978-1-4200-7267-9, MR 2531482.
• Mansour, Toufik (2013), Combinatorics of Set Partitions, Discrete Mathematics and its Applications, Boca Raton, Florida: CRC Press, ISBN 978-1-4398-6333-6, MR 2953184.
• Mansour, Toufik; Schork, Matthias (2015), Commutation Relations, Normal Ordering, and Stirling Numbers, Discrete Mathematics and its Applications, Boca Raton, Florida: CRC Press, ISBN 978-1-4665-7988-0.
See also
• List of Israeli Druze
• Schröder–Hipparchus number
References
1. Israel's first Druze math lecturer does it by the numbers, Israel21c, October 5, 2003.
2. Permutations With Forbidden Patterns, Ph.D. thesis, T. Mansour, 2001, from the Haifa University Theses and Dissertations Collection, retrieved 2014-09-06.
3. Faculty members Archived 2014-10-11 at the Wayback Machine, Mathematics Department, University of Haifa, retrieved 2014-09-06.
External links
• Home page and list of publications
• Google Scholar profile
• ORCID profile
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| Wikipedia |
Tournament (graph theory)
A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a directed edge (often, called an arc) with any one of the two possible orientations.
Tournament
A tournament on 4 vertices
Vertices$n$
Edges${\binom {n}{2}}$
Table of graphs and parameters
Many of the important properties of tournaments were first investigated by H. G. Landau in Landau (1953) to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things.
The name tournament originates from such a graph's interpretation as the outcome of a round-robin tournament in which every player encounters every other player exactly once, and in which no draws occur. In the tournament digraph, the vertices correspond to the players. The edge between each pair of players is oriented from the winner to the loser. If player $a$ beats player $b$, then it is said that $a$ dominates $b$. If every player beats the same number of other players (indegree = outdegree), the tournament is called regular.
Paths and cycles
Theorem — Any tournament on a finite number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): n of vertices contains a Hamiltonian path, i.e., directed path on all $n$ vertices (Rédei 1934).
This is easily shown by induction on $n$: suppose that the statement holds for $n$, and consider any tournament $T$ on $n+1$ vertices. Choose a vertex $v_{0}$ of $T$ and consider a directed path $v_{1},v_{2},\ldots ,v_{n}$ in $T\setminus \{v_{0}\}$. There is some $i\in \{0,\ldots ,n\}$ such that $(i=0\vee v_{i}\rightarrow v_{0})\wedge (v_{0}\rightarrow v_{i+1}\vee i=n)$. (One possibility is to let $i\in \{0,\ldots ,n\}$ be maximal such that for every $j\leq i,v_{j}\rightarrow v_{0}$. Alternatively, let $i$ be minimal such that $\forall j>i,v_{0}\rightarrow v_{j}$.)
$v_{1},\ldots ,v_{i},v_{0},v_{i+1},\ldots ,v_{n}$
is a directed path as desired. This argument also gives an algorithm for finding the Hamiltonian path. More efficient algorithms, that require examining only $O(n\log n)$ of the edges, are known. The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament.[1] Rédei's theorem is the special case for complete graphs of the Gallai–Hasse–Roy–Vitaver theorem, relating the lengths of paths in orientations of graphs to the chromatic number of these graphs.[2]
Another basic result on tournaments is that every strongly connected tournament has a Hamiltonian cycle.[3] More strongly, every strongly connected tournament is vertex pancyclic: for each vertex $v$, and each $k$ in the range from three to the number of vertices in the tournament, there is a cycle of length $k$ containing $v$.[4] A tournament $T$is $k$-strongly connected if for every set $U$ of $k-1$ vertices of $T$, $T-U$ is strongly connected. If the tournament is 4‑strongly connected, then each pair of vertices can be connected with a Hamiltonian path.[5] For every set $B$ of at most $k-1$ arcs of a $k$-strongly connected tournament $T$, we have that $T-B$ has a Hamiltonian cycle.[6] This result was extended by Bang-Jensen, Gutin & Yeo (1997).
Transitivity
A tournament in which $((a\rightarrow b)$ and $(b\rightarrow c))$ $\Rightarrow $ $(a\rightarrow c)$ is called transitive. In other words, in a transitive tournament, the vertices may be (strictly) totally ordered by the edge relation, and the edge relation is the same as reachability.
Equivalent conditions
The following statements are equivalent for a tournament $T$ on $n$ vertices:
1. $T$ is transitive.
2. $T$ is a strict total ordering.
3. $T$ is acyclic.
4. $T$ does not contain a cycle of length 3.
5. The score sequence (set of outdegrees) of $T$ is $\{0,1,2,\ldots ,n-1\}$.
6. $T$ has exactly one Hamiltonian path.
Ramsey theory
Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular, every tournament on $n$ vertices contains a transitive subtournament on $1+\lfloor \log _{2}n\rfloor $ vertices.[7] The proof is simple: choose any one vertex $v$ to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of $v$ or the set of outgoing neighbors of $v$, whichever is larger. For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; the Paley tournament on seven vertices shows that this is the most that can be guaranteed (Erdős & Moser 1964). However, Reid & Parker (1970) showed that this bound is not tight for some larger values of $n$.
Erdős & Moser (1964) proved that there are tournaments on $n$ vertices without a transitive subtournament of size $2+2\lfloor \log _{2}n\rfloor $ Their proof uses a counting argument: the number of ways that a $k$-element transitive tournament can occur as a subtournament of a larger tournament on $n$ labeled vertices is
${\binom {n}{k}}k!2^{{\binom {n}{2}}-{\binom {k}{2}}},$
and when $k$ is larger than $2+2\lfloor \log _{2}n\rfloor $, this number is too small to allow for an occurrence of a transitive tournament within each of the $2^{\binom {n}{2}}$ different tournaments on the same set of $n$ labeled vertices.
Paradoxical tournaments
A player who wins all games would naturally be the tournament's winner. However, as the existence of non-transitive tournaments shows, there may not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament $T=(V,E)$ is called $k$-paradoxical if for every $k$-element subset $S$ of $V$ there is a vertex $v_{0}$ in $V\setminus S$ such that $v_{0}\rightarrow v$ for all $v\in S$. By means of the probabilistic method, Paul Erdős showed that for any fixed value of $k$, if $|V|\geq k^{2}2^{k}\ln(2+o(1))$, then almost every tournament on $V$ is $k$-paradoxical.[8] On the other hand, an easy argument shows that any $k$-paradoxical tournament must have at least $2^{k+1}-1$ players, which was improved to $(k+2)2^{k-1}-1$ by Esther and George Szekeres (1965).[9] There is an explicit construction of $k$-paradoxical tournaments with $k^{2}4^{k-1}(1+o(1))$ players by Graham and Spencer (1971) namely the Paley tournament.
Condensation
The condensation of any tournament is itself a transitive tournament. Thus, even for tournaments that are not transitive, the strongly connected components of the tournament may be totally ordered.[10]
Score sequences and score sets
The score sequence of a tournament is the nondecreasing sequence of outdegrees of the vertices of a tournament. The score set of a tournament is the set of integers that are the outdegrees of vertices in that tournament.
Landau's Theorem (1953) A nondecreasing sequence of integers $(s_{1},s_{2},\ldots ,s_{n})$ is a score sequence if and only if :
1. $0\leq s_{1}\leq s_{2}\leq \cdots \leq s_{n}$
2. $s_{1}+s_{2}+\cdots +s_{i}\geq {i \choose 2},{\text{ for }}i=1,2,\ldots ,n-1$
3. $s_{1}+s_{2}+\cdots +s_{n}={n \choose 2}.$
Let $s(n)$ be the number of different score sequences of size $n$. The sequence $s(n)$ (sequence A000571 in the OEIS) starts as:
1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, ...
Winston and Kleitman proved that for sufficiently large n:
$s(n)>c_{1}4^{n}n^{-5/2},$
where $c_{1}=0.049.$ Takács later showed, using some reasonable but unproven assumptions, that
$s(n)<c_{2}4^{n}n^{-5/2},$
where $c_{2}<4.858.$
Together these provide evidence that:
$s(n)\in \Theta (4^{n}n^{-5/2}).$
Here $\Theta $ signifies an asymptotically tight bound.
Yao showed that every nonempty set of nonnegative integers is the score set for some tournament.
Majority relations
In social choice theory, tournaments naturally arise as majority relations of preference profiles.[11] Let $A$ be a finite set of alternatives, and consider a list $P=(\succ _{1},\dots ,\succ _{n})$ of linear orders over $A$. We interpret each order $\succ _{i}$ as the preference ranking of a voter $i$. The (strict) majority relation $\succ _{\text{maj}}$ of $P$ over $A$ is then defined so that $a\succ _{\text{maj}}b$ if and only if a majority of the voters prefer $a$ to $b$, that is $|\{i\in [n]:a\succ _{i}b\}|>|\{i\in [n]:b\succ _{i}a\}|$. If the number $n$ of voters is odd, then the majority relation forms the dominance relation of a tournament on vertex set $A$.
By a lemma of McGarvey, every tournament on $m$ vertices can be obtained as the majority relation of at most $m(m-1)$ voters.[11][12] Results by Stearns and Erdős & Moser later established that $\Theta (m/\log m)$ voters are needed to induce every tournament on $m$ vertices.[13]
Laslier (1997) studies in what sense a set of vertices can be called the set of "winners" of a tournament. This revealed to be useful in Political Science to study, in formal models of political economy, what can be the outcome of a democratic process.[14]
See also
• Oriented graph
• Paley tournament
• Sumner's conjecture
• Tournament solution
Notes
1. Bar-Noy & Naor (1990).
2. Havet (2013).
3. Camion (1959).
4. Moon (1966), Theorem 1.
5. Thomassen (1980).
6. Fraisse & Thomassen (1987).
7. Erdős & Moser (1964).
8. Erdős (1963)
9. Szekeres, E.; Szekeres, G. (1965). "On a problem of Schütte and Erdős". Math. Gaz. 49: 290–293.
10. Harary & Moser (1966), Corollary 5b.
11. Brandt, Felix and Brill, Markus and Harrenstein, Paul, "Tournament Solutions". Chapter 3 in: Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016). Handbook of Computational Social Choice. Cambridge University Press. ISBN 9781107060432. (free online version)
12. McGarvey, David C. (1953). "A Theorem on the Construction of Voting Paradoxes". Econometrica. 21 (4): 608–610. doi:10.2307/1907926. JSTOR 1907926.
13. Stearns, Richard (1959). "The Voting Problem". The American Mathematical Monthly. 66 (9): 761–763. doi:10.2307/2310461. JSTOR 2310461.
14. Austen-Smith, D. and J. Banks, Positive Political theory, 1999, The University of Michigan Press.
References
• Bang-Jensen, J.; Gutin, G.; Yeo, A. (1997), "Hamiltonian Cycles Avoiding Prescribed Arcs in Tournaments", Combinatorics, Probability and Computing, 6: 255–261.
• Bar-Noy, A.; Naor, J. (1990), "Sorting, Minimal Feedback Sets and Hamilton Paths in Tournaments", SIAM Journal on Discrete Mathematics, 3 (1): 7–20, doi:10.1137/0403002.
• Camion, Paul (1959), "Chemins et circuits hamiltoniens des graphes complets", Comptes Rendus de l'Académie des Sciences de Paris (in French), 249: 2151–2152.
• Erdős, P. (1963), "On a problem in graph theory" (PDF), The Mathematical Gazette, 47: 220–223, JSTOR 3613396, MR 0159319.
• Erdős, P.; Moser, L. (1964), "On the representation of directed graphs as unions of orderings" (PDF), Magyar Tud. Akad. Mat. Kutató Int. Közl., 9: 125–132, MR 0168494.
• Fraisse, P.; Thomassen, C. (1987), "A constructive solution to a tournament problem", Graphs and Combinatorics, 3: 239–250.
• Graham, R. L.; Spencer, J. H. (1971), "A constructive solution to a tournament problem", Canadian Mathematical Bulletin, 14: 45–48, doi:10.4153/cmb-1971-007-1, MR 0292715.
• Harary, Frank; Moser, Leo (1966), "The theory of round robin tournaments", American Mathematical Monthly, 73 (3): 231–246, doi:10.2307/2315334, JSTOR 2315334.
• Havet, Frédéric (2013), "Section 3.1: Gallai–Roy Theorem and related results" (PDF), Orientations and colouring of graphs, Lecture notes for the summer school SGT 2013 in Oléron, France, pp. 15–19
• Landau, H.G. (1953), "On dominance relations and the structure of animal societies. III. The condition for a score structure", Bulletin of Mathematical Biophysics, 15 (2): 143–148, doi:10.1007/BF02476378.
• Laslier, J.-F. (1997), Tournament Solutions and Majority Voting, Springer.
• Moon, J. W. (1966), "On subtournaments of a tournament", Canadian Mathematical Bulletin, 9 (3): 297–301, doi:10.4153/CMB-1966-038-7.
• Rédei, László (1934), "Ein kombinatorischer Satz", Acta Litteraria Szeged, 7: 39–43.
• Reid, K.B.; Parker, E.T. (1970), "Disproof of a conjecture of Erdös and Moser", Journal of Combinatorial Theory, 9 (3): 225–238, doi:10.1016/S0021-9800(70)80061-8.
• Szekeres, E.; Szekeres, G. (1965), "On a problem of Schütte and Erdős", The Mathematical Gazette, 49: 290–293, doi:10.2307/3612854, MR 0186566.
• Takács, Lajos (1991), "A Bernoulli Excursion and Its Various Applications", Advances in Applied Probability, Applied Probability Trust, 23 (3): 557–585, doi:10.2307/1427622, JSTOR 1427622.
• Thomassen, Carsten (1980), "Hamiltonian-Connected Tournaments", Journal of Combinatorial Theory, Series B, 28 (2): 142–163, doi:10.1016/0095-8956(80)90061-1.
• Yao, T.X. (1989), "On Reid conjecture of score sets for tournaments", Chinese Sci. Bull., 34: 804–808.
This article incorporates material from tournament on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Tournament solution
A tournament solution is a function that maps an oriented complete graph to a nonempty subset of its vertices. It can informally be thought of as a way to find the "best" alternatives among all of the alternatives that are "competing" against each other in the tournament. Tournament solutions originate from social choice theory,[1][2][3][4] but have also been considered in sports competition, game theory,[5] multi-criteria decision analysis, biology,[6][7] webpage ranking,[8] and dueling bandit problems.[9]
Part of the Politics series
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By type of representation
• Mixed-member majoritarian
• Mixed-member proportional
Non-compensatory mixed systems
• Parallel voting
• Majority bonus
Compensatory mixed systems
• Additional member system
• Mixed single vote (positive vote transfer)
• Scorporo (negative vote transfer)
• Mixed ballot transferable vote
• Alternative Vote Plus
• Dual-member proportional
• Rural–urban proportional
Other systems and related theory
Semi-proportional representation
• Single non-transferable vote
• Limited voting
• Cumulative voting
• Binomial voting
Other systems
• Multiple non-transferable vote
• Double simultaneous vote
• Proxy voting
• Delegated voting
• Indirect STV
• Liquid democracy
• Random selection (sortition, random ballot)
Social choice theory
• Arrow's theorem
• Gibbard–Satterthwaite theorem
• Public choice theory
List of electoral systems
• List of electoral systems by country
• Comparison of electoral systems
Politics portal
In the context of social choice theory, tournament solutions are closely related to Fishburn's C1 social choice functions,[10] and thus seek to show who the best candidates are among all candidates.
Definition
A tournament (graph) $T$ is a tuple $(A,\succ )$ where $A$ is a set of vertices (called alternatives) and $\succ $ is a connex and asymmetric binary relation over the vertices. In social choice theory, the binary relation typically represents the pairwise majority comparison between alternatives.
A tournament solution is a function $f$ that maps each tournament $T=(A,\succ )$ to a nonempty subset $f(T)$ of the alternatives $A$ (called the choice set[2]) and does not distinguish between isomorphic tournaments:
If $h:A\rightarrow B$ is a graph isomorphism between two tournaments $T=(A,\succ )$ and ${\widetilde {T}}=(B,{\widetilde {\succ }})$, then $a\in f(T)\Leftrightarrow h(a)\in f({\widetilde {T}})$
Examples
Common examples of tournament solutions are:[1][2]
• Copeland's method
• Top cycle
• Slater set
• Bipartisan set
• Uncovered set
• Banks set
• Minimal covering set
• Tournament equilibrium set
References
1. Laslier, J.-F. [in French] (1997). Tournament Solutions and Majority Voting. Springer Verlag.
2. Felix Brandt; Markus Brill; Paul Harrenstein (28 April 2016). "Chapter 3: Tournament Solutions" (PDF). In Felix Brandt; Vincent Conitzer; Ulle Endriss; Jérôme Lang; Ariel D. Procaccia (eds.). Handbook of Computational Social Choice. Cambridge University Press. ISBN 978-1-316-48975-8.
3. Brandt, F. (2009). Tournament Solutions - Extensions of Maximality and Their Applications to Decision-Making. Habilitation Thesis, Faculty for Mathematics, Computer Science, and Statistics, University of Munich.
4. Scott Moser. "Chapter 6: Majority rule and tournament solutions". In J. C. Heckelman; N. R. Miller (eds.). Handbook of Social Choice and Voting. Edgar Elgar.
5. Fisher, D. C.; Ryan, J. (1995). "Tournament games and positive tournaments". Journal of Graph Theory. 19 (2): 217–236. doi:10.1002/jgt.3190190208.
6. Allesina, S.; Levine, J. M. (2011). "A competitive network theory of species diversity". Proceedings of the National Academy of Sciences. 108 (14): 5638–5642. Bibcode:2011PNAS..108.5638A. doi:10.1073/pnas.1014428108. PMC 3078357. PMID 21415368.
7. Landau, H. G. (1951). "On dominance relations and the structure of animal societies: I. Effect of inherent characteristics". Bulletin of Mathematical Biophysics. 13 (1): 1–19. doi:10.1007/bf02478336.
8. Felix Brandt; Felix Fischer (2007). "PageRank as a Weak Tournament Solution" (PDF). Lecture Notes in Computer Science (LNCS). 3rd International Workshop on Internet and Network Economics (WINE). Vol. 4858. San Diego, USA: Springer. pp. 300–305.
9. Siddartha Ramamohan; Arun Rajkumar; Shivani Agarwal (2016). Dueling Bandits: Beyond Condorcet Winners to General Tournament Solutions (PDF). 29th Conference on Neural Information Processing Systems (NIPS 2016). Barcelona, Spain.
10. Fishburn, P. C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030.
| Wikipedia |
Block-stacking problem
In statics, the block-stacking problem (sometimes known as The Leaning Tower of Lire (Johnson 1955), also the book-stacking problem, or a number of other similar terms) is a puzzle concerning the stacking of blocks at the edge of a table.
Statement
The block-stacking problem is the following puzzle:
Place $N$ identical rigid rectangular blocks in a stable stack on a table edge in such a way as to maximize the overhang.
Paterson et al. (2007) provide a long list of references on this problem going back to mechanics texts from the middle of the 19th century.
Variants
Single-wide
The single-wide problem involves having only one block at any given level. In the ideal case of perfectly rectangular blocks, the solution to the single-wide problem is that the maximum overhang is given by $ \sum _{i=1}^{N}{\frac {1}{2i}}$ times the width of a block. This sum is one half of the corresponding partial sum of the harmonic series. Because the harmonic series diverges, the maximal overhang tends to infinity as $N$ increases, meaning that it is possible to achieve any arbitrarily large overhang, with sufficient blocks.
NMaximum overhang
expressed as a fractiondecimalrelative size
11/20.5 0.5
23/40.75 0.75
311/12~0.91667 0.91667
425/24~1.04167 1.04167
5137/120~1.14167 1.14167
649/401.225 1.225
7363/280~1.29643 1.29643
8761/560~1.35893 1.35893
97 129/5 040~1.41448 1.41448
107 381/5 040~1.46448 1.46448
NMaximum overhang
expressed as a fractiondecimalrelative size
1183 711/55 440~1.50994 1.50994
1286 021/55 440~1.55161 1.55161
131 145 993/720 720~1.59007 1.59007
141 171 733/720 720~1.62578 1.62578
151 195 757/720 720~1.65911 1.65911
162 436 559/1 441 440~1.69036 1.69036
1742 142 223/24 504 480~1.71978 1.71978
1814 274 301/8 168 160~1.74755 1.74755
19275 295 799/155 195 040~1.77387 1.77387
2055 835 135/31 039 008~1.79887 1.79887
NMaximum overhang
expressed as a fractiondecimalrelative size
2118 858 053/10 346 336~1.82268 1.82268
2219 093 197/10 346 336~1.84541 1.84541
23444 316 699/237 965 728~1.86715 1.86715
241 347 822 955/713 897 184~1.88798 1.88798
2534 052 522 467/17 847 429 600~1.90798 1.90798
2634 395 742 267/17 847 429 600~1.92721 1.92721
27312 536 252 003/160 626 866 400~1.94573 1.94573
28315 404 588 903/160 626 866 400~1.96359 1.96359
299 227 046 511 387/4 658 179 125 600~1.98083 1.98083
309 304 682 830 147/4 658 179 125 600~1.99749 1.99749
The number of blocks required to reach at least $N$ block-lengths past the edge of the table is 4, 31, 227, 1674, 12367, 91380, ... (sequence A014537 in the OEIS).[1]
Multi-wide
Multi-wide stacks using counterbalancing can give larger overhangs than a single width stack. Even for three blocks, stacking two counterbalanced blocks on top of another block can give an overhang of 1, while the overhang in the simple ideal case is at most 11/12. As Paterson et al. (2007) showed, asymptotically, the maximum overhang that can be achieved by multi-wide stacks is proportional to the cube root of the number of blocks, in contrast to the single-wide case in which the overhang is proportional to the logarithm of the number of blocks. However, it has been shown that in reality this is impossible and the number of blocks that we can move to the right, due to block stress, is not more than a specified number. For example, for a special brick with h = 0.20 m, Young's modulus E = 3000 MPa and density ρ = 1.8×103 kg/m3 and limiting compressive stress 3 MPa,the approximate value of N will be 853 and the maximum tower height becomes 170 m.[2]
Robustness
Hall (2005) discusses this problem, shows that it is robust to nonidealizations such as rounded block corners and finite precision of block placing, and introduces several variants including nonzero friction forces between adjacent blocks.
References in media
In 2018, Michael Stevens, creator of various YouTube channels including Vsauce and D!NG, uploaded a video where Michael and former MythBusters star Adam Savage, discuss and construct a model of the block-stacking problem using plywood.[4]
References
1. Sloane, N. J. A. (ed.). "Sequence A014537 (Number of books required for n book-lengths of overhang in the harmonic book stacking problem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Khoshbin-e-Khoshnazar, M. R. (2007). "Simplifying modelling can mislead students". Physics Education. 42: 14–15. doi:10.1088/0031-9120/42/1/F05. S2CID 250745206.
3. M Paterson et al, Maximum Overhang, The Mathematical Association of America, November 2009
4. The Leaning Tower of Lire, retrieved 2022-08-02
• Hall, J. F. (2005). "Fun with stacking blocks". American Journal of Physics. 73 (12): 1107–1116. Bibcode:2005AmJPh..73.1107H. doi:10.1119/1.2074007..
• Johnson, Paul B. (April 1955). "Leaning Tower of Lire". American Journal of Physics. 23 (4): 240. Bibcode:1955AmJPh..23..240J. doi:10.1119/1.1933957.
• Paterson, Mike; Peres, Yuval; Thorup, Mikkel; Winkler, Peter; Zwick, Uri (2007). "Maximum overhang". arXiv:0707.0093 [math.HO].
External links
• Weisstein, Eric W. "Book Stacking Problem". MathWorld.
• "Building an Infinite Bridge". PBS Infinite Series. 2017-05-04. Retrieved 2018-09-03.
| Wikipedia |
Tower of fields
In mathematics, a tower of fields is a sequence of field extensions
F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ...
The name comes from such sequences often being written in the form
${\begin{array}{c}\vdots \\|\\F_{2}\\|\\F_{1}\\|\\\ F_{0}.\end{array}}$
A tower of fields may be finite or infinite.
Examples
• Q ⊆ R ⊆ C is a finite tower with rational, real and complex numbers.
• The sequence obtained by letting F0 be the rational numbers Q, and letting
$F_{n+1}=F_{n}\!\left(2^{1/2^{n}}\right)$
(i.e. Fn+1 is obtained from Fn by adjoining a 2n th root of 2) is an infinite tower.
• If p is a prime number the p th cyclotomic tower of Q is obtained by letting F0 = Q and Fn be the field obtained by adjoining to Q the pn th roots of unity. This tower is of fundamental importance in Iwasawa theory.
• The Golod–Shafarevich theorem shows that there are infinite towers obtained by iterating the Hilbert class field construction to a number field.
References
• Section 4.1.4 of Escofier, Jean-Pierre (2001), Galois theory, Graduate Texts in Mathematics, vol. 204, Springer-Verlag, ISBN 978-0-387-98765-1
| Wikipedia |
Tower of objects
In category theory, a branch of abstract mathematics, a tower is defined as follows. Let ${\mathcal {I}}$ be the poset
$\cdots \rightarrow 2\rightarrow 1\rightarrow 0$
of whole numbers in reverse order, regarded as a category. A (countable) tower of objects in a category ${\mathcal {A}}$ is a functor from ${\mathcal {I}}$ to ${\mathcal {A}}$.
In other words, a tower (of ${\mathcal {A}}$) is a family of objects $\{A_{i}\}_{i\geq 0}$ in ${\mathcal {A}}$ where there exists a map
$A_{i}\rightarrow A_{j}$ if $i>j$
and the composition
$A_{i}\rightarrow A_{j}\rightarrow A_{k}$
is the map $A_{i}\rightarrow A_{k}$
Example
Let $M_{i}=M$ for some $R$-module $M$. Let $M_{i}\rightarrow M_{j}$ be the identity map for $i>j$. Then $\{M_{i}\}$ forms a tower of modules.
References
• Section 3.5 of Weibel, Charles A. (1994), An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4
| Wikipedia |
Toy problem
In scientific disciplines, a toy problem[1][2] or a puzzlelike problem[3] is a problem that is not of immediate scientific interest, yet is used as an expository device to illustrate a trait that may be shared by other, more complicated, instances of the problem, or as a way to explain a particular, more general, problem solving technique. A toy problem is useful to test and demonstrate methodologies. Researchers can use toy problems to compare the performance of different algorithms. They are also good for game designing.
For instance, while engineering a large system, the large problem is often broken down into many smaller toy problems which have been well understood in detail. Often these problems distill a few important aspects of complicated problems so that they can be studied in isolation. Toy problems are thus often very useful in providing intuition about specific phenomena in more complicated problems.
As an example, in the field of artificial intelligence, classical puzzles, games and problems are often used as toy problems. These include sliding-block puzzles, N-Queens problem, missionaries and cannibals problem, tic-tac-toe, chess,[1] Tower of Hanoi and others.[2][3]
See also
• Blocks world
• Firing squad synchronization problem
• Monkey and banana problem
• Secretary problem
References
1. Stuart J. Russell, Peter Norvig (2010). Artificial Intelligence: A Modern Approach (3 ed.). pp. 70–73, 102–107, 109–110, 115, 162. ISBN 978-0-13-604259-4.
2. Korf, Richard E (2012). "Research challenges in combinatorial search": 2129–2133. {{cite journal}}: Cite journal requires |journal= (help)
3. Pearl, Judea (1984). Heuristics: intelligent search strategies for computer problem solving. p. 4. ISBN 0-201-05594-5.
External links
• "toy problem". The Jargon Lexicon.
| Wikipedia |
Toy theorem
In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem. One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem.
In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the proofs of a toy theorem (derived from a non-trivial theorem) can provide insight that would be hard to obtain otherwise.
Toy theorems can also have educational value as well. For example, after presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem.
Examples
A toy theorem of the Brouwer fixed-point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the intermediate value theorem.
Another example of toy theorem is Rolle's theorem, which is obtained from the mean value theorem by equating the function values at the endpoints.
See also
• Corollary
• Fundamental theorem
• Lemma (mathematics)
• Toy model
References
This article incorporates material from toy theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
| Wikipedia |
Trace class
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators.
In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators.
Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces).
Note that the trace operator studied in partial differential equations is an unrelated concept.
Definition
Suppose $H$ is a Hilbert space and $A:H\to H$ a bounded linear operator on $H$ which is non-negative (I.e., semi—positive-definite) and self-adjoint. The trace of $A$, denoted by $\operatorname {Tr} A,$ is the sum of the series[1]
$\operatorname {Tr} A=\sum _{k}\left\langle Ae_{k},e_{k}\right\rangle ,$
where $\left(e_{k}\right)_{k}$ is an orthonormal basis of $H$. The trace is a sum on non-negative reals and is therefore a non-negative real or infinity. It can be shown that the trace does not depend on the choice of orthonormal basis.
For an arbitrary bounded linear operator $T:H\to H$ on $H,$ we define its absolute value, denoted by $|T|,$ to be the positive square root of $T^{*}T,$ that is, $|T|:={\sqrt {T^{*}T}}$ is the unique bounded positive operator on $H$ such that $|T|\circ |T|=T^{*}\circ T.$ The operator $T:H\to H$ is said to be in the trace class if $\operatorname {Tr} (|T|)<\infty .$ We denote the space of all trace class linear operators on H by $B_{1}(H).$ (One can show that this is indeed a vector space.)
If $T$ is in the trace class, we define the trace of $T$ by
$\operatorname {Tr} T=\sum _{k}\left\langle Te_{k},e_{k}\right\rangle ,$
where $\left(e_{k}\right)_{k}$ is an arbitrary orthonormal basis of $H$. It can be shown that this is an absolutely convergent series of complex numbers whose sum does not depend on the choice of orthonormal basis.
When H is finite-dimensional, every operator is trace class and this definition of trace of T coincides with the definition of the trace of a matrix.
Equivalent formulations
Given a bounded linear operator $T:H\to H$, each of the following statements is equivalent to $T$ being in the trace class:
• $\operatorname {Tr} (|T|)<\infty .$[1]
• For some orthonormal basis $\left(e_{k}\right)_{k}$ of H, the sum of positive terms $ \sum _{k}\left\langle |T|\,e_{k},e_{k}\right\rangle $ is finite.
• For every orthonormal basis $\left(e_{k}\right)_{k}$ of H, the sum of positive terms $ \sum _{k}\left\langle |T|\,e_{k},e_{k}\right\rangle $ is finite.
• T is a compact operator and $ \sum _{i=1}^{\infty }s_{i}<\infty ,$ where $s_{1},s_{2},\ldots $ are the eigenvalues of $|T|$ (also known as the singular values of T) with each eigenvalue repeated as often as its multiplicity.[1]
• There exist two orthogonal sequences $\left(x_{i}\right)_{i=1}^{\infty }$ and $\left(y_{i}\right)_{i=1}^{\infty }$ in $H$ and a sequence $\left(\lambda _{i}\right)_{i=1}^{\infty }$ in $\ell ^{1}$ such that for all $x\in H,$ $ T(x)=\sum _{i=1}^{\infty }\lambda _{i}\left\langle x,x_{i}\right\rangle y_{i}.$[2] Here, the infinite sum means that the sequence of partial sums $ \left(\sum _{i=1}^{N}\lambda _{i}\left\langle x,x_{i}\right\rangle y_{i}\right)_{N=1}^{\infty }$ converges to $T(x)$ in H.
• T is a nuclear operator.
• T is equal to the composition of two Hilbert-Schmidt operators.[1]
• $ {\sqrt {|T|}}$ is a Hilbert-Schmidt operator.[1]
• T is an integral operator.[3]
• There exist weakly closed and equicontinuous (and thus weakly compact) subsets $A^{\prime }$ and $B^{\prime \prime }$ of $H^{\prime }$ and $H^{\prime \prime },$ respectively, and some positive Radon measure $\mu $ on $A^{\prime }\times B^{\prime \prime }$ of total mass $\leq 1$ such that for all $x\in H$ and $y^{\prime }\in H^{\prime }$:
$y^{\prime }(T(x))=\int _{A^{\prime }\times B^{\prime \prime }}x^{\prime }(x)\;y^{\prime \prime }\left(y^{\prime }\right)\,\mathrm {d} \mu \left(x^{\prime },y^{\prime \prime }\right).$
Trace-norm
We define the trace-norm of a trace class operator T to be the value
$\|T\|_{1}:=\operatorname {Tr} (|T|).$
One can show that the trace-norm is a norm on the space of all trace class operators $B_{1}(H)$ and that $B_{1}(H)$, with the trace-norm, becomes a Banach space.
If T is trace class then[4]
$\|T\|_{1}=\sup \left\{|\operatorname {Tr} (CT)|:\|C\|\leq 1{\text{ and }}C:H\to H{\text{ is a compact operator }}\right\}.$
Examples
Every bounded linear operator that has a finite-dimensional range (i.e. operators of finite-rank) is trace class;[1] furthermore, the space of all finite-rank operators is a dense subspace of $B_{1}(H)$ (when endowed with the $\|\cdot \|_{1}$ norm).[4] The composition of two Hilbert-Schmidt operators is a trace class operator.[1]
Given any $x,y\in H,$ define the operator $x\otimes y:H\to H$ by $(x\otimes y)(z):=\langle z,y\rangle x.$ Then $x\otimes y$ is a continuous linear operator of rank 1 and is thus trace class; moreover, for any bounded linear operator A on H (and into H), $\operatorname {Tr} (A(x\otimes y))=\langle Ax,y\rangle .$[4]
Properties
1. If $A:H\to H$ is a non-negative self-adjoint operator, then $A$ is trace-class if and only if $\operatorname {Tr} A<\infty .$ Therefore, a self-adjoint operator $A$ is trace-class if and only if its positive part $A^{+}$ and negative part $A^{-}$ are both trace-class. (The positive and negative parts of a self-adjoint operator are obtained by the continuous functional calculus.)
2. The trace is a linear functional over the space of trace-class operators, that is,
$\operatorname {Tr} (aA+bB)=a\operatorname {Tr} (A)+b\operatorname {Tr} (B).$
The bilinear map
$\langle A,B\rangle =\operatorname {Tr} (A^{*}B)$
is an inner product on the trace class; the corresponding norm is called the Hilbert–Schmidt norm. The completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.
3. $\operatorname {Tr} :B_{1}(H)\to \mathbb {C} $ is a positive linear functional such that if $T$ is a trace class operator satisfying $T\geq 0{\text{ and }}\operatorname {Tr} T=0,$ then $T=0.$[1]
4. If $T:H\to H$ is trace-class then so is $T^{*}$ and $\|T\|_{1}=\left\|T^{*}\right\|_{1}.$[1]
5. If $A:H\to H$ is bounded, and $T:H\to H$ is trace-class, then $AT$ and $TA$ are also trace-class (i.e. the space of trace-class operators on H is an ideal in the algebra of bounded linear operators on H), and[1] [5][1]
$\|AT\|_{1}=\operatorname {Tr} (|AT|)\leq \|A\|\|T\|_{1},\quad \|TA\|_{1}=\operatorname {Tr} (|TA|)\leq \|A\|\|T\|_{1}.$
Furthermore, under the same hypothesis,[1]
$\operatorname {Tr} (AT)=\operatorname {Tr} (TA)$
and $|\operatorname {Tr} (AT)|\leq \|A\|\|T\|.$ The last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.
6. If $\left(e_{k}\right)_{k}$ and $\left(f_{k}\right)_{k}$ are two orthonormal bases of H and if T is trace class then $ \sum _{k}\left|\left\langle Te_{k},f_{k}\right\rangle \right|\leq \|T\|_{1}.$[4]
7. If A is trace-class, then one can define the Fredholm determinant of $I+A$:
$\det(I+A):=\prod _{n\geq 1}[1+\lambda _{n}(A)],$
where $\{\lambda _{n}(A)\}_{n}$ is the spectrum of $A.$ The trace class condition on $A$ guarantees that the infinite product is finite: indeed,
$\det(I+A)\leq e^{\|A\|_{1}}.$
It also implies that $\det(I+A)\neq 0$ if and only if $(I+A)$ is invertible.
8. If $A:H\to H$ is trace class then for any orthonormal basis $\left(e_{k}\right)_{k}$ of $H,$ the sum of positive terms $ \sum _{k}\left|\left\langle A\,e_{k},e_{k}\right\rangle \right|$ is finite.[1]
9. If $A=B^{*}C$ for some Hilbert-Schmidt operators $B$ and $C$ then for any normal vector $e\in H,$ $ |\langle Ae,e\rangle |={\frac {1}{2}}\left(\|Be\|^{2}+\|Ce\|^{2}\right)$ holds.[1]
Lidskii's theorem
Let $A$ be a trace-class operator in a separable Hilbert space $H,$ and let $\{\lambda _{n}(A)\}_{n=1}^{N},$ $N\leq \infty $ be the eigenvalues of $A.$ Let us assume that $\lambda _{n}(A)$ are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of $\lambda $ is $k,$ then $\lambda $ is repeated $k$ times in the list $\lambda _{1}(A),\lambda _{2}(A),\dots $). Lidskii's theorem (named after Victor Borisovich Lidskii) states that
$\operatorname {Tr} (A)=\sum _{n=1}^{N}\lambda _{n}(A)$
Note that the series on the right converges absolutely due to Weyl's inequality
$\sum _{n=1}^{N}\left|\lambda _{n}(A)\right|\leq \sum _{m=1}^{M}s_{m}(A)$
between the eigenvalues $\{\lambda _{n}(A)\}_{n=1}^{N}$ and the singular values $\{s_{m}(A)\}_{m=1}^{M}$ of the compact operator $A.$[6]
Relationship between common classes of operators
One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces, with trace-class operators as the noncommutative analogue of the sequence space $\ell ^{1}(\mathbb {N} ).$
Indeed, it is possible to apply the spectral theorem to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an $\ell ^{1}$ sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of $\ell ^{\infty }(\mathbb {N} ),$ the compact operators that of $c_{0}$ (the sequences convergent to 0), Hilbert–Schmidt operators correspond to $\ell ^{2}(\mathbb {N} ),$ and finite-rank operators to $c_{00}$ (the sequences that have only finitely many non-zero terms). To some extent, the relationships between these classes of operators are similar to the relationships between their commutative counterparts.
Recall that every compact operator $T$ on a Hilbert space takes the following canonical form: there exist orthonormal bases $\left(u_{i}\right)_{i}$ and $\left(v_{i}\right)_{i}$ and a sequence $\left(\alpha _{i}\right)_{i}$ of non-negative numbers with $\alpha _{i}\to 0$ such that
$Tx=\sum _{i}\alpha _{i}\langle x,v_{i}\rangle u_{i}\quad {\text{ for all }}x\in H.$
Making the above heuristic comments more precise, we have that $T$ is trace-class iff the series $ \sum _{i}\alpha _{i}$ is convergent, $T$ is Hilbert–Schmidt iff $ \sum _{i}\alpha _{i}^{2}$ is convergent, and $T$ is finite-rank iff the sequence $\left(\alpha _{i}\right)_{i}$ has only finitely many nonzero terms. This allows to relate these classes of operators. The following inclusions hold and are all proper when $H$ is infinite-dimensional:
$\{{\text{ finite rank }}\}\subseteq \{{\text{ trace class }}\}\subseteq \{{\text{ Hilbert-Schmidt }}\}\subseteq \{{\text{ compact }}\}.$
The trace-class operators are given the trace norm $ \|T\|_{1}=\operatorname {Tr} \left[\left(T^{*}T\right)^{1/2}\right]=\sum _{i}\alpha _{i}.$ The norm corresponding to the Hilbert–Schmidt inner product is
$\|T\|_{2}=\left[\operatorname {Tr} \left(T^{*}T\right)\right]^{1/2}=\left(\sum _{i}\alpha _{i}^{2}\right)^{1/2}.$
Also, the usual operator norm is $ \|T\|=\sup _{i}\left(\alpha _{i}\right).$ By classical inequalities regarding sequences,
$\|T\|\leq \|T\|_{2}\leq \|T\|_{1}$
for appropriate $T.$
It is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.
Trace class as the dual of compact operators
The dual space of $c_{0}$ is $\ell ^{1}(\mathbb {N} ).$ Similarly, we have that the dual of compact operators, denoted by $K(H)^{*},$ is the trace-class operators, denoted by $B_{1}.$ The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let $f\in K(H)^{*},$ we identify $f$ with the operator $T_{f}$ defined by
$\langle T_{f}x,y\rangle =f\left(S_{x,y}\right),$
where $S_{x,y}$ is the rank-one operator given by
$S_{x,y}(h)=\langle h,y\rangle x.$
This identification works because the finite-rank operators are norm-dense in $K(H).$ In the event that $T_{f}$ is a positive operator, for any orthonormal basis $u_{i},$ one has
$\sum _{i}\langle T_{f}u_{i},u_{i}\rangle =f(I)\leq \|f\|,$
where $I$ is the identity operator:
$I=\sum _{i}\langle \cdot ,u_{i}\rangle u_{i}.$
But this means that $T_{f}$ is trace-class. An appeal to polar decomposition extend this to the general case, where $T_{f}$ need not be positive.
A limiting argument using finite-rank operators shows that $\|T_{f}\|_{1}=\|f\|.$ Thus $K(H)^{*}$ is isometrically isomorphic to $C_{1}.$
As the predual of bounded operators
Recall that the dual of $\ell ^{1}(\mathbb {N} )$ is $\ell ^{\infty }(\mathbb {N} ).$ In the present context, the dual of trace-class operators $B_{1}$ is the bounded operators $B(H).$ More precisely, the set $B_{1}$ is a two-sided ideal in $B(H).$ So given any operator $T\in B(H),$ we may define a continuous linear functional $\varphi _{T}$ on $B_{1}$ by $\varphi _{T}(A)=\operatorname {Tr} (AT).$ This correspondence between bounded linear operators and elements $\varphi _{T}$ of the dual space of $B_{1}$ is an isometric isomorphism. It follows that $B(H)$ is the dual space of $C_{1}.$ This can be used to define the weak-* topology on $B(H).$
See also
• Nuclear operator
• Nuclear operators between Banach spaces
• Trace operator
References
1. Conway 1990, p. 267.
2. Trèves 2006, p. 494.
3. Trèves 2006, pp. 502–508.
4. Conway 1990, p. 268.
5. M. Reed and B. Simon, Functional Analysis, Exercises 27, 28, page 218.
6. Simon, B. (2005) Trace ideals and their applications, Second Edition, American Mathematical Society.
Bibliography
• Conway, John (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
• Dixmier, J. (1969). Les Algebres d'Operateurs dans l'Espace Hilbertien. Gauthier-Villars.
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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| Wikipedia |
Trace diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix. Several results in linear algebra, such as Cramer's Rule and the Cayley–Hamilton theorem, have simple diagrammatic proofs. They are closely related to Penrose's graphical notation.
Formal definition
Let V be a vector space of dimension n over a field F (with n≥2), and let Hom(V,V) denote the linear transformations on V. An n-trace diagram is a graph ${\mathcal {D}}=(V_{1}\sqcup V_{2}\sqcup V_{n},E)$, where the sets Vi (i = 1, 2, n) are composed of vertices of degree i, together with the following additional structures:
• a ciliation at each vertex in the graph, which is an explicit ordering of the adjacent edges at that vertex;
• a labeling V2 → Hom(V,V) associating each degree-2 vertex to a linear transformation.
Note that V2 and Vn should be considered as distinct sets in the case n = 2. A framed trace diagram is a trace diagram together with a partition of the degree-1 vertices V1 into two disjoint ordered collections called the inputs and the outputs.
The "graph" underlying a trace diagram may have the following special features, which are not always included in the standard definition of a graph:
• Loops are permitted (a loop is an edge that connects a vertex to itself).
• Edges that have no vertices are permitted, and are represented by small circles.
• Multiple edges between the same two vertices are permitted.
Drawing conventions
• When trace diagrams are drawn, the ciliation on an n-vertex is commonly represented by a small mark between two of the incident edges (in the figure above, a small red dot); the specific ordering of edges follows by proceeding counter-clockwise from this mark.
• The ciliation and labeling at a degree-2 vertex are combined into a single directed node that allows one to differentiate the first edge (the incoming edge) from the second edge (the outgoing edge).
• Framed diagrams are drawn with inputs at the bottom of the diagram and outputs at the top of the diagram. In both cases, the ordering corresponds to reading from left to right.
Correspondence with multilinear functions
Every framed trace diagram corresponds to a multilinear function between tensor powers of the vector space V. The degree-1 vertices correspond to the inputs and outputs of the function, while the degree-n vertices correspond to the generalized Levi-Civita symbol (which is an anti-symmetric tensor related to the determinant). If a diagram has no output strands, its function maps tensor products to a scalar. If there are no degree-1 vertices, the diagram is said to be closed and its corresponding function may be identified with a scalar.
By definition, a trace diagram's function is computed using signed graph coloring. For each edge coloring of the graph's edges by n labels, so that no two edges adjacent to the same vertex have the same label, one assigns a weight based on the labels at the vertices and the labels adjacent to the matrix labels. These weights become the coefficients of the diagram's function.
In practice, a trace diagram's function is typically computed by decomposing the diagram into smaller pieces whose functions are known. The overall function can then be computed by re-composing the individual functions.
Examples
3-Vector diagrams
Several vector identities have easy proofs using trace diagrams. This section covers 3-trace diagrams. In the translation of diagrams to functions, it can be shown that the positions of ciliations at the degree-3 vertices has no influence on the resulting function, so they may be omitted.
It can be shown that the cross product and dot product of 3-dimensional vectors are represented by
In this picture, the inputs to the function are shown as vectors in yellow boxes at the bottom of the diagram. The cross product diagram has an output vector, represented by the free strand at the top of the diagram. The dot product diagram does not have an output vector; hence, its output is a scalar.
As a first example, consider the scalar triple product identity
$(\mathbf {u} \times \mathbf {v} )\cdot \mathbf {w} =\mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} )=(\mathbf {w} \times \mathbf {u} )\cdot \mathbf {v} =\det(\mathbf {u} \mathbf {v} \mathbf {w} ).$
To prove this diagrammatically, note that all of the following figures are different depictions of the same 3-trace diagram (as specified by the above definition):
Combining the above diagrams for the cross product and the dot product, one can read off the three leftmost diagrams as precisely the three leftmost scalar triple products in the above identity. It can also be shown that the rightmost diagram represents det[u v w]. The scalar triple product identity follows because each is a different representation of the same diagram's function.
As a second example, one can show that
(where the equality indicates that the identity holds for the underlying multilinear functions). One can show that this kind of identity does not change by "bending" the diagram or attaching more diagrams, provided the changes are consistent across all diagrams in the identity. Thus, one can bend the top of the diagram down to the bottom, and attach vectors to each of the free edges, to obtain
which reads
$(\mathbf {x} \times \mathbf {u} )\cdot (\mathbf {v} \times \mathbf {w} )=(\mathbf {x} \cdot \mathbf {v} )(\mathbf {u} \cdot \mathbf {w} )-(\mathbf {x} \cdot \mathbf {w} )(\mathbf {u} \cdot \mathbf {v} ),$
a well-known identity relating four 3-dimensional vectors.
Diagrams with matrices
The simplest closed diagrams with a single matrix label correspond to the coefficients of the characteristic polynomial, up to a scalar factor that depends only on the dimension of the matrix. One representation of these diagrams is shown below, where $\propto $ is used to indicate equality up to a scalar factor that depends only on the dimension n of the underlying vector space.
.
Properties
Let G be the group of n×n matrices. If a closed trace diagram is labeled by k different matrices, it may be interpreted as a function from $G^{k}$ to an algebra of multilinear functions. This function is invariant under simultaneous conjugation, that is, the function corresponding to $(g_{1},\ldots ,g_{k})$ is the same as the function corresponding to $(ag_{1}a^{-1},\ldots ,ag_{k}a^{-1})$ for any invertible $a\in G$.
Extensions and applications
Trace diagrams may be specialized for particular Lie groups by altering the definition slightly. In this context, they are sometimes called birdtracks, tensor diagrams, or Penrose graphical notation.
Trace diagrams have primarily been used by physicists as a tool for studying Lie groups. The most common applications use representation theory to construct spin networks from trace diagrams. In mathematics, they have been used to study character varieties.
See also
• Multilinear map
• Gain graph
References
Books:
• Diagram Techniques in Group Theory, G. E. Stedman, Cambridge University Press, 1990
• Group Theory: Birdtracks, Lie's, and Exceptional Groups, Predrag Cvitanović, Princeton University Press, 2008, http://birdtracks.eu/
| Wikipedia |
Trace field of a representation
In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name field of definition.
Fuchsian and Kleinian groups
Trace field and invariant trace fields for Fuchsian groups
Fuchsian groups are discrete subgroups of $\mathrm {PSL} _{2}(\mathbb {R} )$. The trace of an element in $\mathrm {PSL} _{2}(\mathbb {R} )$ is well-defined up to sign (by taking the trace of an arbitrary preimage in $\mathrm {SL} _{2}(\mathbb {R} )$) and the trace field of $\Gamma $ is the field generated over $\mathbb {Q} $ by the traces of all elements of $\Gamma $ (see for example in Maclachlan & Reid (2003)).
The invariant trace field is equal to the trace field of the subgroup $\Gamma ^{(2)}$ generated by all squares of elements of $\Gamma $ (a finite-index subgroup of $\Gamma $).[1]
The invariant trace field of Fuchsian groups is stable under taking commensurable groups. This is not the case for the trace field;[2] in particular the trace field is in general different from the invariant trace field.
Quaternion algebras for Fuchsian groups
Let $\Gamma $ be a Fuchsian group and $k$ its trace field. Let $A$ be the $k$-subalgebra of the matrix algebra $M_{2}(\mathbb {R} )$ generated by the preimages of elements of $\Gamma $. The algebra $A$ is then as simple as possible, more precisely:[3]
If $\Gamma $ is of the first or second type then $A$ is a quaternion algebra over $k$.
The algebra $A$ is called the quaternion algebra of $\Gamma $. The quaternion algebra of $\Gamma ^{(2)}$ is called the invariant quaternion algebra of $\Gamma $, denoted by $A\Gamma $. As for trace fields, the former is not the same for all groups in the same commensurability class but the latter is.
If $\Gamma $ is an arithmetic Fuchsian group then $k\Gamma $ and $A\Gamma $ together are a number field and quaternion algebra from which a group commensurable to $\Gamma $ may be derived.[4]
Kleinian groups
The theory for Kleinian groups (discrete subgroups of $\mathrm {PSL} _{2}(\mathbb {C} )$) is mostly similar as that for Fuchsian groups.[5] One big difference is that the trace field of a group of finite covolume is always a number field.[6]
Trace fields and fields of definition for subgroups of Lie groups
Definition
When considering subgroups of general Lie groups (which are not necessarily defined as a matrix groups) one has to use a linear representation of the group to take traces of elements. The most natural one is the adjoint representation. It turns out that for applications it is better, even for groups which have a natural lower-dimensional linear representation (such as the special linear groups $\mathrm {SL} _{n}(\mathbb {R} )$), to always define the trace field using the adjoint representation. Thus we have the following definition, originally due to Ernest Vinberg,[7] who used the terminology "field of definition".[8]
Let $G$ be a Lie group and $\Gamma \subset G$ a subgroup. Let $\rho $ be the adjoint representation of $G$. The trace field of $\Gamma $ is the field:
$k\Gamma =\mathbb {Q} (\{\operatorname {trace} (\rho (\gamma )):\gamma \in \Gamma \}).$
If two Zariski-dense subgroups of $G$ are commensurable then they have the same trace field in this sense.
The trace field for lattices
Let $G$ be a semisimple Lie group and $\Gamma \subset G$ a lattice. Suppose further that either $\Gamma $ is irreducible and $G$ is not locally isomorphic to $\mathrm {SL} _{2}(\mathbb {R} )$, or that $\Gamma $ has no factor locally isomorphic to $\mathrm {SL} _{2}(\mathbb {R} )$. Then local rigidity implies the following result.
The field $k\Gamma $ is a number field.
Furthermore, there exists an algebraic group $\mathbf {G} $ over $k\Gamma $ such that the group of real points $\mathbf {G} (\mathbb {R} )$ is isomorphic to $\rho (G)$ and $\rho (\Gamma )$ is contained in a conjugate of $\mathbf {G} (k\Gamma )$.[7][9] Thus $k\Gamma $ is a "field of definition" for $\Gamma $ in the sense that it is a field of definition of its Zariski closure in the adjoint representation.
In the case where $\Gamma $ is arithmetic then it is commensurable to the arithmetic group defined by $\mathbf {G} $.
For Fuchsian groups the field $k\Gamma $ defined above is equal to its invariant trace field. For Kleinian groups they are the same if we use the adjoint representation over the complex numbers.[10]
Notes
1. Maclachlan & Reid 2003, Chapter 3.3.
2. Maclachlan & Reid 2003, Example 3.3.1.
3. Maclachlan & Reid 2003, Theorem 3.2.1.
4. Maclachlan & Reid 2003, Chapter 8.4.
5. Maclachlan & Reid 2003, Chapter 3.
6. Maclachlan & Reid 2003, Theorem 3.1.2.
7. Vinberg 1971.
8. Margulis 1991, Chapter VIII.
9. Margulis 1991, Chapter VIII, proposition 3.22.
10. Maclachlan & Reid 2003, p. 321.
References
• Vinberg, Ernest (1971). "Rings of definition of dense subgroups of semisimple linear groups". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). Vol. 35. pp. 45–55. MR 0279206.
• Maclachlan, Colin; Reid, Alan (2003). The arithmetic of hyperbolic 3-manifolds. Springer.
• Margulis, Grigory (1991). Discrete subgroups of semisimple Lie groups. Ergebnisse de Mathematik und ihrer Grenzgebiete. Springer-Verlag. ISBN 3-540-12179-X. MR 1090825.
| Wikipedia |
Field trace
In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.
Definition
Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L,
$m_{\alpha }:L\to L{\text{ given by }}m_{\alpha }(x)=\alpha x$,
is a K-linear transformation of this vector space into itself. The trace, TrL/K(α), is defined as the trace (in the linear algebra sense) of this linear transformation.[1]
For α in L, let σ1(α), ..., σn(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K). Then
$\operatorname {Tr} _{L/K}(\alpha )=[L:K(\alpha )]\sum _{j=1}^{n}\sigma _{j}(\alpha ).$
If L/K is separable then each root appears only once[2] (however this does not mean the coefficient above is one; for example if α is the identity element 1 of K then the trace is [L:K ] times 1).
More particularly, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α,[1] i.e.,
$\operatorname {Tr} _{L/K}(\alpha )=\sum _{\sigma \in \operatorname {Gal} (L/K)}\sigma (\alpha ),$
where Gal(L/K) denotes the Galois group of L/K.
Example
Let $L=\mathbb {Q} ({\sqrt {d}})$ be a quadratic extension of $\mathbb {Q} $. Then a basis of $L/\mathbb {Q} $ is $\{1,{\sqrt {d}}\}.$ If $\alpha =a+b{\sqrt {d}}$ then the matrix of $m_{\alpha }$ is:
$\left[{\begin{matrix}a&bd\\b&a\end{matrix}}\right]$,
and so, $\operatorname {Tr} _{L/\mathbb {Q} }(\alpha )=[L:\mathbb {Q} (\alpha )]\left(\sigma _{1}(\alpha )+\sigma _{2}(\alpha )\right)=1\times \left(\sigma _{1}(\alpha )+{\overline {\sigma _{1}}}(\alpha )\right)=a+b{\sqrt {d}}+a-b{\sqrt {d}}=2a$.[1] The minimal polynomial of α is X2 − 2a X + (a2 − db2).
Properties of the trace
Several properties of the trace function hold for any finite extension.[3]
The trace TrL/K : L → K is a K-linear map (a K-linear functional), that is
$\operatorname {Tr} _{L/K}(\alpha a+\beta b)=\alpha \operatorname {Tr} _{L/K}(a)+\beta \operatorname {Tr} _{L/K}(b){\text{ for all }}\alpha ,\beta \in K$.
If α ∈ K then $\operatorname {Tr} _{L/K}(\alpha )=[L:K]\alpha .$
Additionally, trace behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from M to L with the trace from L to K, i.e.
$\operatorname {Tr} _{M/K}=\operatorname {Tr} _{L/K}\circ \operatorname {Tr} _{M/L}$.
Finite fields
Let L = GF(qn) be a finite extension of a finite field K = GF(q). Since L/K is a Galois extension, if α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.[4]
$\operatorname {Tr} _{L/K}(\alpha )=\alpha +\alpha ^{q}+\cdots +\alpha ^{q^{n-1}}.$
In this setting we have the additional properties:[5]
• $\operatorname {Tr} _{L/K}(a^{q})=\operatorname {Tr} _{L/K}(a){\text{ for }}a\in L$.
• For any $\alpha \in K$, there are exactly $q^{n-1}$ elements $b\in L$ with $\operatorname {Tr} _{L/K}(b)=\alpha $.
Theorem.[6] For b ∈ L, let Fb be the map $a\mapsto \operatorname {Tr} _{L/K}(ba).$ Then Fb ≠ Fc if b ≠ c. Moreover, the K-linear transformations from L to K are exactly the maps of the form Fb as b varies over the field L.
When K is the prime subfield of L, the trace is called the absolute trace and otherwise it is a relative trace.[4]
Application
A quadratic equation, ax2 + bx + c = 0 with a ≠ 0, and coefficients in the finite field $\operatorname {GF} (q)=\mathbb {F} _{q}$ has either 0, 1 or 2 roots in GF(q) (and two roots, counted with multiplicity, in the quadratic extension GF(q2)). If the characteristic of GF(q) is odd, the discriminant Δ = b2 − 4ac indicates the number of roots in GF(q) and the classical quadratic formula gives the roots. However, when GF(q) has even characteristic (i.e., q = 2h for some positive integer h), these formulas are no longer applicable.
Consider the quadratic equation ax2 + bx + c = 0 with coefficients in the finite field GF(2h).[7] If b = 0 then this equation has the unique solution $x={\sqrt {\frac {c}{a}}}$ in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:
$y^{2}+y+\delta =0,{\text{ where }}\delta ={\frac {ac}{b^{2}}}.$
This equation has two solutions in GF(q) if and only if the absolute trace $\operatorname {Tr} _{GF(q)/GF(2)}(\delta )=0.$ In this case, if y = s is one of the solutions, then y = s + 1 is the other. Let k be any element of GF(q) with $\operatorname {Tr} _{GF(q)/GF(2)}(k)=1.$ Then a solution to the equation is given by:
$y=s=k\delta ^{2}+(k+k^{2})\delta ^{4}+\ldots +(k+k^{2}+\ldots +k^{2^{h-2}})\delta ^{2^{h-1}}.$
When h = 2m' + 1, a solution is given by the simpler expression:
$y=s=\delta +\delta ^{2^{2}}+\delta ^{2^{4}}+\ldots +\delta ^{2^{2m}}.$
Trace form
When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (x, y) to TrL/K(xy) is a nondegenerate, symmetric bilinear form called the trace form. If L/K is a Galois extension, the trace form is invariant with respect to the Galois group.
The trace form is used in algebraic number theory in the theory of the different ideal.
The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K.[8] The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.[8]
If L/K is an inseparable extension, then the trace form is identically 0.[9]
See also
• Field norm
• Reduced trace
Notes
1. Rotman 2002, p. 940
2. Rotman 2002, p. 941
3. Roman 2006, p. 151
4. Lidl & Niederreiter 1997, p.54
5. Mullen & Panario 2013, p. 21
6. Lidl & Niederreiter 1997, p.56
7. Hirschfeld 1979, pp. 3-4
8. Lorenz (2008) p.38
9. Isaacs 1994, p. 369 as footnoted in Rotman 2002, p. 943
References
• Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853526-0
• Isaacs, I.M. (1994), Algebra, A Graduate Course, Brooks/Cole Publishing
• Lidl, Rudolf; Niederreiter, Harald (1997) [1983], Finite Fields, Encyclopedia of Mathematics and its Applications, vol. 20 (Second ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6
• Roman, Steven (2006), Field theory, Graduate Texts in Mathematics, vol. 158 (Second ed.), Springer, Chapter 8, ISBN 978-0-387-27677-9, Zbl 1172.12001
• Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 978-0-13-087868-7
Further reading
• Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. Vol. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017.
• Section VI.5 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
| Wikipedia |
Trace identity
In mathematics, a trace identity is any equation involving the trace of a matrix.
Properties
Trace identities are invariant under simultaneous conjugation.
Uses
They are frequently used in the invariant theory of $n\times n$ matrices to find the generators and relations of the ring of invariants, and therefore are useful in answering questions similar to that posed by Hilbert's fourteenth problem.
Examples
• The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy
$\operatorname {tr} \left(A^{n}\right)-c_{n-1}\operatorname {tr} (A)\operatorname {tr} \left(A^{n-1}\right)+\cdots +(-1)^{n}n\det(A)=0\,$
where the coefficients $c_{i}$ are given by the elementary symmetric polynomials of the eigenvalues of A.
• All square matrices satisfy
$\operatorname {tr} (A)=\operatorname {tr} \left(A^{\mathsf {T}}\right).\,$
See also
• Trace inequality – inequalities involving linear operators on Hilbert spacesPages displaying wikidata descriptions as a fallback
References
Rowen, Louis Halle (2008), Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, vol. 2, American Mathematical Society, p. 412, ISBN 9780821841532.
| Wikipedia |
Trace operator
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions (boundary value problems), where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions.
Motivation
On a bounded, smooth domain $ \Omega \subset \mathbb {R} ^{n}$, consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions:
${\begin{alignedat}{2}-\Delta u&=f&\quad &{\text{in }}\Omega ,\\u&=g&&{\text{on }}\partial \Omega \end{alignedat}}$
with given functions $ f$ and $ g$ with regularity discussed in the application section below. The weak solution $ u\in H^{1}(\Omega )$ of this equation must satisfy
$\int _{\Omega }\nabla u\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }f\varphi \,\mathrm {d} x$ for all $ \varphi \in H_{0}^{1}(\Omega )$.
The $ H^{1}(\Omega )$-regularity of $ u$ is sufficient for the well-definedness of this integral equation. It is not apparent, however, in which sense $ u$ can satisfy the boundary condition $ u=g$ on $ \partial \Omega $: by definition, $ u\in H^{1}(\Omega )\subset L^{2}(\Omega )$ is an equivalence class of functions which can have arbitrary values on $ \partial \Omega $ since this is a null set with respect to the n-dimensional Lebesgue measure.
If $ \Omega \subset \mathbb {R} ^{1}$ there holds $ H^{1}(\Omega )\hookrightarrow C^{0}({\bar {\Omega }})$ by Sobolev's embedding theorem, such that $ u$ can satisfy the boundary condition in the classical sense, i.e. the restriction of $ u$ to $ \partial \Omega $ agrees with the function $ g$ (more precisely: there exists a representative of $ u$ in $ C({\bar {\Omega }})$ with this property). For $ \Omega \subset \mathbb {R} ^{n}$ with $ n>1$ such an embedding does not exist and the trace operator $ T$ presented here must be used to give meaning to $ u|_{\partial \Omega }$. Then $ u\in H^{1}(\Omega )$ with $ Tu=g$ is called a weak solution to the boundary value problem if the integral equation above is satisfied. For the definition of the trace operator to be reasonable, there must hold $ Tu=u|_{\partial \Omega }$ for sufficiently regular $ u$.
Trace theorem
The trace operator can be defined for functions in the Sobolev spaces $ W^{1,p}(\Omega )$ with $ 1\leq p<\infty $, see the section below for possible extensions of the trace to other spaces. Let $ \Omega \subset \mathbb {R} ^{n}$ for $ n\in \mathbb {N} $ be a bounded domain with Lipschitz boundary. Then[1] there exists a bounded linear trace operator
$T\colon W^{1,p}(\Omega )\to L^{p}(\partial \Omega )$
such that $ T$ extends the classical trace, i.e.
$Tu=u|_{\partial \Omega }$ for all $ u\in W^{1,p}(\Omega )\cap C({\bar {\Omega }})$.
The continuity of $ T$ implies that
$\|Tu\|_{L^{p}(\partial \Omega )}\leq C\|u\|_{W^{1,p}(\Omega )}$ for all $ u\in W^{1,p}(\Omega )$
with constant only depending on $ p$ and $ \Omega $. The function $ Tu$ is called trace of $ u$ and is often simply denoted by $ u|_{\partial \Omega }$. Other common symbols for $ T$ include $ tr$ and $ \gamma $.
Construction
This paragraph follows Evans,[2] where more details can be found, and assumes that $ \Omega $ has a $ C^{1}$-boundary. A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo.[1] On a $ C^{1}$-domain, the trace operator can be defined as continuous linear extension of the operator
$T:C^{\infty }({\bar {\Omega }})\to L^{p}(\partial \Omega )$
to the space $ W^{1,p}(\Omega )$. By density of $ C^{\infty }({\bar {\Omega }})$ in $ W^{1,p}(\Omega )$ such an extension is possible if $ T$ is continuous with respect to the $ W^{1,p}(\Omega )$-norm. The proof of this, i.e. that there exists $ C>0$ (depending on $ \Omega $ and $ p$) such that
$\|Tu\|_{L^{p}(\partial \Omega )}\leq C\|u\|_{W^{1,p}(\Omega )}$ for all $u\in C^{\infty }({\bar {\Omega }}).$
is the central ingredient in the construction of the trace operator. A local variant of this estimate for $ C^{1}({\bar {\Omega }})$-functions is first proven for a locally flat boundary using the divergence theorem. By transformation, a general $ C^{1}$-boundary can be locally straightened to reduce to this case, where the $ C^{1}$-regularity of the transformation requires that the local estimate holds for $ C^{1}({\bar {\Omega }})$-functions.
With this continuity of the trace operator in $ C^{\infty }({\bar {\Omega }})$ an extension to $ W^{1,p}(\Omega )$ exists by abstract arguments and $ Tu$ for $ u\in W^{1,p}(\Omega )$ can be characterized as follows. Let $ u_{k}\in C^{\infty }({\bar {\Omega }})$ be a sequence approximating $ u\in W^{1,p}(\Omega )$ by density. By the proven continuity of $ T$ in $ C^{\infty }({\bar {\Omega }})$ the sequence $ u_{k}|_{\partial \Omega }$ is a Cauchy sequence in $ L^{p}(\partial \Omega )$ and $ Tu=\lim _{k\to \infty }u_{k}|_{\partial \Omega }$ with limit taken in $ L^{p}(\partial \Omega )$.
The extension property $ Tu=u|_{\partial \Omega }$ holds for $ u\in C^{\infty }({\bar {\Omega }})$ by construction, but for any $ u\in W^{1,p}(\Omega )\cap C({\bar {\Omega }})$ there exists a sequence $ u_{k}\in C^{\infty }({\bar {\Omega }})$ which converges uniformly on $ {\bar {\Omega }}$ to $ u$, verifying the extension property on the larger set $ W^{1,p}(\Omega )\cap C({\bar {\Omega }})$.
The case p = ∞
If $ \Omega $ is bounded and has a $ C^{1}$-boundary then by Morrey's inequality there exists a continuous embedding $ W^{1,\infty }(\Omega )\hookrightarrow C^{0,1}(\Omega )$, where $ C^{0,1}(\Omega )$ denotes the space of Lipschitz continuous functions. In particular, any function $ u\in W^{1,\infty }(\Omega )$ has a classical trace $ u|_{\partial \Omega }\in C(\partial \Omega )$ and there holds
$\|u|_{\partial \Omega }\|_{C(\partial \Omega )}\leq \|u\|_{C^{0,1}(\Omega )}\leq C\|u\|_{W^{1,\infty }(\Omega )}.$
Functions with trace zero
The Sobolev spaces $ W_{0}^{1,p}(\Omega )$ for $ 1\leq p<\infty $ are defined as the closure of the set of compactly supported test functions $ C_{c}^{\infty }(\Omega )$ with respect to the $ W^{1,p}(\Omega )$-norm. The following alternative characterization holds:
$W_{0}^{1,p}(\Omega )=\{u\in W^{1,p}(\Omega )\mid Tu=0\}=\ker(T\colon W^{1,p}(\Omega )\to L^{p}(\partial \Omega )),$
where $ \ker(T)$ is the kernel of $ T$, i.e. $ W_{0}^{1,p}(\Omega )$ is the subspace of functions in $ W^{1,p}(\Omega )$ with trace zero.
Image of the trace operator
For p > 1
The trace operator is not surjective onto $ L^{p}(\partial \Omega )$ if $ p>1$, i.e. not every function in $ L^{p}(\partial \Omega )$ is the trace of a function in $ W^{1,p}(\Omega )$. As elaborated below the image consists of functions which satisfy an $ L^{p}$-version of Hölder continuity.
Abstract characterization
An abstract characterization of the image of $ T$ can be derived as follows. By the isomorphism theorems there holds
$T(W^{1,p}(\Omega ))\cong W^{1,p}(\Omega )/\ker(T\colon W^{1,p}(\Omega )\to L^{p}(\partial \Omega ))=W^{1,p}(\Omega )/W_{0}^{1,p}(\Omega )$
where $ X/N$ denotes the quotient space of the Banach space $ X$ by the subspace $ N\subset X$ and the last identity follows from the characterization of $ W_{0}^{1,p}(\Omega )$ from above. Equipping the quotient space with the quotient norm defined by
$\|u\|_{W^{1,p}(\Omega )/W_{0}^{1,p}(\Omega )}=\inf _{u_{0}\in W_{0}^{1,p}(\Omega )}\|u-u_{0}\|_{W^{1,p}(\Omega )}$
the trace operator $ T$ is then a surjective, bounded linear operator
$T\colon W^{1,p}(\Omega )\to W^{1,p}(\Omega )/W_{0}^{1,p}(\Omega )$.
Characterization using Sobolev–Slobodeckij spaces
A more concrete representation of the image of $ T$ can be given using Sobolev-Slobodeckij spaces which generalize the concept of Hölder continuous functions to the $ L^{p}$-setting. Since $ \partial \Omega $ is a (n-1)-dimensional Lipschitz manifold embedded into $ \mathbb {R} ^{n}$ an explicit characterization of these spaces is technically involved. For simplicity consider first a planar domain $ \Omega '\subset \mathbb {R} ^{n-1}$. For $ v\in L^{p}(\Omega ')$ define the (possibly infinite) norm
$\|v\|_{W^{1-1/p,p}(\Omega ')}=\left(\|v\|_{L^{p}(\Omega ')}^{p}+\int _{\Omega '\times \Omega '}{\frac {|v(x)-v(y)|^{p}}{|x-y|^{(1-1/p)p+(n-1)}}}\,\mathrm {d} (x,y)\right)^{1/p}$
which generalizes the Hölder condition $ |v(x)-v(y)|\leq C|x-y|^{1-1/p}$. Then
$W^{1-1/p,p}(\Omega ')=\left\{v\in L^{p}(\Omega ')\;\mid \;\|v\|_{W^{1-1/p,p}(\Omega ')}<\infty \right\}$
equipped with the previous norm is a Banach space (a general definition of $ W^{s,p}(\Omega ')$ for non-integer $ s>0$ can be found in the article for Sobolev-Slobodeckij spaces). For the (n-1)-dimensional Lipschitz manifold $ \partial \Omega $ define $ W^{1-1/p,p}(\partial \Omega )$ by locally straightening $ \partial \Omega $ and proceeding as in the definition of $ W^{1-1/p,p}(\Omega ')$.
The space $ W^{1-1/p,p}(\partial \Omega )$ can then be identified as the image of the trace operator and there holds[1] that
$T\colon W^{1,p}(\Omega )\to W^{1-1/p,p}(\partial \Omega )$
is a surjective, bounded linear operator.
For p = 1
For $ p=1$ the image of the trace operator is $ L^{1}(\partial \Omega )$ and there holds[1] that
$T\colon W^{1,1}(\Omega )\to L^{1}(\partial \Omega )$
is a surjective, bounded linear operator.
Right-inverse: trace extension operator
The trace operator is not injective since multiple functions in $ W^{1,p}(\Omega )$ can have the same trace (or equivalently, $ W_{0}^{1,p}(\Omega )\neq 0$). The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain. Specifically, for $ 1<p<\infty $ there exists a bounded, linear trace extension operator[3]
$E\colon W^{1-1/p,p}(\partial \Omega )\to W^{1,p}(\Omega )$,
using the Sobolev-Slobodeckij characterization of the trace operator's image from the previous section, such that
$T(Ev)=v$ for all $ v\in W^{1-1/p,p}(\partial \Omega )$
and, by continuity, there exists $ C>0$ with
$\|Ev\|_{W^{1,p}(\Omega )}\leq C\|v\|_{W^{1-1/p,p}(\partial \Omega )}$.
Notable is not the mere existence but the linearity and continuity of the right inverse. This trace extension operator must not be confused with the whole-space extension operators $ W^{1,p}(\Omega )\to W^{1,p}(\mathbb {R} ^{n})$ which play a fundamental role in the theory of Sobolev spaces.
Extension to other spaces
Higher derivatives
Many of the previous results can be extended to $ W^{m,p}(\Omega )$ with higher differentiability $ m=2,3,\ldots $ if the domain is sufficiently regular. Let $ N$ denote the exterior unit normal field on $ \partial \Omega $. Since $ u|_{\partial \Omega }$ can encode differentiability properties in tangential direction only the normal derivative $ \partial _{N}u|_{\partial \Omega }$ is of additional interest for the trace theory for $ m=2$. Similar arguments apply to higher-order derivatives for $ m>2$.
Let $ 1<p<\infty $ and $ \Omega \subset \mathbb {R} ^{n}$ be a bounded domain with $ C^{m,1}$-boundary. Then[3] there exists a surjective, bounded linear higher-order trace operator
$T_{m}\colon W^{m,p}(\Omega )\to \prod _{l=0}^{m-1}W^{m-l-1/p,p}(\partial \Omega )$
with Sobolev-Slobodeckij spaces $ W^{s,p}(\partial \Omega )$ for non-integer $ s>0$ defined on $ \partial \Omega $ through transformation to the planar case $ W^{s,p}(\Omega ')$ for $ \Omega '\subset \mathbb {R} ^{n-1}$, whose definition is elaborated in the article on Sobolev-Slobodeckij spaces. The operator $ T_{m}$ extends the classical normal traces in the sense that
$T_{m}u=\left(u|_{\partial \Omega },\partial _{N}u|_{\partial \Omega },\ldots ,\partial _{N}^{m-1}u|_{\partial \Omega }\right)$ for all $ u\in W^{m,p}(\Omega )\cap C^{m-1}({\bar {\Omega }}).$
Furthermore, there exists a bounded, linear right-inverse of $ T_{m}$, a higher-order trace extension operator[3]
$E_{m}\colon \prod _{l=0}^{m-1}W^{m-l-1/p,p}(\partial \Omega )\to W^{m,p}(\Omega )$.
Finally, the spaces $ W_{0}^{m,p}(\Omega )$, the completion of $ C_{c}^{\infty }(\Omega )$ in the $ W^{m,p}(\Omega )$-norm, can be characterized as the kernel of $ T_{m}$,[3] i.e.
$W_{0}^{m,p}(\Omega )=\{u\in W^{m,p}(\Omega )\mid T_{m}u=0\}$.
No trace in Lp
There is no sensible extension of the concept of traces to $ L^{p}(\Omega )$ for $ 1\leq p<\infty $ since any bounded linear operator which extends the classical trace must be zero on the space of test functions $ C_{c}^{\infty }(\Omega )$, which is a dense subset of $ L^{p}(\Omega )$, implying that such an operator would be zero everywhere.
Generalized normal trace
Let $ \operatorname {div} v$ denote the distributional divergence of a vector field $ v$. For $ 1<p<\infty $ and bounded Lipschitz domain $ \Omega \subset \mathbb {R} ^{n}$ define
$E_{p}(\Omega )=\{v\in (L^{p}(\Omega ))^{n}\mid \operatorname {div} v\in L^{p}(\Omega )\}$
which is a Banach space with norm
$\|v\|_{E_{p}(\Omega )}=\left(\|v\|_{L^{p}(\Omega )}^{p}+\|\operatorname {div} v\|_{L^{p}(\Omega )}^{p}\right)^{1/p}$.
Let $ N$ denote the exterior unit normal field on $ \partial \Omega $. Then[4] there exists a bounded linear operator
$T_{N}\colon E_{p}(\Omega )\to (W^{1-1/q,q}(\partial \Omega ))'$,
where $ q=p/(p-1)$ is the conjugate exponent to $ p$ and $ X'$ denotes the continuous dual space to a Banach space $ X$, such that $ T_{N}$ extends the normal trace $ (v\cdot N)|_{\partial \Omega }$ for $ v\in (C^{\infty }({\bar {\Omega }}))^{n}$ in the sense that
$T_{N}v={\bigl \{}\varphi \in W^{1-1/q,q}(\partial \Omega )\mapsto \int _{\partial \Omega }\varphi v\cdot N\,\mathrm {d} S{\bigr \}}$.
The value of the normal trace operator $ (T_{N}v)(\varphi )$ for $ \varphi \in W^{1-1/q,q}(\partial \Omega )$ is defined by application of the divergence theorem to the vector field $ w=E\varphi \,v$ where $ E$ is the trace extension operator from above.
Application. Any weak solution $ u\in H^{1}(\Omega )$ to $ -\Delta u=f\in L^{2}(\Omega )$ in a bounded Lipschitz domain $ \Omega \subset \mathbb {R} ^{n}$ has a normal derivative in the sense of $ T_{N}\nabla u\in (W^{1/2,2}(\partial \Omega ))^{*}$. This follows as $ \nabla u\in E_{2}(\Omega )$ since $ \nabla u\in L^{2}(\Omega )$ and $ \operatorname {div} (\nabla u)=\Delta u=-f\in L^{2}(\Omega )$. This result is notable since in Lipschitz domains in general $ u\not \in H^{2}(\Omega )$, such that $ \nabla u$ may not lie in the domain of the trace operator $ T$.
Application
The theorems presented above allow a closer investigation of the boundary value problem
${\begin{alignedat}{2}-\Delta u&=f&\quad &{\text{in }}\Omega ,\\u&=g&&{\text{on }}\partial \Omega \end{alignedat}}$
on a Lipschitz domain $ \Omega \subset \mathbb {R} ^{n}$ from the motivation. Since only the Hilbert space case $ p=2$ is investigated here, the notation $ H^{1}(\Omega )$ is used to denote $ W^{1,2}(\Omega )$ etc. As stated in the motivation, a weak solution $ u\in H^{1}(\Omega )$ to this equation must satisfy $ Tu=g$ and
$\int _{\Omega }\nabla u\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }f\varphi \,\mathrm {d} x$ for all $ \varphi \in H_{0}^{1}(\Omega )$,
where the right-hand side must be interpreted for $ f\in H^{-1}(\Omega )=(H_{0}^{1}(\Omega ))'$ as a duality product with the value $ f(\varphi )$.
Existence and uniqueness of weak solutions
The characterization of the range of $ T$ implies that for $ Tu=g$ to hold the regularity $ g\in H^{1/2}(\partial \Omega )$ is necessary. This regularity is also sufficient for the existence of a weak solution, which can be seen as follows. By the trace extension theorem there exists $ Eg\in H^{1}(\Omega )$ such that $ T(Eg)=g$. Defining $ u_{0}$ by $ u_{0}=u-Eg$ we have that $ Tu_{0}=Tu-T(Eg)=0$ and thus $ u_{0}\in H_{0}^{1}(\Omega )$ by the characterization of $ H_{0}^{1}(\Omega )$ as space of trace zero. The function $ u_{0}\in H_{0}^{1}(\Omega )$ then satisfies the integral equation
$\int _{\Omega }\nabla u_{0}\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }\nabla (u-Eg)\cdot \nabla \varphi \,\mathrm {d} x=\int _{\Omega }f\varphi \,\mathrm {d} x-\int _{\Omega }\nabla Eg\cdot \nabla \varphi \,\mathrm {d} x$ for all $ \varphi \in H_{0}^{1}(\Omega )$.
Thus the problem with inhomogeneous boundary values for $ u$ could be reduced to a problem with homogeneous boundary values for $ u_{0}$, a technique which can be applied to any linear differential equation. By the Riesz representation theorem there exists a unique solution $ u_{0}$ to this problem. By uniqueness of the decomposition $ u=u_{0}+Eg$, this is equivalent to the existence of a unique weak solution $ u$ to the inhomogeneous boundary value problem.
Continuous dependence on the data
It remains to investigate the dependence of $ u$ on $ f$ and $ g$. Let $ c_{1},c_{2},\ldots >0$ denote constants independent of $ f$ and $ g$. By continuous dependence of $ u_{0}$ on the right-hand side of its integral equation, there holds
$\|u_{0}\|_{H_{0}^{1}(\Omega )}\leq c_{1}\left(\|f\|_{H^{-1}(\Omega )}+\|Eg\|_{H^{1}(\Omega )}\right)$
and thus, using that $ \|u_{0}\|_{H_{0}^{1}(\Omega )}\leq c_{2}\|u_{0}\|_{H^{1}(\Omega )}$ and $ \|Eg\|_{H^{1}(\Omega )}\leq c_{3}\|g\|_{H^{1/2}(\Omega )}$ by continuity of the trace extension operator, it follows that
${\begin{aligned}\|u\|_{H^{1}(\Omega )}&\leq \|u_{0}\|_{H^{1}(\Omega )}+\|Eg\|_{H^{1}(\Omega )}\leq c_{1}c_{2}\|f\|_{H^{-1}(\Omega )}+(1+c_{1}c_{2})\|Eg\|_{H^{1}(\Omega )}\\&\leq c_{4}\left(\|f\|_{H^{-1}(\Omega )}+\|g\|_{H^{1/2}(\partial \Omega )}\right)\end{aligned}}$
and the solution map
$H^{-1}(\Omega )\times H^{1/2}(\partial \Omega )\ni (f,g)\mapsto u\in H^{1}(\Omega )$
is therefore continuous.
See also
• Trace class
• Nuclear operators between Banach spaces
References
1. Gagliardo, Emilio (1957). "Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili". Rendiconti del Seminario Matematico della Università di Padova. 27: 284–305.
2. Evans, Lawrence (1998). Partial differential equations. Providence, R.I.: American Mathematical Society. pp. 257–261. ISBN 0-8218-0772-2.
3. Nečas, Jindřich (1967). Les méthodes directes en théorie des équations elliptiques. Paris: Masson et Cie, Éditeurs, Prague: Academia, Éditeurs. pp. 90–104.
4. Sohr, Hermann (2001). The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts Basler Lehrbücher. Basel: Birkhäuser. pp. 50–51. doi:10.1007/978-3-0348-8255-2. ISBN 978-3-0348-9493-7.
• Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. ISBN 978-1-4704-2921-8
| Wikipedia |
Trace theory
In mathematics and computer science, trace theory aims to provide a concrete mathematical underpinning for the study of concurrent computation and process calculi. The underpinning is provided by an algebraic definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic foundation, analogous to the way that the free monoid provides the underpinning for formal languages.
The power of trace theory stems from the fact that the algebra of dependency graphs (such as Petri nets) is isomorphic to that of trace monoids, and thus, one can apply both algebraic formal language tools, as well as tools from graph theory.
While the trace monoid had been studied by Pierre Cartier and Dominique Foata for its combinatorics in the 1960s, trace theory was first formulated by Antoni Mazurkiewicz in the 1970s, in an attempt to evade some of the problems in the theory of concurrent computation, including the problems of interleaving and non-deterministic choice with regards to refinement in process calculi.
References
• Volker Diekert, Grzegorz Rozenberg, eds. The Book of Traces, (1995) World Scientific, Singapore ISBN 981-02-2058-8
• Volker Diekert, Yves Metivier, "Partial Commutation and Traces", In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Vol. 3, Beyond Words. Springer-Verlag, Berlin, 1997.
• Volker Diekert, Combinatorics on traces, LNCS 454, Springer, 1990, ISBN 3-540-53031-2
| Wikipedia |
Traced monoidal category
In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback.
A traced symmetric monoidal category is a symmetric monoidal category C together with a family of functions
$\mathrm {Tr} _{X,Y}^{U}:\mathbf {C} (X\otimes U,Y\otimes U)\to \mathbf {C} (X,Y)$
called a trace, satisfying the following conditions:
• naturality in $X$: for every $f:X\otimes U\to Y\otimes U$ and $g:X'\to X$,
$\mathrm {Tr} _{X',Y}^{U}(f\circ (g\otimes \mathrm {id} _{U}))=\mathrm {Tr} _{X,Y}^{U}(f)\circ g$
• naturality in $Y$: for every $f:X\otimes U\to Y\otimes U$ and $g:Y\to Y'$,
$\mathrm {Tr} _{X,Y'}^{U}((g\otimes \mathrm {id} _{U})\circ f)=g\circ \mathrm {Tr} _{X,Y}^{U}(f)$
• dinaturality in $U$: for every $f:X\otimes U\to Y\otimes U'$ and $g:U'\to U$
$\mathrm {Tr} _{X,Y}^{U}((\mathrm {id} _{Y}\otimes g)\circ f)=\mathrm {Tr} _{X,Y}^{U'}(f\circ (\mathrm {id} _{X}\otimes g))$
• vanishing I: for every $f:X\otimes I\to Y\otimes I$, (with $\rho _{X}\colon X\otimes I\cong X$ being the right unitor),
$\mathrm {Tr} _{X,Y}^{I}(f)=\rho _{Y}\circ f\circ \rho _{X}^{-1}$
• vanishing II: for every $f:X\otimes U\otimes V\to Y\otimes U\otimes V$
$\mathrm {Tr} _{X,Y}^{U}(\mathrm {Tr} _{X\otimes U,Y\otimes U}^{V}(f))=\mathrm {Tr} _{X,Y}^{U\otimes V}(f)$
• superposing: for every $f:X\otimes U\to Y\otimes U$ and $g:W\to Z$,
$g\otimes \mathrm {Tr} _{X,Y}^{U}(f)=\mathrm {Tr} _{W\otimes X,Z\otimes Y}^{U}(g\otimes f)$
• yanking:
$\mathrm {Tr} _{X,X}^{X}(\gamma _{X,X})=\mathrm {id} _{X}$
(where $\gamma $ is the symmetry of the monoidal category).
Properties
• Every compact closed category admits a trace.
• Given a traced monoidal category C, the Int construction generates the free (in some bicategorical sense) compact closure Int(C) of C.
References
• Joyal, André; Street, Ross; Verity, Dominic (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society. 119 (3): 447–468. Bibcode:1996MPCPS.119..447J. doi:10.1017/S0305004100074338. S2CID 50511333.
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
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• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
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Colimits
• Initial objects
• Coproducts
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• Cokernels and quotients
• Pushout
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Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
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Constructions on categories
• Free category
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• Kleisli category
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Higher category theory
Key concepts
• Categorification
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n-categories
Weak n-categories
• Bicategory (pseudofunctor)
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| Wikipedia |
Trachette Jackson
Trachette Levon Jackson (born July 24, 1972) is an American mathematician who is a professor of mathematics at the University of Michigan and is known for work in mathematical oncology.[1] She uses many different approaches, including continuous and discrete mathematical models, numerical simulations, and experiments to study tumor growth and treatment. Specifically, her lab is interested in "molecular pathways associated with intratumoral angiogenesis," "cell-tissue interactions associated with tumor-induced angiogenesis," and "tumor heterogeneity and cancer stem cells."[2]
Trachette Jackson
BornJuly 24, 1972
Monroe, Louisiana
Alma materArizona State University,
University of Washington
SpousePatrick Nelson
ChildrenTwo children
Scientific career
FieldsMathematics
InstitutionsUniversity of Michigan,
University of Minnesota,
Duke University
Education and career
Jackson's parents were in the military and traveled frequently through her childhood; as a teenager, she lived in Mesa, Arizona. There, in a summer calculus course, her talent for mathematics brought her to the attention of Arizona State University mathematics professor Joaquín Bustoz, Jr. She went on to undergraduate studies at ASU, originally intending to study engineering, but she was steered to mathematics by Bustoz.[3] From there, her interest in pure math developed into an interest in mathematical biology when she attended a talk by her future PhD advisor, James D. Murray, on the mathematics of pattern formation and "how the leopard got its spots."[4] She graduated in 1994, and she earned her MS and PhD at the University of Washington in 1996 and 1998.[5][6] After postdoctoral research at the University of Minnesota, Environmental Protection Agency, and Duke University, she joined the University of Michigan faculty in 2000, and she was promoted to full professor in 2008.[7]
Awards and recognition
She was awarded a Sloan Research Fellowship in 2003,[8] becoming the second African-American woman after Kathleen Adebola Okikiolu to become a Sloan Fellow in mathematics. She won a James S. McDonnell 21st Century Scientist Grant in 2005,[9] and won the Blackwell-Tapia Prize in 2010.[10] In 2017, she was selected as a fellow of the Association for Women in Mathematics in the inaugural class.[11] Jackson's work also earned her recognition by Mathematically Gifted & Black as a Black History Month 2017 Honoree.[12] She was named a SIAM Fellow in the 2021 class of fellows, "for innovative contributions to mathematical modeling in cancer biology and for the advancement of underrepresented minorities in science".[13] In 2021, she was awarded the University Diversity and Social Transformation Professorship at the University of Michigan,[14] in recognition of her "extraordinary commitment to increasing opportunities for girls, women, and underrepresented minority students in STEM, through her teaching and leadership."[15]
References
1. Seymour, Add, Jr. (January 10, 2008). "Mathematics: Connecting the Dots – Trachette Jackson". Emerging Scholars: The Class of 2008. Diverse Magazine. Retrieved 2015-08-01.{{cite magazine}}: CS1 maint: multiple names: authors list (link)
2. "The Jackson Cancer Modeling Group". University of Michigan Website. Retrieved 2015-08-01.
3. Castillo-Chavez, Carlos (July–August 2010). "Teacher, Research Mathematician, Mentor: A Groundbreaking Career in Computational and Mathematical Biology" (PDF). Expanding our Scope. SIAM News. 43 (6).
4. Lamb, Evelyn (October 9, 2013). "Mathematics, Live: A Conversation with Victoria Booth and Trachette Jackson". Roots of Unity. Scientific American. Retrieved 2015-08-01.
5. "Trachette Jackson". TheHistoryMakers. Retrieved 2015-08-01.
6. Trachette Jackson at the Mathematics Genealogy Project
7. Curriculum Vitae, October 25, 2021. Retrieved 2021-10-26.
8. Past Fellows, Sloan Foundation, retrieved 2019-09-09
9. "Funded Grants". James S. McDonnell Foundation. Retrieved 2022-12-05.
10. "Trachette L. Jackson: "Mathematical Models of Tumor Angiogenesis"". The Michael E. Moody Lecture Series. Harvey Mudd College. Retrieved 2015-08-01.
11. "2018 Inaugural Class of AWM Fellows Program". awm-math.org/awards/awm-fellows/. Association for Women in Mathematics. Retrieved 9 January 2021.
12. "Trachette Jackson". Mathematically Gifted & Black.{{cite web}}: CS1 maint: url-status (link)
13. "SIAM Announces Class of 2021 Fellows". SIAM News. Society for Industrial and Applied Mathematics. March 31, 2021. Retrieved 2021-04-03.
14. "Our U-M UDSTPs | U-M LSA National Center for Institutional Diversity". lsa.umich.edu. Retrieved 2021-10-26.
15. "Trachette L. Jackson | U-M LSA National Center for Institutional Diversity". lsa.umich.edu. Retrieved 2021-10-26.
External links
• Williams, Scott W. "Trachette Jackson". Black Women in Mathematics. State University of New York at Buffalo, Department of Mathematics.
• Meet a mathematician! Video Interview
Authority control
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| Wikipedia |
Trachtenberg system
The Trachtenberg system is a system of rapid mental calculation. The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. It was developed by the Russian engineer Jakow Trachtenberg in order to keep his mind occupied while being in a Nazi concentration camp.
The rest of this article presents some methods devised by Trachtenberg. Some of the algorithms Trachtenberg developed are ones for general multiplication, division and addition. Also, the Trachtenberg system includes some specialised methods for multiplying small numbers between 5 and 13 (but shown here is 2–12).
The section on addition demonstrates an effective method of checking calculations that can also be applied to multiplication.
General multiplication
The method for general multiplication is a method to achieve multiplications $a\times b$ with low space complexity, i.e. as few temporary results as possible to be kept in memory. This is achieved by noting that the final digit is completely determined by multiplying the last digit of the multiplicands. This is held as a temporary result. To find the next to last digit, we need everything that influences this digit: The temporary result, the last digit of $a$ times the next-to-last digit of $b$, as well as the next-to-last digit of $a$ times the last digit of $b$. This calculation is performed, and we have a temporary result that is correct in the final two digits.
In general, for each position $n$ in the final result, we sum for all $i$:
$a{\text{ (digit at }}i{\text{ )}}\times b{\text{ (digit at }}(n-i){\text{)}}.$
People can learn this algorithm and thus multiply four-digit numbers in their head – writing down only the final result. They would write it out starting with the rightmost digit and finishing with the leftmost.
Trachtenberg defined this algorithm with a kind of pairwise multiplication where two digits are multiplied by one digit, essentially only keeping the middle digit of the result. By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held.
Example: $123456\times 789$
To find the first (rightmost) digit of the answer, start at the first digit of the multiplicand:
The units digit of $9\times 6$ is $4.$
The first digit of the answer is $4$. The tens digit $5$ is ignored.
To find the second digit of the answer, start at the second digit of the multiplicand:
The units digit of $9\times 5$ plus the tens digit of $9\times 6$ plus
The units digit of $8\times 6$.
$5+5+8=18$.
The second digit of the answer is $8$ and carry $1$ to the third digit.
To find the third digit of the answer, start at the third digit of the multiplicand:
The units digit of $9\times 4$ plus the tens digit of $9\times 5$ plus
The units digit of $8\times 5$ plus the tens digit of $8\times 6$ plus
The units digit of $7\times 6$
$1+6+4+0+4+2=17$
The third digit of the answer is $7$ and carry $1$ to the next digit.
To find the fourth digit of the answer, start at the fourth digit of the multiplicand:
The units digit of $9\times 3$ plus the tens digit of $9\times 4$ plus
The units digit of $8\times 4$ plus the tens digit of $8\times 5$ plus
The units digit of $7\times 5$ plus the tens digit of $7\times 6$.
$1+7+3+2+4+5+4=26$ carried from the third digit.
The fourth digit of the answer is $6$ and carry $2$ to the next digit.
Continue with the same method to obtain the remaining digits.
Trachtenberg called this the 2 Finger Method. The calculations for finding the fourth digit from the example above are illustrated at right. The arrow from the nine will always point to the digit of the multiplicand directly above the digit of the answer you wish to find, with the other arrows each pointing one digit to the right. Each arrow head points to a UT Pair, or Product Pair. The vertical arrow points to the product where we will get the Units digit, and the sloping arrow points to the product where we will get the Tens digits of the Product Pair. If an arrow points to a space with no digit there is no calculation for that arrow. As you solve for each digit you will move each of the arrows over the multiplicand one digit to the left until all of the arrows point to prefixed zeros.
Division in the Trachtenberg System is done much the same as in multiplication but with subtraction instead of addition. Splitting the dividend into smaller Partial Dividends, then dividing this Partial Dividend by only the left-most digit of the divisor will provide the answer one digit at a time. As you solve each digit of the answer you then subtract Product Pairs (UT pairs) and also NT pairs (Number-Tens) from the Partial Dividend to find the next Partial Dividend. The Product Pairs are found between the digits of the answer so far and the divisor. If a subtraction results in a negative number you have to back up one digit and reduce that digit of the answer by one. With enough practice this method can be done in your head.
General addition
A method of adding columns of numbers and accurately checking the result without repeating the first operation. An intermediate sum, in the form of two rows of digits, is produced. The answer is obtained by taking the sum of the intermediate results with an L-shaped algorithm. As a final step, the checking method that is advocated both removes the risk of repeating any original errors and identifies the precise column in which an error occurs all at once. It is based on check (or digit) sums, such as the nines-remainder method.
For the procedure to be effective, the different operations used in each stage must be kept distinct, otherwise there is a risk of interference.
Other multiplication algorithms
When performing any of these multiplication algorithms the following "steps" should be applied.
The answer must be found one digit at a time starting at the least significant digit and moving left. The last calculation is on the leading zero of the multiplicand.
Each digit has a neighbor, i.e., the digit on its right. The rightmost digit's neighbor is the trailing zero.
The 'halve' operation has a particular meaning to the Trachtenberg system. It is intended to mean "half the digit, rounded down" but for speed reasons people following the Trachtenberg system are encouraged to make this halving process instantaneous. So instead of thinking "half of seven is three and a half, so three" it's suggested that one thinks "seven, three". This speeds up calculation considerably. In this same way the tables for subtracting digits from 10 or 9 are to be memorized.
And whenever the rule calls for adding half of the neighbor, always add 5 if the current digit is odd. This makes up for dropping 0.5 in the next digit's calculation.
Numbers and digits (base 10)
Digits and numbers are two different notions. The number T consists of n digits cn ... c1.
$T=10^{n-1}*c_{n}+...+10^{0}*c_{1}$
Multiplying by 2
Proof
$ {\begin{aligned}R&=T*2\Leftrightarrow \\R&=2*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n-1}*2*c_{n}+\ldots +10^{0}*2*c_{1}\\\\&QED\end{aligned}}$
Rule:
1. Multiply each digit by 2 (with carrying).
Example: 8624 × 2
Working from left to right:
8+8=16,
6+6=12 (carry the 1),
2+2=4
4+4=8;
8624 × 2 = 17248
Example: 76892 × 2
Working from left to right:
7+7=14
6+6=12
8+8=16
9+9=18
2+2=4;
76892 × 2 =153784
Multiplying by 3
Proof
${\begin{aligned}R&=T*3\Leftrightarrow \\R&=3*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=(10/2-2)*(10^{n-1}*c_{n}+10^{n-2}*c_{n-1}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*(c_{n}/2-2)+10^{n}*2+10^{n-1}*(c_{n-1}/2-2)+10^{n-1}*2+\ldots +10^{1}*(c_{1}/2-2)+10^{1}*2\\&-2*(10^{n-1}*c_{n}+10^{n-2}*c_{n-1}+\ldots +10^{1}*c_{2}+10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*(c_{n}/2-2)+10^{n-1}*(c_{n-1}/2+20-2-2*c_{n})+10^{n-2}*(c_{n-2}/2+20-2-2*c_{n-1})\\&+\ldots +10^{1}*(c_{1}/2+20-2-2*c_{2})+10^{0}*(20-2*c_{1})\Leftrightarrow \\R&=10^{n}*(c_{n}/2-2)+10^{n-1}*(2*(9-c_{n})+c_{n-1}/2)+10^{n-2}*(2*(9-c_{n-1})+c_{n-2}/2)\\&+\ldots +10^{1}*(2*(9-c_{2})+c_{1}/2)+10^{0}*(2*(10-c_{1}))\Leftrightarrow \vdots \Re \to \aleph {\text{: a }}=(a{\text{ div }}b)*b+(a{\bmod {b}})\\R&=10^{n}*(((c_{n}{\text{ div }}2)*2+(c_{n}{\bmod {2}}))/2-2)+10^{n-1}*(2*(9-c_{n})+c_{n-1}/2)+10^{n-2}*(2*(9-c_{n-1})+c_{n-2}/2)\\&+\ldots +10^{1}*(2*(9-c_{2})+c_{1}/2)+10^{0}*(2*(10-c_{1}))\Leftrightarrow \\R&=10^{n}*((c_{n}{\text{ div }}2)-2)+10^{n-1}*(10*(c_{n}{\bmod {2}})/2+2*(9-c_{n})+c_{n-1}/2)+10^{n-2}*(2*(9-c_{n-1})+c_{n-2}/2)\\&+\ldots +10^{1}*(2*(9-c_{2})+c_{1}/2)+10^{0}*(2*(10-c_{1}))\Leftrightarrow \\R&=10^{n}*((c_{n}{\text{ div }}2)-2)+10^{n-1}*(2*(9-c_{n})+c_{n-1}/2+(c_{n}{\bmod {2}})*5)+10^{n-2}*(2*(9-c_{n-1})+c_{n-2}/2)\\&+\ldots +10^{1}*(2*(9-c_{2})+c_{1}/2)+10^{0}*(2*(10-c_{1}))\Leftrightarrow \\R&=10^{n}*((c_{n}{\text{ div }}2)-2)+10^{n-1}*(2*(9-c_{n})+(c_{n-1}{\text{ div }}2)+{\text{ if}}(c_{n}{\bmod {2}}<>0;5;0))\\&+\ldots +10^{1}*(2*(9-c_{2})+(c_{1}{\text{ div }}2)+{\text{ if}}(c_{2}{\bmod {2}}<>0;5;0))\\&+10^{0}*(2*(10-c_{1})+{\text{ if}}(c_{1}{\bmod {2}}<>0;5;0))\\\\&QED\end{aligned}}$
Rule:
1. Subtract the rightmost digit from 10.
2. Subtract the remaining digits from 9.
3. Double the result.
4. Add half of the neighbor to the right, plus 5 if the digit is odd.
5. For the leading zero, subtract 2 from half of the neighbor.
Example: 492 × 3 = 1476
Working from right to left:
(10 − 2) × 2 + Half of 0 (0) = 16. Write 6, carry 1.
(9 − 9) × 2 + Half of 2 (1) + 5 (since 9 is odd) + 1 (carried) = 7. Write 7.
(9 − 4) × 2 + Half of 9 (4) = 14. Write 4, carry 1.
Half of 4 (2) − 2 + 1 (carried) = 1. Write 1.
Multiplying by 4
Proof
${\begin{aligned}R&=T*4\Leftrightarrow \\R&=4*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=(10/2-1)*(10^{n-1}*c_{n}+10^{n-2}*c_{n-1}+\ldots +10^{0}*c_{1})\Leftrightarrow \vdots {\mbox{ see proof of method 3}}\\R&=10^{n}*((c_{n}{\text{ div }}2)-1)+10^{n-1}*((9-c_{n})+(c_{n-1}{\text{ div }}2)+{\text{ if}}(c_{n}{\bmod {2}}<>0;5;0))\\&+\ldots +10^{1}*((9-c_{2})+(c_{1}{\text{ div }}2)+{\text{ if}}(c_{2}{\bmod {2}}<>0;5;0))\\&+10^{0}*((10-c_{1})+{\text{ if}}(c_{1}{\bmod {2}}<>0;5;0))\\\\&QED\end{aligned}}$
Rule:
1. Subtract the right-most digit from 10.
2. Subtract the remaining digits from 9.
3. Add half of the neighbor, plus 5 if the digit is odd.
4. For the leading 0, subtract 1 from half of the neighbor.
Example: 346 × 4 = 1384
Working from right to left:
(10 − 6) + Half of 0 (0) = 4. Write 4.
(9 − 4) + Half of 6 (3) = 8. Write 8.
(9 − 3) + Half of 4 (2) + 5 (since 3 is odd) = 13. Write 3, carry 1.
Half of 3 (1) − 1 + 1 (carried) = 1. Write 1.
Multiplying by 5
Proof
${\begin{aligned}R&=T*5\Leftrightarrow \\R&=5*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=(10/2)*(10^{n-1}*c_{n}+10^{n-2}*c_{n-1}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*(c_{n}/2)+10^{n-1}*(c_{n-1}/2)+\ldots +10^{1}*(c_{1}/2)\Leftrightarrow \vdots \Re \to \aleph {\text{: a }}=(a{\text{ div }}b)*b+(a{\bmod {b}})\\R&=10^{n}*((c_{n}{\text{ div }}2)*2+(c_{n}{\bmod {2}}))/2+10^{n-1}*((c_{n-1}{\text{ div }}2)*2+(c_{n-1}{\bmod {2}}))/2\\&+\ldots +10^{2}*((c_{2}{\text{ div }}2)*2+(c_{2}{\bmod {2}}))/2+10^{1}*((c_{1}{\text{ div }}2)*2+(c_{1}{\bmod {2}}))/2\Leftrightarrow \\R&=10^{n}*((c_{n}{\text{ div }}2)+(c_{n}{\bmod {2}})/2)+10^{n-1}*((c_{n-1}{\text{ div }}2)+(c_{n-1}{\bmod {2}})/2)\\&+\ldots +10^{2}*((c_{2}{\text{ div }}2)+(c_{2}{\bmod {2}})/2)+10^{1}*((c_{1}{\text{ div }}2)+(c_{1}{\bmod {2}})/2)\Leftrightarrow \\R&=10^{n}*(c_{n}{\text{ div }}2)+10^{n-1}*10*(c_{n}{\bmod {2}})/2+10^{n-1}*(c_{n-1}{\text{ div }}2)+10^{n-2}*10*(c_{n-1}{\bmod {2}})/2+10^{n-2}*(c_{n-2}{\text{ div }}2)\\&+\ldots +10^{2}*(c_{2}{\text{ div }}2)+10^{1}*10*(c_{2}{\bmod {2}})/2+(c_{1}{\text{ div }}2))+10^{0}*10*(c_{1}{\bmod {2}})/2\Leftrightarrow \\R&=10^{n}*(c_{n}{\text{ div }}2)+10^{n-1}*(c_{n-1}{\text{ div }}2)+10^{n-1}*(c_{n}{\bmod {2}})*5+10^{n-2}*(c_{n-2}{\text{ div }}2)+10^{n-2}*(c_{n-1}{\bmod {2}})*5\\&+\ldots +10^{2}*(c_{2}{\text{ div }}2)+10^{2}*(c_{3}{\bmod {2}})*5+10^{1}*(c_{1}{\text{ div }}2)+10^{1}*(c_{2}{\bmod {2}})*5+10^{0}*(c_{1}{\bmod {2}})*5\Leftrightarrow \\R&=10^{n}*(c_{n}{\text{ div }}2)+10^{n-1}*((c_{n-1}{\text{ div }}2)+{\text{ if}}(c_{n}{\bmod {2}}<>0;5;0))+10^{n-2}*((c_{n-2}{\text{ div }}2)+{\text{ if}}(c_{n-1}{\bmod {2}}<>0;5;0))\\&+\ldots +10^{2}*((c_{2}{\text{ div }}2)+{\text{ if}}(c_{3}{\bmod {2}}<>0;5;0))+10^{1}*((c_{1}{\text{ div }}2)+{\text{ if}}(c_{2}{\bmod {2}}<>0;5;0))+10^{0}*{\text{ if}}(c_{1}{\bmod {2}}<>0;5;0)\\\\&QED\end{aligned}}$
Rule:
1. Take half of the neighbor, then, if the current digit is odd, add 5.
Example: 42×5=210
Half of 2's neighbor, the trailing zero, is 0.
Half of 4's neighbor is 1.
Half of the leading zero's neighbor is 2.
43×5 = 215
Half of 3's neighbor is 0, plus 5 because 3 is odd, is 5.
Half of 4's neighbor is 1.
Half of the leading zero's neighbor is 2.
93×5=465
Half of 3's neighbor is 0, plus 5 because 3 is odd, is 5.
Half of 9's neighbor is 1, plus 5 because 9 is odd, is 6.
Half of the leading zero's neighbor is 4.
Multiplying by 6
Proof
${\begin{aligned}R&=T*6\Leftrightarrow \\R&=6*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=(10/2+1)*(10^{n-1}*c_{n}+10^{n-2}*c_{n-1}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*c_{n}/2+1*10^{n-1}*c_{n}+10^{n-1}*c_{n-1}/2+1*10^{n-2}*c_{n-1}+\ldots +10^{1}*c_{1}/2+1*10^{0}*c_{1}\Leftrightarrow \\R&=10^{n}*c_{n}/2+10^{n-1}*(c_{n}+c_{n-1}/2)+\ldots +10^{1}*c_{1}/2+c_{1}\Leftrightarrow \vdots \Re \to \aleph {\text{: a }}=(a{\text{ div }}b)*b+(a{\bmod {b}})\\R&=10^{n}*((c_{n}{\text{ div }}2)*2+(c_{n}{\bmod {2}}))/2+10^{n-1}*(c_{n}+c_{n-1}/2)+\ldots +10^{1}*c_{1}/2+c_{1}\Leftrightarrow \\R&=10^{n}*(c_{n}{\text{ div }}2)+10^{n-1}*(c_{n}{\bmod {2}})*5+10^{n-1}*c_{n}+10^{n-1}*((c_{n-1}{\text{ div }}2)*2+(c_{n-1}{\bmod {2}}))/2+\ldots +10^{1}*c_{1}/2+c_{1}\Leftrightarrow \\R&=10^{n}*(c_{n}{\text{ div }}2)+10^{n-1}*(c_{n}+(c_{n-1}{\text{ div }}2)+{\text{ if}}((c_{n}{\bmod {2}})<>0;5;0))+10^{n-2}*(c_{n-1}{\bmod {2}})*5+\ldots +10^{1}*c_{1}/2+c_{1}\Leftrightarrow \\R&=10^{n}*(c_{n}{\text{ div }}2)+10^{n-1}*(c_{n}+(c_{n-1}{\text{ div }}2)+{\text{ if}}((c_{n}{\bmod {2}})<>0;5;0))\\&+10^{n-2}*(c_{n-1}+(c_{n-2}{\text{ div }}2)+{\text{ if}}((c_{n-1}{\bmod {2}})<>0;5;0))\\&+\ldots +10^{0}*(c_{1}+{\text{ if}}((c_{1}{\bmod {2}})<>0;5;0))\\\\&QED\end{aligned}}$
Rule:
1. Add half of the neighbor to each digit. If the current digit is odd, add 5.
Example: 357 × 6 = 2142
Working right to left:
7 has no neighbor, add 5 (since 7 is odd) = 12. Write 2, carry the 1.
5 + half of 7 (3) + 5 (since the starting digit 5 is odd) + 1 (carried) = 14. Write 4, carry the 1.
3 + half of 5 (2) + 5 (since 3 is odd) + 1 (carried) = 11. Write 1, carry 1.
0 + half of 3 (1) + 1 (carried) = 2. Write 2.
Multiplying by 7
Proof
${\begin{aligned}R&=T*7\Leftrightarrow \\R&=7*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=(10/2+2)*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \vdots {\mbox{ see proof of method 6}}\\R&=10^{n}*(c_{n}{\text{ div }}2)+10^{n-1}*(2*c_{n}+(c_{n-1}{\text{ div }}2)+{\text{ if}}(c_{n}{\bmod {2}}<>0;5;0))\\&+10^{n-2}*(2*c_{n-1}+(c_{n-2}{\text{ div }}2)+{\text{ if}}(c_{n-1}{\bmod {2}}<>0;5;0))\\&+\ldots +10^{1}*(2*c_{2}+(c_{1}{\text{ div }}2)+{\text{ if}}(c_{2}{\bmod {2}}<>0;5;0))+2*c_{1}+{\text{ if}}(c_{1}{\bmod {2}}<>0;5;0)\\\\&QED\end{aligned}}$
Rule:
1. Double each digit.
2. Add half of its neighbor to the right (dropping decimals, if any). The neighbor of the units position is 0.
3. If the base-digit is even add 0 otherwise add 5.
4. Add in any carryover from the previous step.
Example: 693 × 7 = 4,851
Working from right to left:
(3×2) + 0 + 5 + 0 = 11 = carryover 1, result 1.
(9×2) + 1 + 5 + 1 = 25 = carryover 2, result 5.
(6×2) + 4 + 0 + 2 = 18 = carryover 1, result 8.
(0×2) + 3 + 0 + 1 = 4 = result 4.
Multiplying by 8
Proof
${\begin{aligned}R&=T*8\Leftrightarrow \\R&=T*4*2\Leftrightarrow \vdots {\mbox{ see proof of method 4}}\\R&=10^{n}*2*(c_{n}/2-1)+10^{n-1}*2*((9-c_{n})+c_{n-1}/2)+10^{n-2}*2*((9-c_{n-1})+c_{n-2}/2)\\&+\ldots +10^{1}*2*((9-c_{2})+c_{1}/2)+10^{0}*2*(10-c_{1})\Leftrightarrow \\R&=10^{n}*(c_{n}-2)+10^{n-1}*(2*(9-c_{n})+c_{n-1})+\ldots +10^{2}*(2*(9-c_{3})+c_{2})+10^{1}*(2*(9-c_{2})+c_{1})+2*(10-c_{1})\\\\&QED\end{aligned}}$
Rule:
1. Subtract right-most digit from 10.
1. Subtract the remaining digits from 9.
2. Double the result.
3. Add the neighbor.
4. For the leading zero, subtract 2 from the neighbor.
Example: 456 × 8 = 3648
Working from right to left:
(10 − 6) × 2 + 0 = 8. Write 8.
(9 − 5) × 2 + 6 = 14, Write 4, carry 1.
(9 − 4) × 2 + 5 + 1 (carried) = 16. Write 6, carry 1.
4 − 2 + 1 (carried) = 3. Write 3.
Multiplying by 9
Proof
${\begin{aligned}R&=T*9\Leftrightarrow \\R&=(10-1)*T\Leftrightarrow \\R&=10^{n}*(c_{n}-1)+10^{n}+10^{n-1}*(c_{n-1}-1)+10^{n-1}+\ldots +10^{1}*(c_{1}-1)+10^{1}\\&-(10^{n-1}*c_{n}+10^{n-2}*c_{n-1}+\ldots +10^{1}*c_{2}+10^{0}*c_{1})\Leftrightarrow \vdots {\mbox{ see proof of method 4}}\\R&=10^{n}*(c_{n}-1)+10^{n-1}*(9-c_{n}+c_{n-1})+10^{n-2}*(9-c_{n-1}+c_{n-2})+\ldots +10^{1}*(9-c_{2}+c_{1})+10^{0}*(10-c_{1})\\\\&QED\end{aligned}}$
Rule:
1. Subtract the right-most digit from 10.
1. Subtract the remaining digits from 9.
2. Add the neighbor to the sum
3. For the leading zero, subtract 1 from the neighbor.
For rules 9, 8, 4, and 3 only the first digit is subtracted from 10. After that each digit is subtracted from nine instead.
Example: 2,130 × 9 = 19,170
Working from right to left:
(10 − 0) + 0 = 10. Write 0, carry 1.
(9 − 3) + 0 + 1 (carried) = 7. Write 7.
(9 − 1) + 3 = 11. Write 1, carry 1.
(9 − 2) + 1 + 1 (carried) = 9. Write 9.
2 − 1 = 1. Write 1.
Multiplying by 10
Add 0 (zero) as the rightmost digit.
Proof
${\begin{aligned}R&=T*10\Leftrightarrow \\R&=10*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*c_{n}+\ldots +10^{1}*c_{1}\\\\&QED\end{aligned}}$
Multiplying by 11
Proof
${\begin{aligned}R&=T*11\Leftrightarrow \\R&=T*(10+1)\\R&=10*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})+(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*c_{n}+10^{n-1}*(c_{n}+c_{n-1})+\ldots +10^{1}*(c_{2}+c_{1})+c_{1}\\\\&QED\end{aligned}}$
Rule:
1. Add the digit to its neighbor. (By "neighbor" we mean the digit on the right.)
Example: $3,425\times 11=37,675$
(0 + 3) (3 + 4) (4 + 2) (2 + 5) (5 + 0)
3 7 6 7 5
To illustrate:
11=10+1
Thus,
$3425\times 11=3425\times (10+1)$
$\rightarrow 37675=34250+3425$
Multiplying by 12
Proof
${\begin{aligned}R&=T*12\Leftrightarrow \\R&=T*(10+2)\\R&=10*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})+2*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*c_{n}+10^{n-1}*(2*c_{n}+c_{n-1})+\ldots +10^{1}*(2*c_{2}+c_{1})+2*c_{1}\\\\&QED\end{aligned}}$
Rule: to multiply by 12:
Starting from the rightmost digit, double each digit and add the neighbor. (The "neighbor" is the digit on the right.)
If the answer is greater than a single digit, simply carry over the extra digit (which will be a 1 or 2) to the next operation. The remaining digit is one digit of the final result.
Example: $316\times 12$
Determine neighbors in the multiplicand 0316:
• digit 6 has no right neighbor
• digit 1 has neighbor 6
• digit 3 has neighbor 1
• digit 0 (the prefixed zero) has neighbor 3
${\begin{aligned}6\times 2&=12{\text{ (2 carry 1) }}\\1\times 2+6+1&=9\\3\times 2+1&=7\\0\times 2+3&=3\\0\times 2+0&=0\\[10pt]316\times 12&=3,792\end{aligned}}$
Multiplying by 13
Proof
${\begin{aligned}R&=T*13\Leftrightarrow \\R&=T*(10+3)\\R&=10*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})+3*(10^{n-1}*c_{n}+\ldots +10^{0}*c_{1})\Leftrightarrow \\R&=10^{n}*c_{n}+10^{n-1}*(3*c_{n}+c_{n-1})+\ldots +10^{1}*(3*c_{2}+c_{1})+3*c_{1}\\\\&QED\end{aligned}}$
Publications
• Rushan Ziatdinov, Sajid Musa. Rapid mental computation system as a tool for algorithmic thinking of elementary school students development. European Researcher 25(7): 1105–1110, 2012 .
• The Trachtenberg Speed System of Basic Mathematics by Jakow Trachtenberg, A. Cutler (Translator), R. McShane (Translator), was published by Doubleday and Company, Inc. Garden City, New York in 1960.[1]
The book contains specific algebraic explanations for each of the above operations.
Most of the information in this article is from the original book.
The algorithms/operations for multiplication, etc., can be expressed in other more compact ways that the book does not specify, despite the chapter on algebraic description.[lower-alpha 1]
In popular culture
The 2017 American film Gifted revolves around a child prodigy who at the age of 7 impresses her teacher by doing calculations in her head using the Trachtenberg system.[2]
Other systems
There are many other methods of calculation in mental mathematics. The list below shows a few other methods of calculating, though they may not be entirely mental.
• Bharati Krishna Tirtha's book "Vedic Mathematics"
• Mental abacus – As students become used to manipulating the abacus with their fingers, they are typically asked to do calculation by visualizing abacus in their head. Almost all proficient abacus users are adept at doing arithmetic mentally.
• Chisanbop
Notes
1. All of this information is from an original book published and printed in 1960. The original book has seven full Chapters and is 270 pages long. The chapter titles are as follows. The numerous sub-categories in each chapter are not listed. The Trachtenberg speed system of basic mathematics
• Chapter 1 Tables or no tables
• Chapter 2 Rapid multiplication by the direct method
• Chapter 3 Speed multiplication-"two-finger" method
• Chapter 4 Addition and the right answer
• Chapter 5 Division – Speed and accuracy
• Chapter 6 Squares and square roots
• Chapter 7 Algebraic description of the method
Quotes:
• "A revolutionary new method for high-speed multiplication, division, addition, subtraction and square root." (1960)
• "The best selling method for high-speed multiplication, division, addition, subtraction and square root – without a calculator." (Reprinted 2009)
• Multiplication is done without multiplication tables
• "Can you multiply 5132437201 times 4522736502785 in seventy seconds?" "One young boy (grammar school-no calculator) did--successfully--by using The Trachtenberg Speed System of Basic Mathematics"
• Jakow Trachtenberg (its founder) escaped from Hitler's Germany from an active institution toward the close of WWII. Professor Trachtenberg fled to Germany when the czarist regime was overthrown in his homeland, Russia, and lived there peacefully until his mid-thirties when his anti-Hitler attitudes forced him to flee again. He was a fugitive and when captured spent a total of seven years in various concentration camps. It was during these years that Professor Trachtenberg devised the system of speed mathematics. Most of his work was done without pen or paper. Therefore most of the techniques can be performed mentally.
References
1. Trachtenberg, Jakow (1960). Cutler, Ann (ed.). The Trachtenberg Speed System of Basic Mathematics. Translated by A. Cutler, R. McShane. Doubleday and Company, Inc. p. 270. 1962 edition: ISBN 9780285629165.
2. @GiftedtheMovie (9 March 2017). "Hobbies include playing with legos and learning the Trachtenberg system 👷♀️📚✏️ @McKennaGraceful is Mary // #GiftedMovie" (Tweet) – via Twitter.
Further reading
• Trachtenberg, J. (1960). The Trachtenberg Speed System of Basic Mathematics. Doubleday and Company, Inc., Garden City, NY, USA.
• Катлер Э., Мак-Шейн Р.Система быстрого счёта по Трахтенбергу, 1967 (in Russian).
• Rushan Ziatdinov, Sajid Musa. "Rapid Mental Computation System as a Tool for Algorithmic Thinking of Elementary School Students Development", European Researcher 25(7): 1105–1110, 2012.
External links
• Chandrashekhar, Kiran. "[Learn All about] Mathematical Shortcuts", SapnaEdu.in at the Wayback Machine (archived 30 May 2018)
• Gifted (2017 film), This film is more about the Trachtenberg system, with Mckenna Grace, a young artist who has learned this technique, playing the lead role.
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State (functional analysis)
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both §§ Mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M .
In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number).
Jordan decomposition
States can be viewed as noncommutative generalizations of probability measures. By Gelfand representation, every commutative C*-algebra A is of the form C0(X) for some locally compact Hausdorff X. In this case, S(A) consists of positive Radon measures on X, and the § pure states are the evaluation functionals on X.
More generally, the GNS construction shows that every state is, after choosing a suitable representation, a vector state.
A bounded linear functional on a C*-algebra A is said to be self-adjoint if it is real-valued on the self-adjoint elements of A. Self-adjoint functionals are noncommutative analogues of signed measures.
The Jordan decomposition in measure theory says that every signed measure can be expressed as the difference of two positive measures supported on disjoint sets. This can be extended to the noncommutative setting.
Theorem — Every self-adjoint f in A* can be written as f = f+ − f− where f+ and f− are positive functionals and ||f|| = ||f+|| + ||f−||.
Proof
A proof can be sketched as follows: Let Ω be the weak*-compact set of positive linear functionals on A with norm ≤ 1, and C(Ω) be the continuous functions on Ω. A can be viewed as a closed linear subspace of C(Ω) (this is Kadison's function representation). By Hahn–Banach, f extends to a g in C(Ω)* with
It follows from the above decomposition that A* is the linear span of states.
Some important classes of states
Pure states
By the Krein-Milman theorem, the state space of M has extreme points. The extreme points of the state space are termed pure states and other states are known as mixed states.
Vector states
For a Hilbert space H and a vector x in H, the equation ωx(A) := ⟨Ax,x⟩ (for A in B(H) ), defines a positive linear functional on B(H). Since ωx(1)=||x||2, ωx is a state if ||x||=1. If A is a C*-subalgebra of B(H) and M an operator system in A, then the restriction of ωx to M defines a positive linear functional on M. The states of M that arise in this manner, from unit vectors in H, are termed vector states of M.
Faithful states
A state $\tau $ is faithful, if it is injective on the positive elements, that is, $\tau (a^{*}a)=0$ implies $a=0$.
Normal states
A state $\tau $ is called normal, iff for every monotone, increasing net $H_{\alpha }$ of operators with least upper bound $H$, $\tau (H_{\alpha })\;$ converges to $\tau (H)\;$.
Tracial states
A tracial state is a state $\tau $ such that
$\tau (AB)=\tau (BA)\;.$
For any separable C*-algebra, the set of tracial states is a Choquet simplex.
Factorial states
A factorial state of a C*-algebra A is a state such that the commutant of the corresponding GNS representation of A is a factor.
See also
• Quantum state
• Gelfand–Naimark–Segal construction
• Quantum mechanics
• Quantum state
• Density matrix
References
• Lin, H. (2001), An Introduction to the Classification of Amenable C*-algebras, World Scientific
Functional analysis (topics – glossary)
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