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Turing jump
In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem X a successively harder decision problem X′ with the property that X′ is not decidable by an oracle machine with an oracle for X.
The operator is called a jump operator because it increases the Turing degree of the problem X. That is, the problem X′ is not Turing-reducible to X. Post's theorem establishes a relationship between the Turing jump operator and the arithmetical hierarchy of sets of natural numbers.[1] Informally, given a problem, the Turing jump returns the set of Turing machines that halt when given access to an oracle that solves that problem.
Definition
The Turing jump of X can be thought of as an oracle to the halting problem for oracle machines with an oracle for X.[1]
Formally, given a set X and a Gödel numbering φiX of the X-computable functions, the Turing jump X′ of X is defined as
$X'=\{x\mid \varphi _{x}^{X}(x)\ {\mbox{is defined}}\}.$
The nth Turing jump X(n) is defined inductively by
$X^{(0)}=X,$
$X^{(n+1)}=(X^{(n)})'.$
The ω jump X(ω) of X is the effective join of the sequence of sets X(n) for n ∈ N:
$X^{(\omega )}=\{p_{i}^{k}\mid i\in \mathbb {N} {\text{ and }}k\in X^{(i)}\},$
where pi denotes the ith prime.
The notation 0′ or ∅′ is often used for the Turing jump of the empty set. It is read zero-jump or sometimes zero-prime.
Similarly, 0(n) is the nth jump of the empty set. For finite n, these sets are closely related to the arithmetic hierarchy,[2] and is in particular connected to Post's theorem.
The jump can be iterated into transfinite ordinals: there are jump operators $j^{\delta }$ for sets of natural numbers when $\delta $ is an ordinal that has a code in Kleene's ${\mathcal {O}}$ (regardless of code, the resulting jumps are the same by a theorem of Spector),[2] in particular the sets 0(α) for α < ω1CK, where ω1CK is the Church–Kleene ordinal, are closely related to the hyperarithmetic hierarchy.[1] Beyond ω1CK, the process can be continued through the countable ordinals of the constructible universe, using Jensen's work on fine structure theory of Godel's L.[3][2] The concept has also been generalized to extend to uncountable regular cardinals.[4]
Examples
• The Turing jump 0′ of the empty set is Turing equivalent to the halting problem.[5]
• For each n, the set 0(n) is m-complete at level $\Sigma _{n}^{0}$ in the arithmetical hierarchy (by Post's theorem).
• The set of Gödel numbers of true formulas in the language of Peano arithmetic with a predicate for X is computable from X(ω).[6]
Properties
• X′ is X-computably enumerable but not X-computable.
• If A is Turing-equivalent to B, then A′ is Turing-equivalent to B′. The converse of this implication is not true.
• (Shore and Slaman, 1999) The function mapping X to X′ is definable in the partial order of the Turing degrees.[5]
Many properties of the Turing jump operator are discussed in the article on Turing degrees.
References
1. Ambos-Spies, Klaus; Fejer, Peter A. (2014), "Degrees of Unsolvability", Handbook of the History of Logic, Elsevier, vol. 9, pp. 443–494, doi:10.1016/b978-0-444-51624-4.50010-1, ISBN 9780444516244.
2. S. G. Simpson, The Hierarchy Based on the Jump Operator, p.269. The Kleene Symposium (North-Holland, 1980)
3. Hodes, Harold T. (June 1980). "Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees". Journal of Symbolic Logic. Association for Symbolic Logic. 45 (2): 204–220. doi:10.2307/2273183. JSTOR 2273183. S2CID 41245500.
4. Lubarsky, Robert S. (December 1987). "Uncountable master codes and the jump hierarchy". The Journal of Symbolic Logic. 52 (4): 952–958. doi:10.2307/2273829. ISSN 0022-4812. JSTOR 2273829. S2CID 46113113.
5. Shore, Richard A.; Slaman, Theodore A. (1999). "Defining the Turing Jump". Mathematical Research Letters. 6 (6): 711–722. doi:10.4310/MRL.1999.v6.n6.a10.
6. Hodes, Harold T. (June 1980). "Jumping through the transfinite: the master code hierarchy of Turing degrees". The Journal of Symbolic Logic. 45 (2): 204–220. doi:10.2307/2273183. ISSN 0022-4812. JSTOR 2273183. S2CID 41245500.
• Ambos-Spies, K. and Fejer, P. Degrees of Unsolvability. Unpublished. http://www.cs.umb.edu/~fejer/articles/History_of_Degrees.pdf
• Lerman, M. (1983). Degrees of unsolvability: local and global theory. Berlin; New York: Springer-Verlag. ISBN 3-540-12155-2.
• Lubarsky, Robert S. (Dec 1987). "Uncountable Master Codes and the Jump Hierarchy". Journal of Symbolic Logic. Vol. 52, no. 4. pp. 952–958. JSTOR 2273829.
• Rogers Jr, H. (1987). Theory of recursive functions and effective computability. MIT Press, Cambridge, MA, USA. ISBN 0-07-053522-1.
• Shore, R.A.; Slaman, T.A. (1999). "Defining the Turing jump" (PDF). Mathematical Research Letters. 6 (5–6): 711–722. doi:10.4310/mrl.1999.v6.n6.a10. Retrieved 2008-07-13.
• Soare, R.I. (1987). Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer. ISBN 3-540-15299-7.
| Wikipedia |
Zero-product property
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, ${\text{if }}ab=0,{\text{ then }}a=0{\text{ or }}b=0.$
For the product of zero factors, see empty product.
This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties.[1] All of the number systems studied in elementary mathematics — the integers $\mathbb {Z} $, the rational numbers $\mathbb {Q} $, the real numbers $\mathbb {R} $, and the complex numbers $\mathbb {C} $ — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.
Algebraic context
Suppose $A$ is an algebraic structure. We might ask, does $A$ have the zero-product property? In order for this question to have meaning, $A$ must have both additive structure and multiplicative structure.[2] Usually one assumes that $A$ is a ring, though it could be something else, e.g. the set of nonnegative integers $\{0,1,2,\ldots \}$ with ordinary addition and multiplication, which is only a (commutative) semiring.
Note that if $A$ satisfies the zero-product property, and if $B$ is a subset of $A$, then $B$ also satisfies the zero product property: if $a$ and $b$ are elements of $B$ such that $ab=0$, then either $a=0$ or $b=0$ because $a$ and $b$ can also be considered as elements of $A$.
Examples
• A ring in which the zero-product property holds is called a domain. A commutative domain with a multiplicative identity element is called an integral domain. Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zero-product property holds for any subring of a skew field.
• If $p$ is a prime number, then the ring of integers modulo $p$ has the zero-product property (in fact, it is a field).
• The Gaussian integers are an integral domain because they are a subring of the complex numbers.
• In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.
• The set of nonnegative integers $\{0,1,2,\ldots \}$ is not a ring (being instead a semiring), but it does satisfy the zero-product property.
Non-examples
• Let $\mathbb {Z} _{n}$ denote the ring of integers modulo $n$. Then $\mathbb {Z} _{6}$ does not satisfy the zero product property: 2 and 3 are nonzero elements, yet $2\cdot 3\equiv 0{\pmod {6}}$.
• In general, if $n$ is a composite number, then $\mathbb {Z} _{n}$ does not satisfy the zero-product property. Namely, if $n=qm$ where $0<q,m<n$, then $m$ and $q$ are nonzero modulo $n$, yet $qm\equiv 0{\pmod {n}}$.
• The ring $\mathbb {Z} ^{2\times 2}$ of 2×2 matrices with integer entries does not satisfy the zero-product property: if
$M={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}$
and $N={\begin{pmatrix}0&1\\0&1\end{pmatrix}},$ then
$MN={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}=0,$
yet neither $M$ nor $N$ is zero.
• The ring of all functions $f:[0,1]\to \mathbb {R} $, from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any n ≥ 2, functions $f_{1},\ldots ,f_{n}$, none of which is identically zero, such that $f_{i}\,f_{j}$ is identically zero whenever $i\neq j$.
• The same is true even if we consider only continuous functions, or only even infinitely smooth functions. On the other hand, analytic functions have the zero-product property.
Application to finding roots of polynomials
Suppose $P$ and $Q$ are univariate polynomials with real coefficients, and $x$ is a real number such that $P(x)Q(x)=0$. (Actually, we may allow the coefficients and $x$ to come from any integral domain.) By the zero-product property, it follows that either $P(x)=0$ or $Q(x)=0$. In other words, the roots of $PQ$ are precisely the roots of $P$ together with the roots of $Q$.
Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial $x^{3}-2x^{2}-5x+6$ factorizes as $(x-3)(x-1)(x+2)$; hence, its roots are precisely 3, 1, and −2.
In general, suppose $R$ is an integral domain and $f$ is a monic univariate polynomial of degree $d\geq 1$ with coefficients in $R$. Suppose also that $f$ has $d$ distinct roots $r_{1},\ldots ,r_{d}\in R$. It follows (but we do not prove here) that $f$ factorizes as $f(x)=(x-r_{1})\cdots (x-r_{d})$. By the zero-product property, it follows that $r_{1},\ldots ,r_{d}$ are the only roots of $f$: any root of $f$ must be a root of $(x-r_{i})$ for some $i$. In particular, $f$ has at most $d$ distinct roots.
If however $R$ is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial $x^{3}+3x^{2}+2x$ has six roots in $\mathbb {Z} _{6}$ (though it has only three roots in $\mathbb {Z} $).
See also
• Fundamental theorem of algebra
• Integral domain and domain
• Prime ideal
• Zero divisor
Notes
1. The other being a⋅0 = 0⋅a = 0. Mustafa A. Munem and David J. Foulis, Algebra and Trigonometry with Applications (New York: Worth Publishers, 1982), p. 4.
2. There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication.
References
• David S. Dummit and Richard M. Foote, Abstract Algebra (3d ed.), Wiley, 2003, ISBN 0-471-43334-9.
External links
• PlanetMath: Zero rule of product
| Wikipedia |
Zero-sum Ramsey theory
In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics. It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group $A$), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in $A$). It combines tools from number theory, algebra, linear algebra, graph theory, discrete analysis, and other branches of mathematics.
The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv:[1] for any $2m-1$ elements of $\mathbb {Z} _{m}$, there is a subset of size $m$ that sums to zero.[2] (This bound is tight, as a sequence of $m-1$ zeroes and $m-1$ ones cannot have any subset of size $m$ summing to zero.[2]) There are known proofs of this result using the Cauchy-Davenport theorem, Fermat's little theorem, or the Chevalley–Warning theorem.[2]
Generalizing this result, one can define for any abelian group G the minimum quantity $EGZ(G)$ of elements of G such that there must be a subsequence of $o(G)$ elements (where $o(G)$ is the order of the group) which adds to zero. It is known that $EGZ(G)\leq 2o(G)-1$, and that this bound is strict if and only if $G=\mathbb {Z} _{m}$.[2]
See also
• Zero-sum problem
References
1. Erdős, Paul; Ginzburg, A.; Ziv, A. (1961). "Theorem in the additive number theory". Bull. Res. Council Israel. 10F: 41–43. Zbl 0063.00009.
2. "Erdös-Ginzburg-Ziv theorem - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2023-05-22.
Further reading
• Zero-sum problems - A survey (open-access journal article)
• Zero-Sum Ramsey Theory: Graphs, Sequences and More (workshop homepage)
• A. Bialostocki, "Zero-sum trees: a survey of results and open problems" N.W. Sauer (ed.) R.E. Woodrow (ed.) B. Sands (ed.), Finite and Infinite Combinatorics in Sets and Logic, Nato ASI Ser., Kluwer Acad. Publ. (1993) pp. 19–29
• Y. Caro, "Zero-sum problems: a survey" Discrete Math., 152 (1996) pp. 93–113
| Wikipedia |
Zero-symmetric graph
In the mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge.[1]
The smallest zero-symmetric graph, with 18 vertices and 27 edges
The truncated cuboctahedron, a zero-symmetric polyhedron
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive
→ edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric
The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter.[2] In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.[3]
Examples
The smallest zero-symmetric graph is a nonplanar graph with 18 vertices.[4] Its LCF notation is [5,−5]9.
Among planar graphs, the truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric.[5]
These examples are all bipartite graphs. However, there exist larger examples of zero-symmetric graphs that are not bipartite.[6]
These examples also have three different symmetry classes (orbits) of edges. However, there exist zero-symmetric graphs with only two orbits of edges. The smallest such graph has 20 vertices, with LCF notation [6,6,-6,-6]5.[7]
Properties
Every finite zero-symmetric graph is a Cayley graph, a property that does not always hold for cubic vertex-transitive graphs more generally and that helps in the solution of combinatorial enumeration tasks concerning zero-symmetric graphs. There are 97687 zero-symmetric graphs on up to 1280 vertices. These graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices.[8]
Unsolved problem in mathematics:
Does every finite zero-symmetric graph contain a Hamiltonian cycle?
(more unsolved problems in mathematics)
All known finite connected zero-symmetric graphs contain a Hamiltonian cycle, but it is unknown whether every finite connected zero-symmetric graph is necessarily Hamiltonian.[9] This is a special case of the Lovász conjecture that (with five known exceptions, none of which is zero-symmetric) every finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian.
See also
• Semi-symmetric graph, graphs that have symmetries between every two edges but not between every two vertices (reversing the roles of edges and vertices in the definition of zero-symmetric graphs)
References
1. Coxeter, Harold Scott MacDonald; Frucht, Roberto; Powers, David L. (1981), Zero-symmetric graphs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, ISBN 0-12-194580-4, MR 0658666
2. Coxeter, Frucht & Powers (1981), p. ix.
3. Lauri, Josef; Scapellato, Raffaele (2003), Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts, Cambridge University Press, p. 66, ISBN 9780521529037.
4. Coxeter, Frucht & Powers (1981), Figure 1.1, p. 5.
5. Coxeter, Frucht & Powers (1981), pp. 75 and 80.
6. Coxeter, Frucht & Powers (1981), p. 55.
7. Conder, Marston D. E.; Pisanski, Tomaž; Žitnik, Arjana (2017), "Vertex-transitive graphs and their arc-types", Ars Mathematica Contemporanea, 12 (2): 383–413, arXiv:1505.02029, doi:10.26493/1855-3974.1146.f96, MR 3646702
8. Potočnik, Primož; Spiga, Pablo; Verret, Gabriel (2013), "Cubic vertex-transitive graphs on up to 1280 vertices", Journal of Symbolic Computation, 50: 465–477, arXiv:1201.5317, doi:10.1016/j.jsc.2012.09.002, MR 2996891.
9. Coxeter, Frucht & Powers (1981), p. 10.
| Wikipedia |
Zero sharp
In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscovered by Solovay (1967, p.52), who considered it as a subset of the natural numbers and introduced the notation O# (with a capital letter O; this later changed to the numeral '0').
Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets.
Definition
Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols c1, c2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with ci interpreted as the uncountable cardinal $\aleph _{i}$. (Here $\aleph _{i}$ means $\aleph _{i}$ in the full universe, not the constructible universe.)
If there is in V an uncountable set of Silver order-indiscernibles in the constructible universe L, then 0# is the set of Gödel numbers of formulas θ of set theory such that
$L_{\omega _{\omega }}\models \theta (\omega _{1},\omega _{2},...\omega _{n})$
where ω1, ... ωω are the "small" uncountable initial ordinals in V, but have all large cardinal properties consistent with V=L relative to L.
There is a subtlety about this definition: by Tarski's undefinability theorem it is not, in general, possible to define the truth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible to define the truth of statements about the constructible universe. More generally, the definition of 0# works provided that there is an uncountable set of indiscernibles for some Lα, and the phrase "0# exists" is used as a shorthand way of saying this.
There are several minor variations of the definition of 0#, which make no significant difference to its properties. There are many different choices of Gödel numbering, and 0# depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode 0# as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number.
Statements implying existence
The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened. The existence of ω1-Erdős cardinals implies the existence of 0#. This is close to being best possible, because the existence of 0# implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0#.
Chang's conjecture implies the existence of 0#.
Statements equivalent to existence
Kunen showed that 0# exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe L into itself.
Donald A. Martin and Leo Harrington have shown that the existence of 0# is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#.
It follows from Jensen's covering theorem that the existence of 0# is equivalent to ωω being a regular cardinal in the constructible universe L.
Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of 0#.
Consequences of existence and non-existence
Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existence of 0# contradicts the axiom of constructibility: V = L.
If 0# exists, then it is an example of a non-constructible Δ1
3
set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ1
2
and Π1
2
sets of integers are constructible.
On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered. In that case, Jensen's covering lemma holds:
For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the same cardinality as x.
This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves $\omega _{1}$ and collapses $\omega _{2}$ to an ordinal of cofinality $\omega $. Let $G$ be an $\omega $-sequence cofinal on $\omega _{2}^{L}$ and generic over L. Then no set in L of L-size smaller than $\omega _{2}^{L}$ (which is uncountable in V, since $\omega _{1}$ is preserved) can cover $G$, since $\omega _{2}$ is a regular cardinal.
Other sharps
If x is any set, then x# is defined analogously to 0# except that one uses L[x] instead of L. See the section on relative constructibility in constructible universe.
See also
• 0†, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurable cardinal.
References
• 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.
• Harrington, Leo (1978), "Analytic determinacy and 0#", The Journal of Symbolic Logic, 43 (4): 685–693, doi:10.2307/2273508, ISSN 0022-4812, JSTOR 2273508, MR 0518675, S2CID 46061318
• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
• Martin, Donald A. (1970), "Measurable cardinals and analytic games", Polska Akademia Nauk. Fundamenta Mathematicae, 66 (3): 287–291, doi:10.4064/fm-66-3-287-291, ISSN 0016-2736, MR 0258637
• Silver, Jack H. (1971) [1966], "Some applications of model theory in set theory", Annals of Pure and Applied Logic, 3 (1): 45–110, doi:10.1016/0003-4843(71)90010-6, ISSN 0168-0072, MR 0409188
• Solovay, Robert M. (1967), "A nonconstructible Δ1
3
set of integers", Transactions of the American Mathematical Society, 127: 50–75, doi:10.2307/1994631, ISSN 0002-9947, JSTOR 1994631, MR 0211873
| Wikipedia |
Zero bias transform
The zero-bias transform is a transform from one probability distribution to another. The transform arises in applications of Stein's method in probability and statistics.
Formal definition
The zero bias transform may be applied to both discrete and continuous random variables. The zero bias transform of a density function f(t), defined for all real numbers t ≥ 0, is the function g(s), defined by
$g(s)=\int _{s}^{\infty }tf(t)1(t>s)\,dt$
where s and t are real numbers and f(t) is the density or mass function of the random variable T.[1]
An equivalent but alternative approach is to deduce the nature of the transformed random variable by evaluating the expected value
$\operatorname {E} (TH(T))=\sigma ^{2}E(h(T^{z}))$
where the right-side superscript denotes a zero biased random variable whereas the left hand side expectation represents the original random variable. An example from each approach is given in the examples section beneath.
If the random variable is discrete the integral becomes a sum from positive infinity to s. The zero bias transform is taken for a mean zero, variance 1 random variable which may require a location-scale transform to the random variable.
Applications
The zero bias transformation arises in applications where a normal approximation is desired. Similar to Stein's method the zero bias transform is often applied to sums of random variables with each summand having finite variance an mean zero.
The zero bias transform has been applied to CDO tranche pricing.[2]
Examples
1. Consider a Bernoulli(p) random variable B with Pr(B = 0) = 1 − p. The zero bias transform of T = (B − p) is:
${\begin{aligned}\operatorname {E} (TH(T))&=-p(1-p)H(-p)+(1-p)pH(1-p)\\&=p(1-p)[H(1-p)-H(-p)]\\&=p(1-p)\int _{-p}^{1-p}h(s)\,ds\end{aligned}}$
where h is the derivative of H. From there it follows that the random variable S is a continuous uniform random variable on the support (−p, 1 − p). This example shows how the zero bias transform smooths a discrete distribution into a continuous distribution.
2. Consider the continuous uniform on the support $(-{\sqrt {3}},{\sqrt {3}})$.
$\int _{s}^{\sqrt {3}}t1(t>s)f(t)\,dt=\int _{s}^{\sqrt {3}}{\frac {t}{2{\sqrt {3}}}}\,dt={\frac {\sqrt {3}}{4}}-{\frac {s^{2}}{{\sqrt {3}}\,4}}{\text{ where }}-{\sqrt {3}}<s<{\sqrt {3}}$
This example shows that the zero bias transform takes continuous symmetric distributions and makes them unimodular.
References
1. Goldstein, Larry; Reinert, Gesine (1997), "Stein's Method and the Zero Bias Transformation with Application to Simple Random Sampling" (PDF), The Annals of Applied Probability, 7 (4): 935–952
2. Karoui, N. El; Jiao, Y. (2009). "Stein's method and zero bias transformation for CDO tranche pricing". Finance and Stochastics. 13 (2): 151–180. doi:10.1007/s00780-008-0084-6.
| Wikipedia |
Zero crossing
A zero-crossing is a point where the sign of a mathematical function changes (e.g. from positive to negative), represented by an intercept of the axis (zero value) in the graph of the function. It is a commonly used term in electronics, mathematics, acoustics, and image processing.
In electronics
In alternating current, the zero-crossing is the instantaneous point at which there is no voltage present. In a sine wave or other simple waveform, this normally occurs twice during each cycle. It is a device for detecting the point where the voltage crosses zero in either direction.
The zero-crossing is important for systems that send digital data over AC circuits, such as modems, X10 home automation control systems, and Digital Command Control type systems for Lionel and other AC model trains.
Counting zero-crossings is also a method used in speech processing to estimate the fundamental frequency of speech.
In a system where an amplifier with digitally controlled gain is applied to an input signal, artifacts in the non-zero output signal occur when the gain of the amplifier is abruptly switched between its discrete gain settings. At audio frequencies, such as in modern consumer electronics like digital audio players, these effects are clearly audible, resulting in a 'zipping' sound when rapidly ramping the gain or a soft 'click' when a single gain change is made. Artifacts are disconcerting and clearly not desirable. If changes are made only at zero-crossings of the input signal, then no matter how the amplifier gain setting changes, the output also remains at zero, thereby minimizing the change. (The instantaneous change in gain will still produce distortion, but it will not produce a click.)
If electrical power is to be switched, no electrical interference is generated if switched at an instant when there is no current—a zero crossing. Early light dimmers and similar devices generated interference; later versions were designed to switch at the zero crossing.
In image processing
In the field of digital image Processing, great emphasis is placed on operators that seek out edges within an image. They are called edge detection or gradient filters. A gradient filter is a filter that seeks out areas of rapid change in pixel value. These points usually mark an edge or a boundary. A Laplace filter is a filter that fits in this family, though it sets about the task in a different way. It seeks out points in the signal stream where the digital signal of an image passes through a pre-set '0' value, and marks this out as a potential edge point. Because the signal has crossed through the point of zero, it is called a zero-crossing. An example can be found here, including the source in Java.
In the field of industrial radiography, it is used as a simple method for the segmentation of potential defects.[1]
See also
• Reconstruction from zero crossings
• Zero crossing control
• Zero-crossing rate
• Zero of a function (a root)
• Sign function
References
1. Mery, Domingo (2015). Computer Vision for X-Ray Testing. Switzerland: Springer International Publishing. p. 271. ISBN 978-3319207469.
| Wikipedia |
Zero dagger
In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) The definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions. Specifically, if ZFC is consistent, then ZFC + "0† does not exist" is consistent. ZFC + "0† exists" is not known to be inconsistent (and most set theorists believe that it is consistent). In other words, it is believed to be independent (see large cardinal for a discussion). It is usually formulated as follows:
0† exists if and only if there exists a non-trivial elementary embedding j : L[U] → L[U] for the relativized Gödel constructible universe L[U], where U is an ultrafilter witnessing that some cardinal κ is measurable.
If 0† exists, then a careful analysis of the embeddings of L[U] into itself reveals that there is a closed unbounded subset of κ, and a closed unbounded proper class of ordinals greater than κ, which together are indiscernible for the structure $(L,\in ,U)$, and 0† is defined to be the set of Gödel numbers of the true formulas about the indiscernibles in L[U].
Solovay showed that the existence of 0† follows from the existence of two measurable cardinals. It is traditionally considered a large cardinal axiom, although it is not a large cardinal, nor indeed a cardinal at all.
See also
• 0#: a set of formulas (or subset of the integers) defined in a similar fashion, but simpler.
References
• Kanamori, Akihiro; Awerbuch-Friedlander, Tamara (1990). "The compleat 0†". Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. 36 (2): 133–141. doi:10.1002/malq.19900360206. ISSN 0044-3050. MR 1068949.
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.
External links
• Definition by "Zentralblatt math database" (PDF)
| Wikipedia |
Zero dynamics
In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems.[1]
History
The idea was introduced thirty years ago as the nonlinear approach to the concept of transmission of zeros. The original purpose of introducing the concept was to develop an asymptotic stabilization with a set of guaranteed regions of attraction (semi-global stabilizability), to make the overall system stable.[2]
Initial working
Given the internal dynamics of any system, zero dynamics refers to the control action chosen in which the output variables of the system are kept identically zero.[3] While, various systems have an equally distinctive set of zeros, such as decoupling zeros, invariant zeros, and transmission zeros. Thus, the reason for developing this concept was to control the non-minimum phase and nonlinear systems effectively.[4]
Applications
The concept is widely utilized in SISO mechanical systems, whereby applying a few heuristic approaches, zeros can be identified for various linear systems.[5] Zero dynamics adds an essential feature to the overall system’s analysis and the design of the controllers. Mainly its behavior plays a significant role in measuring the performance limitations of specific feedback systems. In a Single Input Single Output system, the zero dynamics can be identified by using junction structure patterns. In other words, using concepts like bond graph models can help to point out the potential direction of the SISO systems.[6]
Apart from its application in nonlinear standardized systems, similar controlled results can be obtained by using zero dynamics on nonlinear discrete-time systems. In this scenario, the application of zero dynamics can be an interesting tool to measure the performance of nonlinear digital design systems (nonlinear discrete-time systems).[7]
Before the advent of zero dynamics, the problem of acquiring non-interacting control systems by using internal stability was not specifically discussed. However, with the asymptotic stability present within the zero dynamics of a system, static feedback can be ensured. Such results make zero dynamics an interesting tool to guarantee the internal stability of non-interacting control systems.[8]
References
1. Van de Straete, H.J.; Youcef-Toumi, K. (June 1996). "Physical Meaning of Zeros and Transmission Zeros from Bond Graph Models". IFAC Proceedings Volumes. 29 (1): 4422–4427. doi:10.1016/s1474-6670(17)58377-9. hdl:1721.1/11140. ISSN 1474-6670.
2. Isidori, Alberto (September 2013). "The zero dynamics of a nonlinear system: From the origin to the latest progresses of a long successful story". European Journal of Control. 19 (5): 369–378. doi:10.1016/j.ejcon.2013.05.014. ISSN 0947-3580. S2CID 15277067.
3. Youcef-Toumi, K.; Wu, S-T (June 1991). "Input/Output Linearization using Time Delay Control". 1991 American Control Conference. IEEE: 2601–2606. doi:10.23919/acc.1991.4791872. ISBN 0-87942-565-2. S2CID 20562917.
4. "Control Theory", Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics, vol. 1768, Springer Berlin Heidelberg, 2001, pp. 91–149, doi:10.1007/3-540-45436-5_5, ISBN 978-3-540-42626-4
5. Miu, D. K. (1991-09-01). "Physical Interpretation of Transfer Function Zeros for Simple Control Systems With Mechanical Flexibilities". Journal of Dynamic Systems, Measurement, and Control. 113 (3): 419–424. doi:10.1115/1.2896426. ISSN 0022-0434.
6. Huang, S.Y.; Youcef-Toumi, K. (June 1996). "Zero Dynamics of Nonlinear MIMO Systems from System Configurations - A Bond Graph Approach". IFAC Proceedings Volumes. 29 (1): 4392–4397. doi:10.1016/s1474-6670(17)58372-x. ISSN 1474-6670.
7. Monaco, S.; Normand-Cyrot, D. (September 1988). "Zero dynamics of sampled nonlinear systems". Systems & Control Letters. 11 (3): 229–234. doi:10.1016/0167-6911(88)90063-1. ISSN 0167-6911.
8. Isidori, A.; Grizzle, J.W. (October 1988). "Fixed modes and nonlinear noninteracting control with stability". IEEE Transactions on Automatic Control. 33 (10): 907–914. doi:10.1109/9.7244. ISSN 0018-9286.
| Wikipedia |
ZPP (complexity)
In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these properties:
• It always returns the correct YES or NO answer.
• The running time is polynomial in expectation for every input.
In other words, if the algorithm is allowed to flip a truly-random coin while it is running, it will always return the correct answer and, for a problem of size n, there is some polynomial p(n) such that the average running time will be less than p(n), even though it might occasionally be much longer. Such an algorithm is called a Las Vegas algorithm.
Alternatively, ZPP can be defined as the class of problems for which a probabilistic Turing machine exists with these properties:
• It always runs in polynomial time.
• It returns an answer YES, NO or DO NOT KNOW.
• The answer is always either DO NOT KNOW or the correct answer.
• It returns DO NOT KNOW with probability at most 1/2 for every input (and the correct answer otherwise).
The two definitions are equivalent.
The definition of ZPP is based on probabilistic Turing machines, but, for clarity, note that other complexity classes based on them include BPP and RP. The class BQP is based on another machine with randomness: the quantum computer.
Intersection definition
The class ZPP is exactly equal to the intersection of the classes RP and co-RP. This is often taken to be the definition of ZPP. To show this, first note that every problem which is in both RP and co-RP has a Las Vegas algorithm as follows:
• Suppose we have a language L recognized by both the RP algorithm A and the (possibly completely different) co-RP algorithm B.
• Given an input, run A on the input for one step. If it returns YES, the answer must be YES. Otherwise, run B on the input for one step. If it returns NO, the answer must be NO. If neither occurs, repeat this step.
Note that only one machine can ever give a wrong answer, and the chance of that machine giving the wrong answer during each repetition is at most 50%. This means that the chance of reaching the kth round shrinks exponentially in k, showing that the expected running time is polynomial. This shows that RP intersect co-RP is contained in ZPP.
To show that ZPP is contained in RP intersect co-RP, suppose we have a Las Vegas algorithm C to solve a problem. We can then construct the following RP algorithm:
• Run C for at least double its expected running time. If it gives an answer, give that answer. If it doesn't give any answer before we stop it, give NO.
By Markov's Inequality, the chance that it will yield an answer before we stop it is at least 1/2. This means the chance we'll give the wrong answer on a YES instance, by stopping and yielding NO, is at most 1/2, fitting the definition of an RP algorithm. The co-RP algorithm is identical, except that it gives YES if C "times out".
Witness and proof
The classes NP, RP and ZPP can be thought of in terms of proof of membership in a set.
Definition: A verifier V for a set X is a Turing machine such that:
• if x is in X then there exists a string w such that V(x,w) accepts;
• if x is not in X, then for all strings w, V(x,w) rejects.
The string w can be thought of as the proof of membership. In the case of short proofs (of length bounded by a polynomial in the size of the input) which can be efficiently verified (V is a polynomial-time deterministic Turing machine), the string w is called a witness.
Notes:
• The definition is very asymmetric. The proof of x being in X is a single string. The proof of x not being in X is the collection of all strings, none of which is a proof of membership.
• For all x in X there must be a witness to its membership in X.
• The witness need not be a traditionally construed proof. If V is a probabilistic Turing machine which could possibly accept x if x is in X, then the proof is the string of coin flips which leads the machine to accept x (provided all members in X have some witness and the machine never accepts a non-member).
• The co-concept is a proof of non-membership, or membership in the complement set.
The classes NP, RP and ZPP are sets which have witnesses for membership. The class NP requires only that witnesses exist. They may be very rare. Of the 2f(|x|) possible strings, with f a polynomial, only one need cause the verifier to accept (if x is in X. If x is not in X, no string will cause the verifier to accept).
For the classes RP and ZPP any string chosen at random will likely be a witness.
The corresponding co-classes have witness for non-membership. In particular, co-RP is the class of sets for which, if x is not in X, any randomly chosen string is likely to be a witness for non-membership. ZPP is the class of sets for which any random string is likely to be a witness of x in X, or x not in X, which ever the case may be.
Connecting this definition with other definitions of RP, co-RP and ZPP is easy. The probabilistic polynomial-time Turing Machine V*w(x) corresponds to the deterministic polynomial-time Turing Machine V(x, w) by replacing the random tape of V* with a second input tape for V on which is written the sequence of coin flips. By selecting the witness as a random string, the verifier is a probabilistic polynomial-time Turing Machine whose probability of accepting x when x is in X is large (greater than 1/2, say), but zero if x ∉ X (for RP); of rejecting x when x is not in X is large but zero if x ∈ X (for co-RP); and of correctly accepting or rejecting x as a member of X is large, but zero of incorrectly accepting or rejecting x (for ZPP).
By repeated random selection of a possible witness, the large probability that a random string is a witness gives an expected polynomial time algorithm for accepting or rejecting an input. Conversely, if the Turing Machine is expected polynomial-time (for any given x), then a considerable fraction of the runs must be polynomial-time bounded, and the coin sequence used in such a run will be a witness.
ZPP should be contrasted with BPP. The class BPP does not require witnesses, although witnesses are sufficient (hence BPP contains RP, co-RP and ZPP). A BPP language has V(x,w) accept on a (clear) majority of strings w if x is in X, and conversely reject on a (clear) majority of strings w if x is not in X. No single string w need be definitive, and therefore they cannot in general be considered proofs or witnesses.
Complexity-theoretic properties
It is known that ZPP is closed under complement; that is, ZPP = co-ZPP.
ZPP is low for itself, meaning that a ZPP machine with the power to solve ZPP problems instantly (a ZPP oracle machine) is not any more powerful than the machine without this extra power. In symbols, ZPPZPP = ZPP.
ZPPNPBPP = ZPPNP.
NPBPP is contained in ZPPNP.
Connection to other classes
Since ZPP = RP ∩ coRP, ZPP is obviously contained in both RP and coRP.
The class P is contained in ZPP, and some computer scientists have conjectured that P = ZPP, i.e., every Las Vegas algorithm has a deterministic polynomial-time equivalent.
There exists an oracle relative to which ZPP = EXPTIME. A proof for ZPP = EXPTIME would imply that P ≠ ZPP, as P ≠ EXPTIME (see time hierarchy theorem).
See also
• BPP
• RP
External links
• Complexity Zoo: ZPP Class ZPP
Important complexity classes
Considered feasible
• DLOGTIME
• AC0
• ACC0
• TC0
• L
• SL
• RL
• NL
• NL-complete
• NC
• SC
• CC
• P
• P-complete
• ZPP
• RP
• BPP
• BQP
• APX
• FP
Suspected infeasible
• UP
• NP
• NP-complete
• NP-hard
• co-NP
• co-NP-complete
• AM
• QMA
• PH
• ⊕P
• PP
• #P
• #P-complete
• IP
• PSPACE
• PSPACE-complete
Considered infeasible
• EXPTIME
• NEXPTIME
• EXPSPACE
• 2-EXPTIME
• ELEMENTARY
• PR
• R
• RE
• ALL
Class hierarchies
• Polynomial hierarchy
• Exponential hierarchy
• Grzegorczyk hierarchy
• Arithmetical hierarchy
• Boolean hierarchy
Families of classes
• DTIME
• NTIME
• DSPACE
• NSPACE
• Probabilistically checkable proof
• Interactive proof system
List of complexity classes
| Wikipedia |
Trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: $0,1,$ or $e$ depending on the context. If the group operation is denoted $\,\cdot \,$ then it is defined by $e\cdot e=e.$
The similarly defined trivial monoid is also a group since its only element is its own inverse, and is hence the same as the trivial group.
The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group.
Definitions
Given any group $G,$ the group consisting of only the identity element is a subgroup of $G,$ and, being the trivial group, is called the trivial subgroup of $G.$
The term, when referred to "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/":): G has no nontrivial proper subgroups" refers to the only subgroups of $G$ being the trivial group $\{e\}$ and the group $G$ itself.
Properties
The trivial group is cyclic of order $1$; as such it may be denoted $\mathrm {Z} _{1}$ or $\mathrm {C} _{1}.$ If the group operation is called addition, the trivial group is usually denoted by $0.$ If the group operation is called multiplication then 1 can be a notation for the trivial group. Combining these leads to the trivial ring in which the addition and multiplication operations are identical and $0=1.$
The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.
The trivial group can be made a (bi-)ordered group by equipping it with the trivial non-strict order $\,\leq .$
See also
• Zero object (algebra) – Algebraic structure with only one element
• List of small groups
References
• Rowland, Todd & Weisstein, Eric W. "Trivial Group". MathWorld.
Groups
Basic notions
• Subgroup
• Normal subgroup
• Commutator subgroup
• Quotient group
• Group homomorphism
• (Semi-) direct product
• direct sum
Types of groups
• Finite groups
• Abelian groups
• Cyclic groups
• Infinite group
• Simple groups
• Solvable groups
• Symmetry group
• Space group
• Point group
• Wallpaper group
• Trivial group
Discrete groups
Classification of finite simple groups
Cyclic group Zn
Alternating group An
Sporadic groups
Mathieu group M11..12,M22..24
Conway group Co1..3
Janko groups J1, J2, J3, J4
Fischer group F22..24
Baby monster group B
Monster group M
Other finite groups
Symmetric group Sn
Dihedral group Dn
Rubik's Cube group
Lie groups
• General linear group GL(n)
• Special linear group SL(n)
• Orthogonal group O(n)
• Special orthogonal group SO(n)
• Unitary group U(n)
• Special unitary group SU(n)
• Symplectic group Sp(n)
Exceptional Lie groups
G2
F4
E6
E7
E8
• Circle group
• Lorentz group
• Poincaré group
• Quaternion group
Infinite dimensional groups
• Conformal group
• Diffeomorphism group
• Loop group
• Quantum group
• O(∞)
• SU(∞)
• Sp(∞)
• History
• Applications
• Abstract algebra
| Wikipedia |
Locus (mathematics)
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.[1][2]
The set of the points that satisfy some property is often called the locus of a point satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be located or may move.
History and philosophy
Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is the set of points that are at a given distance from the center.[3]
In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as avoiding the actual infinite was an important philosophical position of earlier mathematicians.[4][5]
Once set theory became the universal basis over which the whole mathematics is built,[6] the term of locus became rather old-fashioned.[7] Nevertheless, the word is still widely used, mainly for a concise formulation, for example:
• Critical locus, the set of the critical points of a differentiable function.
• Zero locus or vanishing locus, the set of points where a function vanishes, in that it takes the value zero.
• Singular locus, the set of the singular points of an algebraic variety.
• Connectedness locus, the subset of the parameter set of a family of rational functions for which the Julia set of the function is connected.
More recently, techniques such as the theory of schemes, and the use of category theory instead of set theory to give a foundation to mathematics, have returned to notions more like the original definition of a locus as an object in itself rather than as a set of points.[5]
Examples in plane geometry
Examples from plane geometry include:
• The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points.[8]
• The set of points equidistant from two intersecting lines is the union of their two angle bisectors.
• All conic sections are loci:[9]
• Circle: the set of points at constant distance (the radius) from a fixed point (the center).
• Parabola: the set of points equidistant from a fixed point (the focus) and a line (the directrix).
• Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
• Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant
Other examples of loci appear in various areas of mathematics. For example, in complex dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps.
Proof of a locus
To prove a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages: the proof that all the points that satisfy the conditions are on the given shape, and the proof that all the points on the given shape satisfy the conditions.[10]
Examples
First example
Find the locus of a point P that has a given ratio of distances k = d1/d2 to two given points.
In this example k = 3, A(−1, 0) and B(0, 2) are chosen as the fixed points.
P(x, y) is a point of the locus
$\Leftrightarrow |PA|=3|PB|$
$\Leftrightarrow |PA|^{2}=9|PB|^{2}$
$\Leftrightarrow (x+1)^{2}+(y-0)^{2}=9(x-0)^{2}+9(y-2)^{2}$
$\Leftrightarrow 8(x^{2}+y^{2})-2x-36y+35=0$
$\Leftrightarrow \left(x-{\frac {1}{8}}\right)^{2}+\left(y-{\frac {9}{4}}\right)^{2}={\frac {45}{64}}.$
This equation represents a circle with center (1/8, 9/4) and radius ${\tfrac {3}{8}}{\sqrt {5}}$. It is the circle of Apollonius defined by these values of k, A, and B.
Second example
A triangle ABC has a fixed side [AB] with length c. Determine the locus of the third vertex C such that the medians from A and C are orthogonal.
Choose an orthonormal coordinate system such that A(−c/2, 0), B(c/2, 0). C(x, y) is the variable third vertex. The center of [BC] is M((2x + c)/4, y/2). The median from C has a slope y/x. The median AM has slope 2y/(2x + 3c).
C(x, y) is a point of the locus
$\Leftrightarrow $ the medians from A and C are orthogonal
$\Leftrightarrow {\frac {y}{x}}\cdot {\frac {2y}{2x+3c}}=-1$
$\Leftrightarrow 2y^{2}+2x^{2}+3cx=0$
$\Leftrightarrow x^{2}+y^{2}+(3c/2)x=0$
$\Leftrightarrow (x+3c/4)^{2}+y^{2}=9c^{2}/16.$
The locus of the vertex C is a circle with center (−3c/4, 0) and radius 3c/4.
Third example
A locus can also be defined by two associated curves depending on one common parameter. If the parameter varies, the intersection points of the associated curves describe the locus.
In the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is perpendicular to k. The angle $\alpha $ between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.
Fourth example
A locus of points need not be one-dimensional (as a circle, line, etc.). For example,[1] the locus of the inequality 2x + 3y – 6 < 0 is the portion of the plane that is below the line of equation 2x + 3y – 6 = 0.
See also
• Algebraic variety
• Curve
• Line (geometry)
• Set-builder notation
• Shape (geometry)
References
1. James, Robert Clarke; James, Glenn (1992), Mathematics Dictionary, Springer, p. 255, ISBN 978-0-412-99041-0.
2. Whitehead, Alfred North (1911), An Introduction to Mathematics, H. Holt, p. 121, ISBN 978-1-103-19784-2.
3. Cooke, Roger L. (2012), "38.3 Topology", The History of Mathematics: A Brief Course (3rd ed.), John Wiley & Sons, ISBN 9781118460290, The word locus is one that we still use today to denote the path followed by a point moving subject to stated constraints, although, since the introduction of set theory, a locus is more often thought of statically as the set of points satisfying a given collection.
4. Bourbaki, N. (2013), Elements of the History of Mathematics, translated by J. Meldrum, Springer, p. 26, ISBN 9783642616938, the classical mathematicians carefully avoided introducing into their reasoning the 'actual infinity'.
5. Borovik, Alexandre (2010), "6.2.4 Can one live without actual infinity?", Mathematics Under the Microscope: Notes on Cognitive Aspects of Mathematical Practice, American Mathematical Society, p. 124, ISBN 9780821847619.
6. Mayberry, John P. (2000), The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications, vol. 82, Cambridge University Press, p. 7, ISBN 9780521770347, set theory provides the foundations for all mathematics.
7. Ledermann, Walter; Vajda, S. (1985), Combinatorics and Geometry, Part 1, Handbook of Applicable Mathematics, vol. 5, Wiley, p. 32, ISBN 9780471900238, We begin by explaining a slightly old-fashioned term.
8. George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975.
9. Hamilton, Henry Parr (1834), An Analytical System of Conic Sections: Designed for the Use of Students, Springer.
10. G. P. West, The new geometry: form 1.
| Wikipedia |
Zero matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of $m\times n$ matrices, and is denoted by the symbol $O$ or $0$ followed by subscripts corresponding to the dimension of the matrix as the context sees fit.[1][2][3] Some examples of zero matrices are
$0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}}.\ $
Properties
The set of $m\times n$ matrices with entries in a ring K forms a ring $K_{m,n}$. The zero matrix $0_{K_{m,n}}\,$ in $K_{m,n}\,$ is the matrix with all entries equal to $0_{K}\,$, where $0_{K}$ is the additive identity in K.
$0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &\ddots &\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}_{m\times n}$
The zero matrix is the additive identity in $K_{m,n}\,$.[4] That is, for all $A\in K_{m,n}\,$ it satisfies the equation
$0_{K_{m,n}}+A=A+0_{K_{m,n}}=A.$
There is exactly one zero matrix of any given dimension m×n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
The zero matrix also represents the linear transformation which sends all the vectors to the zero vector.[5] It is idempotent, meaning that when it is multiplied by itself, the result is itself.
The zero matrix is the only matrix whose rank is 0.
Occurrences
The mortal matrix problem is the problem of determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices.[6]
In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.
See also
• Identity matrix, the multiplicative identity for matrices
• Matrix of ones, a matrix where all elements are one
• Nilpotent matrix
• Single-entry matrix, a matrix where all but one element is zero
References
1. Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126, We have a zero matrix in which aij = 0 for all i, j. ... We shall write it O.
2. "Intro to zero matrices (article) | Matrices". Khan Academy. Retrieved 2020-08-13.
3. Weisstein, Eric W. "Zero Matrix". mathworld.wolfram.com. Retrieved 2020-08-13.
4. Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418, The neutral element for addition is called the zero matrix, for all of its entries are zero.
5. Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842, The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.
6. Cassaigne, Julien; Halava, Vesa; Harju, Tero; Nicolas, Francois (2014). "Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More". arXiv:1404.0644 [cs.DM].
Matrix classes
Explicitly constrained entries
• Alternant
• Anti-diagonal
• Anti-Hermitian
• Anti-symmetric
• Arrowhead
• Band
• Bidiagonal
• Bisymmetric
• Block-diagonal
• Block
• Block tridiagonal
• Boolean
• Cauchy
• Centrosymmetric
• Conference
• Complex Hadamard
• Copositive
• Diagonally dominant
• Diagonal
• Discrete Fourier Transform
• Elementary
• Equivalent
• Frobenius
• Generalized permutation
• Hadamard
• Hankel
• Hermitian
• Hessenberg
• Hollow
• Integer
• Logical
• Matrix unit
• Metzler
• Moore
• Nonnegative
• Pentadiagonal
• Permutation
• Persymmetric
• Polynomial
• Quaternionic
• Signature
• Skew-Hermitian
• Skew-symmetric
• Skyline
• Sparse
• Sylvester
• Symmetric
• Toeplitz
• Triangular
• Tridiagonal
• Vandermonde
• Walsh
• Z
Constant
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| Wikipedia |
Null set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
For the set with no elements, see Empty set.
For the set of zeros of a function, see Zero set.
The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.
More generally, on a given measure space $M=(X,\Sigma ,\mu )$ a null set is a set $S\in \Sigma $ such that $\mu (S)=0.$
Examples
Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers.
The Cantor set is an example of an uncountable null set.
Definition
Suppose $A$ is a subset of the real line $\mathbb {R} $ such that for every $\varepsilon >0,$ there exists a sequence $U_{1},U_{2},\ldots $ of open intervals (where interval $U_{n}=(a_{n},b_{n})\subseteq \mathbb {R} $ has length $\operatorname {length} (U_{n})=b_{n}-a_{n}$) such that
$A\subseteq \bigcup _{n=1}^{\infty }U_{n}\ ~{\textrm {and}}~\ \sum _{n=1}^{\infty }\operatorname {length} (U_{n})<\varepsilon \,,$
then $A$ is a null set,[1] also known as a set of zero-content.
In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of $A$ for which the limit of the lengths of the covers is zero.
Properties
The empty set is always a null set. More generally, any countable union of null sets is null. Any subset of a null set is itself a null set. Together, these facts show that the 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/":): m -null sets of $X$ form a sigma-ideal on $X.$ Similarly, the measurable $m$-null sets form a sigma-ideal of the sigma-algebra of measurable sets. Thus, null sets may be interpreted as negligible sets, defining a notion of almost everywhere.
Lebesgue measure
The Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space.
A subset $N$ of $\mathbb {R} $ has null Lebesgue measure and is considered to be a null set in $\mathbb {R} $ if and only if:
Given any positive number $\varepsilon ,$ there is a sequence $I_{1},I_{2},\ldots $ of intervals in $\mathbb {R} $ such that $N$ is contained in the union of the $I_{1},I_{2},\ldots $ and the total length of the union is less than $\varepsilon .$
This condition can be generalised to $\mathbb {R} ^{n},$ using $n$-cubes instead of intervals. In fact, the idea can be made to make sense on any manifold, even if there is no Lebesgue measure there.
For instance:
• With respect to $\mathbb {R} ^{n},$ all singleton sets are null, and therefore all countable sets are null. In particular, the set $\mathbb {Q} $ of rational numbers is a null set, despite being dense in $\mathbb {R} .$
• The standard construction of the Cantor set is an example of a null uncountable set in $\mathbb {R} ;$ ;} however other constructions are possible which assign the Cantor set any measure whatsoever.
• All the subsets of $\mathbb {R} ^{n}$ whose dimension is smaller than $n$ have null Lebesgue measure in $\mathbb {R} ^{n}.$ For instance straight lines or circles are null sets in $\mathbb {R} ^{2}.$
• Sard's lemma: the set of critical values of a smooth function has measure zero.
If $\lambda $ is Lebesgue measure for $\mathbb {R} $ and π is Lebesgue measure for $\mathbb {R} ^{2}$, then the product measure $\lambda \times \lambda =\pi .$ In terms of null sets, the following equivalence has been styled a Fubini's theorem:[2]
• For $A\subset \mathbb {R} ^{2}$ and $A_{x}=\{y:(x,y)\in A\},$
$\pi (A)=0\iff \lambda \left(\left\{x:\lambda \left(A_{x}\right)>0\right\}\right)=0.$
Uses
Null sets play a key role in the definition of the Lebesgue integral: if functions $f$ and $g$ are equal except on a null set, then $f$ is integrable if and only if $g$ is, and their integrals are equal. This motivates the formal definition of $L^{p}$ spaces as sets of equivalence classes of functions which differ only on null sets.
A measure in which all subsets of null sets are measurable is complete. Any non-complete measure can be completed to form a complete measure by asserting that subsets of null sets have measure zero. Lebesgue measure is an example of a complete measure; in some constructions, it is defined as the completion of a non-complete Borel measure.
A subset of the Cantor set which is not Borel measurable
The Borel measure is not complete. One simple construction is to start with the standard Cantor set $K,$ which is closed hence Borel measurable, and which has measure zero, and to find a subset $F$ of $K$ which is not Borel measurable. (Since the Lebesgue measure is complete, this $F$ is of course Lebesgue measurable.)
First, we have to know that every set of positive measure contains a nonmeasurable subset. Let $f$ be the Cantor function, a continuous function which is locally constant on $K^{c},$ and monotonically increasing on $[0,1],$ with $f(0)=0$ and $f(1)=1.$ Obviously, $f(K^{c})$ is countable, since it contains one point per component of $K^{c}.$ Hence $f(K^{c})$ has measure zero, so $f(K)$ has measure one. We need a strictly monotonic function, so consider $g(x)=f(x)+x.$ Since $f(x)$ is strictly monotonic and continuous, it is a homeomorphism. Furthermore, $g(K)$ has measure one. Let $E\subseteq g(K)$ be non-measurable, and let $F=g^{-1}(E).$ Because $g$ is injective, we have that $F\subseteq K,$ and so $F$ is a null set. However, if it were Borel measurable, then $f(F)$ would also be Borel measurable (here we use the fact that the preimage of a Borel set by a continuous function is measurable; $g(F)=(g^{-1})^{-1}(F)$ is the preimage of $F$ through the continuous function $h=g^{-1}.$) Therefore, $F$ is a null, but non-Borel measurable set.
Haar null
In a separable Banach space $(X,+),$ the group operation moves any subset $A\subseteq X$ to the translates $A+x$ for any $x\in X.$ When there is a probability measure μ on the σ-algebra of Borel subsets of $X,$ such that for all $x,$ $\mu (A+x)=0,$ then $A$ is a Haar null set.[3]
The term refers to the null invariance of the measures of translates, associating it with the complete invariance found with Haar measure.
Some algebraic properties of topological groups have been related to the size of subsets and Haar null sets.[4] Haar null sets have been used in Polish groups to show that when A is not a meagre set then $A^{-1}A$ contains an open neighborhood of the identity element.[5] This property is named for Hugo Steinhaus since it is the conclusion of the Steinhaus theorem.
See also
• Cantor function – Continuous function that is not absolutely continuous
• Empty set – Mathematical set containing no elements
• Measure (mathematics) – Generalization of mass, length, area and volume
• Nothing – Complete absence of anything; the opposite of everything
References
1. Franks, John (2009). A (Terse) Introduction to Lebesgue Integration. The Student Mathematical Library. Vol. 48. American Mathematical Society. p. 28. doi:10.1090/stml/048. ISBN 978-0-8218-4862-3.
2. van Douwen, Eric K. (1989). "Fubini's theorem for null sets". American Mathematical Monthly. 96 (8): 718–21. doi:10.1080/00029890.1989.11972270. JSTOR 2324722. MR 1019152.
3. Matouskova, Eva (1997). "Convexity and Haar Null Sets" (PDF). Proceedings of the American Mathematical Society. 125 (6): 1793–1799. doi:10.1090/S0002-9939-97-03776-3. JSTOR 2162223.
4. Solecki, S. (2005). "Sizes of subsets of groups and Haar null sets". Geometric and Functional Analysis. 15: 246–73. CiteSeerX 10.1.1.133.7074. doi:10.1007/s00039-005-0505-z. MR 2140632. S2CID 11511821.
5. Dodos, Pandelis (2009). "The Steinhaus property and Haar-null sets". Bulletin of the London Mathematical Society. 41 (2): 377–44. arXiv:1006.2675. Bibcode:2010arXiv1006.2675D. doi:10.1112/blms/bdp014. MR 4296513. S2CID 119174196.
Further reading
• Capinski, Marek; Kopp, Ekkehard (2005). Measure, Integral and Probability. Springer. p. 16. ISBN 978-1-85233-781-0.
• Jones, Frank (1993). Lebesgue Integration on Euclidean Spaces. Jones & Bartlett. p. 107. ISBN 978-0-86720-203-8.
• Oxtoby, John C. (1971). Measure and Category. Springer-Verlag. p. 3. ISBN 978-0-387-05349-3.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
| Wikipedia |
Zero mode
In physics, a zero mode is an eigenvector with a vanishing eigenvalue.
In various subfields of physics zero modes appear whenever a physical system possesses a certain symmetry. For example, normal modes of multidimensional harmonic oscillator (e.g. a system of beads arranged around the circle, connected with springs) corresponds to elementary vibrational modes of the system. In such a system zero modes typically occur and are related with a rigid rotation around the circle.
The kernel of an operator consists of left zero modes, and the cokernel consists of the right zero modes.
| Wikipedia |
Constant problem
In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero.
The problem
This problem is also referred to as the identity problem[1] or the method of zero estimates. It has no formal statement as such but refers to a general problem prevalent in transcendental number theory. Often proofs in transcendence theory are proofs by contradiction. Specifically, they use some auxiliary function to create an integer n ≥ 0, which is shown to satisfy n < 1. Clearly, this means that n must have the value zero, and so a contradiction arises if one can show that in fact n is not zero.
In many transcendence proofs, proving that n ≠ 0 is very difficult, and hence a lot of work has been done to develop methods that can be used to prove the non-vanishing of certain expressions. The sheer generality of the problem is what makes it difficult to prove general results or come up with general methods for attacking it. The number n that arises may involve integrals, limits, polynomials, other functions, and determinants of matrices.
Results
In certain cases, algorithms or other methods exist for proving that a given expression is non-zero, or of showing that the problem is undecidable. For example, if x1, ..., xn are real numbers, then there is an algorithm[2] for deciding whether there are integers a1, ..., an such that
$a_{1}x_{1}+\cdots +a_{n}x_{n}=0\,.$
If the expression we are interested in contains an oscillating function, such as the sine or cosine function, then it has been shown that the problem is undecidable, a result known as Richardson's theorem. In general, methods specific to the expression being studied are required to prove that it cannot be zero.
See also
• Integer relation algorithm
References
1. Richardson, Daniel (1968). "Some Unsolvable Problems Involving Elementary Functions of a Real Variable". Journal of Symbolic Logic. 33: 514–520. doi:10.2307/2271358. JSTOR 2271358.
2. Bailey, David H. (January 1988). "Numerical Results on the Transcendence of Constants Involving π, e, and Euler's Constant" (PDF). Mathematics of Computation. 50 (20): 275–281. doi:10.1090/S0025-5718-1988-0917835-1.
| Wikipedia |
Null semigroup
In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero.[1] If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.[2] According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."[1]
Null semigroup
Let S be a semigroup with zero element 0. Then S is called a null semigroup if xy = 0 for all x and y in S.
Cayley table for a null semigroup
Let S = {0, a, b, c} be (the underlying set of) a null semigroup. Then the Cayley table for S is as given below:
Cayley table for a null semigroup
0 a b c
0 0 0 0 0
a 0 0 0 0
b 0 0 0 0
c 0 0 0 0
Left zero semigroup
A semigroup in which every element is a left zero element is called a left zero semigroup. Thus a semigroup S is a left zero semigroup if xy = x for all x and y in S.
Cayley table for a left zero semigroup
Let S = {a, b, c} be a left zero semigroup. Then the Cayley table for S is as given below:
Cayley table for a left zero semigroup
a b c
a a a a
b b b b
c c c c
Right zero semigroup
A semigroup in which every element is a right zero element is called a right zero semigroup. Thus a semigroup S is a right zero semigroup if xy = y for all x and y in S.
Cayley table for a right zero semigroup
Let S = {a, b, c} be a right zero semigroup. Then the Cayley table for S is as given below:
Cayley table for a right zero semigroup
a b c
a a b c
b a b c
c a b c
Properties
A non-trivial null (left/right zero) semigroup does not contain an identity element. It follows that the only null (left/right zero) monoid is the trivial monoid.
The class of null semigroups is:
• closed under taking subsemigroups
• closed under taking quotient of subsemigroup
• closed under arbitrary direct products.
It follows that the class of null (left/right zero) semigroups is a variety of universal algebra, and thus a variety of finite semigroups. The variety of finite null semigroups is defined by the identity ab = cd.
See also
• Right group
References
1. A H Clifford; G B Preston (1964). The algebraic theory of semigroups Vol I. mathematical Surveys. Vol. 1 (2 ed.). American Mathematical Society. pp. 3–4. ISBN 978-0-8218-0272-4.
2. M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 19
| Wikipedia |
Zero object (algebra)
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as {0}. One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism).
This article is about trivial or zero algebraic structures. For zero elements in algebraic structures, see Zero element. For the zero object in a category, see Initial and terminal objects.
Instances of the zero object include, but are not limited to the following:
• As a group, the zero group or trivial group.
• As a ring, the zero ring or trivial ring.
• As an algebra over a field or algebra over a ring, the trivial algebra.
• As a module (over a ring R), the zero module. The term trivial module is also used, although it may be ambiguous, as a trivial G-module is a G-module with a trivial action.
• As a vector space (over a field R), the zero vector space, zero-dimensional vector space or just zero space.
These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties.
In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as:
κ0 = 0 , where κ ∈ R.
The most general of them, the zero module, is a finitely-generated module with an empty generating set.
For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, 0 × 0 = 0, because there are no non-zero elements. This structure is associative and commutative. A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R,
$r=r\times 1=r\times 0=0.$
In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see § Unital structures below.
Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module.
The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra.
The zero-dimensional vector space is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above.
Properties
2↕ ${\begin{bmatrix}0\\0\end{bmatrix}}$ = ${\begin{bmatrix}\,\\\,\end{bmatrix}}$ [ ] ‹0
↔
1
^
0
↔
1
Element of the zero space, written as empty column vector (rightmost one), is multiplied by 2×0 empty matrix to obtain 2-dimensional zero vector (leftmost). Rules of matrix multiplication are respected.
The zero ring, zero module and zero vector space are the zero objects of, respectively, the category of pseudo-rings, the category of modules and the category of vector spaces. However, the zero ring is not a zero object in the category of rings, since there is no ring homomorphism of the zero ring in any other ring.
The zero object, by definition, must be a terminal object, which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0.
The zero object, also by definition, must be an initial object, which means that a morphism {0} → A must exist and be unique for an arbitrary object A. This morphism maps 0, the only element of {0}, to the zero element 0 ∈ A, called the zero vector in vector spaces. This map is a monomorphism, and hence its image is isomorphic to {0}. For modules and vector spaces, this subset {0} ⊂ A is the only empty-generated submodule (or 0-dimensional linear subspace) in each module (or vector space) A.
Unital structures
The {0} object is a terminal object of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an initial object (and hence, a zero object in the category-theoretical sense) depend on exact definition of the multiplicative identity 1 in a specified structure.
If the definition of 1 requires that 1 ≠ 0, then the {0} object cannot exist because it may contain only one element. In particular, the zero ring is not a field. If mathematicians sometimes talk about a field with one element, this abstract and somewhat mysterious mathematical object is not a field.
In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist. But not as initial object because identity-preserving morphisms from {0} to any object where 1 ≠ 0 do not exist. For example, in the category of rings Ring the ring of integers Z is the initial object, not {0}.
If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor 1 ≠ 0, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.
Notation
Zero vector spaces and zero modules are usually denoted by 0 (instead of {0}). This is always the case when they occur in an exact sequence.
See also
• Nildimensional space
• Triviality (mathematics)
• Examples of vector spaces
• Field with one element
• Empty semigroup
• Zero element
• List of zero terms
External links
• David Sharpe (1987). Rings and factorization. Cambridge University Press. p. 10 : trivial ring. ISBN 0-521-33718-6.
• Barile, Margherita. "Trivial Module". MathWorld.
• Barile, Margherita. "Zero Module". MathWorld.
| Wikipedia |
Zero-sum game
Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two sides, where the result is an advantage for one side and an equivalent loss for the other.[1] In other words, player one's gain is equivalent to player two's loss, with the result that the net improvement in benefit of the game is zero.[2]
Not to be confused with Empty sum or Zero game.
If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally. Other examples of zero-sum games in daily life include games like poker, chess, and bridge where one person gains and another person loses, which results in a zero-net benefit for every player.[3] In the markets and financial instruments, futures contracts and options are zero-sum games as well.[4]
In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. A zero-sum game is also called a strictly competitive game, while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality,[5] or with Nash equilibrium. Prisoner's Dilemma is a classic non-zero-sum game.[6]
Definition
Choice 1 Choice 2
Choice 1 −A, A B, −B
Choice 2 C, −C −D, D
Generic zero-sum game
Option 1 Option 2
Option 1 2, −2 −2, 2
Option 2 −2, 2 2, −2
Another example of the classic zero-sum game
The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal. Generally, any game where all strategies are Pareto optimal is called a conflict game.[7][8]
Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero.[9] Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.
In situation where one decision maker's gain (or loss) does not necessarily result in the other decision makers' loss (or gain), they are referred to as non-zero-sum.[10] Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players is sometimes more or less than what they began with.
The idea of Pareto optimal payoff in a zero-sum game gives rise to a generalized relative selfish rationality standard, the punishing-the-opponent standard, where both players always seek to minimize the opponent's payoff at a favourable cost to themselves rather than prefer more over less. The punishing-the-opponent standard can be used in both zero-sum games (e.g. warfare game, chess) and non-zero-sum games (e.g. pooling selection games).[11] The player in the game has a simple enough desire to maximise the profit for them, and the opponent wishes to minimise it.[12]
Solution
For two-player finite zero-sum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium.
Example
A zero-sum game (Two person)
Blue
Red
A B C
1
−30
30
10
−10
−20
20
2
10
−10
−20
20
20
−20
A game's payoff matrix is a convenient representation. Consider these situations as an example, the two-player zero-sum game pictured at right or above.
The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.
Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.
In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.
Émile Borel and John von Neumann had the fundamental insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute probably optimal strategies for all two-player zero-sum games.
For the example given above, it turns out that Red should choose action 1 with probability 4/7 and action 2 with probability 3/7, and Blue should assign the probabilities 0, 4/7, and 3/7 to the three actions A, B, and C. Red will then win 20/7 points on average per game.
Solving
The Nash equilibrium for a two-player, zero-sum game can be found by solving a linear programming problem. Suppose a zero-sum game has a payoff matrix M where element Mi,j is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of M is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (Raghavan 1994, p. 740) by solving the following linear program to find a vector u:
Minimize:
$\sum _{i}u_{i}$
Subject to the constraints:
u ≥ 0
M u ≥ 1.
The first constraint says each element of the u vector must be nonnegative, and the second constraint says each element of the M u vector must be at least 1. For the resulting u vector, the inverse of the sum of its elements is the value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each possible pure strategy.
If the game matrix does not have all positive elements, add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will not affect the equilibrium mixed strategies for the equilibrium.
The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Alternatively, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of M (adding a constant so it is positive), then solving the resulting game.
If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations and thus such games are equivalent to linear programs, in general.[13]
Universal solution
If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or non-credible after the play is started, such as poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games concerning starting the game or not.[14]
The most common or simple example from the subfield of social psychology is the concept of "social traps". In some cases pursuing individual personal interest can enhance the collective well-being of the group, but in other situations, all parties pursuing personal interest results in mutually destructive behaviour.
Copeland's review notes that an n-player non-zero-sum game can be converted into an (n+1)-player zero-sum game, where the n+1st player, denoted the fictitious player, receives the negative of the sum of the gains of the other n-players (the global gain / loss).[15]
Zero-sum three-person games
It is clear that there are manifold relationships between players in a zero-sum three-person game, in a zero-sum two-person game, anything one player wins is necessarily lost by the other and vice versa; therefore, there is always an absolute antagonism of interests, and that is similar in the three-person game.[16] A particular move of a player in a zero-sum three-person game would be assumed to be clearly beneficial to him and may disbenefits to both other players, or benefits to one and disbenefits to the other opponent.[16] Particularly, parallelism of interests between two players makes a cooperation desirable; it may happen that a player has a choice among various policies: Get into a parallelism interest with another player by adjusting his conduct, or the opposite; that he can choose with which of other two players he prefers to build such parallelism, and to what extent.[16] The picture on the left shows that a typical example of a zero-sum three-person game. If Player 1 chooses to defence, but Player 2 & 3 chooses to offence, both of them will gain one point. At the same time, Player 2 will lose two-point because points are taken away by other players, and it is evident that Player 2 & 3 has parallelism of interests.
Economic benefits of low-cost airlines in saturated markets - net benefits or a zero-sum game [17]
Studies show that the entry of low-cost airlines into the Hong Kong market brought in $671 million in revenue and resulted in an outflow of $294 million.
Therefore, the replacement effect should be considered when introducing a new model, which will lead to economic leakage and injection. Thus introducing new models requires caution. For example, if the number of new airlines departing from and arriving at the airport is the same, the economic contribution to the host city may be a zero-sum game. Because for Hong Kong, the consumption of overseas tourists in Hong Kong is income, while the consumption of Hong Kong residents in opposite cities is outflow. In addition, the introduction of new airlines can also have a negative impact on existing airlines.
Consequently, when a new aviation model is introduced, feasibility tests need to be carried out in all aspects, taking into account the economic inflow and outflow and displacement effects caused by the model.
Zero-sum Games in Financial Markets
Derivatives trading may be considered a zero-sum game, as each dollar gained by one party in a transaction must be lost by the other, hence yielding a net transfer of wealth of zero.[18]
An options contract - whereby a buyer purchases a derivative contract which provides them with the right to buy an underlying asset from a seller at a specified strike price before a specified expiration date – is an example of a zero-sum game. A futures contract – whereby a buyer purchases a derivative contract to buy an underlying asset from the seller for a specified price on a specified date – is also an example of a zero-sum game.[19] This is because the fundamental principle of these contracts is that they are agreements between two parties, and any gain made by one party must be matched by a loss sustained by the other.
If the price of the underlying asset increases before the expiration date the buyer may exercise/ close the options/ futures contract. The buyers gain and corresponding sellers loss will be the difference between the strike price and value of the underlying asset at that time. Hence, the net transfer of wealth is zero.
Swaps, which involve the exchange of cash flows from two different financial instruments, are also considered a zero-sum game.[20] Consider a standard interest rate swap whereby Firm A pays a fixed rate and receives a floating rate; correspondingly Firm B pays a floating rate and receives a fixed rate. If rates increase, then Firm A will gain, and Firm B will lose by the rate differential (floating rate – fixed rate). If rates decrease, then Firm A will lose, and Firm B will gain by the rate differential (fixed rate – floating rate).
Whilst derivatives trading may be considered a zero-sum game, it is important to remember that this is not an absolute truth. The financial markets are complex and multifaceted, with a range of participants engaging in a variety of activities. While some trades may result in a simple transfer of wealth from one party to another, the market as a whole is not purely competitive, and many transactions serve important economic functions.
The stock market is an excellent example of a positive-sum game, often erroneously labelled as a zero-sum game. This is a zero-sum fallacy: the perception that one trader in the stock market may only increase the value of their holdings if another trader decreases their holdings.[21]
The primary goal of the stock market is to match buyers and sellers, but the prevailing price is the one which equilibrates supply and demand. Stock prices generally move according to changes in future expectations, such as acquisition announcements, upside earnings surprises, or improved guidance.[22]
For instance, if Company C announces a deal to acquire Company D, and investors believe that the acquisition will result in synergies and hence increased profitability for Company C, there will be an increased demand for Company C stock. In this scenario, all existing holders of Company C stock will enjoy gains without incurring any corresponding measurable losses to other players.
Furthermore, in the long run, the stock market is a positive-sum game. As economic growth occurs, demand increases, output increases, companies grow, and company valuations increase, leading to value creation and wealth addition in the market.
Complexity
It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.
Extensions
In 1944, John von Neumann and Oskar Morgenstern proved that any non-zero-sum game for n players is equivalent to a zero-sum game with n + 1 players; the (n + 1)th player representing the global profit or loss.[23]
Misunderstandings
Zero-sum games and particularly their solutions are commonly misunderstood by critics of game theory, usually with respect to the independence and rationality of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational games.[5]
Politics is sometimes called zero sum[24][25][26] because in common usage the idea of a stalemate is perceived to be "zero sum"; politics and macroeconomics are not zero sum games, however, because they do not constitute conserved systems.
Zero-sum thinking
In psychology, zero-sum thinking refers to the perception that a given situation is like a zero-sum game, where one person's gain is equal to another person's loss.
See also
• Bimatrix game
• Comparative advantage
• Dutch disease
• Gains from trade
• Lump of labour fallacy
• Positive-sum game
• No-win situation
References
1. Cambridge business English dictionary. Cambridge: Cambridge University Press. 2011. ISBN 978-0-521-12250-4. OCLC 741548935.
2. Blakely, Sara. "Zero-Sum Game Meaning: Examples of Zero-Sum Games". Master Class. Master Class. Retrieved 2022-04-28.
3. Von Neumann, John; Oskar Morgenstern (2007). Theory of games and economic behavior (60th anniversary ed.). Princeton: Princeton University Press. ISBN 978-1-4008-2946-0. OCLC 830323721.
4. Kenton, Will. "Zero-Sum Game". Investopedia. Retrieved 2021-04-25.
5. Ken Binmore (2007). Playing for real: a text on game theory. Oxford University Press US. ISBN 978-0-19-530057-4., chapters 1 & 7
6. Chiong, Raymond; Jankovic, Lubo (2008). "Learning game strategy design through iterated Prisoner's Dilemma". International Journal of Computer Applications in Technology. 32 (3): 216. doi:10.1504/ijcat.2008.020957. ISSN 0952-8091.
7. Bowles, Samuel (2004). Microeconomics: Behavior, Institutions, and Evolution. Princeton University Press. pp. 33–36. ISBN 0-691-09163-3.
8. "Two-Person Zero-Sum Games: Basic Concepts". Neos Guide. Neos Guide. Retrieved 2022-04-28.
9. Washburn, Alan (2014). Two-Person Zero-Sum Games. International Series in Operations Research & Management Science. Vol. 201. Boston, MA: Springer US. doi:10.1007/978-1-4614-9050-0. ISBN 978-1-4614-9049-4.
10. "Non Zero Sum Game". Monash Business School. Retrieved 2021-04-25.
11. Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 978-1507658246. Chapter 1 and Chapter 4.
12. Von Neumann, John; Oskar Morgenstern (2007). Theory of games and economic behavior (60th anniversary ed.). Princeton: Princeton University Press. p. 98. ISBN 978-1-4008-2946-0. OCLC 830323721.
13. Ilan Adler (2012) The equivalence of linear programs and zero-sum games. Springer
14. Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 978-1507658246. Chapter 4.
15. Arthur H. Copeland (July 1945) Book review, Theory of games and economic behavior. By John von Neumann and Oskar Morgenstern (1944). Review published in the Bulletin of the American Mathematical Society 51(7) pp 498-504 (July 1945)
16. Von Neumann, John; Oskar Morgenstern (2007). Theory of games and economic behavior (60th anniversary ed.). Princeton: Princeton University Press. pp. 220–223. ISBN 978-1-4008-2946-0. OCLC 830323721.
17. Pratt, Stephen; Schucker, Markus (March 2018). "Economic impact of low-cost carrier in a saturated transport market: Net benefits or zero-sum game?". Tourism Economics: The Business and Finance of Tourism and Recreation. 25 (2): 149–170.
18. Levitt, Steven D. (February 2004). "Why are Gambling Markets Organized so Differently from Financial Markets?". The Economic Journal. 114 (10): 223–246. doi:10.1111/j.1468-0297.2004.00207.x. S2CID 2289856 – via RePEc.
19. "Options vs. Futures: What's the Difference?". Investopedia. Retrieved 2023-04-24.
20. Turnbull, Stuart M. (1987). "Swaps: A Zero Sum Game?". Financial Management. 16 (1): 15–21. doi:10.2307/3665544. ISSN 0046-3892. JSTOR 3665544.
21. Engle, Eric (September 2008). "The Stock Market as a Game: An Agent Based Approach to Trading in Stocks". Quantitative Finance Papers – via RePEc.
22. Olson, Erika S. (2010-10-26). Zero-Sum Game: The Rise of the World's Largest Derivatives Exchange. John Wiley & Sons. ISBN 978-0-470-62420-3.
23. Theory of Games and Economic Behavior. Princeton University Press (1953). June 25, 2005. ISBN 9780691130613. Retrieved 2018-02-25.
24. Rubin, Jennifer (2013-10-04). "The flaw in zero sum politics". The Washington Post. Retrieved 2017-03-08.
25. "Lexington: Zero-sum politics". The Economist. 2014-02-08. Retrieved 2017-03-08.
26. "Zero-sum game | Define Zero-sum game at". Dictionary.com. Retrieved 2017-03-08.
Further reading
• Misstating the Concept of Zero-Sum Games within the Context of Professional Sports Trading Strategies, series Pardon the Interruption (2010-09-23) ESPN, created by Tony Kornheiser and Michael Wilbon, performance by Bill Simmons
• Handbook of Game Theory – volume 2, chapter Zero-sum two-person games, (1994) Elsevier Amsterdam, by Raghavan, T. E. S., Edited by Aumann and Hart, pp. 735–759, ISBN 0-444-89427-6
• Power: Its Forms, Bases and Uses (1997) Transaction Publishers, by Dennis Wrong
External links
• Play zero-sum games online by Elmer G. Wiens.
• Game Theory & its Applications – comprehensive text on psychology and game theory. (Contents and Preface to Second Edition.)
• A playable zero-sum game and its mixed strategy Nash equilibrium.
Topics in game theory
Definitions
• Congestion game
• Cooperative game
• Determinacy
• Escalation of commitment
• Extensive-form game
• First-player and second-player win
• Game complexity
• Graphical game
• Hierarchy of beliefs
• Information set
• Normal-form game
• Preference
• Sequential game
• Simultaneous game
• Simultaneous action selection
• Solved game
• Succinct game
Equilibrium
concepts
• Bayesian Nash equilibrium
• Berge equilibrium
• Core
• Correlated equilibrium
• Epsilon-equilibrium
• Evolutionarily stable strategy
• Gibbs equilibrium
• Mertens-stable equilibrium
• Markov perfect equilibrium
• Nash equilibrium
• Pareto efficiency
• Perfect Bayesian equilibrium
• Proper equilibrium
• Quantal response equilibrium
• Quasi-perfect equilibrium
• Risk dominance
• Satisfaction equilibrium
• Self-confirming equilibrium
• Sequential equilibrium
• Shapley value
• Strong Nash equilibrium
• Subgame perfection
• Trembling hand
Strategies
• Backward induction
• Bid shading
• Collusion
• Forward induction
• Grim trigger
• Markov strategy
• Dominant strategies
• Pure strategy
• Mixed strategy
• Strategy-stealing argument
• Tit for tat
Classes
of games
• Bargaining problem
• Cheap talk
• Global game
• Intransitive game
• Mean-field game
• Mechanism design
• n-player game
• Perfect information
• Large Poisson game
• Potential game
• Repeated game
• Screening game
• Signaling game
• Stackelberg competition
• Strictly determined game
• Stochastic game
• Symmetric game
• Zero-sum game
Games
• Go
• Chess
• Infinite chess
• Checkers
• Tic-tac-toe
• Prisoner's dilemma
• Gift-exchange game
• Optional prisoner's dilemma
• Traveler's dilemma
• Coordination game
• Chicken
• Centipede game
• Lewis signaling game
• Volunteer's dilemma
• Dollar auction
• Battle of the sexes
• Stag hunt
• Matching pennies
• Ultimatum game
• Rock paper scissors
• Pirate game
• Dictator game
• Public goods game
• Blotto game
• War of attrition
• El Farol Bar problem
• Fair division
• Fair cake-cutting
• Cournot game
• Deadlock
• Diner's dilemma
• Guess 2/3 of the average
• Kuhn poker
• Nash bargaining game
• Induction puzzles
• Trust game
• Princess and monster game
• Rendezvous problem
Theorems
• Arrow's impossibility theorem
• Aumann's agreement theorem
• Folk theorem
• Minimax theorem
• Nash's theorem
• Negamax theorem
• Purification theorem
• Revelation principle
• Sprague–Grundy theorem
• Zermelo's theorem
Key
figures
• Albert W. Tucker
• Amos Tversky
• Antoine Augustin Cournot
• Ariel Rubinstein
• Claude Shannon
• Daniel Kahneman
• David K. Levine
• David M. Kreps
• Donald B. Gillies
• Drew Fudenberg
• Eric Maskin
• Harold W. Kuhn
• Herbert Simon
• Hervé Moulin
• John Conway
• Jean Tirole
• Jean-François Mertens
• Jennifer Tour Chayes
• John Harsanyi
• John Maynard Smith
• John Nash
• John von Neumann
• Kenneth Arrow
• Kenneth Binmore
• Leonid Hurwicz
• Lloyd Shapley
• Melvin Dresher
• Merrill M. Flood
• Olga Bondareva
• Oskar Morgenstern
• Paul Milgrom
• Peyton Young
• Reinhard Selten
• Robert Axelrod
• Robert Aumann
• Robert B. Wilson
• Roger Myerson
• Samuel Bowles
• Suzanne Scotchmer
• Thomas Schelling
• William Vickrey
Miscellaneous
• All-pay auction
• Alpha–beta pruning
• Bertrand paradox
• Bounded rationality
• Combinatorial game theory
• Confrontation analysis
• Coopetition
• Evolutionary game theory
• First-move advantage in chess
• Game Description Language
• Game mechanics
• Glossary of game theory
• List of game theorists
• List of games in game theory
• No-win situation
• Solving chess
• Topological game
• Tragedy of the commons
• Tyranny of small decisions
Authority control: National
• Germany
• Israel
• United States
| Wikipedia |
Zero suppression
Zero suppression is the removal of redundant zeroes from a number. This can be done for storage, page or display space constraints or formatting reasons, such as making a letter more legible.[1][2][3]
Examples
• 00049823 → 49823
• 7.678600000 → 7.6786
• 0032.3231000 → 32.3231
• 2.45000×1010 → 2.45×1010
• 0.0045×1010 → 4.5×107
One must be careful; in physics and related disciplines, trailing zeros are used to indicate the precision of the number, as an error of ±1 in the last place is assumed. Examples:
• 4.5981 is 4.5981 ± 0.0001
• 4.59810 is 4.5981 ± 0.00001
• 4.598100 is 4.5981 ± 0.000001
Data compression
It is also a way to store a large array of numbers, where many of the entries are zero. By omitting the zeroes, and instead storing the indices along with the values of the non-zero items, less space may be used in total. It only makes sense if the extra space used for storing the indices (on average) is smaller than the space saved by not storing the zeroes. This is sometimes used in a sparse array.
Example:
• Original array: 0, 1, 0, 0, 2, 5, 0, 0, 0, 4, 0, 0, 0, 0, 0
• Pairs of index and data: {2,1}, {5,2}, {6,5}, {10,4}
See also
• Run-length encoding – Form of lossless data compression
• Zero code suppression – Digital telecommunications techniquePages displaying short descriptions of redirect targets
• Zero-suppressed decision diagram – Kind of binary decision diagram
References
1. "Telecom Glossary 2000: Zero Suppression". U.S.: Institute for Telecommunication Sciences, NTIA. Archived from the original on 2008-09-25.
2. Parr, E. A. (1999). Industrial Control Handbook (3 ed.). Industrial Press, Inc. p. 582. ISBN 978-0-8311-3085-5.
3. Grabowski, Ralph (2010). Using AutoCAD 2011. Autodesk Press. p. 648. ISBN 978-1-111-12514-1.
| Wikipedia |
Symbols for zero
The modern numerical digit 0 is usually written as a circle, an ellipse or a rounded square or rectangle.
Glyphs
In most modern typefaces, the height of the 0 character is the same as the other digits. However, in typefaces with text figures, the character is often shorter (x-height).
Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0.[1] Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.[1]
The digit 0 with a dot in the centre seems to have originated as an option on IBM 3270 displays. Its appearance has continued with Taligent's command line typeface Andalé Mono. One variation used a short vertical bar instead of the dot. This could be confused with the Greek letter Theta on a badly focused display, but in practice there was no confusion because theta was not (then) a displayable character and very little used anyway.
An alternative, the slashed zero (looking similar to the letter O except for the slash), was primarily used in hand-written coding sheets before transcription to punched cards or tape, and is also used in old-style ASCII graphic sets descended from the default typewheel on the Teletype Model 33 ASR. This form is similar to the symbol $\emptyset $, or "∅" (Unicode character U+2205), representing the empty set, as well as to the letter Ø used in several Scandinavian languages. Some Burroughs/Unisys equipment displays a digit 0 with a reversed slash.
The opposing convention that has the letter O with a slash and the digit 0 without was advocated by SHARE, a prominent IBM user group,[1] and recommended by IBM for writing FORTRAN programs,[2] and by a few other early mainframe makers; this is even more problematic for Scandinavians because it means two of their letters collide. Others advocated the opposite convention,[1] including IBM for writing Algol programs.[2] Another convention used on some early line printers left digit 0 unornamented but added a tail or hook to the capital O so that it resembled an inverted Q (like U+213A ℺) or cursive capital letter-O (${\mathcal {O}}$).[1]
Some fonts designed for use with computers made one of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). The TI-99/4A computer has a more angular capital O and a more rounded digit 0, whereas others made the choice the other way around.
The typeface used on most European vehicle registration plates distinguishes the two symbols partially in this manner (having a more rectangular or wider shape for the capital O than the digit 0), but in several countries a further distinction is made by slitting open the digit 0 on the upper right side (as in German plates using the fälschungserschwerende Schrift, "forgery-impeding typeface").
Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether. For example, confirmation numbers[3] used by Southwest Airlines use only the capital letters O and I instead of the digits 0 and 1, while Canadian postal codes use only the digits 1 and 0 and never the capital letters O and I, although letters and numbers always alternate.
Other
Other representations of zero
• Usual appearance of zero on seven-segment displays
• Unusual smaller appearance of zero on seven-segment displays
• International maritime signal flag for 0
On the seven-segment displays of calculators, watches, and household appliances, 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments.
The international maritime signal flag has five plus signs in an X arrangement.
Zero symbols in Unicode
• U+0030 0 DIGIT ZERO
• U+0660 ٠ ARABIC-INDIC DIGIT ZERO
• U+06DF ۟ ARABIC SMALL HIGH ROUNDED ZERO
• U+06E0 ۠ ARABIC SMALL HIGH UPRIGHT RECTANGULAR ZERO
• U+06F0 ۰ EXTENDED ARABIC-INDIC DIGIT ZERO
• U+07C0 ߀ NKO DIGIT ZERO
• U+0966 ० DEVANAGARI DIGIT ZERO
• U+09E6 ০ BENGALI DIGIT ZERO
• U+0A66 ੦ GURMUKHI DIGIT ZERO
• U+0AE6 ૦ GUJARATI DIGIT ZERO
• U+0B66 ୦ ORIYA DIGIT ZERO
• U+0BE6 ௦ TAMIL DIGIT ZERO
• U+0C66 ౦ TELUGU DIGIT ZERO
• U+0C78 ౸ TELUGU FRACTION DIGIT ZERO FOR ODD POWERS OF FOUR
• U+0CE6 ೦ KANNADA DIGIT ZERO
• U+0D66 ൦ MALAYALAM DIGIT ZERO
• U+0DE6 ෦ SINHALA LITH DIGIT ZERO
• U+0E50 ๐ THAI DIGIT ZERO
• U+0ED0 ໐ LAO DIGIT ZERO
• U+0F20 ༠ TIBETAN DIGIT ZERO
• U+0F33 ༳ TIBETAN DIGIT HALF ZERO
• U+1040 ၀ MYANMAR DIGIT ZERO
• U+1090 ႐ MYANMAR SHAN DIGIT ZERO
• U+17E0 ០ KHMER DIGIT ZERO
• U+1810 ᠐ MONGOLIAN DIGIT ZERO
• U+1946 ᥆ LIMBU DIGIT ZERO
• U+19D0 ᧐ NEW TAI LUE DIGIT ZERO
• U+1A80 ᪀ TAI THAM HORA DIGIT ZERO
• U+1A90 ᪐ TAI THAM THAM DIGIT ZERO
• U+1B50 ᭐ BALINESE DIGIT ZERO
• U+1BB0 ᮰ SUNDANESE DIGIT ZERO
• U+1C40 ᱀ LEPCHA DIGIT ZERO
• U+1C50 ᱐ OL CHIKI DIGIT ZERO
• U+2070 ⁰ SUPERSCRIPT ZERO
• U+2080 ₀ SUBSCRIPT ZERO
• U+2189 ↉ VULGAR FRACTION ZERO THIRDS
• U+24EA ⓪ CIRCLED DIGIT ZERO
• U+24FF ⓿ NEGATIVE CIRCLED DIGIT ZERO
• U+3007 〇 IDEOGRAPHIC NUMBER ZERO
• U+3358 ㍘ IDEOGRAPHIC TELEGRAPH SYMBOL FOR HOUR ZERO
• U+A620 ꘠ VAI DIGIT ZERO
• U+A8D0 ꣐ SAURASHTRA DIGIT ZERO
• U+A8E0 ꣠ COMBINING DEVANAGARI DIGIT ZERO
• U+A900 ꤀ KAYAH LI DIGIT ZERO
• U+A9D0 ꧐ JAVANESE DIGIT ZERO
• U+A9F0 ꧰ MYANMAR TAI LAING DIGIT ZERO
• U+AA50 ꩐ CHAM DIGIT ZERO
• U+ABF0 ꯰ MEETEI MAYEK DIGIT ZERO
• U+FF10 ０ FULLWIDTH DIGIT ZERO
• U+1018A 𐆊 GREEK ZERO SIGN
• U+104A0 𐒠 OSMANYA DIGIT ZERO
• U+10D30 𐴰 HANIFI ROHINGYA DIGIT ZERO
• U+11066 𑁦 BRAHMI DIGIT ZERO
• U+110F0 𑃰 SORA SOMPENG DIGIT ZERO
• U+11136 𑄶 CHAKMA DIGIT ZERO
• U+111D0 𑇐 SHARADA DIGIT ZERO
• U+112F0 𑋰 KHUDAWADI DIGIT ZERO
• U+11366 𑍦 COMBINING GRANTHA DIGIT ZERO
• U+11450 𑑐 NEWA DIGIT ZERO
• U+114D0 𑓐 TIRHUTA DIGIT ZERO
• U+11650 𑙐 MODI DIGIT ZERO
• U+116C0 𑛀 TAKRI DIGIT ZERO
• U+11730 𑜰 AHOM DIGIT ZERO
• U+118E0 𑣠 WARANG CITI DIGIT ZERO
• U+11950 𑥐 DIVES AKURU DIGIT ZERO
• U+11C50 𑱐 BHAIKSUKI DIGIT ZERO
• U+11D50 𑵐 MASARAM GONDI DIGIT ZERO
• U+11DA0 𑶠 GUNJALA GONDI DIGIT ZERO
• U+16A60 𖩠 MRO DIGIT ZERO
• U+16B50 𖭐 PAHAWH HMONG DIGIT ZERO
• U+16E80 𖺀 MEDEFAIDRIN DIGIT ZERO
• U+1D2E0 𝋠 MAYAN NUMERAL ZERO
• U+1D7CE 𝟎 MATHEMATICAL BOLD DIGIT ZERO
• U+1D7D8 𝟘 MATHEMATICAL DOUBLE-STRUCK DIGIT ZERO
• U+1D7E2 𝟢 MATHEMATICAL SANS-SERIF DIGIT ZERO
• U+1D7EC 𝟬 MATHEMATICAL SANS-SERIF BOLD DIGIT ZERO
• U+1D7F6 𝟶 MATHEMATICAL MONOSPACE DIGIT ZERO
• U+1E140 𞅀 NYIAKENG PUACHUE HMONG DIGIT ZERO
• U+1E2F0 𞋰 WANCHO DIGIT ZERO
• U+1E950 𞥐 ADLAM DIGIT ZERO
• U+1F100 🄀 DIGIT ZERO FULL STOP
• U+1F101 🄁 DIGIT ZERO COMMA
• U+1F10B 🄋 DINGBAT CIRCLED SANS-SERIF DIGIT ZERO
• U+1F10C 🄌 DINGBAT NEGATIVE CIRCLED SANS-SERIF DIGIT ZERO
• U+1F10D 🄍 CIRCLED ZERO WITH SLASH
• U+1FBF0 🯰 SEGMENTED DIGIT ZERO
• U+E0030 TAG DIGIT ZERO
See also
• Arabic numeral variations § Slashed zero
• Regional handwriting variation § Arabic numerals
• Ø (disambiguation)
References
1. Bemer, Robert William (August 1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh". Communications of the ACM. 10 (8): 513–518. doi:10.1145/363534.363563. S2CID 294510.
2. Einarsson, Bo; Shokin, Yurij (2007-05-24). "Fortran 90 for the Fortran 77 Programmer". Appendix 7: "The historical development of Fortran. Archived from the original on 2017-02-28.
3. "Check in for your Flight Reservation".
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| Wikipedia |
Zero of a function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function $f$, is a member $x$ of the domain of $f$ such that $f(x)$ vanishes at $x$; that is, the function $f$ attains the value of 0 at $x$, or equivalently, $x$ is the solution to the equation $f(x)=0$.[1] A "zero" of a function is thus an input value that produces an output of 0.[2]
"Root of a function" redirects here. For a half iterate of a function, see Functional square root.
A graph of the function $\cos(x)$ for $x$ in $\left[-2\pi ,2\pi \right]$, with zeros at $-{\tfrac {3\pi }{2}},\;-{\tfrac {\pi }{2}},\;{\tfrac {\pi }{2}}$, and ${\tfrac {3\pi }{2}},$ marked in red.
A root of a polynomial is a zero of the corresponding polynomial function.[1] The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities.[3] For example, the polynomial $f$ of degree two, defined by $f(x)=x^{2}-5x+6$ has the two roots (or zeros) that are 2 and 3.
$f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.$
If the function maps real numbers to real numbers, then its zeros are the $x$-coordinates of the points where its graph meets the x-axis. An alternative name for such a point $(x,0)$ in this context is an $x$-intercept.
Solution of an equation
Every equation in the unknown $x$ may be rewritten as
$f(x)=0$
by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function $f$. In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.
Polynomial roots
Main article: Properties of polynomial roots
Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).
Fundamental theorem of algebra
Main article: Fundamental theorem of algebra
The fundamental theorem of algebra states that every polynomial of degree $n$ has $n$ complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs.[2] Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.
Computing roots
Main articles: Root-finding algorithm, Real-root isolation, and Equation solving
Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see algebraic solution).
Zero set
In various areas of mathematics, the zero set of a function is the set of all its zeros. More precisely, if $f:X\to \mathbb {R} $ is a real-valued function (or, more generally, a function taking values in some additive group), its zero set is $f^{-1}(0)$, the inverse image of $\{0\}$ in $X$.
Under the same hypothesis on the codomain of the function, a level set of a function $f$ is the zero set of the function $f-c$ for some $c$ in the codomain of $f.$
The zero set of a linear map is also known as its kernel.
The cozero set of the function $f:X\to \mathbb {R} $ is the complement of the zero set of $f$ (i.e., the subset of $X$ on which $f$ is nonzero).
Applications
In algebraic geometry, the first definition of an algebraic variety is through zero sets. Specifically, an affine algebraic set is the intersection of the zero sets of several polynomials, in a polynomial ring $k\left[x_{1},\ldots ,x_{n}\right]$ over a field. In this context, a zero set is sometimes called a zero locus.
In analysis and geometry, any closed subset of $\mathbb {R} ^{n}$ is the zero set of a smooth function defined on all of $\mathbb {R} ^{n}$. This extends to any smooth manifold as a corollary of paracompactness.
In differential geometry, zero sets are frequently used to define manifolds. An important special case is the case that $f$ is a smooth function from $\mathbb {R} ^{p}$ to $\mathbb {R} ^{n}$. If zero is a regular value of $f$, then the zero set of $f$ is a smooth manifold of dimension $m=p-n$ by the regular value theorem.
For example, the unit $m$-sphere in $\mathbb {R} ^{m+1}$ is the zero set of the real-valued function $f(x)=\Vert x\Vert ^{2}-1$.
See also
• Marden's theorem
• Root-finding algorithm
• Sendov's conjecture
• Vanish at infinity
• Zero crossing
• Zeros and poles
References
1. "Algebra - Zeroes/Roots of Polynomials". tutorial.math.lamar.edu. Retrieved 2019-12-15.
2. Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 535. ISBN 0-13-165711-9.
3. "Roots and zeros (Algebra 2, Polynomial functions)". Mathplanet. Retrieved 2019-12-15.
Further reading
• Weisstein, Eric W. "Root". MathWorld.
| Wikipedia |
Zerosumfree monoid
In abstract algebra, an additive monoid $(M,0,+)$ is said to be zerosumfree, conical, centerless or positive if nonzero elements do not sum to zero. Formally:
$(\forall a,b\in M)\ a+b=0\implies a=b=0\!$
This means that the only way zero can be expressed as a sum is as $0+0$.
References
• Wehrung, Friedrich (1996). "Tensor products of structures with interpolation". Pacific Journal of Mathematics. 176 (1): 267–285. doi:10.2140/pjm.1996.176.267. Zbl 0865.06010.
| Wikipedia |
Zeroth-order logic
Zeroth-order logic is a branch of logic without variables or quantifiers. Some authors use the phrase "zeroth-order logic" as a synonym for the propositional calculus,[1] but an alternative definition extends propositional logic by adding constants, operations, and relations on non-Boolean values.[2] Every zeroth-order language in this broader sense is complete and compact.[2]
References
1. Andrews, Peter B. (2002), An introduction to mathematical logic and type theory: to truth through proof, Applied Logic Series, vol. 27 (Second ed.), Kluwer Academic Publishers, Dordrecht, p. 201, doi:10.1007/978-94-015-9934-4, ISBN 1-4020-0763-9, MR 1932484.
2. Tao, Terence (2010), "1.4.2 Zeroth-order logic", An epsilon of room, II, American Mathematical Society, Providence, RI, pp. 27–31, doi:10.1090/gsm/117, ISBN 978-0-8218-5280-4, MR 2780010.
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Zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.[1] A graphical illustration of a nildimensional space is a point.[2]
This article is about zero dimension in topology. For several kinds of zero space in algebra, see zero object (algebra).
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• A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement which is a cover by disjoint open sets.
• A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.
• A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.
The three notions above agree for separable, metrisable spaces.
Properties of spaces with small inductive dimension zero
• A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel'skii & Tkachenko 2008, Proposition 3.1.7, p.136) for the non-trivial direction.)
• Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space.
• Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers $2^{I}$ where $2=\{0,1\}$ is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, $2^{I}$ is the Cantor space.
Manifolds
All points of a zero-dimensional manifold are isolated.
In particular, the zero-dimensional hypersphere is a pair of points, and the zero-dimensional ball is a single point.
Notes
• Arhangel'skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures. Atlantis Studies in Mathematics. Vol. 1. Atlantis Press. ISBN 978-90-78677-06-2.
• Engelking, Ryszard (1977). General Topology. PWN, Warsaw.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
References
1. Hazewinkel, Michiel (1989). Encyclopaedia of Mathematics, Volume 3. Kluwer Academic Publishers. p. 190. ISBN 9789400959941.
2. Wolcott, Luke; McTernan, Elizabeth (2012). "Imagining Negative-Dimensional Space" (PDF). In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza (eds.). Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona, USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 10 July 2015.
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| Wikipedia |
1 + 1 + 1 + 1 + ⋯
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written $\sum _{n=1}^{\infty }n^{0}$, $\sum _{n=1}^{\infty }1^{n}$, or simply $\sum _{n=1}^{\infty }1$, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1n can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the p-adic numbers for some p. In the context of the extended real number line
$\sum _{n=1}^{\infty }1=+\infty \,,$
The series 1 + 1 + 1 + 1 + ⋯
After smoothing
since its sequence of partial sums increases monotonically without bound.
Where the sum of n0 occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function:
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1-2^{1-s}}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}\,.$
The two formulas given above are not valid at zero however, but the analytic continuation is.
$\zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\!,$
Using this one gets (given that Γ(1) = 1),
$\zeta (0)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \sin \left({\frac {\pi s}{2}}\right)\ \zeta (1-s)={\frac {1}{\pi }}\lim _{s\rightarrow 0}\ \left({\frac {\pi s}{2}}-{\frac {\pi ^{3}s^{3}}{48}}+...\right)\ \left(-{\frac {1}{s}}+...\right)=-{\frac {1}{2}}$
where the power series expansion for ζ(s) about s = 1 follows because ζ(s) has a simple pole of residue one there. In this sense 1 + 1 + 1 + 1 + ⋯ = ζ(0) = −1/2.
Emilio Elizalde presents a comment from others about the series:
In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + ⋯ = −1/2.' Implying maybe: If you do not know this, it is no use to continue listening.[2]
See also
• Grandi's series
• 1 − 2 + 3 − 4 + · · ·
• 1 + 2 + 3 + 4 + · · ·
• 1 + 2 + 4 + 8 + · · ·
• 1 − 2 + 4 − 8 + ⋯
• 1 − 1 + 2 − 6 + 24 − 120 + · · ·
• Harmonic series
Notes
1. Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
2. Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc.....9076E.
External links
• OEIS sequence A000012 (The simplest sequence of positive numbers: the all 1's sequence)
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• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
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• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
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• 1 − 2 + 3 − 4 + ⋯
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• Infinite arithmetic series
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| Wikipedia |
List of zeta functions
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.$
Zeta functions include:
• Airy zeta function, related to the zeros of the Airy function
• Arakawa–Kaneko zeta function
• Arithmetic zeta function
• Artin–Mazur zeta function of a dynamical system
• Barnes zeta function or double zeta function
• Beurling zeta function of Beurling generalized primes
• Dedekind zeta function of a number field
• Duursma zeta function of error-correcting codes
• Epstein zeta function of a quadratic form
• Goss zeta function of a function field
• Hasse–Weil zeta function of a variety
• Height zeta function of a variety
• Hurwitz zeta function, a generalization of the Riemann zeta function
• Igusa zeta function
• Ihara zeta function of a graph
• L-function, a "twisted" zeta function
• Lefschetz zeta function of a morphism
• Lerch zeta function, a generalization of the Riemann zeta function
• Local zeta function of a characteristic-p variety
• Matsumoto zeta function
• Minakshisundaram–Pleijel zeta function of a Laplacian
• Motivic zeta function of a motive
• Multiple zeta function, or Mordell–Tornheim zeta function of several variables
• p-adic zeta function of a p-adic number
• Prime zeta function, like the Riemann zeta function, but only summed over primes
• Riemann zeta function, the archetypal example
• Ruelle zeta function
• Selberg zeta function of a Riemann surface
• Shimizu L-function
• Shintani zeta function
• Subgroup zeta function
• Witten zeta function of a Lie group
• Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function.
• Zeta function of an operator or spectral zeta function
See also
Other functions called zeta functions, but not analogous to the Riemann zeta function
• Jacobi zeta function
• Weierstrass zeta function
Topics related to zeta functions
• Artin conjecture
• Birch and Swinnerton-Dyer conjecture
• Riemann hypothesis and the generalized Riemann hypothesis.
• Selberg class S
• Explicit formulae for L-functions
• Trace formula
External links
• A directory of all known zeta functions
| Wikipedia |
Zeta function regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.
Renormalization and regularization
Renormalization
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Minimal subtraction scheme
Regularization
Dimensional regularization
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Lattice regularization
Zeta function regularization
Causal perturbation theory
Hadamard regularization
Point-splitting regularization
Definition
There are several different summation methods called zeta function regularization for defining the sum of a possibly divergent series a1 + a2 + ....
One method is to define its zeta regularized sum to be ζA(−1) if this is defined, where the zeta function is defined for large Re(s) by
$\zeta _{A}(s)={\frac {1}{a_{1}^{s}}}+{\frac {1}{a_{2}^{s}}}+\cdots $
if this sum converges, and by analytic continuation elsewhere.
In the case when an = n, the zeta function is the ordinary Riemann zeta function. This method was used by Euler to "sum" the series 1 + 2 + 3 + 4 + ... to ζ(−1) = −1/12.
Hawking (1977) showed that in flat space, in which the eigenvalues of Laplacians are known, the zeta function corresponding to the partition function can be computed explicitly. Consider a scalar field φ contained in a large box of volume V in flat spacetime at the temperature T = β−1. The partition function is defined by a path integral over all fields φ on the Euclidean space obtained by putting τ = it which are zero on the walls of the box and which are periodic in τ with period β. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field φ. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed.
Another method defines the possibly divergent infinite product a1a2.... to be exp(−ζ′A(0)). Ray & Singer (1971) used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace of A−s. Minakshisundaram & Pleijel (1949) showed that if A is the Laplacian of a compact Riemannian manifold then the Minakshisundaram–Pleijel zeta function converges and has an analytic continuation as a meromorphic function to all complex numbers, and Seeley (1967) extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization. See "analytic torsion."
Hawking (1977) suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta function regularization in order to calculate the partition functions for thermal graviton and matter's quanta in curved background such as on the horizon of black holes and on de Sitter background using the relation by the inverse Mellin transformation to the trace of the kernel of heat equations.
Example
The first example in which zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at –3, which diverges explicitly. However, it can be analytically continued to s = –3 where hopefully there is no pole, thus giving a finite value to the expression. A detailed example of this regularization at work is given in the article on the detail example of the Casimir effect, where the resulting sum is very explicitly the Riemann zeta-function (and where the seemingly legerdemain analytic continuation removes an additive infinity, leaving a physically significant finite number).
An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. More generally, the zeta-function approach can be used to regularize the whole energy–momentum tensor both in flat and in curved spacetime.
The unregulated value of the energy is given by a summation over the zero-point energy of all of the excitation modes of the vacuum:
$\langle 0|T_{00}|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}$
Here, $T_{00}$ is the zeroth component of the energy–momentum tensor and the sum (which may be an integral) is understood to extend over all (positive and negative) energy modes $\omega _{n}$; the absolute value reminding us that the energy is taken to be positive. This sum, as written, is usually infinite ($\omega _{n}$ is typically linear in n). The sum may be regularized by writing it as
$\langle 0|T_{00}(s)|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}|\omega _{n}|^{-s}$
where s is some parameter, taken to be a complex number. For large, real s greater than 4 (for three-dimensional space), the sum is manifestly finite, and thus may often be evaluated theoretically.
The zeta-regularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. Zeta-function regularization is used in conformal field theory, renormalization and in fixing the critical spacetime dimension of string theory.
Relation to other regularizations
Zeta function regularization is equivalent to dimensional regularization, see. However, the main advantage of the zeta regularization is that it can be used whenever the dimensional regularization fails, for example if there are matrices or tensors inside the calculations $\epsilon _{i,j,k}$
Relation to Dirichlet series
Zeta-function regularization gives an analytic structure to any sums over an arithmetic function f(n). Such sums are known as Dirichlet series. The regularized form
${\tilde {f}}(s)=\sum _{n=1}^{\infty }f(n)n^{-s}$
converts divergences of the sum into simple poles on the complex s-plane. In numerical calculations, the zeta-function regularization is inappropriate, as it is extremely slow to converge. For numerical purposes, a more rapidly converging sum is the exponential regularization, given by
$F(t)=\sum _{n=1}^{\infty }f(n)e^{-tn}.$
This is sometimes called the Z-transform of f, where z = exp(−t). The analytic structure of the exponential and zeta-regularizations are related. By expanding the exponential sum as a Laurent series
$F(t)={\frac {a_{N}}{t^{N}}}+{\frac {a_{N-1}}{t^{N-1}}}+\cdots $
one finds that the zeta-series has the structure
${\tilde {f}}(s)={\frac {a_{N}}{s-N}}+\cdots .$
The structure of the exponential and zeta-regulators are related by means of the Mellin transform. The one may be converted to the other by making use of the integral representation of the Gamma function:
$\Gamma (s)=\int _{0}^{\infty }t^{s-1}e^{-t}\,dt$
which leads to the identity
$\Gamma (s){\tilde {f}}(s)=\int _{0}^{\infty }t^{s-1}F(t)\,dt$
relating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurent series.
Heat kernel regularization
The sum
$f(s)=\sum _{n}a_{n}e^{-s|\omega _{n}|}$
is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the $\omega _{n}$ can sometimes be understood as eigenvalues of the heat kernel. In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the Laplace–Stieltjes transform, in that
$f(s)=\int _{0}^{\infty }e^{-st}\,d\alpha (t)$
where $\alpha (t)$ is a step function, with steps of $a_{n}$ at $t=|\omega _{n}|$. A number of theorems for the convergence of such a series exist. For example, by the Hardy-Littlewood Tauberian theorem, if
$L=\limsup _{n\to \infty }{\frac {\log \vert \sum _{k=1}^{n}a_{k}\vert }{|\omega _{n}|}}$
then the series for $f(s)$ converges in the half-plane $\Re (s)>L$ and is uniformly convergent on every compact subset of the half-plane $\Re (s)>L$. In almost all applications to physics, one has $L=0$
History
Much of the early work establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods was done by G. H. Hardy and J. E. Littlewood in 1916 and is based on the application of the Cahen–Mellin integral. The effort was made in order to obtain values for various ill-defined, conditionally convergent sums appearing in number theory.
In terms of application as the regulator in physical problems, before Hawking (1977), J. Stuart Dowker and Raymond Critchley in 1976 proposed a zeta-function regularization method for quantum physical problems. Emilio Elizalde and others have also proposed a method based on the zeta regularization for the integrals $\int _{a}^{\infty }x^{m-s}dx$, here $x^{-s}$ is a regulator and the divergent integral depends on the numbers $\zeta (s-m)$ in the limit $s\to 0$ see renormalization. Also unlike other regularizations such as dimensional regularization and analytic regularization, zeta regularization has no counterterms and gives only finite results.
See also
• Generating function – Formal power series; coefficients encode information about a sequence indexed by natural numbers
• Perron's formula – Formula to calculate the sum of an arithmetic function in analytic number theory
• Renormalization – Method in physics used to deal with infinities
• 1 + 1 + 1 + 1 + ⋯ – Divergent series
• 1 + 2 + 3 + 4 + ⋯ – Divergent series
• Analytic torsion – Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds
• Ramanujan summation – Mathematical techniques for summing divergent infinite series
• Minakshisundaram–Pleijel zeta function
• Zeta function (operator)
References
• ^ Tom M. Apostol, "Modular Functions and Dirichlet Series in Number Theory", "Springer-Verlag New York. (See Chapter 8.)"
• ^ A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini, "Analytic Aspects of Quantum Fields", World Scientific Publishing, 2003, ISBN 981-238-364-6
• ^ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp. 119–196. (See, for example, theorem 2.12)
• Hawking, S. W. (1977), "Zeta function regularization of path integrals in curved spacetime", Communications in Mathematical Physics, 55 (2): 133–148, Bibcode:1977CMaPh..55..133H, doi:10.1007/BF01626516, ISSN 0010-3616, MR 0524257, S2CID 121650064
• ^ V. Moretti, "Direct z-function approach and renormalization of one-loop stress tensor in curved spacetimes, Phys. Rev.D 56, 7797 (1997).
• Minakshisundaram, S.; Pleijel, Å. (1949), "Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds", Canadian Journal of Mathematics, 1 (3): 242–256, doi:10.4153/CJM-1949-021-5, ISSN 0008-414X, MR 0031145
• Ray, D. B.; Singer, I. M. (1971), "R-torsion and the Laplacian on Riemannian manifolds", Advances in Mathematics, 7 (2): 145–210, doi:10.1016/0001-8708(71)90045-4, MR 0295381
• "Zeta-function method for regularization", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Seeley, R. T. (1967), "Complex powers of an elliptic operator", in Calderón, Alberto P. (ed.), Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proceedings of Symposia in Pure Mathematics, vol. 10, Providence, R.I.: Amer. Math. Soc., pp. 288–307, ISBN 978-0-8218-1410-9, MR 0237943
• ^ Dowker, J. S.; Critchley, R. (1976), "Effective Lagrangian and energy–momentum tensor in de Sitter space", Physical Review D, 13 (12): 3224–3232, Bibcode:1976PhRvD..13.3224D, doi:10.1103/PhysRevD.13.3224
• ^ D. Fermi, L. Pizzocchero, "Local zeta regularization and the scalar Casimir effect. A general approach based on integral kernels", World Scientific Publishing, ISBN: 978-981-3224-99-5 (hardcover), ISBN: 978-981-3225-01-5 (ebook). DOI: 10.1142/10570 (2017).
| Wikipedia |
ZetaGrid
ZetaGrid was at one time the largest distributed computing project, designed to explore the non-trivial roots of the Riemann zeta function, checking over one billion roots a day.
Roots of the zeta function are of particular interest in mathematics; a single root out of alignment would disprove the Riemann hypothesis, with far-reaching consequences for all of mathematics. As of June, 2023 no counterexample to the Riemann hypothesis has been found.
The project ended in November 2005 due to instability of the hosting provider.[1] The first more than 1013 zeroes were checked.[2] The project administrator stated that after the results were analyzed, they would be posted on the American Mathematical Society website.[3] The official status remains unclear, however, as it was never published nor independently verified. This is likely because there was no evidence that each zero was actually computed, as there was no process implemented to check each one as it was calculated.[4][5]
References
1. Zeta Finished – Free-DC Forum
2. Ed Pegg Jr. «Ten Trillion Zeta Zeros»
3. "ZetaGrid - News". 2010-11-18. Archived from the original on 2010-11-18. Retrieved 2023-06-04.
4. Yannick Saouter, Xavier Gourdon and Patrick Demichel. An improved lower bound for the de Bruijn-Newman constant. Math. Comp. 80 (2011) 2283. MR 2813360.
5. Yannick Saouter and Patrick Demichel. A sharp region where π(x)−li(x) is positive. Math. Comp. 79 (2010) 2398. MR 2684372.
External links
• Home page (Web archive)
| Wikipedia |
Particular values of the Riemann zeta function
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ(s) and is named after the mathematician Bernhard Riemann. When the argument s is a real number greater than one, the zeta function satisfies the equation
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\,.$
It can therefore provide the sum of various convergent infinite series, such as $ \zeta (2)={\frac {1}{1^{2}}}+$$ {\frac {1}{2^{2}}}+$$ {\frac {1}{3^{2}}}+\ldots \,.$ Explicit or numerically efficient formulae exist for ζ(s) at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
The same equation in s above also holds when s is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at s = 1. The complex derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values of s, for which the corresponding summation would diverge. For example, the full zeta function exists at s = −1 (and is therefore finite there), but the corresponding series would be $ 1+2+3+\ldots \,,$ whose partial sums would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.
The Riemann zeta function at 0 and 1
At zero, one has
$\zeta (0)={B_{1}^{-}}=-{B_{1}^{+}}=-{\tfrac {1}{2}}\!$
At 1 there is a pole, so ζ(1) is not finite but the left and right limits are:
$\lim _{\varepsilon \to 0^{\pm }}\zeta (1+\varepsilon )=\pm \infty $
Since it is a pole of first order, it has a complex residue
$\lim _{\varepsilon \to 0}\varepsilon \zeta (1+\varepsilon )=1\,.$
Positive integers
Even positive integers
For the even positive integers $n$, one has the relationship to the Bernoulli numbers:
$\zeta (n)=(-1)^{{\tfrac {n}{2}}+1}{\frac {(2\pi )^{n}B_{n}}{2(n!)}}\,.$
The computation of ζ(2) is known as the Basel problem. The value of ζ(4) is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:
${\begin{aligned}\zeta (2)&=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\\[4pt]\zeta (4)&=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\\[4pt]\zeta (6)&=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}\\[4pt]\zeta (8)&=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}\\[4pt]\zeta (10)&=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}\\[4pt]\zeta (12)&=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}\\[4pt]\zeta (14)&=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}\\[4pt]\zeta (16)&=1+{\frac {1}{2^{16}}}+{\frac {1}{3^{16}}}+\cdots ={\frac {3617\pi ^{16}}{325641566250}}\,.\end{aligned}}$
Taking the limit $n\rightarrow \infty $, one obtains $\zeta (\infty )=1$.
Selected values for even integers
Value Decimal expansion Source
$\zeta (2)$ 1.6449340668482264364... OEIS: A013661
$\zeta (4)$ 1.0823232337111381915... OEIS: A013662
$\zeta (6)$ 1.0173430619844491397... OEIS: A013664
$\zeta (8)$ 1.0040773561979443393... OEIS: A013666
$\zeta (10)$ 1.0009945751278180853... OEIS: A013668
$\zeta (12)$ 1.0002460865533080482... OEIS: A013670
$\zeta (14)$ 1.0000612481350587048... OEIS: A013672
$\zeta (16)$ 1.0000152822594086518... OEIS: A013674
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
$A_{n}\zeta (2n)=\pi ^{2n}B_{n}$
where $A_{n}$ and $B_{n}$ are integers for all even $n$. These are given by the integer sequences OEIS: A002432 and OEIS: A046988, respectively, in OEIS. Some of these values are reproduced below:
coefficients
n A B
1 6 1
2 90 1
3 945 1
4 9450 1
5 93555 1
6 638512875 691
7 18243225 2
8 325641566250 3617
9 38979295480125 43867
10 1531329465290625 174611
11 13447856940643125 155366
12 201919571963756521875 236364091
13 11094481976030578125 1315862
14 564653660170076273671875 6785560294
15 5660878804669082674070015625 6892673020804
16 62490220571022341207266406250 7709321041217
17 12130454581433748587292890625 151628697551
If we let $\eta _{n}=B_{n}/A_{n}$ be the coefficient of $\pi ^{2n}$ as above,
$\zeta (2n)=\sum _{\ell =1}^{\infty }{\frac {1}{\ell ^{2n}}}=\eta _{n}\pi ^{2n}$
then we find recursively,
${\begin{aligned}\eta _{1}&=1/6\\\eta _{n}&=\sum _{\ell =1}^{n-1}(-1)^{\ell -1}{\frac {\eta _{n-\ell }}{(2\ell +1)!}}+(-1)^{n+1}{\frac {n}{(2n+1)!}}\end{aligned}}$
This recurrence relation may be derived from that for the Bernoulli numbers.
Also, there is another recurrence:
$\zeta (2n)={\frac {1}{n+{\frac {1}{2}}}}\sum _{k=1}^{n-1}\zeta (2k)\zeta (2n-2k)\quad {\text{ for }}\quad n>1$
which can be proved, using that ${\frac {d}{dx}}\cot(x)=-1-\cot ^{2}(x)$
The values of the zeta function at non-negative even integers have the generating function:
$\sum _{n=0}^{\infty }\zeta (2n)x^{2n}=-{\frac {\pi x}{2}}\cot(\pi x)=-{\frac {1}{2}}+{\frac {\pi ^{2}}{6}}x^{2}+{\frac {\pi ^{4}}{90}}x^{4}+{\frac {\pi ^{6}}{945}}x^{6}+\cdots $
Since
$\lim _{n\rightarrow \infty }\zeta (2n)=1$
The formula also shows that for $n\in \mathbb {N} ,n\rightarrow \infty $,
$\left|B_{2n}\right|\sim {\frac {(2n)!\,2}{\;~(2\pi )^{2n}\,}}$
Odd positive integers
The sum of the harmonic series is infinite.
$\zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty \!$
The value ζ(3) is also known as Apéry's constant and has a role in the electron's gyromagnetic ratio. The value ζ(3) also appears in Planck's law. These and additional values are:
Selected values for odd integers
Value Decimal expansion Source
$\zeta (3)$ 1.2020569031595942853... OEIS: A02117
$\zeta (5)$ 1.0369277551433699263... OEIS: A013663
$\zeta (7)$ 1.0083492773819228268... OEIS: A013665
$\zeta (9)$ 1.0020083928260822144... OEIS: A013667
$\zeta (11)$ 1.0004941886041194645... OEIS: A013669
$\zeta (13)$ 1.0001227133475784891... OEIS: A013671
$\zeta (15)$ 1.0000305882363070204... OEIS: A013673
It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n ∈ $\mathbb {N} $ , are irrational.[1] There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ(5), ζ(7), ζ(9), or ζ(11) is irrational.[2]
The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.[3]
Most of the identities following below are provided by Simon Plouffe. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.
Plouffe stated the following identities without proof.[4] Proofs were later given by other authors.[5]
ζ(5)
${\begin{aligned}\zeta (5)&={\frac {1}{294}}\pi ^{5}-{\frac {72}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {2}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\\\zeta (5)&=12\sum _{n=1}^{\infty }{\frac {1}{n^{5}\sinh(\pi n)}}-{\frac {39}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}+{\frac {1}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\end{aligned}}$
ζ(7)
$\zeta (7)={\frac {19}{56700}}\pi ^{7}-2\sum _{n=1}^{\infty }{\frac {1}{n^{7}(e^{2\pi n}-1)}}\!$
Note that the sum is in the form of a Lambert series.
ζ(2n + 1)
By defining the quantities
$S_{\pm }(s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}(e^{2\pi n}\pm 1)}}$
a series of relationships can be given in the form
$0=A_{n}\zeta (n)-B_{n}\pi ^{n}+C_{n}S_{-}(n)+D_{n}S_{+}(n)$
where An, Bn, Cn and Dn are positive integers. Plouffe gives a table of values:
coefficients
n A B C D
3 180 7 360 0
5 1470 5 3024 84
7 56700 19 113400 0
9 18523890 625 37122624 74844
11 425675250 1453 851350500 0
13 257432175 89 514926720 62370
15 390769879500 13687 781539759000 0
17 1904417007743250 6758333 3808863131673600 29116187100
19 21438612514068750 7708537 42877225028137500 0
21 1881063815762259253125 68529640373 3762129424572110592000 1793047592085750
These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.
A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.[6][7][8]
Negative integers
In general, for negative integers (and also zero), one has
$\zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}$
The so-called "trivial zeros" occur at the negative even integers:
$\zeta (-2n)=0$
(Ramanujan summation)
The first few values for negative odd integers are
${\begin{aligned}\zeta (-1)&=-{\frac {1}{12}}\\[4pt]\zeta (-3)&={\frac {1}{120}}\\[4pt]\zeta (-5)&=-{\frac {1}{252}}\\[4pt]\zeta (-7)&={\frac {1}{240}}\\[4pt]\zeta (-9)&=-{\frac {1}{132}}\\[4pt]\zeta (-11)&={\frac {691}{32760}}\\[4pt]\zeta (-13)&=-{\frac {1}{12}}\end{aligned}}$
However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on the first value, see 1 + 2 + 3 + 4 + · · ·.
So ζ(m) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.
Derivatives
The derivative of the zeta function at the negative even integers is given by
$\zeta ^{\prime }(-2n)=(-1)^{n}{\frac {(2n)!}{2(2\pi )^{2n}}}\zeta (2n+1)\,.$
The first few values of which are
${\begin{aligned}\zeta ^{\prime }(-2)&=-{\frac {\zeta (3)}{4\pi ^{2}}}\\[4pt]\zeta ^{\prime }(-4)&={\frac {3}{4\pi ^{4}}}\zeta (5)\\[4pt]\zeta ^{\prime }(-6)&=-{\frac {45}{8\pi ^{6}}}\zeta (7)\\[4pt]\zeta ^{\prime }(-8)&={\frac {315}{4\pi ^{8}}}\zeta (9)\,.\end{aligned}}$
One also has
${\begin{aligned}\zeta ^{\prime }(0)&=-{\frac {1}{2}}\ln(2\pi )\\[4pt]\zeta ^{\prime }(-1)&={\frac {1}{12}}-\ln A\\[4pt]\zeta ^{\prime }(2)&={\frac {1}{6}}\pi ^{2}(\gamma +\ln 2-12\ln A+\ln \pi )\end{aligned}}$
where A is the Glaisher–Kinkelin constant. The first of these identities implies that the regularized product of the reciprocals of the positive integers is $1/{\sqrt {2\pi }}$, thus the amusing "equation" $\infty !={\sqrt {2\pi }}$ !={\sqrt {2\pi }}} .[9]
From the logarithmic derivative of the functional equation,
$2{\frac {\zeta '(1/2)}{\zeta (1/2)}}=\log(2\pi )+{\frac {\pi \cos(\pi /4)}{2\sin(\pi /4)}}-{\frac {\Gamma '(1/2)}{\Gamma (1/2)}}=\log(2\pi )+{\frac {\pi }{2}}+2\log 2+\gamma \,.$
Selected derivatives
Value Decimal expansion Source
$\zeta '(3)$ −0.19812624288563685333... OEIS: A244115
$\zeta '(2)$ −0.93754825431584375370... OEIS: A073002
$\zeta '(0)$ −0.91893853320467274178... OEIS: A075700
$\zeta '(-{\tfrac {1}{2}})$ −0.36085433959994760734... OEIS: A271854
$\zeta '(-1)$ −0.16542114370045092921... OEIS: A084448
$\zeta '(-2)$ −0.030448457058393270780... OEIS: A240966
$\zeta '(-3)$ +0.0053785763577743011444... OEIS: A259068
$\zeta '(-4)$ +0.0079838114502686242806... OEIS: A259069
$\zeta '(-5)$ −0.00057298598019863520499... OEIS: A259070
$\zeta '(-6)$ −0.0058997591435159374506... OEIS: A259071
$\zeta '(-7)$ −0.00072864268015924065246... OEIS: A259072
$\zeta '(-8)$ +0.0083161619856022473595... OEIS: A259073
Series involving ζ(n)
The following sums can be derived from the generating function:
$\sum _{k=2}^{\infty }\zeta (k)x^{k-1}=-\psi _{0}(1-x)-\gamma $
where ψ0 is the digamma function.
${\begin{aligned}\sum _{k=2}^{\infty }(\zeta (k)-1)&=1\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k)-1)&={\frac {3}{4}}\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k+1)-1)&={\frac {1}{4}}\\[4pt]\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)&={\frac {1}{2}}\end{aligned}}$
Series related to the Euler–Mascheroni constant (denoted by γ) are
${\begin{aligned}\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=\gamma \\[4pt]\sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}&=1-\gamma \\[4pt]\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2+\gamma -1\end{aligned}}$
and using the principal value
$\zeta (k)=\lim _{\varepsilon \to 0}{\frac {\zeta (k+\varepsilon )+\zeta (k-\varepsilon )}{2}}$
which of course affects only the value at 1, these formulae can be stated as
${\begin{aligned}\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }{\frac {\zeta (k)-1}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2\end{aligned}}$
and show that they depend on the principal value of ζ(1) = γ .
Nontrivial zeros
Main article: Riemann hypothesis
Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The Riemann hypothesis states that the real part of every nontrivial zero must be 1/2. In other words, all known nontrivial zeros of the Riemann zeta are of the form z = 1/2 + yi where y is a real number. The following table contains the decimal expansion of Im(z) for the first few nontrivial zeros:
Selected nontrivial zeros
Decimal expansion of Im(z) Source
14.134725141734693790... OEIS: A058303
21.022039638771554992... OEIS: A065434
25.010857580145688763... OEIS: A065452
30.424876125859513210... OEIS: A065453
32.935061587739189690... OEIS: A192492
37.586178158825671257... OEIS: A305741
40.918719012147495187... OEIS: A305742
43.327073280914999519... OEIS: A305743
48.005150881167159727... OEIS: A305744
49.773832477672302181... OEIS: A306004
Andrew Odlyzko computed the first 2 million nontrivial zeros accurate to within 4×10−9, and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.[10][11]
Ratios
Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting particular values of the gamma function into the functional equation
$\zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)$
We have simple relations for half-integer arguments
${\begin{aligned}{\frac {\zeta (3/2)}{\zeta (-1/2)}}&=-4\pi \\{\frac {\zeta (5/2)}{\zeta (-3/2)}}&=-{\frac {16\pi ^{2}}{3}}\\{\frac {\zeta (7/2)}{\zeta (-5/2)}}&={\frac {64\pi ^{3}}{15}}\\{\frac {\zeta (9/2)}{\zeta (-7/2)}}&={\frac {256\pi ^{4}}{105}}\end{aligned}}$
Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation
$\Gamma \left({\tfrac {3}{4}}\right)=\left({\tfrac {\pi }{2}}\right)^{\tfrac {1}{4}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}^{\tfrac {1}{2}}$
is the zeta ratio relation
${\frac {\zeta (3/4)}{\zeta (1/4)}}=2{\sqrt {\frac {\pi }{(2-{\sqrt {2}})\operatorname {AGM} \left({\sqrt {2}},1\right)}}}$
where AGM is the arithmetic–geometric mean. In a similar vein, it is possible to form radical relations, such as from
${\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}$
the analogous zeta relation is
${\frac {\zeta (1/5)^{2}\zeta (7/10)\zeta (9/10)}{\zeta (1/10)\zeta (3/10)\zeta (4/5)^{2}}}={\frac {(5-{\sqrt {5}})\left({\sqrt {10}}+{\sqrt {5+{\sqrt {5}}}}\right)}{10\cdot 2^{\tfrac {3}{10}}}}$
References
1. Rivoal, T. (2000). "La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs". Comptes Rendus de l'Académie des Sciences, Série I. 331 (4): 267–270. arXiv:math/0008051. Bibcode:2000CRASM.331..267R. doi:10.1016/S0764-4442(00)01624-4. S2CID 119678120.
2. W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/rm2001v056n04abeh000427. S2CID 250734661.
3. Boos, H.E.; Korepin, V.E.; Nishiyama, Y.; Shiroishi, M. (2002). "Quantum correlations and number theory". J. Phys. A. 35 (20): 4443–4452. arXiv:cond-mat/0202346. Bibcode:2002JPhA...35.4443B. doi:10.1088/0305-4470/35/20/305. S2CID 119143600..
4. "Identities for Zeta(2*n+1)".
5. "Formulas for Odd Zeta Values and Powers of Pi".
6. Karatsuba, E. A. (1995). "Fast calculation of the Riemann zeta function ζ(s) for integer values of the argument s". Probl. Perdachi Inf. 31 (4): 69–80. MR 1367927.
7. E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).
8. E. A. Karatsuba: Fast evaluation of ζ(3). Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).
9. Muñoz García, E.; Pérez Marco, R. (2008), "The Product Over All Primes is $4\pi ^{2}$", Commun. Math. Phys. (277): 69–81.
10. Odlyzko, Andrew. "Tables of zeros of the Riemann zeta function". Retrieved 7 September 2022.
11. Odlyzko, Andrew. "Papers on Zeros of the Riemann Zeta Function and Related Topics". Retrieved 7 September 2022.
Further reading
• Ciaurri, Óscar; Navas, Luis M.; Ruiz, Francisco J.; Varona, Juan L. (May 2015). "A Simple Computation of ζ(2k)". The American Mathematical Monthly. 122 (5): 444–451. doi:10.4169/amer.math.monthly.122.5.444. JSTOR 10.4169/amer.math.monthly.122.5.444. S2CID 207521195.
• Simon Plouffe, "Identities inspired from Ramanujan Notebooks Archived 2009-01-30 at the Wayback Machine", (1998).
• Simon Plouffe, "Identities inspired by Ramanujan Notebooks part 2 PDF Archived 2011-09-26 at the Wayback Machine" (2006).
• Vepstas, Linas (2006). "On Plouffe's Ramanujan identities" (PDF). The Ramanujan Journal. 27 (3): 387–408. arXiv:math.NT/0609775. doi:10.1007/s11139-011-9335-9. S2CID 8789411.
• Zudilin, Wadim (2001). "One of the Numbers ζ(5), ζ(7), ζ(9), ζ(11) Is Irrational". Russian Mathematical Surveys. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/RM2001v056n04ABEH000427. MR 1861452. S2CID 250734661. PDF PDF Russian PS Russian
• Nontrival zeros reference by Andrew Odlyzko:
• Bibliography
• Tables
| Wikipedia |
Zeta distribution
In probability theory and statistics, the zeta distribution is a discrete probability distribution. If X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function
$f_{s}(k)=k^{-s}/\zeta (s)\,$
zeta
Probability mass function
Plot of the Zeta PMF on a log-log scale. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Parameters $s\in (1,\infty )$
Support $k\in \{1,2,\ldots \}$
PMF ${\frac {1/k^{s}}{\zeta (s)}}$
CDF ${\frac {H_{k,s}}{\zeta (s)}}$
Mean ${\frac {\zeta (s-1)}{\zeta (s)}}~{\textrm {for}}~s>2$
Mode $1\,$
Variance ${\frac {\zeta (s)\zeta (s-2)-\zeta (s-1)^{2}}{\zeta (s)^{2}}}~{\textrm {for}}~s>3$
Entropy $\sum _{k=1}^{\infty }{\frac {1/k^{s}}{\zeta (s)}}\log(k^{s}\zeta (s)).\,\!$
MGF does not exist
CF ${\frac {\operatorname {Li} _{s}(e^{it})}{\zeta (s)}}$
where ζ(s) is the Riemann zeta function (which is undefined for s = 1).
The multiplicities of distinct prime factors of X are independent random variables.
The Riemann zeta function being the sum of all terms $k^{-s}$ for positive integer k, it appears thus as the normalization of the Zipf distribution. The terms "Zipf distribution" and the "zeta distribution" are often used interchangeably. But while the Zeta distribution is a probability distribution by itself, it is not associated to the Zipf's law with same exponent. See also Yule–Simon distribution
Definition
The Zeta distribution is defined for positive integers $k\geq 1$, and its probability mass function is given by
$P(x=k)={\frac {1}{\zeta (s)}}k^{-s}$,
where $s>1$ is the parameter, and $\zeta (s)$ is the Riemann zeta function.
The cumulative distribution function is given by
$P(x\leq k)={\frac {H_{k,s}}{\zeta (s)}},$
where $H_{k,s}$ is the generalized harmonic number
$H_{k,s}=\sum _{i=1}^{k}{\frac {1}{i^{s}}}.$
Moments
The nth raw moment is defined as the expected value of Xn:
$m_{n}=E(X^{n})={\frac {1}{\zeta (s)}}\sum _{k=1}^{\infty }{\frac {1}{k^{s-n}}}$
The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of $s-n$ that are greater than unity. Thus:
$m_{n}=\left\{{\begin{matrix}\zeta (s-n)/\zeta (s)&{\textrm {for}}~n<s-1\\\infty &{\textrm {for}}~n\geq s-1\end{matrix}}\right.$
The ratio of the zeta functions is well-defined, even for n > s − 1 because the series representation of the zeta function can be analytically continued. This does not change the fact that the moments are specified by the series itself, and are therefore undefined for large n.
Moment generating function
The moment generating function is defined as
$M(t;s)=E(e^{tX})={\frac {1}{\zeta (s)}}\sum _{k=1}^{\infty }{\frac {e^{tk}}{k^{s}}}.$
The series is just the definition of the polylogarithm, valid for $e^{t}<1$ so that
$M(t;s)={\frac {\operatorname {Li} _{s}(e^{t})}{\zeta (s)}}{\text{ for }}t<0.$
Since this does not converge on an open interval containing $t=0$, the moment generating function does not exist.
The case s = 1
ζ(1) is infinite as the harmonic series, and so the case when s = 1 is not meaningful. However, if A is any set of positive integers that has a density, i.e. if
$\lim _{n\to \infty }{\frac {N(A,n)}{n}}$
exists where N(A, n) is the number of members of A less than or equal to n, then
$\lim _{s\to 1^{+}}P(X\in A)\,$
is equal to that density.
The latter limit can also exist in some cases in which A does not have a density. For example, if A is the set of all positive integers whose first digit is d, then A has no density, but nonetheless the second limit given above exists and is proportional to
$\log(d+1)-\log(d)=\log \left(1+{\frac {1}{d}}\right),\,$
which is Benford's law.
Infinite divisibility
The Zeta distribution can be constructed with a sequence of independent random variables with a Geometric distribution. Let $p$ be a prime number and $X(p^{-s})$ be a random variable with a Geometric distribution of parameter $p^{-s}$, namely
$\quad \quad \quad \mathbb {P} \left(X(p^{-s})=k\right)=p^{-ks}(1-p^{-s})$
If the random variables $(X(p^{-s}))_{p\in {\mathcal {P}}}$ are independent, then, the random variable $Z_{s}$ defined by
$\quad \quad \quad Z_{s}=\prod _{p\in {\mathcal {P}}}p^{X(p^{-s})}$
has the Zeta distribution : $\mathbb {P} \left(Z_{s}=n\right)={\frac {1}{n^{s}\zeta (s)}}$.
Stated differently, the random variable $\log(Z_{s})=\sum _{p\in {\mathcal {P}}}X(p^{-s})\,\log(p)$ is infinitely divisible with Lévy measure given by the following sum of Dirac masses :
$\quad \quad \quad \Pi _{s}(dx)=\sum _{p\in {\mathcal {P}}}\sum _{k\geqslant 1}{\frac {p^{-ks}}{k}}\delta _{k\log(p)}(dx)$
See also
Other "power-law" distributions
• Cauchy distribution
• Lévy distribution
• Lévy skew alpha-stable distribution
• Pareto distribution
• Zipf's law
• Zipf–Mandelbrot law
• Infinitely divisible distribution
External links
• Gut, Allan. "Some remarks on the Riemann zeta distribution". CiteSeerX 10.1.1.66.3284. What Gut calls the "Riemann zeta distribution" is actually the probability distribution of −log X, where X is a random variable with what this article calls the zeta distribution.
• Weisstein, Eric W. "Zipf Distribution". MathWorld.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
| Wikipedia |
Zeta function (operator)
The zeta function of a mathematical operator ${\mathcal {O}}$ is a function defined as
$\zeta _{\mathcal {O}}(s)=\operatorname {tr} \;{\mathcal {O}}^{-s}$
for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace.
The zeta function may also be expressible as a spectral zeta function[1] in terms of the eigenvalues $\lambda _{i}$ of the operator ${\mathcal {O}}$ by
$\zeta _{\mathcal {O}}(s)=\sum _{i}\lambda _{i}^{-s}$.
It is used in giving a rigorous definition to the functional determinant of an operator, which is given by
$\det {\mathcal {O}}:=e^{-\zeta '_{\mathcal {O}}(0)}\;.$
The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold.
One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.[2]
See also
• Quillen metric
References
1. Lapidus & van Frankenhuijsen (2006) p.23
2. Soulé, C.; with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer (1992), Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge: Cambridge University Press, pp. viii+177, ISBN 0-521-41669-8, MR 1208731
• Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006), Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, Springer Monographs in Mathematics, New York, NY: Springer-Verlag, ISBN 0-387-33285-5, Zbl 1119.28005
• Fursaev, Dmitri; Vassilevich, Dmitri (2011), Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory, Theoretical and Mathematical Physics, Springer-Verlag, p. 98, ISBN 978-94-007-0204-2
| Wikipedia |
Riemann–Hurwitz formula
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
Statement
For a compact, connected, orientable surface $S$, the Euler characteristic $\chi (S)$ is
$\chi (S)=2-2g$,
where g is the genus (the number of handles), since the Betti numbers are $1,2g,1,0,0,\dots $. In the case of an (unramified) covering map of surfaces
$\pi \colon S'\to S$
that is surjective and of degree $N$, we have the formula
$\chi (S')=N\cdot \chi (S).$
That is because each simplex of $S$ should be covered by exactly $N$ in $S'$, at least if we use a fine enough triangulation of $S$, as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together).
Now assume that $S$ and $S'$ are Riemann surfaces, and that the map $\pi $ is complex analytic. The map $\pi $ is said to be ramified at a point P in S′ if there exist analytic coordinates near P and π(P) such that π takes the form π(z) = zn, and n > 1. An equivalent way of thinking about this is that there exists a small neighborhood U of P such that π(P) has exactly one preimage in U, but the image of any other point in U has exactly n preimages in U. The number n is called the ramification index at P and also denoted by eP. In calculating the Euler characteristic of S′ we notice the loss of eP − 1 copies of P above π(P) (that is, in the inverse image of π(P)). Now let us choose triangulations of S and S′ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then S′ will have the same number of d-dimensional faces for d different from zero, but fewer than expected vertices. Therefore, we find a "corrected" formula
$\chi (S')=N\cdot \chi (S)-\sum _{P\in S'}(e_{P}-1)$
or as it is also commonly written, using that $\chi (X)=2-2g(X)$ and multiplying through by -1:
$2g(S')-2=N\cdot (2g(S)-2)+\sum _{P\in S'}(e_{P}-1)$
(all but finitely many P have eP = 1, so this is quite safe). This formula is known as the Riemann–Hurwitz formula and also as Hurwitz's theorem.
Another useful form of the formula is:
$\chi (S')-r=N\cdot (\chi (S)-b)$
where r is the number points in S' at which the cover has nontrivial ramification (ramification points) and b is the number of points in S that are images of such points (branch points). Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from S and disjoint disc neighborhoods of the ramification points in S' so that the restriction of $\pi $ is a covering. Then apply the general degree formula to the restriction, use the fact that the Euler characteristic of the disc equals 1, and use the additivity of the Euler characteristic under connected sums.
Examples
The Weierstrass $\wp $-function, considered as a meromorphic function with values in the Riemann sphere, yields a map from an elliptic curve (genus 1) to the projective line (genus 0). It is a double cover (N = 2), with ramification at four points only, at which e = 2. The Riemann–Hurwitz formula then reads
$0=2\cdot 2-4\cdot (2-1)$
with the summation taken over four ramification points.
The formula may also be used to calculate the genus of hyperelliptic curves.
As another example, the Riemann sphere maps to itself by the function zn, which has ramification index n at 0, for any integer n > 1. There can only be other ramification at the point at infinity. In order to balance the equation
$2=n\cdot 2-(n-1)-(e_{\infty }-1)$
we must have ramification index n at infinity, also.
Consequences
Several results in algebraic topology and complex analysis follow.
Firstly, there are no ramified covering maps from a curve of lower genus to a curve of higher genus – and thus, since non-constant meromorphic maps of curves are ramified covering spaces, there are no non-constant meromorphic maps from a curve of lower genus to a curve of higher genus.
As another example, it shows immediately that a curve of genus 0 has no cover with N > 1 that is unramified everywhere: because that would give rise to an Euler characteristic > 2.
Generalizations
For a correspondence of curves, there is a more general formula, Zeuthen's theorem, which gives the ramification correction to the first approximation that the Euler characteristics are in the inverse ratio to the degrees of the correspondence.
An orbifold covering of degree N between orbifold surfaces S' and S is a branched covering, so the Riemann–Hurwitz formula implies the usual formula for coverings
$\chi (S')=N\cdot \chi (S)\,$
denoting with $\chi \,$ the orbifold Euler characteristic.
References
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052, section IV.2.
Topics in algebraic curves
Rational curves
• Five points determine a conic
• Projective line
• Rational normal curve
• Riemann sphere
• Twisted cubic
Elliptic curves
Analytic theory
• Elliptic function
• Elliptic integral
• Fundamental pair of periods
• Modular form
Arithmetic theory
• Counting points on elliptic curves
• Division polynomials
• Hasse's theorem on elliptic curves
• Mazur's torsion theorem
• Modular elliptic curve
• Modularity theorem
• Mordell–Weil theorem
• Nagell–Lutz theorem
• Supersingular elliptic curve
• Schoof's algorithm
• Schoof–Elkies–Atkin algorithm
Applications
• Elliptic curve cryptography
• Elliptic curve primality
Higher genus
• De Franchis theorem
• Faltings's theorem
• Hurwitz's automorphisms theorem
• Hurwitz surface
• Hyperelliptic curve
Plane curves
• AF+BG theorem
• Bézout's theorem
• Bitangent
• Cayley–Bacharach theorem
• Conic section
• Cramer's paradox
• Cubic plane curve
• Fermat curve
• Genus–degree formula
• Hilbert's sixteenth problem
• Nagata's conjecture on curves
• Plücker formula
• Quartic plane curve
• Real plane curve
Riemann surfaces
• Belyi's theorem
• Bring's curve
• Bolza surface
• Compact Riemann surface
• Dessin d'enfant
• Differential of the first kind
• Klein quartic
• Riemann's existence theorem
• Riemann–Roch theorem
• Teichmüller space
• Torelli theorem
Constructions
• Dual curve
• Polar curve
• Smooth completion
Structure of curves
Divisors on curves
• Abel–Jacobi map
• Brill–Noether theory
• Clifford's theorem on special divisors
• Gonality of an algebraic curve
• Jacobian variety
• Riemann–Roch theorem
• Weierstrass point
• Weil reciprocity law
Moduli
• ELSV formula
• Gromov–Witten invariant
• Hodge bundle
• Moduli of algebraic curves
• Stable curve
Morphisms
• Hasse–Witt matrix
• Riemann–Hurwitz formula
• Prym variety
• Weber's theorem (Algebraic curves)
Singularities
• Acnode
• Crunode
• Cusp
• Delta invariant
• Tacnode
Vector bundles
• Birkhoff–Grothendieck theorem
• Stable vector bundle
• Vector bundles on algebraic curves
| Wikipedia |
Zeuthen–Segre invariant
In algebraic geometry, the Zeuthen–Segre invariant I is an invariant of a projective surface found in a complex projective space which was introduced by Zeuthen (1871) and rediscovered by Corrado Segre (1896).
The invariant I is defined to be d – 4g – b if the surface has a pencil of curves, non-singular of genus g except for d curves with 1 ordinary node, and with b base points where the curves are non-singular and transverse.
Alexander (1914) showed that the Zeuthen–Segre invariant I is χ–4, where χ is the topological Euler–Poincaré characteristic introduced by Poincaré (1895), which is equal to the Chern number c2 of the surface.
References
• Alexander, J. W. (1914), "Sur les cycles des surfaces algébriques et sur une définition topologique de l'invariant de Zeuthen-Segre", Atti della Accademia Nazionale dei Lincei. Rend. V (2), 23: 55–62
• Baker, Henry Frederick (1933), Principles of geometry. Volume 6. Introduction to the theory of algebraic surfaces and higher loci., Cambridge Library Collection, Cambridge University Press, ISBN 978-1-108-01782-4, MR 2850141 Reprinted 2010
• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
• Poincaré, Henri (1895), "Analysis Situs", Journal de l'École Polytechnique, 1: 1–123
• Segre, C. (1896), "Intorno ad un carattere delle superficie e delle varietà superiori algebriche.", Atti della Accademia delle Scienze di Torino (in Italian), 31: 485–501
• Zeuthen, H. G. (1871), "Études géométriques de quelques-unes des propriétés de deux surfaces dont les points se correspondent un-à-un", Mathematische Annalen, Springer Berlin / Heidelberg, 4: 21–49, doi:10.1007/BF01443296, ISSN 0025-5831, S2CID 121840169
| Wikipedia |
Sign extension
Sign extension (abbreviated as sext) is the operation, in computer arithmetic, of increasing the number of bits of a binary number while preserving the number's sign (positive/negative) and value. This is done by appending digits to the most significant side of the number, following a procedure dependent on the particular signed number representation used.
For example, if six bits are used to represent the number "00 1010" (decimal positive 10) and the sign extend operation increases the word length to 16 bits, then the new representation is simply "0000 0000 0000 1010". Thus, both the value and the fact that the value was positive are maintained.
If ten bits are used to represent the value "11 1111 0001" (decimal negative 15) using two's complement, and this is sign extended to 16 bits, the new representation is "1111 1111 1111 0001". Thus, by padding the left side with ones, the negative sign and the value of the original number are maintained.
In the Intel x86 instruction set, for example, there are two ways of doing sign extension:
• using the instructions cbw, cwd, cwde, and cdq: convert byte to word, word to doubleword, word to extended doubleword, and doubleword to quadword, respectively (in the x86 context a byte has 8 bits, a word 16 bits, a doubleword and extended doubleword 32 bits, and a quadword 64 bits);
• using one of the sign extended moves, accomplished by the movsx ("move with sign extension") family of instructions.
Zero extension
A similar concept is zero extension (abbreviated as zext). In a move or convert operation, zero extension refers to setting the high bits of the destination to zero, rather than setting them to a copy of the most significant bit of the source. If the source of the operation is an unsigned number, then zero extension is usually the correct way to move it to a larger field while preserving its numeric value, while sign extension is correct for signed numbers.
In the x86 and x64 instruction sets, the movzx instruction ("move with zero extension") performs this function. For example, movzx ebx, al copies a byte from the al register to the low-order byte of ebx and then fills the remaining bytes of ebx with zeroes.
On x64, most instructions that write to the entirety of lower 32 bits of any of the general-purpose registers will zero the upper half of the destination register. For example, the instruction mov eax, 1234 will clear the upper 32 bits of the rax[lower-alpha 1] register.
See also
• Arithmetic shift and logical shift
References
• Mano, Morris M.; Kime, Charles R. (2004). Logic and Computer Design Fundamentals (3rd ed.), pp 453. Pearson Prentice Hall. ISBN 0-13-140539-X.
Notes
1. RAX - 64 bit accumulator
| Wikipedia |
Yitang Zhang
Yitang Zhang (Chinese: 张益唐; born February 5, 1955)[3] is a Chinese-American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015.[4]
Yitang Zhang
Zhang in 2014
Born (1955-02-05) February 5, 1955
Shanghai, China
CitizenshipUnited States
Alma materPeking University (BS, MA)
Purdue University (PhD)
Known forEstablishing the existence of an infinitely repeatable prime 2-tuple[1]
AwardsOstrowski Prize (2013)
Cole Prize (2014)
Rolf Schock Prize (2014)
MacArthur Fellowship (2014)
Scientific career
FieldsNumber theory
InstitutionsUniversity of New Hampshire
University of California, Santa Barbara
ThesisThe Jacobian conjecture and the degree of field extension (1992)
Doctoral advisorTzuong-Tsieng Moh (莫宗堅)[2]
Previously working at the University of New Hampshire as a lecturer, Zhang submitted a paper to the Annals of Mathematics in 2013 which established the first finite bound on the least gap between consecutive primes that is attained infinitely often. This work led to a 2013 Ostrowski Prize, a 2014 Cole Prize, a 2014 Rolf Schock Prize, and a 2014 MacArthur Fellowship. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015.[5][6][7][8]
Early life and education
Zhang was born in Shanghai, China, with his ancestral home in Pinghu, Zhejiang. He lived in Shanghai with his grandmother until he went to Peking University. At around the age of nine, he found a proof of the Pythagorean theorem.[9] He first learned about Fermat's Last Theorem and the Goldbach conjecture when he was 10.[9] During the Cultural Revolution, he and his mother were sent to the countryside to work in the fields. He worked as a laborer for 10 years and was unable to attend high school.[9] After the Cultural Revolution ended, Zhang entered Peking University in 1978 as an undergraduate student and received a bachelor of science in mathematics in 1982. He became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained a master of science in mathematics in 1984.[10]
After receiving his master's degree in mathematics, with recommendations from Professor Ding Shisun, the President of Peking University, and Professor Deng Donggao, Chair of the university's Math Department,[11] Zhang was granted a full scholarship at Purdue University. Zhang arrived at Purdue in January 1985, studied there for six and a half years, and obtained his PhD in mathematics in December 1991.
Career
Zhang's PhD work was on the Jacobian conjecture. After graduation, Zhang had trouble finding an academic position. In a 2013 interview with Nautilus magazine, Zhang said he did not get a job after graduation. "During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor [Tzuong-Tsieng Moh] did not write me letters of recommendation."[12] Zhang made this claim again in George Csicsery's documentary film "Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture"[13] while discussing his difficulties at Purdue and in the years that followed.[9] Moh claimed that Zhang never came back to him requesting recommendation letters.[11] In a detailed profile published in The New Yorker magazine in February 2015, Alec Wilkinson wrote Zhang "parted unhappily" with Moh, and that Zhang "left Purdue without Moh's support, and, having published no papers, was unable to find an academic job".[7] In 2018, responding to reports of his treatment of Zhang, Moh posted an update on his website. Moh wrote that Zhang "failed miserably" in proving the Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time".[14]
After some years, Zhang managed to find a position as a lecturer at the University of New Hampshire, where he was hired by Kenneth Appel in 1999. Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop.[1] A profile published in the Quanta Magazine reports that Zhang used to live in his car during the initial job-hunting days.[9] He served as lecturer at UNH from 1999[15] until around January 2014, when UNH appointed him to a full professorship as a result of his breakthrough on prime numbers.[16] Zhang stayed for a semester at The Institute For Advanced Study in Princeton, NJ, in 2014, and he joined the University of California, Santa Barbara in fall 2015.[17]
Research
On April 17, 2013, Zhang announced a proof that there are infinitely many pairs of prime numbers that differ by less than 70 million. This result implies the existence of an infinitely repeatable prime 2-tuple,[1] thus establishing a theorem akin to the twin prime conjecture. Zhang's paper was accepted by Annals of Mathematics in early May 2013,[6] his first publication since his last paper in 2001.[18] The proof was refereed by leading experts in analytic number theory.[7] Zhang's result set off a flurry of activity in the field, such as the Polymath8 project.
If P(N) stands for the proposition that there is an infinitude of pairs of prime numbers (not necessarily consecutive primes) that differ by exactly N, then Zhang's result is equivalent to the statement that there exists at least one even integer k < 70,000,000 such that P(k) is true. The classical form of the twin prime conjecture is equivalent to P(2); and in fact it has been conjectured that P(k) holds for all even integers k.[19][20] While these stronger conjectures remain unproven, a result due to James Maynard in November 2013, employing a different technique, showed that P(k) holds for some k ≤ 600.[21] Subsequently, in April 2014, the Polymath project 8 lowered the bound to k ≤ 246.[22] With current methods k ≤ 6 is the best attainable, and in fact k ≤ 12 and k ≤ 6 follow using current methods if the Elliott–Halberstam conjecture and its generalization, respectively, hold.[7][22]
Honors and awards
Zhang was awarded the 2013 Morningside Special Achievement Award in Mathematics,[23] the 2013 Ostrowski Prize,[24] the 2014 Frank Nelson Cole Prize in Number Theory,[16][25] and the 2014 Rolf Schock Prize[26] in Mathematics.
He is a recipient of the 2014 MacArthur award,[27] and was elected as an Academia Sinica Fellow during the same year.[10] He was an invited speaker at the 2014 International Congress of Mathematicians.
Political views
In 1989 Zhang joined a group interested in Chinese democracy (中国民联). In a 2013 interview, he affirmed that his political views on the subject had not changed since.[7][28]
Publications
• Zhang, Yitang (2007). "On the Landau-Siegel Zeros Conjecture". arXiv:0705.4306 [math.NT].
• Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7.
• Zhang, Yitang (2022). "Discrete mean estimates and the Landau-Siegel zero". arXiv:2211.02515 [math.NT].
References
1. Klarreich, Erica (May 19, 2013). "Unheralded Mathematician Bridges the Prime Gap". Quanta Magazine. Retrieved May 19, 2013.
2. Yitang Zhang at the Mathematics Genealogy Project
3. Zhang, Yitang (1991). The Jacobian conjecture and the degree of field extension. Purdue University. pp. 1–24. Retrieved March 4, 2021.
4. "Yitang (Tom) Zhang | Department of Mathematics – UC Santa Barbara". math.ucsb.edu. Retrieved October 19, 2022.
5. Yitang Zhang, Mathematician, MacArthur Fellows Program, MacArthur Foundation, September 17, 2014
6. Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761. Zbl 1290.11128. (subscription required)
7. Wilkinson, Alec. "The Pursuit of Beauty". The New Yorker. No. February 2, 2015.
8. "Yitang (Tom) Zhang | Department of Mathematics – UC Santa Barbara". math.ucsb.edu. Retrieved February 15, 2018.
9. Thomas Lin (April 2, 2015). "After Prime Proof, an Unlikely Star Rises". Quanta Magazine.
10. "Mathematics and Physical Sciences Yitang Zhang". sinica.edu.tw. 2014.
11. Moh, Tzuong-Tsieng. "Zhang, Yitang's life at Purdue (Jan. 1985-Dec, 1991)" (PDF). Retrieved May 24, 2013.
12. "The Twin Prime Hero". Nautilus.
13. Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture on IMdB
14. "Bio" (PDF). math.purdue.edu. 2013. Retrieved August 9, 2021.
15. Macalaster, Gretyl (December 14, 2013). "Math world stunned by UNH lecturer's find". New Hampshire Union Leader.
16. "January 2014 AMS-MAA Prize booklet" (PDF). p. 7.
17. "Celebrity Mathematician Joins UCSB Faculty | The Daily Nexus". September 17, 2015.
18. Jordan Ellenberg (May 22, 2013). "The Beauty of Bounded Gaps". Slate. Retrieved January 23, 2017.
19. McKee, Maggie (May 14, 2013). "First proof that infinitely many prime numbers come in pairs". Nature. Retrieved May 21, 2013.
20. Chang, Kenneth (May 20, 2013). "Solving a Riddle of Primes". The New York Times. Retrieved May 21, 2013.
21. Klarreich, Erica (November 20, 2013). "Together and Alone, Closing the Prime Gap". Retrieved November 20, 2013.
22. "Bounded gaps between primes". Polymath.
23. "ICCM 2013: Morningside Awards".
24. "The 2013 Ostrowski Prize".
25. "Yitang Zhang Receives 2014 AMS Cole Prize in Number Theory".
26. "The 2014 Rolf Schock Prize".
27. Lee, Felicia R. (September 17, 2014). "MacArthur Awards Go to 21 Diverse Fellows". The New York Times.
28. "张益唐问答录" (in Chinese). July 1, 2013. Retrieved June 30, 2015.
External links
• Alec Wilkinson, The Pursuit of Beauty, Yitang Zhang solves a pure-math mystery, The New Yorker, Profiles, February 2, 2015, issue
• Discover Magazine article by Steve Nadis, "Prime Solver"
• Gaps between Primes – Numberphile – University of Nottingham video (shorter version)
• Gaps between Primes (extra footage) – Numberphile (longer version)
Rolf Schock Prize laureates
Logic and philosophy
• Willard Van Orman Quine (1993)
• Michael Dummett (1995)
• Dana Scott (1997)
• John Rawls (1999)
• Saul Kripke (2001)
• Solomon Feferman (2003)
• Jaakko Hintikka (2005)
• Thomas Nagel (2008)
• Hilary Putnam (2011)
• Derek Parfit (2014)
• Ruth Millikan (2017)
• Saharon Shelah (2018)
• Dag Prawitz / Per Martin-Löf (2020)
• David Kaplan (2022)
Mathematics
• Elias M. Stein (1993)
• Andrew Wiles (1995)
• Mikio Sato (1997)
• Yuri I. Manin (1999)
• Elliott H. Lieb (2001)
• Richard P. Stanley (2003)
• Luis Caffarelli (2005)
• Endre Szemerédi (2008)
• Michael Aschbacher (2011)
• Yitang Zhang (2014)
• Richard Schoen (2017)
• Ronald Coifman (2018)
• Nikolai G. Makarov (2020)
• Jonathan Pila (2022)
Visual arts
• Rafael Moneo (1993)
• Claes Oldenburg (1995)
• Torsten Andersson (1997)
• Herzog & de Meuron (1999)
• Giuseppe Penone (2001)
• Susan Rothenberg (2003)
• SANAA / Kazuyo Sejima + Ryue Nishizawa (2005)
• Mona Hatoum (2008)
• Marlene Dumas (2011)
• Anne Lacaton / Jean-Philippe Vassal (2014)
• Doris Salcedo (2017)
• Andrea Branzi (2018)
• Francis Alÿs (2020)
• Rem Koolhaas (2022)
Musical arts
• Ingvar Lidholm (1993)
• György Ligeti (1995)
• Jorma Panula (1997)
• Kronos Quartet (1999)
• Kaija Saariaho (2001)
• Anne Sofie von Otter (2003)
• Mauricio Kagel (2005)
• Gidon Kremer (2008)
• Andrew Manze (2011)
• Herbert Blomstedt (2014)
• Wayne Shorter (2017)
• Barbara Hannigan (2018)
• György Kurtág (2020)
• Víkingur Ólafsson (2022)
Authority control
International
• ISNI
• VIAF
National
• Germany
• United States
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Zhang Luping
Zhang Luping (simplified Chinese: 张鹭平; traditional Chinese: 張鹭平; 1945-1998) was a Chinese martial artist and mathematician born in Jiaxing, Zhejiang province. He was best known in China for his exceptional skill at tai chi's push hands, and for an incident in his hometown in which he accidentally broke a weightlifting champion’s forearm during an arm wrestling match. He was a student of Cai Hong Xiang (蔡鸿祥),[1]: 2–7 Wang Zi-Ping (王子平),[1]: 6 and Fu Zhong Wen (傅鍾文) .[1] He was also a descendant of Zhang Jun (traditional Chinese: 張浚; simplified Chinese: 张浚; pinyin: Zhāng Jùn, 1097–1164) and of Zhang Jiugao (simplified Chinese: 張九皋), who was the brother of Zhang Jiuling (simplified Chinese: 张九龄). He was noted for his deep knowledge of the five styles of tai chi, his superb application of the principles, and his highly developed internal power. In an age when many great martial arts teachers remained reluctant to share their highest insights and techniques, Zhang championed in his teaching an attitude of openness and a strong desire to ensure the continuation of Chinese martial traditions.
Zhang Luping
Master Zhang Lu Ping demonstrating tai chi at the Tai Chi Farm (Zhang San Feng Festival)
Martial arts career
Zhang started learning Shaolin Kung Fu when he was 13 years old from Shaolin and Jin Woo grandmaster Fang Nan Tang (方南堂). Zhang was captain of the Wushu team [2] at East Normal University in Shanghai, where he studied under three-time Chinese Wushu national champion (1953-1960) and five-time Chinese national martial arts competition gold medalist Cai Hong Xiang. Because of his excellent performance and dedication to learning the arts, Cai Hong Xiang arranged for Zhang to study under the famous grandmaster Wang Zi-Ping. Zhang also had the privilege at that time to study alongside The Magic Fist Dragon, Cai Long Yun.
Eventually, Zhang developed an interest in tai chi. He learned Chen-style taijiquan from many Chen lineage holders, including Dong Xiang Gen (董祥根)[1]: 3 and Du Wen Cai (都文才).[1]: 2 whom was the last student of Chen Zhao Kui(陈照奎). His form was also corrected by grandmaster Gu Liu Xin (顾留馨) (a former student of Chen Fake, Sun Lutang, and Chen Weiming, who corrected his style with a level of detail that would set him apart from other Chen style practitioners). Zhang also studied Wu style tai chi with master Sun Ren Zhi (孙润志)[1]: 2 and Xin Yi, another internal style similar to tai chi, with the well-known Shang Hai-based master, "Little Tiger" Zhang Hai Sheng (小老虎章海深),[1]: 5 who was highly respected for his skill in combat.
Scholarship and emigration to the United States
Zhang earned his master's degree in mathematics at East Normal University in Shang Hai under Prof. Cheng Chang Ping (陈昌平教授), who had done extensive mathematics work with Prof. Wolf Von Wahl at the University of Bayreuth in West Germany. Zhang also studied under professor Wang Guang Yin (王光寅) at the math research department of Chinese Academy of Sciences in Bei Jing. In 1983, Zhang published an article in The Mathematical Journal (数学学报) entitled “Hλ solutions of the 1st class of Fuchs type equations with operator coefficients” (一类具算子系数的Fuchs型方程的Hλ解) with his colleague Wang Ju Yan (王继延).
Zhang Luping was known in the academic community for his ground-breaking work in differential equations. He came to the United States in 1985 for a master's degree in mathematics at Carnegie-Mellon University in Pittsburgh, PA, following which he completed a doctorate degree and a post doctorate degree at the University of Massachusetts, Amherst under Prof. M. S. Berger. In November of 1994, he co-authored with professor Berger a paper published by International Publications for the PanAmerican Mathematical Journal entitled “A New Method for Large Quasiperiodic Nonlinear Oscillations with Fixed Frequencies for the Non-dissipative Second Order Conservative System of the Second Type” about the communication of applied nonlinear analysis. After completing his post-doctorate work, he taught mathematics at the University of California, Irvine and the University of Massachusetts, Amherst.
Legacy
In 1975, Zhang became the Zhejiang Province chen style tai chi champion.[2] MA. In 1998, Zhang defeated several local martial artists in Pittsburgh, PA and was invited to teach seminars at the Zhang San Feng Festival at the Tai Chi Farm owned by late master Jou Tsung Hwa. He held seminars all across the U.S. and judged many U.S. competitions, including the Houston 1990 United States National Chinese Martial Arts Competition. He was a Special Master for Taste of China and many similar martial arts events. He was twice pictured on the cover of Tai Chi International Magazine, as well as Inside Kung Fu magazine and the Pa Kua Zhang Newsletter. The Australian magazine "Tai Chi Combat and Health" called Master Zhang the "Real Thing"[3] In Tai Chi for Dummies, author Therese Ikoian wrote, "The late tai chi Master Zhang Lu Ping knew the spiraling technique well. Manny Fuentes had the 'privilege of being thrown around by him'. No matter how well Manny thought that he'd prepared a pending movement, he said that he was moved as easily as if he were a leaf, not a 175-pound man! By the time Zhang manifested the spiraling force up into his arms and hands, it contained an irresistible momentum."[4]
Zhang died in 1998 in Amherst, Massachusetts. His son Huan Zhang, a practitioner of tai chi and a scholar like his father, wrote a biographical article for China’s Premier Tai Chi - The spirit of Kung Fu Magazine（太极武魂杂志)[5] in his father's memory.
References
• Huan Zhang, (June 2018), "In memory of my father, Zhang Lu Ping", Tai Chi - The spirit of Kung Fu Magazine, Beijing Physical Education Press , China
1. B. Jones, (April 1990). "A LOOK at T'ai Chi Teachers in China", T'ai Chi Magazine, The Leading International Magazine of Tai Chi Chuan
2. Teri Lynn Breier, (May 1989).P-34,"China's Lu Ping Zhang: Teaching While Learning", Inside Kung Fu Magazine
3. Leroy Clark, (June, 1991). P7-8, "Tai Chi Combat and Health", Taiji Publications, Murwillumbah, Australia
4. Therese Ikoian, Manny Fuentes, (2001), "Tai Chi for Dummies", Wiley Publishing Inc, ISBN 0-7645-5351-8 p 71.
5. William Phillips, (2019), "In the presence of Cheng Man-Ching", My life and lessons with the Master of Five Excellences, Floating World Press, ISBN 978-0648283126 p ii
| Wikipedia |
Sun-Yung Alice Chang
Sun-Yung Alice Chang (Chinese: 張聖容; pinyin: Zhāng Shèngróng, Hakka: Chông Sṳn-yùng, [t͡soŋ sɨn juŋ]; born 1948) is a Taiwanese American mathematician specializing in aspects of mathematical analysis ranging from harmonic analysis and partial differential equations to differential geometry. She is the Eugene Higgins Professor of Mathematics at Princeton University.[1]
Sun-Yung Alice Chang
Sun-Yung Alice Chang, 2007
Born1948
Xian, China
NationalityAmerican
Other namesAlice Chang
Alma materNational Taiwan University
University of California, Berkeley
SpousePaul C. Yang
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Los Angeles
Princeton University
Doctoral advisorDonald Sarason
Life
Chang was born in Xian, China in 1948 and grew up in Taiwan. She received her Bachelor of Science degree in 1970 from National Taiwan University, and her doctorate in 1974 from the University of California, Berkeley.[2] At Berkeley, Chang wrote her thesis on the study of bounded analytic functions. Chang became a full professor at UCLA in 1980 before moving to Princeton in 1998.[3]
Career and research
Chang's research interests include the study of geometric types of nonlinear partial differential equations and problems in isospectral geometry. Working with her husband Paul Yang and others, she produced contributions to differential equations in relation to geometry and topology.[3]
She teaches at Princeton University as of 1998. Before that, she held visiting positions at University of California-Berkeley; Institute for Advanced Study, Princeton, N.J.; and Swiss Federal Institute of Technology, Zurich, Switzerland.[3] She served at Swiss Federal Institute of Technology as a visiting professor in 2015.[4]
In 2004,[5] she was interviewed by Yu Kiang Leong for Creative Minds, Charmed Lives: Interviews at Institute for Mathematical Sciences, National University of Singapore, and she declared:
«In the mathematical community, we should leave room for people who want to do work in their own way. Mathematical research is not just a scientific approach; the nature of mathematics is sometimes close to that of art. Some people want individual character and an individual way of working things out. They should be appreciated too. There should be room for single research and collaborative research».[6]
Chang's life was profiled in the 2017 documentary film Girls who fell in love with Math.[7]
Service and honors
• Sloan Foundation Research Fellowship, 1979–1981[8]
• Invited speaker at the International Congress of Mathematicians in Berkeley, 1986 [8]
• Vice president of the American Mathematical Society, 1989-1991[8]
• Ruth Lyttle Satter Prize in Mathematics of the American Mathematical Society, 1995 [8]
• Guggenheim Fellowship, 1998 [9]
• Plenary Speaker at the International Congress of Mathematicians in Beijing, 2002[10]
• Member, American Academy of Arts and Sciences, 2008 [11]
• Honorary Degree, UPMC, 2013 [12]
• Fellow, National Academy of Sciences, 2009 [13]
• Fellow, Academia Sinica, 2012[14]
• Fellow, American Mathematical Society, 2015[15]
• Fellow, Association for Women in Mathematics, 2019[16]
• MSRI Simons Professor, 2015-2016[17]
Publications
• Chang, Sun-Yung A.; Yang, Paul C. Conformal deformation of metrics on $S^{2}$. J. Differential Geom. 27 (1988), no. 2, 259–296.
• Chang, Sun-Yung Alice; Yang, Paul C. Prescribing Gaussian curvature on $S^{2}$. Acta Math. 159 (1987), no. 3–4, 215–259.
• Chang, Sun-Yung A.; Yang, Paul C. Extremal metrics of zeta function determinants on 4-manifolds. Ann. of Math. (2) 142 (1995), no. 1, 171–212.
• Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C. The scalar curvature equation on 2- and 3-spheres. Calc. Var. Partial Differential Equations 1 (1993), no. 2, 205–229.
• Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C. An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math. (2) 155 (2002), no. 3, 709–787.
• Chang, S.-Y. A.; Wilson, J. M.; Wolff, T. H. Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60 (1985), no. 2, 217–246.
• Carleson, Lennart; Chang, Sun-Yung A. On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. (2) 110 (1986), no. 2, 113–127.
• Chang, Sun-Yung A.; Fefferman, Robert Some recent developments in Fourier analysis and $H^{p}$-theory on product domains. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 1–43.
• Chang, Sun-Yung A.; Fefferman, Robert A continuous version of duality of $H^{1}$ with BMO on the bidisc. Ann. of Math. (2) 112 (1980), no. 1, 179–201.
References
• O'Connor, John J.; Robertson, Edmund F., "Sun-Yung Alice Chang", MacTutor History of Mathematics Archive, University of St Andrews
1. "Sun-Yung Alice Chang". web.math.princeton.edu. Retrieved 2020-11-12.
2. Gursky, Matthew J.; Wang, Yi (March 2020). "Sun-Yung Alice Chang and Geometric Analysis" (PDF). Notices of the American Mathematical Society. 67 (3): 318. doi:10.1090/noti2037.
3. "Sun-Yung Alice Chang". Faculty Profiles. Princeton University. Retrieved 23 February 2014.
4. "List of guests". Swiss Federal Institute of Technology. Retrieved 23 February 2014.
5. Leong, Y K (Summer 2012). "An Interview with Sun-Yung Alice Chang" (PDF). Asia Pacific Mathematics Newsletter. 2: 25–29.
6. Leong, Yu Kiang (2010). Creative Minds, Charmed Lives: Interviews at Institute for Mathematical Sciences, National University of Singapore. World Scientific. ISBN 9789814317580. Retrieved 2017-09-10.
7. "Girls who fell in love with Math". Taiwan Film Institute. 31 August 2017. Retrieved 2018-02-04.
8. Oakes, Elizabeth H. (2002). International encyclopedia of women scientists. New York, NY: Facts on File. p. 58. ISBN 0816043817.
9. "Sun-Yung Alice Chang". Guggenheim Foundation. Archived from the original on 27 February 2014. Retrieved 23 February 2014.
10. "Plenary Speakers". International Congress of Mathematics. Archived from the original on 23 February 2014. Retrieved 23 February 2014.
11. "Members of the American Academy of Arts & Sciences: 1780-2013" (PDF). American Academy of Arts and Sciences. Retrieved 23 February 2014.
12. "Professor Alice Chang awarded Doctor Honoris Causa of Pierre and Marie Curie University". Princeton University. Retrieved 26 January 2019.
13. "Sun-Yung Alice Chang". National Academy of Sciences. Retrieved 23 February 2014.
14. 2012 Academicians Announced
15. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-16.
16. 2019 Class of AWM Fellows, Association for Women in Mathematics, retrieved 2019-01-19
17. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved 2021-06-07.
External links
• Leong, Y K (July 2012). "An Interview with Sun-Yung Alice Chang" (PDF). Asia Pacific Mathematics Newsletter. pp. 25–29.
• AWM Fellows List 2019
• Gursky, Matthew J.; Wang, Yi (March 2020). "Sun-Yung Alice Chang and Geometric Analysis" (PDF). Notices of the American Mathematical Society. 67 (3): 318–326. doi:10.1090/noti2037.
Ruth Lyttle Satter Prize in Mathematics recipients
• 1991 Dusa McDuff
• 1993 Lai-Sang Young
• 1995 Sun-Yung Alice Chang
• 1997 Ingrid Daubechies
• 1999 Bernadette Perrin-Riou
• 2001 Karen E. Smith & Sijue Wu
• 2003 Abigail Thompson
• 2005 Svetlana Jitomirskaya
• 2007 Claire Voisin
• 2009 Laure Saint-Raymond
• 2011 Amie Wilkinson
• 2013 Maryam Mirzakhani
• 2015 Hee Oh
• 2017 Laura DeMarco
• 2019 Maryna Viazovska
• 2021 Kaisa Matomäki
• 2023 Panagiota Daskalopoulos & Nataša Šešum
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• IdRef
| Wikipedia |
Shou-Wu Zhang
Shou-Wu Zhang (Chinese: 张寿武; pinyin: Zhāng Shòuwǔ; born October 9, 1962) is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.
Shou-Wu Zhang
Shou-Wu Zhang in 2014
Born (1962-10-09) October 9, 1962
Hexian, Anhui, China
NationalityAmerican
Alma mater
• Columbia University
• Chinese Academy of Sciences
• Sun Yat-sen University
Known for
• Arakelov theory
• Arithmetic dynamics
• Bogomolov conjecture
• Gross–Zagier theorem
Awards
List of Awards
• Sloan Fellowship (1997)
• Morningside Medal (1998)
• Clay Prize Fellow (2003)
• Guggenheim Fellow (2009)
• American Academy of Arts and Sciences Fellow (2011)
• American Mathematical Society Fellow (2016)
Scientific career
FieldsMathematics
Institutions
• Princeton University
• Columbia University
• Institute for Advanced Study
ThesisPositive Line Bundles on Arithmetic Surfaces (1991)
Doctoral advisorLucien Szpiro
Other academic advisorsWang Yuan
Doctoral students
• Wei Zhang
• Xinyi Yuan
• Tian Ye
• Yifeng Liu
Other notable students
• Bhargav Bhatt
Influences
• Gerd Faltings
• Dorian M. Goldfeld
• Benedict Gross
Biography
Early life
Shou-Wu Zhang was born in Hexian, Ma'anshan, Anhui, China on October 9, 1962.[1][2][3] Zhang grew up in a poor farming household and could not attend school until eighth grade due to the Cultural Revolution.[1] He spent most of his childhood raising ducks in the countryside and self-studying mathematics textbooks that he acquired from sent-down youth in trades for frogs.[1][2] By the time he entered junior high school at the age of fourteen, he had self-learned calculus and had become interested in number theory after reading about Chen Jingrun's proof of Chen's theorem which made substantial progress on Goldbach's conjecture.[1][2][4]
Education
Zhang was admitted to the Sun Yat-sen University chemistry department in 1980 after scoring poorly on his mathematics entrance examinations, but he later transferred to the mathematics department after feigning color blindness and received his bachelor's degree in mathematics in 1983.[5][1][2][3][4][6] He then studied under analytic number theorist Wang Yuan at the Chinese Academy of Sciences where he received his master's degree in 1986.[1][4][3][6] In 1986, Zhang was brought to the United States to pursue his doctoral studies at Columbia University by Dorian M. Goldfeld.[1][2] He then studied under Goldfeld, Hervé Jacquet, Lucien Szpiro, and Gerd Faltings, and then completed his PhD at Columbia University under Szpiro in 1991.[7][1][2][4][3][6]
Career
Zhang was a member of the Institute for Advanced Study and an assistant professor at Princeton University from 1991 to 1996.[3][6] In 1996, Zhang moved back to Columbia University where he was a tenured professor until 2013.[1][5][3][6] He has been a professor at Princeton University since 2011[5][6] and is an Eugene Higgins Professor since 2021. [8]
Zhang is on the editorial boards of: Acta Mathematica Sinica, Algebra & Number Theory, Forum of Mathematics, Journal of Differential Geometry, National Science Review, Pure and Applied Mathematics Quarterly, Science in China, and Research in Number Theory.[5] He has previously served on the editorial boards of: Journal of Number Theory, Journal of the American Mathematical Society, Journal of Algebraic Geometry, and International Journal of Number Theory.[5]
Research
Zhang's doctoral thesis Positive line bundles on Arithmetic Surfaces (Zhang 1992) proved a Nakai–Moishezon type theorem in intersection theory using a result from differential geometry already proved in Tian Gang's doctoral thesis.[5] In a series of subsequent papers (Zhang 1993, 1995a, 1995b, Szpiro, Ullmo & Zhang 1997), he further developed his theory of 'positive line bundles' in Arakelov theory which culminated in a proof (with Emmanuel Ullmo) of the Bogomolov conjecture (Zhang 1998).[5]
In a series of works in the 2000s (Zhang 2001b, 2004, Yuan, Zhang & W. Zhang 2009), Zhang proved a generalization of the Gross–Zagier theorem from elliptic curves over rationals to modular abelian varieties of GL(2) type over totally real fields.[5] In particular, the latter result led him to a proof of the rank one Birch-Swinnerton-Dyer conjecture for modular abelian varieties of GL(2) type over totally real fields through his work relating the Néron–Tate height of Heegner points to special values of L-functions in (Zhang 1997, 2001a).[5][9] Eventually, Yuan, Zhang, and W. Zhang (2013) established a full generalization of the Gross–Zagier theorem to all Shimura curves.
In arithmetic dynamics, Zhang (1995a, 2006) posed conjectures on the Zariski density of non-fibered endomorphisms of quasi-projective varieties and Ghioca, Tucker, and Zhang (2011) proposed a dynamical analogue of the Manin–Mumford conjecture.[10][5]
In 2018, Yuan and Zhang (2018) proved the averaged Colmez conjecture which was shown to imply the André–Oort conjecture for Siegel modular varieties by Jacob Tsimerman.[11]
Awards
Zhang has received a Sloan Foundation Research Fellowship (1997) and a Morningside Gold Medal of Mathematics (1998). He is also a Clay Foundation Prize Fellow (2003), Guggenheim Foundation Fellow (2009), Fellow of the American Academy of Arts and Sciences (2011), and Fellow of the American Mathematical Society (2016).[12][13][5] He was also an invited speaker at the International Congress of Mathematicians in 1998.[14][5][6][15]
Selected publications
Arakelov theory
• Zhang, Shou-Wu (1993), "Admissible pairing on a curve", Inventiones Mathematicae, 112 (1): 421–432, Bibcode:1993InMat.112..171Z, doi:10.1007/BF01232429, S2CID 120229374.
• Zhang, Shou-Wu (1995a), "Small points and adelic metrics", Journal of Algebraic Geometry, 8 (1): 281–300.
• Zhang, Shou-Wu (1995b), "Positive line bundles on arithmetic varieties", Journal of the American Mathematical Society, 136 (3): 187–221, doi:10.1090/S0894-0347-1995-1254133-7.
• Zhang, Shou-Wu (1996), "Heights and reductions of semi-stable varieties", Compositio Mathematica, 104 (1): 77–105.
• Zhang, Shou-Wu (2010), "Gross–Schoen cycles and Dualising sheaves", Invent. Math., 179 (1): 1–73, Bibcode:2010InMat.179....1Z, doi:10.1007/s00222-009-0209-3, S2CID 5698835.
• Yuan, Xinyi; Zhang, Shou-Wu (2017), "The arithmetic Hodge index theorem for adelic line bundles", Math. Ann., 367 (3–4): 1123–1171, doi:10.1007/s00208-016-1414-1, S2CID 2813125.
Bogomolov Conjecture
• Szpiro, Lucien; Ullmo, Emmanuel; Zhang, Shou-Wu (1997), "Equirépartition des petits points", Inventiones Mathematicae, 127 (2): 337–347, Bibcode:1997InMat.127..337S, doi:10.1007/s002220050123, S2CID 119668209.
• Zhang, Shou-Wu (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics, 147 (1): 159–165, doi:10.2307/120986, JSTOR 120986.
Gross--Zagier formulae
• Zhang, Shou-Wu (1997), "Heights of Heegner cycles and derivatives of L-series", Inventiones Mathematicae, 130 (1): 99–152, Bibcode:1997InMat.130...99Z, doi:10.1007/s002220050179, S2CID 10537873.
• Zhang, Shou-Wu (2001), "Heights of Heegner points on Shimura curves", Annals of Mathematics, 153 (1): 27–147, arXiv:math/0101269, doi:10.2307/2661372, JSTOR 2661372, S2CID 119624920.
• Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei (2009), "The Gross–Kohnen–Zagier Theorem over Totally Real Fields", Compositio Mathematica, 145 (5): 1147–1162, doi:10.1112/S0010437X08003734.
• Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei (2013), The Gross–Zagier formula on Shimura curves, Annals of Mathematics Studies, vol. 184.
• Liu, Yifeng; Zhang, Shou-Wu; Zhang, Wei (2018a), "A p-adic Waldspurger formula", Duke Mathematical Journal, 167 (4): 743–833, arXiv:1511.08172, doi:10.1215/00127094-2017-0045, S2CID 4867572.
• Yuan, Xinyi; Zhang, Shou-Wu (2018b), "On the averaged Colmez conjecture", Annals of Mathematics, 187 (2): 553–638, arXiv:1507.06903, doi:10.4007/annals.2018.187.2.4, S2CID 118916754.
Arithmetic dynamics
• Zhang, Shou-Wu (2006), "Distributions in algebraic dynamics", in Yau, Shing-Tung (ed.), Essays in geometry in memory of S.S. Chern, Surveys in Differential Geometry, vol. 10, Somerville, MA: International Press, pp. 381–430, doi:10.4310/SDG.2005.v10.n1.a9, MR 2408228.
• Ghioca, Dragos; Tucker, Thomas J.; Zhang, Shou-Wu (2011), "Towards a dynamical Manin-Mumford conjecture", International Mathematics Research Notices, 22: 5109–5122, doi:10.1093/imrn/rnq283, MR 2854724.
References
1. "从放鸭娃到数学大师" [From ducklings to mathematics master] (in Chinese). Academy of Mathematics and Systems Science. 11 November 2011. Retrieved 5 May 2019.
2. "專訪張壽武：在數學殿堂里，依然懷抱小學四年級的夢想" [Interview with Zhang Shou-Wu: In the mathematics department, he still has his dream from fourth grade of elementary school] (in Chinese). Beijing Sina Net. 3 May 2019. Retrieved 5 May 2019.
3. "旅美青年数学家张寿武" [Zhang Shouwu, a young mathematician in the United States] (in Chinese). He County Government. 2 November 2017. Retrieved 5 May 2019.
4. "专访数学家张寿武：要让别人解中国人出的数学题" [Interview with mathematician Zhang Shouwu: Let others solve the math problems of Chinese people] (in Chinese). Sina Education. 4 May 2019. Retrieved 5 May 2019.
5. Leong, Y. K. (July–December 2018). "Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry" (PDF). Imprints. No. 32. The Institute for Mathematical Sciences, National University of Singapore. pp. 32–36. Retrieved 5 May 2019.
6. "专访数学家张寿武：数学苍穹闪烁中国新星" [Interview with mathematician Zhang Shouwu: A new Chinese star flashing in the mathematical sky] (in Chinese). Zhishi Fenzi. 4 December 2017. Retrieved 5 May 2019.
7. Shou-Wu Zhang at the Mathematics Genealogy Project
8. Office of Communications. "Faculty members named to endowed professorships". Princeton University. Retrieved May 26, 2021.
9. Zhang, Wei (2013). "The Birch–Swinnerton-Dyer conjecture and Heegner points: a survey". Current Developments in Mathematics. 2013: 169–203. doi:10.4310/CDM.2013.v2013.n1.a3..
10. Benedetto, Robert; Ingram, Patrick; Jones, Rafe; Manes, Michelle; Silverman, Joseph H.; Tucker, Thomas J. (2019). "Current trends and open problems in arithmetic dynamics". Bulletin of the American Mathematical Society. 56 (4): 611–685. arXiv:1806.04980. doi:10.1090/bull/1665. S2CID 53550119.
11. "February 2018". Notices of the American Mathematical Society. 65 (2): 191. 2018. ISSN 1088-9477.
12. 2016 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2015-11-16
13. "Shou-Wu Zhang". John Simon Guggenheim Foundation. Retrieved 31 January 2019.
14. "ICM Plenary and Invited Speakers". Retrieved 31 January 2019.
15. Zhang, Shou-Wu (1998). "Small points and Arakelov theory". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 217–225.
External links
• Princeton home page
• Shou-Wu Zhang at the Mathematics Genealogy Project
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Wei Zhang (mathematician)
Wei Zhang (Chinese: 张伟; born 1981) is a Chinese mathematician specializing in number theory. He is currently a Professor of Mathematics at the Massachusetts Institute of Technology.[1]
Wei Zhang
Zhang at the Mathematical Research Institute of Oberwolfach in 2017
Born1981 (age 41–42)
Alma materColumbia University
Peking University
Awards
• SASTRA Ramanujan Prize (2010)
• Sloan Research Fellowship (2013)
• Morningside Gold Medal (2016)
• New Horizons In Mathematics Prize (2018)
• Clay Research Award (2019)
• Fellow of the American Mathematical Society (2019)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
Columbia University
ThesisModularity of Generating Functions of Special Cycles on Shimura Varieties (2009)
Doctoral advisorShou-Wu Zhang
Education
Zhang grew up in Sichuan province in China and attended Chengdu No.7 High School. [2] He earned his B.S. in Mathematics from Peking University in 2004 and his Ph.D. from Columbia University in 2009 under the supervision of Shou-Wu Zhang.[3][4]
Career
Zhang was a postdoctoral researcher and Benjamin Peirce Fellow at Harvard University from 2009 to 2011. He was a member of the mathematics faculty at Columbia University from 2011 to 2017, initially as an assistant professor before becoming a full professor in 2015. He has been a full professor at the Massachusetts Institute of Technology since 2017.[4][5]
Work
His collaborations with Zhiwei Yun, Xinyi Yuan and Xinwen Zhu have received attention in publications such as Quanta Magazine and Business Insider.[6][7] In particular, his work with Zhiwei Yun on the Taylor expansion of L-functions is "already being hailed as one of the most exciting breakthroughs in an important area of number theory in the last 30 years."[6]
Zhang has also made substantial contributions to the global Gan–Gross–Prasad conjecture.
Awards
He was a recipient of the SASTRA Ramanujan Prize in 2010, for "far-reaching contributions by himself and in collaboration with others to a broad range of areas in mathematics, including number theory, automorphic forms, L-functions, trace formulas, representation theory, and algebraic geometry.”[8] In 2013, Zhang received a Sloan Research Fellowship; in 2016 Zhang was awarded the Morningside Gold Medal of Mathematics.[4][9] In December 2017 he was awarded 2018 New Horizons In Mathematics Prize together with Zhiwei Yun, Aaron Naber and Maryna Viazovska. In 2019 he received the Clay Research Award.[10]
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to number theory, algebraic geometry and geometric representation theory".[11] He was elected to the American Academy of Arts and Sciences in 2023.[12]
Publications (selected)
• "Automorphic period and the central value of Rankin-Selberg L-function", J. Amer. Math. Soc. 27 (2014), 541–612.
• "On arithmetic fundamental lemmas", Invent. Math., 188 (2012), No. 1, 197–252.
• "Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups", Annals of Mathematics 180 (2014), No. 3, 971–1049.
• "Selmer groups and the indivisibility of Heegner points", Cambridge Journal of Mathematics 2 (2014), no. 2, 191–253.
• (with Michael Rapoport, Ulrich Terstiege) "On the Arithmetic Fundamental Lemma in the minuscule case", Compositio Mathematica 149 (2013), no. 10, 1631–1666.
• (with Xinyi Yuan, Shou-Wu Zhang) "The Gross–Kohnen–Zagier theorem over totally real fields", Compositio Mathematica 145 (2009), no. 5, 1147–1162.
• (with Xinyi Yuan, Shou-Wu Zhang) "The Gross–Zagier formula on Shimura curves", Annals of Mathematics Studies vol. 184, Princeton University Press, 2012.
• (with Manjul Bhargava, Christopher Skinner) "A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture", preprint.
• (with Zhiwei Yun) "Shtukas and the Taylor expansion of L-functions", Annals of Mathematics 186 (2017), No. 3, 767–911.
• (with Xinyi Yuan, Shou-Wu Zhang) "Triple product L-series and Gross–Kudla–Schoen cycles", preprint.
• (with Yifeng Liu, Shou-Wu Zhang) "On p-adic Waldspurger formula", preprint.
References
1. "Wei Zhang | MIT Mathematics".
2. "成都七中2000届校友张伟获2019年克雷研究奖，首位华人数学家". 15 September 2019.
3. "Wei Zhang", Mathematics Genealogy Project.
4. "Curriculum Vitae" (PDF). Wei Zhang. Retrieved September 4, 2020.
5. "Wei Zhang". Massachusetts Institute of Technology Department of Mathematics. Retrieved September 4, 2020.
6. "Math Quartet Joins Forces on Unified Theory", Quanta Magazine. Retrieved on 4 December 2016.
7. "Math Quartet Joins Forces on Unified Theory", Business Insider. Retrieved on 4 December 2016.
8. Notices of the AMS, January 2011, American Mathematical Society.
9. "Wei Zhang awarded the 2016 ICCM Morningside Gold Medal", Columbia University. Published 18 August 2016.
10. Clay Research Award 2019
11. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2018-11-07
12. New members, American Academy of Arts and Sciences, 2023, retrieved 2023-04-21
Recipients of SASTRA Ramanujan Prize
• Manjul Bhargava (2005)
• Kannan Soundararajan (2005)
• Terence Tao (2006)
• Ben Green (2007)
• Akshay Venkatesh (2008)
• Kathrin Bringmann (2009)
• Wei Zhang (2010)
• Roman Holowinsky (2011)
• Zhiwei Yun (2012)
• Peter Scholze (2013)
• James Maynard (2014)
• Jacob Tsimerman (2015)
• Kaisa Matomäki (2016)
• Maksym Radziwill (2016)
• Maryna Viazovska (2017)
• Yifeng Liu (2018)
• Jack Thorne (2018)
• Adam Harper (2019)
• Shai Evra (2020)
• Will Sawin (2021)
• Yunqing Tang (2022)
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Hongkai Zhao
Hongkai Zhao is a Chinese mathematician and Ruth F. DeVarney Distinguished Professor of Mathematics at Duke University. He was formerly the Chancellor's Professor in the Department of Mathematics at the University of California, Irvine. He is known for his work in scientific computing, imaging and numerical analysis, such as the fast sweeping method for Hamilton-Jacobi equation[1] and numerical methods for moving interface problems.[2]
Zhao had obtained his Bachelor of Science degree in the applied mathematics from the Peking University in 1990 and two years later got his Master's in the same field from the University of Southern California. From 1992 to 1996 he attended University of California, Los Angeles where he got his Ph.D. in mathematics. From 1996 to 1998 Zhao was a Gábor Szegő Assistant Professor at the Department of Mathematics of Stanford University and then got promoted to Research Associate which he kept till 1999. He has been at the University of California, Irvine since. At the same time he is also a member of the Institute for Mathematical Behavioral Sciences and the Department of Computer Science of UCI. From 2010 to 2013 and 2016 to 2019, Zhao was the chairman of the Department of Mathematics and since 2016 serves as Chancellor's Professor of mathematics.[3]
Hongkai Zhao received Alfred P. Sloan Fellowship in 2002 and the Feng Kang Prize[4] in Scientific Computing in 2007. He was elected as a Fellow of the Society for Industrial and Applied Mathematics, in the 2022 Class of SIAM Fellows, "for seminal contributions to scientific computation, numerical analysis, and applications in science and engineering".[5]
When it comes to free time he likes to watch and play sports games.[6]
References
1. Zhao, Hongkai (2005). "A fast sweeping method for Eikonal equations". Mathematics of Computation. 74 (250): 603–627. doi:10.1090/S0025-5718-04-01678-3. ISSN 0025-5718.
2. Zhao, Hong-Kai; Chan, T.; Merriman, B.; Osher, S. (1996-08-01). "A Variational Level Set Approach to Multiphase Motion". Journal of Computational Physics. 127 (1): 179–195. Bibcode:1996JCoPh.127..179Z. doi:10.1006/jcph.1996.0167. ISSN 0021-9991.
3. "Hongkai Zhao CV" (PDF). Retrieved 2019-01-29.
4. "Hongkai Zhao awarded the 2007 Feng Kang Prize". UCI. 2007-10-05. Retrieved 2019-01-29.
5. "SIAM Announces Class of 2022 Fellows". SIAM News. March 31, 2022. Retrieved 2022-03-31.
6. Jennifer Fitzenberger (2009-01-22). "Mathematics in the real world". UCI. Retrieved 2019-01-29.
External links
• Hongkai Zhao publications indexed by Google Scholar
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Zhao Youqin's π algorithm
Zhao Youqin's π algorithm was an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (赵友钦, ? – 1330) to calculate the value of π in his book Ge Xiang Xin Shu (革象新书).
Algorithm
Zhao Youqin started with an inscribed square in a circle with radius r.[1]
If $\ell $ denotes the length of a side of the square, draw a perpendicular line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram:
$d={\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}$
$e=r-d=r-{\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}.$
Extend the perpendicular line d to dissect the circle into an octagon; $\ell _{2}$ denotes the length of one side of octagon.
$\ell _{2}={\sqrt {\left({\frac {\ell }{2}}\right)^{2}+e^{2}}}$
$\ell _{2}={\frac {1}{2}}{\sqrt {\ell ^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell ^{2}}}\right)^{2}}}$
Let $l_{3}$ denotes the length of a side of hexadecagon
$\ell _{3}={\frac {1}{2}}{\sqrt {\ell _{2}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{2}^{2}}}\right)^{2}}}$
similarly
$\ell _{n+1}={\frac {1}{2}}{\sqrt {\ell _{n}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{n}^{2}}}\right)^{2}}}$
Proceeding in this way, he at last calculated the side of a 16384-gon, multiplying it by 16384 to obtain 3141.592 for a circle with diameter = 1000 units, or
$\pi =3.141592.\,$
He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of π, that is 3, 3.14, 22/7 and 355/113, the last is the most exact.[2]
See also
• Liu Hui's π algorithm
References
1. Yoshio Mikami, Development of Mathematics in China and Japan, Chapter 20, The Studies about the Value of π etc., pp 135–138
2. Yoshio Mikami, p136
| Wikipedia |
Zhen Luan
Zhen Luan (甄鸾) (535 – 566) was a Chinese mathematician, astronomer and daoist who was active during the Northern Zhou (557-581) of the Southern and Northern Dynasties period.
Born in the Wuji County of the present day Hubei Province, he is primarily known for the comments on the ancient mathematical treatises. Proceeding from them, he paid special attention to the "Nine Palaces" calculation technique; his description of the Luo shu represents an early example of textual comment on this scheme.
Zhen Luan developed the Tianhe calendar which was implemented in 566 and was current for the next 18 years.
Zhen trained in a Daoist congregation,[1] but converted to Buddhism out of disgust with Daoist sexual practices.[2][3] He wrote the anti-Daoist text Xiaodao Lun in 570 for Emperor Wu of Northern Zhou.[4][5]
His solid scholarship was commended by Yan Yuan (1635 - 1704).
Literature
• Schuyler Cammann, "The Magic Square of Three in Old Chinese Philosophy and Religion" in History of Religions, Vol. 1, No. 1 (Summer, 1961), pp. 37–80.
References
1. Wile, Douglas (1992). "Sexual Practices and Taoism". Art of the Bedchamber: The Chinese Sexual Yoga Classics Including Women's Solo Meditation Texts. SUNY Press. pp. 25–26.
2. Gulik, Robert H Van (1974). Sexual Life in Ancient China. Brill Archive. p. 89.
3. Farzeen Baldrian-Hussein, Farzeen (October 1996). "Laughing at the Tao: Debates among Buddhists and Taoists in Medieval China by Livia Kohn (review)". Asian Folklore Studies. 55 (2): 361–363. doi:10.2307/1178836. JSTOR 1178836.
4. Bokenkamp, Stephen R (1990). "Stages of Transcendence". In Buswell, Robert E (ed.). Chinese Buddhist Apocrypha. University of Hawaii Press.
5. Komjathy, Louis (2012). "The Daoist Tradition in China". In Nadeau, Randall L (ed.). The Wiley-Blackwell Companion to Chinese Religions. John Wiley & Sons. pp. 179–180.
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Zhi-Ming Ma
Zhi-Ming Ma. (Chinese: 馬志明; pinyin: Zhi-Ming Ma) is a Chinese mathematics professor of Chinese Academy of Sciences.[1] Ma is a former Vice Chairman of the Executive Committee for International Mathematical Union.[2], a two times president of Chinese Mathematical Society, an elected member of World Academy of Sciences and the Chairman of Graduate Degree Committee of Academy of Math and Systems Science, Chinese Academy of Sciences.[3][1]
Zhi-Ming Ma
馬志明
Born1948
Chengdu, Sichuan
Citizenship China
Alma materChongqing Normal University,
China Science and Technology University
Beijing Municipality,
Chinese Academy of Sciences
Known forHe discovered a new framework of quasi-regular Dirichlet forms which correspond to right processes in one-to-one manner.
Scientific career
FieldsMathematics
InstitutionsChinese Academy of Sciences
International Mathematical Union]]
Biography
Ma was born in January, 1948 at Chengdu, Sichuan while Jiaocheng County in Shanxi Province was his native origin. He obtained his first degree in Mathematics from Chongqing Normal University in 1978.[1] In 1981, he received his Master's degree from China Science and Technology University, Graduate School of Science Beijing Municipality. In 1984 he received his doctorate degree in Applied Mathematics from Chinese Academy of Sciences.[4]
Contributions
Ma's contribution in the theory of Dirichlet forms and Markov processes brought an end to a twenty years puzzle in the field. Ma and his team discovered a new framework of quasi-regular Dirichlet forms which correspond to right processes in one-to-one manner. His book, written in collaboration with Michael Rockner, An Introduction to the Theory of (Non-symmetric) Dirichlet Forms, has become a notable text in this field. His work on the proof of the Feynman-Kac probabilistic representation of mixed boundary problems of Schrodinger operators with measure-valued potentials is an important contribution to the theory of the Schrodinger equation.[1]
Selected publications
• Zhi-Ming Ma, M. Röckner Introduction to the theory of (non-symmetric) Dirichlet forms.1 November, 1992[5]
• Guan, QY., Ma, ZM. Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian. Probab. Theory Relat. Fields 134, 649–694 (2006). https://doi.org/10.1007/s00440-005-0438-3[6]
• Albeverio, Sergio, Ma, Zhiming Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms. https://doi.org/10.18910/8386[7]
Awards
Ma's contributions to science have led to his recognition with various awards including; the First Class Prize for Natural Sciences by the Chinese Academy of Sciences, the Max-Planck Research Award by the Max-Planck Society and Alexander von Humboldt Foundation, the Chinese National Natural Sciences Prize, the Shiing Shen. Chern Mathematics Prize, the Qiu-Shi Outstanding Young Scholars Prize, the He-Liang-He-Li Sciences and Technology Progress Prize，and the Hua Loo-Keng Mathematics Prize.[1]
Memberships
Ma is a member of different recognised scientific organisations. He was elected as an Academician of the Chinese Academy of Sciences in 1995 and he became a fellow of the Third World Academy of Sciences in 1999.[8] He was the Chairman of the organizing committee for the International Congress of Mathematicians that was held in Beijing (2002). He was elected as a member of the Executive Committee for International Mathematical Union in 2003 and he became the Vice president in 2007. He was elected as a member of Chinese Mathematical Society and he became the president in 2003.[1][8]
References
1. "Ma Zhiming". en.nankai.edu.cn. Retrieved 2022-05-28.
2. "IMU Executive Committee | International Mathematical Union (IMU)". www.mathunion.org. Retrieved 2022-05-28.
3. "Ma, Zhi-Ming". TWAS. Retrieved 2022-05-28.
4. "China Vitae : Biography of Ma Zhiming". www.chinavitae.com. Retrieved 2022-05-28.
5. Ma, Zhi-Ming; Röckner, M. (1992). Introduction to the theory of (non-symmetric) Dirichlet forms. Universitext. doi:10.1007/978-3-642-77739-4. ISBN 978-3-540-55848-4. S2CID 122441749.
6. Guan, Qing-Yang; Ma, Zhi-Ming (2006-04-01). "Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian". Probability Theory and Related Fields. 134 (4): 649–694. doi:10.1007/s00440-005-0438-3. ISSN 1432-2064. S2CID 18733207.
7. Albeverio, Sergio; Ma, Zhiming (1992). "Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms". Osaka Journal of Mathematics. 29 (2): 247–265. doi:10.18910/8386. hdl:11094/8386.
8. "Zhi-Ming Ma curriculum vitae" (PDF).
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Zhihong Xia
Zhihong "Jeff" Xia (Chinese: 夏志宏; pinyin: Xià Zhìhóng; born 20 September 1962, in Dongtai, Jiangsu, China) is a Chinese-American mathematician.
Education and career
Xia received, in 1982, from Nanjing University a bachelor's degree in astronomy and in 1988, a PhD in mathematics from Northwestern University with thesis advisor Donald G. Saari, for his thesis, The Existence of the Non-Collision Singularities.[1] From 1988 to 1990, Xia was an assistant professor at Harvard University and from 1990 to 1994, an associate professor at Georgia Institute of Technology (and Institute Fellow). In 1994, he became a full professor at Northwestern University and since 2000, he has been the Arthur and Gladys Pancoe Professor of Mathematics.[2]
His research deals with celestial mechanics, dynamical systems, Hamiltonian dynamics, and ergodic theory. In his dissertation, he solved the Painlevé conjecture, a long-standing problem posed in 1895 by Paul Painlevé. The problem concerns the existence of singularities of non-collision character in the $N$-body problem in three-dimensional space; Xia proved the existence for $N\geq 5$. For the existence proof, he constructed an example of five masses, of which four are separated into two pairs which revolve around each other in eccentric elliptical orbits about the z-axis of symmetry, and a fifth mass moves along the z-axis. For selected initial conditions, the fifth mass can be accelerated to an infinite velocity in a finite time interval (without any collision between the bodies involved in the example).[3] The case $N=4$ was open until 2014,[4] when it was solved by Jinxin Xue.[5][6] For $N=3$, Painlevé had proven that the singularities (points of the orbit in which accelerations become infinite in a finite time interval) must be of the collision type. However, Painlevé's proof did not extend to the case $N>3$.
In 1993, Xia was the inaugural winner of the Blumenthal Award of the American Mathematical Society. From 1989 to 1991, he was a Sloan Fellow. From 1993 to 1998, he received the National Young Investigator Award from the National Science Foundation. In 1995, he received the Monroe H. Martin Prize in Applied Mathematics from the University of Maryland.[7] In 1998, he was an Invited Speaker of the International Congress of Mathematicians in Berlin.[8]
Selected publications
• Xia, Zhihong (1992). "The Existence of Noncollision Singularities in Newtonian Systems". Annals of Mathematics. Series 2. 135 (3): 411–468. doi:10.2307/2946572. JSTOR 2946572.
• Xia, Zhihong (1992). "Existence of invariant tori in volume-preserving diffeomorphisms". Ergodic Theory and Dynamical Systems. 12 (3): 621–631. doi:10.1017/S0143385700006969. S2CID 122761956.
• Xia, Zhihong (1992). "Melnikov method and transversal homoclinic points in the restricted three-body problem" (PDF). Journal of Differential Equations. 96 (1): 170–184. Bibcode:1992JDE....96..170X. doi:10.1016/0022-0396(92)90149-H.
• Saari, Donald G.; Xia, Zhihong (1993). "Off to Infinity in Finite Time" (PDF). Notices of the AMS. 42: 538–546.
• Xia, Z (1994). "Arnold diffusion and oscillatory solutions in the planar three-body problem". Journal of Differential Equations. 110 (2): 289–321. Bibcode:1994JDE...110..289X. doi:10.1006/jdeq.1994.1069.
• Saari, Donald G; Xia, Zhihong (1996). "Singularities in the Newtonian 𝑛-body problem". Hamiltonian Dynamics and Celestial Mechanics. Contemporary Mathematics. Vol. 198. American Mathematical Society. pp. 21–30. CiteSeerX 10.1.1.24.1325. doi:10.1090/conm/198/02493. ISBN 978-0-8218-0566-4.
• Xia, Zhihong (1996). "Homoclinic points in symplectic and volume-preserving diffeomorphisms". Communications in Mathematical Physics. 177 (2): 435–449. Bibcode:1996CMaPh.177..435X. doi:10.1007/BF02101901. S2CID 17732615.
• Zhu, Deming; Xia, Zhihong (1998). "Bifurcations of heteroclinic loops". Science in China Series A: Mathematics. 41 (8): 837–848. Bibcode:1998ScChA..41..837Z. doi:10.1007/BF02871667. S2CID 120519869.
• Xia, Zhihong (2004). "Convex central configurations for the n-body problem". Journal of Differential Equations. 200 (2): 185–190. Bibcode:2004JDE...200..185X. doi:10.1016/j.jde.2003.10.001.
• Xia, Zhihong (2006). "Area-preserving surface diffeomorphisms". Communications in Mathematical Physics. 263 (3): 723–735. arXiv:math/0503223. Bibcode:2006CMaPh.263..723X. CiteSeerX 10.1.1.235.4920. doi:10.1007/s00220-005-1514-3. S2CID 14540760.
• Saghin, Radu; Xia, Zhihong (2009). "Geometric expansion, Lyapunov exponents and foliations" (PDF). Annales de l'Institut Henri Poincaré C. 26 (2): 689–704. Bibcode:2009AIHPC..26..689S. doi:10.1016/j.anihpc.2008.07.001. S2CID 119147899.
• Xia, Zhihong; Zhang, Pengfei (2014). "Homoclinic points for convex billiards". Nonlinearity. 27 (6): 1181–1192. arXiv:1310.5279. Bibcode:2014Nonli..27.1181X. doi:10.1088/0951-7715/27/6/1181. S2CID 119627854.
• Xia, Zhihong; Zhang, Pengfei (2017). "Homoclinic intersections for geodesic flows on convex spheres". Dynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov. Contemporary Mathematics. Vol. 698. American Mathematical Society. pp. 221–238.
References
1. Zhi-Hong Xia at the Mathematics Genealogy Project
2. "Zhihong Jeff Xia". Northwestern University.
3. In 1908 Edvard Hugo von Zeipel proved the surprising fact that the existence of a non-collision singularity in the $N$-body problem necessarily causes the velocity of at least one particle to become unbounded.
4. Already in 2003, Joseph L. Gerver gave arguments (a heuristic model) for the existence of a non-collision singularity for the planar Newtonian 4-body problem — however, at the time there was still no rigorous proof. See Gerver, Joseph L. (2003). "Noncollision Singularities: Do Four Bodies Suffice?". Exp. Math. 12 (2): 187–198. doi:10.1080/10586458.2003.10504491. S2CID 23816314.
5. Xue, Jinxin (2014). "Noncollision Singularities in a Planar Four-body Problem". arXiv:1409.0048 [math.DS].
6. Xue, Jinxin (2020). "Non-collision singularities in a planar 4-body problem". Acta Mathematica. 224 (2): 253–388. doi:10.4310/ACTA.2020.v224.n2.a2.
7. "Monroe H. Martin Prize". Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park.
8. Xia, Zhihong (1998). "Arnold diffusion: a variational construction". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 867–877.
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Zhiwei Yun
Zhiwei Yun (Chinese: 恽之玮; pinyin: Yùn Zhīwěi; born September 1982) is a Professor of Mathematics at MIT specializing in number theory, algebraic geometry and representation theory, with a particular focus on the Langlands program.
Zhiwei Yun
Born
Yun Zhiwei (恽之玮)
September 1982 (age 40)
Changzhou, China[1]
Alma materPeking University
Princeton University
Known forcontributions to number theory, representation theory and algebraic geometry
AwardsGold Medal, IMO (2000)[2]
SASTRA Ramanujan Prize (2012)[3]
2018 New Horizons In Mathematics Prize (2018)
Morningside Medal (2019)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
Stanford University
Yale University
Doctoral advisorRobert MacPherson
He was previously a C. L. E. Moore instructor at Massachusetts Institute of Technology from 2010 to 2012, assistant professor then associate professor at Stanford University from 2012 to 2016, and professor at Yale University from 2016 to 2017.
Education
Yun was born in Changzhou, China.[1] As a high schooler, he participated in the International Mathematical Olympiad in 2000; he received a gold medal with a perfect score.[2] Yun received his bachelor's degree from Peking University in 2004. In 2009, he received his Ph.D. from Princeton University, under the direction of Robert MacPherson.[4][5]
Work
His collaborations with Wei Zhang, Xinyi Yuan and Xinwen Zhu have received attention in publications such as Quanta Magazine and Business Insider.[6][7] In particular, his work with Wei Zhang on the Taylor expansion of L-functions is "already being hailed as one of the most exciting breakthroughs in an important area of number theory in the last 30 years."
Yun also made substantial contributions towards the global Gan–Gross–Prasad conjecture.
Awards
Yun was awarded the SASTRA Ramanujan Prize in 2012 for his "fundamental contributions to several areas that lie at the interface of representation theory, algebraic geometry and number theory."[3]
In December 2017, he was awarded 2018 New Horizons In Mathematics Prize together with Wei Zhang, Aaron Naber and Maryna Viazovska.
He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to geometry, number theory, and representation theory, including his construction of motives with exceptional Galois groups".[8] In 2019 he received the Morningside Medal jointly with Xinwen Zhu.[9]
Selected publications
• Yun, Zhiwei; Vincent, Christelle (2015). "Galois representations attached to moments of Kloosterman sums and conjectures of Evans". Compositio Mathematica. 151 (1): 68–120. arXiv:1308.3920. doi:10.1112/S0010437X14007593.
• (with Davesh Maulik) Maulik, Davesh; Yun, Zhiwei (2014). "Macdonald formula for curves with planar singularities". Journal für die reine und angewandte Mathematik. 2014 (694): 27–48. arXiv:1107.2175. doi:10.1515/crelle-2012-0093. S2CID 5317997.
• Yun, Zhiwei (2014). "Motives with exceptional Galois groups and the inverse Galois problem". Inventiones Mathematicae. 196 (2): 267–337. arXiv:1112.2434. Bibcode:2014InMat.196..267Y. doi:10.1007/s00222-013-0469-9.
• (with Roman Bezrukavnikov) Bezrukavnikov, Roman; Yun, Zhiwei (2013). "On Koszul duality for Kac–Moody groups". Representation Theory. 17: 1–98. arXiv:1101.1253. doi:10.1090/S1088-4165-2013-00421-1.
• (with Ngô Bảo Châu and Jochen Heinloth) Heinloth, Jochen; Ngô, Bao-Châu; Yun, Zhiwei (2013). "Kloosterman sheaves for reductive groups". Annals of Mathematics. 177 (1): 241–310. doi:10.4007/annals.2013.177.1.5.
• Yun, Zhiwei (2012). "Langlands duality and global Springer theory". Compositio Mathematica. 148 (3): 835–867. doi:10.1112/S0010437X11007433.
• Yun, Zhiwei (2011). "Global Springer theory". Advances in Mathematics. 228 (1): 266–328. doi:10.1016/j.aim.2011.05.012.
• Gordon, Julia; Yun, Zhiwei (2011). "The fundamental lemma of Jacquet and Rallis". Duke Mathematical Journal. 156 (2): 167–227. arXiv:0901.0900. doi:10.1215/00127094-2010-210. S2CID 14295843.
• Yun, Zhiwei (2009). "Weights of mixed tilting sheaves and geometric Ringel duality". Selecta Mathematica. New Series. 14 (2): 299–320. arXiv:0805.1495. doi:10.1007/s00029-008-0066-8.
• (with Alexei Oblomkov) Oblomkov, Alexei; Yun, Zhiwei (2016). "Geometric representations of graded and rational Cherednik algebras". Advances in Mathematics. 292: 601–706. arXiv:1407.5685. doi:10.1016/j.aim.2016.01.015.
• (with Wei Zhang) Yun, Zhiwei; Zhang, Wei (2017). "Shtukas and the Taylor expansion of L-functions". Annals of Mathematics. 186 (3): 767–911. arXiv:1512.02683. doi:10.4007/annals.2017.186.3.2.
References
1. "北京大学校友恽之玮获2012年"拉马努金"奖". 30 August 2012.
2. "International Mathematical Olympiad".
3. "ZHIWEI YUN TO RECEIVE 2012 SASTRA RAMANUJAN PRIZE". Sastra University. Retrieved 2 September 2016.
4. "Zhiwei Yun" (PDF). Stanford University. Retrieved 2 September 2016.
5. "Zhiwei Yun", Mathematics Genealogy Project. Retrieved on 4 December 2016.
6. "Math Quartet Joins Forces on Unified Theory", Quanta Magazine. Retrieved on 4 December 2016.
7. "Math Quartet Joins Forces on Unified Theory", Business Insider. Retrieved on 4 December 2016.
8. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2018-11-07
9. Morningside Medal 2019
Recipients of SASTRA Ramanujan Prize
• Manjul Bhargava (2005)
• Kannan Soundararajan (2005)
• Terence Tao (2006)
• Ben Green (2007)
• Akshay Venkatesh (2008)
• Kathrin Bringmann (2009)
• Wei Zhang (2010)
• Roman Holowinsky (2011)
• Zhiwei Yun (2012)
• Peter Scholze (2013)
• James Maynard (2014)
• Jacob Tsimerman (2015)
• Kaisa Matomäki (2016)
• Maksym Radziwill (2016)
• Maryna Viazovska (2017)
• Yifeng Liu (2018)
• Jack Thorne (2018)
• Adam Harper (2019)
• Shai Evra (2020)
• Will Sawin (2021)
• Yunqing Tang (2022)
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Zhoubi Suanjing
The Zhoubi Suanjing, also known by many other names, is an ancient Chinese astronomical and mathematical work. The Zhoubi is most famous for its presentation of Chinese cosmology and a form of the Pythagorean theorem. It claims to present 246 problems worked out by the early Zhou culture hero Ji Dan and members of his court, placing its contents in the 11th century BC. However, the present form of the book does not seem to be earlier than the 2nd century Eastern Han, with some additions and commentaries continuing to be added for several more centuries.
The Gougu Theorem diagram added to the Zhoubi by Zhao Shuang
Zhoubi Suanjing
Traditional Chinese《周髀算經》
Simplified Chinese《周髀算经》
Transcriptions
Standard Mandarin
Hanyu PinyinZhōubì suànjīng
Wade–GilesChou-pi Suan-ching
Zhoubi
Chinese《周髀》
Literal meaningThe Zhou Gnomon
On Gnomons and Circular Paths
Transcriptions
Standard Mandarin
Hanyu PinyinZhōubì
Wade–GilesChou-pi
Zhoubi
Traditional Chinese《算經》
Simplified Chinese《算经》
Literal meaningThe Classic of Computation
The Arithmetic Classic
Transcriptions
Standard Mandarin
Hanyu PinyinSuànjīng
Wade–GilesSuan-ching
Names
Zhoubi Suanjing is the atonal pinyin romanization of the modern standard Mandarin pronunciation of the work's Classical Chinese name, 《周髀算經》. The same name has been variously romanized as the Chou Pei Suan Ching,[1] the Tcheou-pi Souane,[2] &c. Its original title was simply the Zhoubi. The character 髀 is a literary term for the femur or thighbone but in context only refers to one or more gnomons, large sticks whose shadows were used for Chinese calendrical and astronomical calculations.[3] Because of the ambiguous nature of the character 周, it has been alternately understood and translated as "On the Gnomon and the Circular Paths of Heaven",[3] the "Zhou Shadow Gauge Manual",[4] "The Gnomon of the Zhou Sundial",[5] and "Gnomon of the Zhou Dynasty".[6] The honorific Suanjing—"Arithmetical Classic",[1] "Sacred Book of Arithmetic",[7] "Mathematical Canon",[6] "Classic of Computations",[8] &c.—was added later.
Dating
Examples of the gnomon described in the work have been found from as early as 2300 BC and Ji Dan, better known as the Duke of Zhou, was an 11th century BC regent and noble during the first generation of the Zhou dynasty. The Zhoubi was traditionally dated to Ji Dan's own life[9] and considered to be the oldest Chinese mathematical treatise.[3] However, although some passages seem to come from the Warring States Period or earlier,[9] the current text of the work mentions Lü Bowei and is believed to have received its current form no earlier than the Eastern Han, during the 1st or 2nd century. It does not appear at all in the Book of Han's account of calendrical, astronomical, and mathematical works, although Joseph Needham allows that this may have been from its current contents having previously been provided in several different works listed in the Han history which are otherwise unknown.[3]
Contents
The Zhoubi is an anonymous collection of 246 problems encountered by the Duke of Zhou and figures in his court, including the astrologer Shang Gao. Each problem includes an answer and a corresponding arithmetic algorithm.
It is an important source on early Chinese cosmology, glossing the ancient idea of a round heaven over a square earth (天圆地方, tiānyuán dìfāng) as similar to the round parasol suspended over some ancient Chinese chariots[10] or a Chinese chessboard.[11] All things measurable were considered variants of the square, while the expansion of a polygon to infinite sides approaches the immeasurable circle.[4] This concept of a "canopy heaven" (蓋天, gàitiān) had earlier produced the jade bì (璧) and cóng (琮) objects and myths about Gonggong, Mount Buzhou, Nüwa, and repairing the sky. Although this eventually developed into an idea of a "spherical heaven" (渾天, hùntiān),[12] the Zhoubi offers numerous explorations of the geometric relationships of simple circles circumscribed by squares and squares circumscribed by circles.[13] A large part of this involves analysis of solar declination in the Northern Hemisphere at various points throughout the year.[3]
At one point during its discussion of the shadows cast by gnomons, the work presents a form of the Pythagorean theorem known as the gougu theorem (勾股定理, gōugǔ dìnglǐ)[14] from the Chinese names—lit. "hook" and "thigh"—of the two sides of the carpenter or try square.[15] In the 3rd century, Zhao Shuang's commentary on the Zhoubi included a diagram effectively proving the theorem[16] for the case of a 3-4-5 triangle,[17] whence it can be generalized to all right triangles. The original text being ambiguous on its own, there is disagreement as to whether this proof was established by Zhao or merely represented an illustration of a previously understood concept earlier than Pythagoras.[18][14] Shang Gao concludes the gougu problem saying "He who understands the earth is a wise man, and he who understands the heavens is a sage. Knowledge is derived from the shadow [straight line], and the shadow is derived from the gnomon [right angle]. The combination of the gnomon with numbers is what guides and rules the ten thousand things."[19]
Commentaries
The Zhoubi has had a prominent place in Chinese mathematics and was the subject of specific commentaries by Zhao Shuang in the 3rd century, Liu Hui in 263, by Zu Gengzhi in the early 6th century, Li Chunfeng in the 7th century, and Yang Hui in 1270.
See also
• Tsinghua Bamboo Slips
• Dunhuang Star Chart
References
Citations
1. Needham & al. (1959), p. 815.
2. EB, 1st ed. (1771), p. 188.
3. Needham & al. (1959), p. 19.
4. Zou (2011), p. 104.
5. Pang-White (2018), p. 464.
6. Cullen (2018), p. 758.
7. Davis & al. (1995), p. 28.
8. Elman (2015), p. 240.
9. Needham & al. (1959), p. 20.
10. Tseng (2011), pp. 45–49.
11. Ding (2020), p. 172.
12. Tseng (2011), p. 50.
13. Tseng (2011), p. 51.
14. Cullen (1996), p. 82.
15. Gamwell (2016), p. 39.
16. Cullen (1996), p. 208.
17. Chemla (2005), p. .
18. Chemla (2005).
19. Gamwell (2016), p. 41.
Works cited
• "Chinese", Encyclopaedia Britannica, vol. II (1st ed.), Edinburgh: Colin Macfarquhar, 1771, pp. 184–192.
• Chemla, Karine (2005), Geometrical Figures and Generality in Ancient China and Beyond, Science in Context, ISBN 0-521-55089-0.
• Cullen, Christopher (1996), Astronomy and Mathematics in Ancient China, Cambridge University Press, ISBN 0-521-55089-0.
• Cullen, Christopher (2018), "Chinese Astronomy in the Early Imperial Age", The Cambridge History of Science, Vol. I: Ancient Science, Cambridge University Press, ISBN 978-110868262-6.
• Davis, Philip J.; et al., eds. (1995), "Brief Chronological Table to 1910", The Mathematical Experience, Modern Birkhäuser Classics, Boston: Birkhäuser, pp. 26–29, ISBN 978-081768294-1.
• Ding, D.X. Daniel (2020), The Historical Roots of Technical Communication in the Chinese Tradition, Newcastle-upon-Tyne: Cambridge Scholars, ISBN 978-152755989-9.
• Elman, Benjamin (2015), "Early Modern or Late Imperial? The Crisis of Classical Philology in Eighteenth-Century China", World Philology, Cambridge: Harvard University Press, pp. 225–244.
• Gamwell, Lynn (2016), Mathematics + Art: A Cultural History, Princeton University Press, ISBN 978-069116528-8.
• Needham, Joseph; et al. (1959), Science & Civilisation in China, Vol. III: Mathematics and the Sciences of the Heavens and the Earth, Cambridge University Press, ISBN 978-052105801-8.
• Pang-White, A. Ann (2018), The Confucian Four Books for Women, Oxford University Press, ISBN 978-0-19-046091-4.
• Tseng, L.Y. Lillian (2011), Picturing Heaven in Early China, East Asian Monographs, Cambridge: Harvard University Asia Center, ISBN 978-0-674-06069-2.
• Zou Hui (2011), A Jesuit Garden in Beijing and Early Modern Chinese Culture, West Lafayette: Purdue University Press, ISBN 978-155753583-2.
Further reading
• 《周髀算經》 (in Chinese), Chinese Text Project.
• 《周髀算經》 (in Chinese), Project Gutenberg.
• Boyer, Carl B. (1991), A History of Mathematics, John Wiley & Sons, ISBN 0-471-54397-7.
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M. D. Chow
M. D. Chow (1878 – February 13, 1949 Shanghai), also known by the Chinese names Zhou Jinjue (Chinese: 周今覺) and Zhou Mingda (Chinese: 周明达), was a Chinese philatelist and mathematician. He was nicknamed the "king of Chinese philately".[1]
Names
Having multiple names was the custom. He was also known as Zhou Meiquan (Chinese: 周美权 or 周梅泉), Zhou Jinjue (Chinese: 周今觉; formerly romanised Chow Chin Tso).
Early life
He was born into a salt merchant family in Yangzhou and moved to Shanghai in 1912. He was home schooled.
Philately
He was most noted as the founding father of Chinese philately and was crowned the King of Chinese Stamps after his acquisition of the rarest stamp, the block of four Red Revenue stamps from the original owner R. A. de Villard in 1927.[2][3][4] He championed the study of the Red Revenues.[5]
To entertain his sick son Wei-Liang Chow in 1923, he brought home many colorful foreign stamps during his recuperation. Soon they both caught the bug and began learning and collecting stamps.[6] He found Chinese Philatelic Society on November 15, 1925.[7] His bi-lingual Philatelic Bulletin won a Special Bronze Medal at the International Philatelic Exhibition in New York in 1926.[8] Chinese stamps eventually became a gold medal contender in 1927 at the Strasbourg International Exhibition in France.[3][4] He's the first Chinese to be granted a fellow of F.R.P.S.L., the Royal Philatelic Society London.[3][4]
Math clubs
In Yangzhou in 1900, he created Zhixin Math Club (Chinese: 知新算社) with Bao Mofen (Chinese: 包墨芬) and Yu Yudong (Chinese: 余雨东). As one of the finest mathematicians in China, he was highly praised by Japanese scholars.[9] In the 1920s, he created Science Society of China (Chinese: 中国科学社) with Ren Hongjuan (Chinese: 任鸿隽) and Hu Mingxia (Chinese: 胡明夏), and was named co-honorary president with Zhang Jian.
Awards and honors
His bi-lingual Philatelic Bulletin won a Special Bronze Medal at the International Philatelic Exhibition in New York in 1926.
References
1. Chen Tse-chuan (1 January 1962). "The Story of Chinese Stamps". Taiwan Today.
2. Ma, Runsheng (1947.) Shanghai: Ma’s Illustrated Catalogue of the Stamps of China
3. Woo, L.Y. (Chinese: 吳樂園) (1983). Taipei: Red Revenue Surcharges Stamp Collection (Chinese: 紅印花加蓋郵票專集)
4. Ministry of Transportation Post Office (Chinese: 交通部郵政總局) (1984). Taipei: Red Revenue Surcharge, Part I of 2 (Chinese: 紅印花郵票上編)
5. Matthew Bennett, Inc. (2001) The "Sun" Collection of the 1897 Red Revenue Surcharge of China. New York: Matthew Bennett, Inc.
6. Philatelic Bulletin, 1923, 3:3, page 27
7. Illustriertes Briefmarken Journal, No. 2, Jan 16, 1926
8. New York Times, World Stamp Show to Give 582 Awards 1926.04.18 retrieved 2015.09.10
9. Sotheby’s catalog titled Postage Stamps of the Far East, April 29, 1996
External links
• (in Chinese) Zhou Jinjue on CNKI
| Wikipedia |
Xinwen Zhu
Xinwen Zhu (Chinese: 朱歆文; born 1982 in Sichuan) is a Chinese mathematician and professor at Stanford University. His work deals primarily with geometric representation theory and in particular the Langlands program, tying number theory to algebraic geometry and quantum physics.[1][2]
Biography
Zhu obtained his A.B. in mathematics from Peking University in 2004 and his Ph.D. in mathematics from the University of California, Berkeley in 2009 under the direction of Edward Frenkel.[1] He taught at Harvard University as a Benjamin Peirce Lecturer and at Northwestern University as an assistant professor before joining the Caltech faculty in 2014. According to the American Mathematical Society, "[Zhu] studies the geometry and topology of flag varieties of loop groups and applies techniques from the geometric Langlands program to arithmetic geometry."[3]
The awards Zhu has received include an AMS Centennial Fellowship in 2013 and a Sloan Fellowship in 2015.[4] His research has been published in Annals of Mathematics and Inventiones mathematicae, among other mathematics journals. Zhu, Wei Zhang, Xinyi Yuan and Zhiwei Yun are frequent collaborators.[5] In 2019 he received the Morningside Medal jointly with Zhiwei Yun.[6] Zhu won the 2020 New Horizons in Mathematics Breakthrough Prize "For work in arithmetic algebraic geometry including applications to the theory of Shimura varieties and the Riemann-Hilbert problem for p-adic varieties."
Publications (selected)
• (with Edward Frenkel) "Gerbal Representations of Double Loop Groups", International Mathematics Research Notices 2012 (2012), No. 17, 3929–4013.
• (with George Pappas]) "Local models of Shimura varieties and a conjecture of Kottwitz", Inventiones mathematicae 194 (2013), No. 1, 147–254.
• "On the coherence conjecture of Pappas and Rapoport", Annals of Mathematics 180 (2014), No. 1, 1–85.
• (with Denis Osipov) "A categorical proof of the Parshin reciprocity laws on algebraic surfaces", Algebra & Number Theory 5 (2011), No. 3, 289–337.
• "Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian", Advances in Mathematics 221 (2009), No. 2, 570–600.
• "Affine Grassmannians and the geometric Satake in mixed characteristic", Annals of Mathematics 185 (2017), No. 2, 403–492.
• (with Edward Frenkel) "Any flat bundle on a punctured disc has an oper structure", Mathematical Research Letters 17 (2010), no. 1, 27–37.
• "The geometric Satake correspondence for ramified groups", Annales Scientifiques de l'École Normale Supérieure 48 (2015), no. 2, 409–451.
• (with Zhiwei Yun) "Integral homology of loop groups via Langlands dual groups", Representation Theory 15 (2011), 347–369.
• (with An Huang, Bong H. Lian) "Period integrals and the Riemann–Hilbert correspondence", Journal of Differential Geometry 104 (2016), No. 2, 325–369.
• (with Tsao-Hsien Chen) "Geometric Langlands in prime characteristic", Compositio Mathematica 153 (2017), No. 2, 395–452.
References
1. "Prime Numbers, Quantum Fields, and Donuts: An Interview with Xinwen Zhu", Caltech. Retrieved on 3 December 2016.
2. 北大数学校友创新合作: 统一数论与几何 [New collaboration among Peking University mathematics alumni: unifying number theory and geometry]. Peking University. 15 December 2015. Retrieved 7 August 2017.
3. "Mathematics People", Notices of the AMS. Retrieved on 3 December 2016.
4. "Caltech Professors Awarded 2015 Sloan Fellowships", Caltech. Retrieved on 3 December 2016.
5. "Math Quartet Joins Forces on Unified Theory", Quanta Magazine. Retrieved on 3 December 2016.
6. Morningside Medal 2019
Authority control
International
• VIAF
National
• Germany
• United States
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Zhuo Qun Song
Zhuo Qun Song (Chinese: 宋卓群; pinyin: Sòng Zhuōqún; born 1997), also called Alex Song, is a Chinese-Canadian who is currently the most highly decorated International Mathematical Olympiad (IMO) contestant, with five gold medals and one bronze medal.
Zhuo Qun Song
宋卓群
Song in 2015
Born1997 (age 25–26)
Tianjin, People's Republic of China
NationalityCanadian
Other namesAlex
Alma materPrinceton University
Phillips Exeter Academy
Known forMost highly decorated IMO contestant with 5 golds and 1 bronze medal
Zhuo Qun Song
Chinese宋卓群
Transcriptions
Standard Mandarin
Hanyu PinyinSòng Zhuōqún
IPA[sʊ̂ŋ ʈʂwó.tɕʰy̌n]
Early life
Song was born in Tianjin, China in 1997.[1] He and his parents moved to Canada in 2002.[1] Song was brought up in Waterloo, Ontario.[2][3]
Song was interested in mathematics at a very young age where he started participating in competitions in first grade. By fourth grade, Song was participating in competitions such as the Canadian Open Mathematics Challenge and the American Mathematics Competitions. In fifth grade, Song became interested in solving Olympiad type questions and started training to solve them.[1]
In 2011, Song moved to the United States to attend Phillips Exeter Academy.[3]
International Mathematical Olympiad
In 2010, when Song was in the seventh grade, he represented Vincent Massey Secondary School in the Canadian Mathematical Olympiad where he finished first place.[1][4]
In the same year, Song represented Canada in the 2010 IMO where he won a Bronze Medal.[4] He would continue to represent Canada for 5 subsequent IMOs where he obtained a gold medal each time. He obtained a perfect score on his final run in 2015, the only contestant to do so that year.[2][3][5] The performances made Song the most decorated contestant of all time.[2][3][6] In 2015, Song was also one of the twelve top scorers of the United States of America Mathematical Olympiad, representing Phillips Exeter Academy.[7]
Results
Year Venue Result
2015 Chiang Mai Gold medal (P)[8]
2014 Cape Town Gold medal[9]
2013 Santa Marta Gold medal[10]
2012 Mar del Plata Gold medal[11]
2011 Amsterdam Gold medal[12]
2010 Astana Bronze medal[13]
Post-IMO
Song graduated from Phillips Exeter Academy in 2015.[2][3]
Song attended Princeton University where he graduated in 2019 with a Bachelor of Arts in Mathematics.[14]
During his time at Princeton, Song was part of the team that participated in the Putnam Competition. His team won second place in 2016[15] and third place in 2017.[16]
Song was previously a Quantitative Researcher at Citadel LLC.[17] He is currently a PhD student at the University of Illinois Urbana–Champaign.[18] He also has been lead coach for the Canadian IMO team since 2020. [19]
Publications
• Kaushansky, Vadim; Reisinger, Christoph; Shkolnikov, Mykhaylo; Song, Zhuo Qun (11 October 2020). "Convergence of a time-stepping scheme to the free boundary in the supercooled Stefan problem". arXiv:2010.05281 [math.PR].
• Song, Zhuo Qun (26 July 2019). "The Convergence of a Time-Stepping Scheme for a McKean-Vlasov Equation with Blow-Ups". {{cite journal}}: Cite journal requires |journal= (help)
• Liu, Yang; Park, Peter S.; Song, Zhuo Qun (1 December 2017). "Bounded gaps between products of distinct primes". Research in Number Theory. 3 (1): 26. doi:10.1007/s40993-017-0089-3. ISSN 2363-9555. S2CID 37218431.
• Liu, Yang; Park, Peter S.; Song, Zhuo Qun (11 December 2016). "The "Riemann Hypothesis" is true for period polynomials of almost all newforms". Research in the Mathematical Sciences. 3 (1): 31. arXiv:1607.04699. doi:10.1186/s40687-016-0081-x. ISSN 2197-9847. S2CID 44531385.
See also
• List of International Mathematical Olympiad participants
References
1. "Team Biographies" (PDF). Canadian Mathematical Society. 2012.{{cite web}}: CS1 maint: url-status (link)
2. Casey, Liam (27 July 2015). "Canadian math whiz wins international competition". CTVNews. Retrieved 25 November 2021.
3. International, Radio Canada (27 July 2015). "Alex Song tops International Math Olympiad". RCI | English. Retrieved 25 November 2021.
4. "Calgary mathlete brings home gold". CBC News. 15 July 2010.{{cite web}}: CS1 maint: url-status (link)
5. Kilkenny, Carmel (29 July 2015). "Alex Song and the Canadian Math Team". Radio Canada International. Retrieved 27 November 2021.
6. "Hall of Fame". International Mathematics Olympiad. Retrieved 27 November 2021.{{cite web}}: CS1 maint: url-status (link)
7. "Winners of the 2015 USA Mathematical Olympiad Announced". Mathematics Association of America. 27 May 2015. Retrieved 27 November 2021.
8. "Alex Song '15 Breaks IMO Record with Five Golds". Phillips Exeter Academy. 21 July 2015. Retrieved 28 November 2021.
9. Hammer, Kate (16 July 2014). "Canada's mathletes ninth in the world at math Olympiad in South Africa". The Globe and Mail. Retrieved 28 November 2021.
10. Hickey, Walt (14 August 2013). "Three American High Schoolers Swept An International Competition By Crushing These Math Problems". Business Insider. Retrieved 28 November 2021.
11. "53rd International Mathematical Olympiad Mar del Plata, Argentina — July 4 – 16, 2012". Canadian Mathematical Society. Retrieved 28 November 2021.
12. "52nd International Mathematical Olympiad Amsterdam, Netherlands — July 16 – 24, 2011". Canadian Mathematical Society. Retrieved 28 November 2021.
13. "51st International Mathematical Olympiad Astana, Kazakhstan — July 5 – 14, 2010". Canadian Mathematical Society. Retrieved 28 November 2021.
14. "Congratulations Class of 2019! | Math". www.math.princeton.edu. Retrieved 25 November 2021.
15. "2016 results".{{cite web}}: CS1 maint: url-status (link)
16. "2017 results".{{cite web}}: CS1 maint: url-status (link)
17. "Six Top Mathletes Selected for Math Team Canada 2021". CMS-SMC. Retrieved 27 November 2021.
18. "IDEA MATH". ideamath.education. Retrieved 19 November 2022.
19. "International Mathematical Olympiad". www.imo-official.org. Retrieved 19 November 2022.
External links
• Zhuo Qun Song's results at International Mathematical Olympiad
| Wikipedia |
Chinese Zhusuan
Zhusuan (Chinese: 珠算; pinyin: zhūsuàn; literally: "bead calculation") is the knowledge and practices of arithmetic calculation through the suanpan or Chinese abacus. In the year 2013, it has been inscribed on the UNESCO Representative List of the Intangible Cultural Heritage of Humanity.[1] While deciding on the inscription, the Intergovernmental Committee noted that "Zhusuan is considered by Chinese people as a cultural symbol of their identity as well as a practical tool; transmitted from generation to generation, it is a calculating technique adapted to multiple aspects of daily life, serving multiform socio-cultural functions and offering the world an alternative knowledge system."[2] The movement to get Chinese Zhusuan inscribed in the list was spearheaded by Chinese Abacus and Mental Arithmetic Association.
History
Zhusuan was a abacus invented in China at the end of the 2nd century CE and reached its peak during the period from the 13th to the 16th century CE. In the 13th century, Guo Shoujing (郭守敬) used Zhusuan to calculate the length of each orbital year and found it to be 365.2425 days. In the 16th century, Zhu Zaiyu (朱載堉) calculated the musical Twelve-interval Equal Temperament using Zhusuan. And again in the 16th century, Wang Wensu (王文素) and Cheng Dawei (程大位) wrote respectively Principles of Algorithms and General Rules of Calculation, summarizing and refining the mathematical algorithms of Zhusuan, thus further boosting the popularity and promotion of Zhusuan. At the end of the 16th century, Zhusuan was introduced to neighboring countries and regions.[3]
In culture
Zhusuan is an important part of the traditional Chinese culture. Zhusuan has a far-reaching effect on various fields of Chinese society, like Chinese folk custom, language, literature, sculpture, architecture, etc., creating a Zhusuan-related cultural phenomenon. For example, ‘Iron Abacus’ (鐵算盤) refers to someone good at calculating; ‘Plus three equals plus five and minus two’ (三下五除二; +3 = +5 − 2) means quick and decisive; ‘3 times 7 equals 21’ indicates quick and rash; and in some places of China, there is a custom of telling children's fortune by placing various daily necessities before them on their first birthday and letting them choose one to predict their future lives. Among the items is an abacus, which symbolizes wisdom and wealth.[3]
References
1. "Chinese Zhusuan, knowledge and practices of mathematical calculation through the abacus". www.unesco.org. UNESCO. Retrieved 29 November 2016.
2. "Decision of the Intergovernmental Committee: 8.COM 8.8". www.unesco.org. UNESCO. Retrieved 29 November 2016.
3. "Nomination File No. 00426". www.unesco.org. UNESCO. Retrieved 29 November 2016.
External links
Look up chinese zhusuan in Wiktionary, the free dictionary.
• UNESCO video on Chinese Zhusuan on YouTube (Published on Dec 4, 2013): Zhusuan
Authority control: National
• Japan
| Wikipedia |
Ziegler spectrum
In mathematics, the (right) Ziegler spectrum of a ring R is a topological space whose points are (isomorphism classes of) indecomposable pure-injective right R-modules. Its closed subsets correspond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegler, who first defined and studied them in 1984.[1]
Definition
Let R be a ring (associative, with 1, not necessarily commutative). A (right) pp-n-formula is a formula in the language of (right) R-modules of the form
$\exists {\overline {y}}\ ({\overline {y}},{\overline {x}})A=0$
where $\ell ,n,m$ are natural numbers, $A$ is an $(\ell +n)\times m$ matrix with entries from R, and ${\overline {y}}$ is an $\ell $-tuple of variables and ${\overline {x}}$ is an $n$-tuple of variables.
The (right) Ziegler spectrum, $\operatorname {Zg} _{R}$, of R is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by $\operatorname {pinj} _{R}$, and the topology has the sets
$(\varphi /\psi )=\{N\in \operatorname {pinj} _{R}\mid \varphi (N)\supsetneq \psi (N)\cap \varphi (N)\}$
as subbasis of open sets, where $\varphi ,\psi $ range over (right) pp-1-formulae and $\varphi (N)$ denotes the subgroup of $N$ consisting of all elements that satisfy the one-variable formula $\varphi $. One can show that these sets form a basis.
Properties
Ziegler spectra are rarely Hausdorff and often fail to have the $T_{0}$-property. However they are always compact and have a basis of compact open sets given by the sets $(\varphi /\psi )$ where $\varphi ,\psi $ are pp-1-formulae.
When the ring R is countable $\operatorname {Zg} _{R}$ is sober.[2] It is not currently known if all Ziegler spectra are sober.
Generalization
Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.[3]
References
1. Ziegler, Martin (1984-04-01). "Model theory of modules" (PDF). Annals of Pure and Applied Logic. SPECIAL ISSUE. 26 (2): 149–213. doi:10.1016/0168-0072(84)90014-9.
2. Ivo Herzog (1993). Elementary duality of modules. Trans. Amer. Math. Soc., 340:1 37–69
3. Herzog, I. (1997). "The Ziegler Spectrum of a Locally Coherent Grothendieck Category". Proceedings of the London Mathematical Society. 74 (3): 503–558. doi:10.1112/S002461159700018X.
| Wikipedia |
Zig-zag product
In graph theory, the zig-zag product of regular graphs $G,H$, denoted by $G\circ H$, is a binary operation which takes a large graph ($G$) and a small graph ($H$) and produces a graph that approximately inherits the size of the large one but the degree of the small one. An important property of the zig-zag product is that if $H$ is a good expander, then the expansion of the resulting graph is only slightly worse than the expansion of $G$.
Roughly speaking, the zig-zag product $G\circ H$ replaces each vertex of $G$ with a copy (cloud) of $H$, and connects the vertices by moving a small step (zig) inside a cloud, followed by a big step (zag) between two clouds, and finally performs another small step inside the destination cloud.
The zigzag product was introduced by Reingold, Vadhan & Wigderson (2000). When the zig-zag product was first introduced, it was used for the explicit construction of constant degree expanders and extractors. Later on, the zig-zag product was used in computational complexity theory to prove that symmetric logspace and logspace are equal (Reingold 2008).
Definition
Let $G$ be a $D$-regular graph on $[N]$ with rotation map $\mathrm {Rot} _{G}$ and let $H$ be a $d$-regular graph on $[D]$ with rotation map $\mathrm {Rot} _{H}$. The zig-zag product $G\circ H$ is defined to be the $d^{2}$-regular graph on $[N]\times [D]$ whose rotation map $\mathrm {Rot} _{G\circ H}$ is as follows:
$\mathrm {Rot} _{G\circ H}((v,a),(i,j))$:
1. Let $(a',i')=\mathrm {Rot} _{H}(a,i)$.
2. Let $(w,b')=\mathrm {Rot} _{G}(v,a')$.
3. Let $(b,j')=\mathrm {Rot} _{H}(b',j)$.
4. Output $((w,b),(j',i'))$.
Properties
Reduction of the degree
It is immediate from the definition of the zigzag product that it transforms a graph $G$ to a new graph which is $d^{2}$-regular. Thus if $G$ is a significantly larger than $H$, the zigzag product will reduce the degree of $G$. Roughly speaking, by amplifying each vertex of $G$ into a cloud of the size of $H$ the product in fact splits the edges of each original vertex between the vertices of the cloud that replace it.
Spectral gap preservation
The expansion of a graph can be measured by its spectral gap, with an important property of the zigzag product the preservation of the spectral gap. That is, if $H$ is a “good enough” expander (has a large spectral gap) then the expansion of the zigzag product is close to the original expansion of $G$.
Formally: Define a $(N,D,\lambda )$-graph as any $D$-regular graph on $N$ vertices, whose second largest eigenvalue (of the associated random walk) has absolute value at most $\lambda $.
Let $G_{1}$ be a $(N_{1},D_{1},\lambda _{1})$-graph and $G_{2}$ be a $(D_{1},D_{2},\lambda _{2})$-graph, then $G_{1}\circ G_{2}$ is a $(N_{1}\cdot D_{1},D_{2}^{2},f(\lambda _{1},\lambda _{2}))$-graph, where $f(\lambda _{1},\lambda _{2})<\lambda _{1}+\lambda _{2}+\lambda _{2}^{2}$.
Connectivity preservation
The zigzag product $G\circ H$ operates separately on each connected component of $G$.
Formally speaking, given two graphs: $G$, a $D$-regular graph on $[N]$ and $H$, a $d$-regular graph on $[D]$ - if $S\subseteq [N]$ is a connected component of $G$ then $G|_{S}\circ H=G\circ H|_{S\times D}$, where $G|_{S}$ is the subgraph of $G$ induced by $S$ (i.e., the graph on $S$ which contains all of the edges in $G$ between vertices in $S$).
Applications
Construction of constant degree expanders
In 2002 Omer Reingold, Salil Vadhan, and Avi Wigderson gave a simple, explicit combinatorial construction of constant-degree expander graphs. The construction is iterative, and needs as a basic building block a single, expander of constant size. In each iteration the zigzag product is used in order to generate another graph whose size is increased but its degree and expansion remains unchanged. This process continues, yielding arbitrarily large expanders.
From the properties of the zigzag product mentioned above, we see that the product of a large graph with a small graph, inherits a size similar to the large graph, and degree similar to the small graph, while preserving its expansion properties from both, thus enabling to increase the size of the expander without deleterious effects.
Solving the undirected s-t connectivity problem in logarithmic space
In 2005 Omer Reingold introduced an algorithm that solves the undirected st-connectivity problem, the problem of testing whether there is a path between two given vertices in an undirected graph, using only logarithmic space. The algorithm relies heavily on the zigzag product.
Roughly speaking, in order to solve the undirected s-t connectivity problem in logarithmic space, the input graph is transformed, using a combination of powering and the zigzag product, into a constant-degree regular graph with a logarithmic diameter. The power product increases the expansion (hence reduces the diameter) at the price of increasing the degree, and the zigzag product is used to reduce the degree while preserving the expansion.
See also
• Graph operations
References
• Reingold, O.; Vadhan, S.; Wigderson, A. (2000), "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors", Proc. 41st IEEE Symposium on Foundations of Computer Science (FOCS), pp. 3–13, arXiv:math/0406038, doi:10.1109/SFCS.2000.892006.
• Reingold, O (2008), "Undirected connectivity in log-space", Journal of the ACM, 55 (4): Article 17, 24 pages, doi:10.1145/1391289.1391291.
• Reingold, O.; Trevisan, L.; Vadhan, S. (2006), "Pseudorandom walks on regular digraphs and the RL vs. L problem", Proc. 38th ACM Symposium on Theory of Computing (STOC), pp. 457–466, doi:10.1145/1132516.1132583.
| Wikipedia |
Ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number generator, as well as precomputed tables. The algorithm is used to generate values from a monotonically decreasing probability distribution. It can also be applied to symmetric unimodal distributions, such as the normal distribution, by choosing a value from one half of the distribution and then randomly choosing which half the value is considered to have been drawn from. It was developed by George Marsaglia and others in the 1960s.
A typical value produced by the algorithm only requires the generation of one random floating-point value and one random table index, followed by one table lookup, one multiply operation and one comparison. Sometimes (2.5% of the time, in the case of a normal or exponential distribution when using typical table sizes) more computations are required. Nevertheless, the algorithm is computationally much faster than the two most commonly used methods of generating normally distributed random numbers, the Marsaglia polar method and the Box–Muller transform, which require at least one logarithm and one square root calculation for each pair of generated values. However, since the ziggurat algorithm is more complex to implement it is best used when large quantities of random numbers are required.
The term ziggurat algorithm dates from Marsaglia's paper with Wai Wan Tsang in 2000; it is so named because it is conceptually based on covering the probability distribution with rectangular segments stacked in decreasing order of size, resulting in a figure that resembles a ziggurat.
Theory of operation
The ziggurat algorithm is a rejection sampling algorithm; it randomly generates a point in a distribution slightly larger than the desired distribution, then tests whether the generated point is inside the desired distribution. If not, it tries again. Given a random point underneath a probability density curve, its x coordinate is a random number with the desired distribution.
The distribution the ziggurat algorithm chooses from is made up of n equal-area regions; n − 1 rectangles that cover the bulk of the desired distribution, on top of a non-rectangular base that includes the tail of the distribution.
Given a monotone decreasing probability density function f(x), defined for all x ≥ 0, the base of the ziggurat is defined as all points inside the distribution and below y1 = f(x1). This consists of a rectangular region from (0, 0) to (x1, y1), and the (typically infinite) tail of the distribution, where x > x1 (and y < y1).
This layer (call it layer 0) has area A. On top of this, add a rectangular layer of width x1 and height A/x1, so it also has area A. The top of this layer is at height y2 = y1 + A/x1, and intersects the density function at a point (x2, y2), where y2 = f(x2). This layer includes every point in the density function between y1 and y2, but (unlike the base layer) also includes points such as (x1, y2) which are not in the desired distribution.
Further layers are then stacked on top. To use a precomputed table of size n (n = 256 is typical), one chooses x1 such that xn = 0, meaning that the top box, layer n − 1, reaches the distribution's peak at (0, f(0)) exactly.
Layer i extends vertically from yi to yi+1, and can be divided into two regions horizontally: the (generally larger) portion from 0 to xi+1 which is entirely contained within the desired distribution, and the (small) portion from xi+1 to xi, which is only partially contained.
Ignoring for a moment the problem of layer 0, and given uniform random variables U0 and U1 ∈ [0,1), the ziggurat algorithm can be described as:
1. Choose a random layer 0 ≤ i < n.
2. Let x = U0xi.
3. If x < xi+1, return x.
4. Let y = yi + U1(yi+1 − yi).
5. Compute f(x). If y < f(x), return x.
6. Otherwise, choose new random numbers and go back to step 1.
Step 1 amounts to choosing a low-resolution y coordinate. Step 3 tests if the x coordinate is clearly within the desired density function without knowing more about the y coordinate. If it is not, step 4 chooses a high-resolution y coordinate, and step 5 does the rejection test.
With closely spaced layers, the algorithm terminates at step 3 a very large fraction of the time. For the top layer n − 1, however, this test always fails, because xn = 0.
Layer 0 can also be divided into a central region and an edge, but the edge is an infinite tail. To use the same algorithm to check if the point is in the central region, generate a fictitious x0 = A/y1. This will generate points with x < x1 with the correct frequency, and in the rare case that layer 0 is selected and x ≥ x1, use a special fallback algorithm to select a point at random from the tail. Because the fallback algorithm is used less than one time in a thousand, speed is not essential.
Thus, the full ziggurat algorithm for one-sided distributions is:
1. Choose a random layer 0 ≤ i < n.
2. Let x = U0xi
3. If x < xi+1, return x.
4. If i = 0, generate a point from the tail using the fallback algorithm.
5. Let y = yi + U1(yi+1 − yi).
6. Compute f(x). If y < f(x), return x.
7. Otherwise, choose new random numbers and go back to step 1.
For a two-sided distribution, the result must be negated 50% of the time. This can often be done conveniently by choosing U0 ∈ (−1,1) and, in step 3, testing if |x| < xi+1.
Fallback algorithms for the tail
Because the ziggurat algorithm only generates most outputs very rapidly, and requires a fallback algorithm whenever x > x1, it is always more complex than a more direct implementation. The specific fallback algorithm depends on the distribution.
For an exponential distribution, the tail looks just like the body of the distribution. One way is to fall back to the most elementary algorithm E = −ln(U1) and let x = x1 − ln(U1). Another is to call the ziggurat algorithm recursively and add x1 to the result.
For a normal distribution, Marsaglia suggests a compact algorithm:
1. Let x = −ln(U1)/x1.
2. Let y = −ln(U2).
3. If 2y > x2, return x + x1.
4. Otherwise, go back to step 1.
Since x1 ≈ 3.5 for typical table sizes, the test in step 3 is almost always successful.
Optimizations
The algorithm can be performed efficiently with precomputed tables of xi and yi = f(xi), but there are some modifications to make it even faster:
• Nothing in the ziggurat algorithm depends on the probability distribution function being normalized (integral under the curve equal to 1), removing normalizing constants can speed up the computation of f(x).
• Most uniform random number generators are based on integer random number generators which return an integer in the range [0, 232 − 1]. A table of 2−32xi lets you use such numbers directly for U0.
• When computing two-sided distributions using a two-sided U0 as described earlier, the random integer can be interpreted as a signed number in the range [−231, 231 − 1], and a scale factor of 2−31 can be used.
• Rather than comparing U0xi to xi+1 in step 3, it is possible to precompute xi+1/xi and compare U0 with that directly. If U0 is an integer random number generator, these limits may be premultiplied by 232 (or 231, as appropriate) so an integer comparison can be used.
• With the above two changes, the table of unmodified xi values is no longer needed and may be deleted.
• When generating IEEE 754 single-precision floating point values, which only have a 24-bit mantissa (including the implicit leading 1), the least-significant bits of a 32-bit integer random number are not used. These bits may be used to select the layer number. (See the references below for a detailed discussion of this.)
• The first three steps may be put into an inline function, which can call an out-of-line implementation of the less frequently needed steps.
Generating the tables
It is possible to store the entire table precomputed, or just include the values n, y1, A, and an implementation of f −1(y) in the source code, and compute the remaining values when initializing the random number generator.
As previously described, you can find xi = f −1(yi) and yi+1 = yi + A/xi. Repeat n − 1 times for the layers of the ziggurat. At the end, you should have yn = f(0). There will be some round-off error, but it is a useful sanity test to see that it is acceptably small.
When actually filling in the table values, just assume that xn = 0 and yn = f(0), and accept the slight difference in layer n − 1's area as rounding error.
Finding x1 and A
Given an initial (guess at) x1, you need a way to compute the area t of the tail for which x > x1. For the exponential distribution, this is just e−x1, while for the normal distribution, assuming you are using the unnormalized f(x) = e−x2/2, this is √π/2 erfc(x/√2). For more awkward distributions, numerical integration may be required.
With this in hand, from x1, you can find y1 = f(x1), the area t in the tail, and the area of the base layer A = x1y1 + t.
Then compute the series yi and xi as above. If yi > f(0) for any i < n, then the initial estimate x1 was too low, leading to too large an area A. If yn < f(0), then the initial estimate x1 was too high.
Given this, use a root-finding algorithm (such as the bisection method) to find the value x1 which produces yn−1 as close to f(0) as possible. Alternatively, look for the value which makes the area of the topmost layer, xn−1(f(0) − yn−1), as close to the desired value A as possible. This saves one evaluation of f −1(x) and is actually the condition of greatest interest.
References
• George Marsaglia; Wai Wan Tsang (2000). "The Ziggurat Method for Generating Random Variables". Journal of Statistical Software. 5 (8). Retrieved 2007-06-20. This paper numbers the layers from 1 starting at the top, and makes layer 0 at the bottom a special case, while the explanation above numbers layers from 0 at the bottom.
• C implementation of the ziggurat method for the normal density function and the exponential density function, that is essentially a copy of the code in the paper. (Potential users should be aware that this C code assumes 32-bit integers.)
• A C# implementation of the ziggurat algorithm and overview of the method.
• Jurgen A. Doornik (2005). "An Improved Ziggurat Method to Generate Normal Random Samples" (Document). Nuffield College, Oxford. {{cite document}}: Unknown parameter |access-date= ignored (help); Unknown parameter |url= ignored (help) Describes the hazards of using the least-significant bits of the integer random number generator to choose the layer number.
• Normal Behavior By Cleve Moler, MathWorks, describing the ziggurat algorithm introduced in MATLAB version 5, 2001.
• The Ziggurat Random Normal Generator Blogs of MathWorks, posted by Cleve Moler, May 18, 2015.
• David B. Thomas; Philip H.W. Leong; Wayne Luk; John D. Villasenor (October 2007). "Gaussian Random Number Generators" (PDF). ACM Computing Surveys. 39 (4): 11:1–38. doi:10.1145/1287620.1287622. ISSN 0360-0300. S2CID 10948255. Retrieved 2009-07-27. [W]hen maintaining extremely high statistical quality is the first priority, and subject to that constraint, speed is also desired, the Ziggurat method will often be the most appropriate choice. Comparison of several algorithms for generating Gaussian random numbers.
• Nadler, Boaz (2006). "Design Flaws in the Implementation of the Ziggurat and Monty Python methods (And some remarks on Matlab randn)". arXiv:math/0603058.. Illustrates problems with underlying uniform pseudo-random number generators and how those problems affect the ziggurat algorithm's output.
• Edrees, Hassan M.; Cheung, Brian; Sandora, McCullen; Nummey, David; Stefan, Deian (13–16 July 2009). Hardware-Optimized Ziggurat Algorithm for High-Speed Gaussian Random Number Generators (PDF). 2009 International Conference on Engineering of Reconfigurable Systems & Algorithms. Las Vegas.
• Marsaglia, George (September 1963). Generating a Variable from the Tail of the Normal Distribution (Technical report). Boeing Scientific Research Labs. Mathematical Note No. 322, DTIC accession number AD0423993. Archived from the original on September 10, 2014 – via Defense Technical Information Center.
| Wikipedia |
Zig-zag lemma
In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.
Statement
In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let $({\mathcal {A}},\partial _{\bullet }),({\mathcal {B}},\partial _{\bullet }')$ and $({\mathcal {C}},\partial _{\bullet }'')$ be chain complexes that fit into the following short exact sequence:
$0\longrightarrow {\mathcal {A}}\mathrel {\stackrel {\alpha }{\longrightarrow }} {\mathcal {B}}\mathrel {\stackrel {\beta }{\longrightarrow }} {\mathcal {C}}\longrightarrow 0$
Such a sequence is shorthand for the following commutative diagram:
where the rows are exact sequences and each column is a chain complex.
The zig-zag lemma asserts that there is a collection of boundary maps
$\delta _{n}:H_{n}({\mathcal {C}})\longrightarrow H_{n-1}({\mathcal {A}}),$
that makes the following sequence exact:
The maps $\alpha _{*}^{}$ and $\beta _{*}^{}$ are the usual maps induced by homology. The boundary maps $\delta _{n}^{}$ are explained below. The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence. A variant version of the zig-zag lemma is commonly known as the "snake lemma" (it extracts the essence of the proof of the zig-zag lemma given below).
Construction of the boundary maps
The maps $\delta _{n}^{}$ are defined using a standard diagram chasing argument. Let $c\in C_{n}$ represent a class in $H_{n}({\mathcal {C}})$, so $\partial _{n}''(c)=0$. Exactness of the row implies that $\beta _{n}^{}$ is surjective, so there must be some $b\in B_{n}$ with $\beta _{n}^{}(b)=c$. By commutativity of the diagram,
$\beta _{n-1}\partial _{n}'(b)=\partial _{n}''\beta _{n}(b)=\partial _{n}''(c)=0.$
By exactness,
$\partial _{n}'(b)\in \ker \beta _{n-1}=\mathrm {im} \;\alpha _{n-1}.$
Thus, since $\alpha _{n-1}^{}$ is injective, there is a unique element $a\in A_{n-1}$ such that $\alpha _{n-1}(a)=\partial _{n}'(b)$. This is a cycle, since $\alpha _{n-2}^{}$ is injective and
$\alpha _{n-2}\partial _{n-1}(a)=\partial _{n-1}'\alpha _{n-1}(a)=\partial _{n-1}'\partial _{n}'(b)=0,$
since $\partial ^{2}=0$. That is, $\partial _{n-1}(a)\in \ker \alpha _{n-2}=\{0\}$. This means $a$ is a cycle, so it represents a class in $H_{n-1}({\mathcal {A}})$. We can now define
$\delta _{}^{}[c]=[a].$
With the boundary maps defined, one can show that they are well-defined (that is, independent of the choices of c and b). The proof uses diagram chasing arguments similar to that above. Such arguments are also used to show that the sequence in homology is exact at each group.
See also
• Mayer–Vietoris sequence
References
• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.
• 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
• Munkres, James R. (1993). Elements of Algebraic Topology. New York: Westview Press. ISBN 0-201-62728-0.
| Wikipedia |
Fence (mathematics)
In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations:
$a<b>c<d>e<f>h<i\cdots $
or
$a>b<c>d<e>f<h>i\cdots $
A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences.
A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century.[1] The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are:
$1,1,2,4,10,32,122,544,2770,15872,101042.$
(sequence A001250 in the OEIS).
The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube.[2]
A partially ordered set is series-parallel if and only if it does not have four elements forming a fence.[3]
Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.[4]
An up-down poset Q(a,b) is a generalization of a zigzag poset in which there are a downward orientations for every upward one and b total elements.[5] For instance, Q(2,9) has the elements and relations
$a>b>c<d>e>f<g>h>i.$
In this notation, a fence is a partially ordered set of the form Q(1,n).
Equivalent conditions
The following conditions are equivalent for a poset P:
1. P is a disjoint union of zigzag posets.
2. If a ≤ b ≤ c in P, either a = b or b = c.
3. $<\circ <\;=\emptyset $, i.e. it is never the case that a < b and b < c, so that < is vacuously transitive.
4. P has dimension at most one (defined analogously to the Krull dimension of a commutative ring).
5. Every element of P is either maximal or minimal.
6. The slice category Pos/P is cartesian closed.[lower-alpha 1]
The prime ideals of a commutative ring R, ordered by inclusion, satisfy the equivalent conditions above if and only if R has Krull dimension at most one.
Notes
1. Here, Pos denotes the category of partially ordered sets.
References
1. André (1881).
2. Gansner (1982) calls the fact that this lattice has a Fibonacci number of elements a “well known fact,” while Stanley (1986) asks for a description of it in an exercise. See also Höft & Höft (1985), Beck (1990), and Salvi & Salvi (2008).
3. Valdes, Tarjan & Lawler (1982).
4. Currie & Visentin (1991); Duffus et al. (1992); Rutkowski (1992a); Rutkowski (1992b); Farley (1995).
5. Gansner (1982).
• André, Désiré (1881), "Sur les permutations alternées", J. Math. Pures Appl., (Ser. 3), 7: 167–184.
• Beck, István (1990), "Partial orders and the Fibonacci numbers", Fibonacci Quarterly, 28 (2): 172–174, MR 1051291.
• Currie, J. D.; Visentin, T. I. (1991), "The number of order-preserving maps of fences and crowns", Order, 8 (2): 133–142, doi:10.1007/BF00383399, hdl:10680/1724, MR 1137906, S2CID 122356472.
• Duffus, Dwight; Rödl, Vojtěch; Sands, Bill; Woodrow, Robert (1992), "Enumeration of order preserving maps", Order, 9 (1): 15–29, doi:10.1007/BF00419036, MR 1194849, S2CID 84180809.
• Farley, Jonathan David (1995), "The number of order-preserving maps between fences and crowns", Order, 12 (1): 5–44, doi:10.1007/BF01108588, MR 1336535, S2CID 120372679.
• Gansner, Emden R. (1982), "On the lattice of order ideals of an up-down poset", Discrete Mathematics, 39 (2): 113–122, doi:10.1016/0012-365X(82)90134-0, MR 0675856.
• Höft, Hartmut; Höft, Margret (1985), "A Fibonacci sequence of distributive lattices", Fibonacci Quarterly, 23 (3): 232–237, MR 0806293.
• Kelly, David; Rival, Ivan (1974), "Crowns, fences, and dismantlable lattices", Canadian Journal of Mathematics, 26 (5): 1257–1271, doi:10.4153/cjm-1974-120-2, MR 0417003.
• Rutkowski, Aleksander (1992a), "The number of strictly increasing mappings of fences", Order, 9 (1): 31–42, doi:10.1007/BF00419037, MR 1194850, S2CID 120965362.
• Rutkowski, Aleksander (1992b), "The formula for the number of order-preserving self-mappings of a fence", Order, 9 (2): 127–137, doi:10.1007/BF00814405, MR 1199291, S2CID 121879635.
• Salvi, Rodolfo; Salvi, Norma Zagaglia (2008), "Alternating unimodal sequences of Whitney numbers", Ars Combinatoria, 87: 105–117, MR 2414008.
• Stanley, Richard P. (1986), Enumerative Combinatorics, Wadsworth, Inc. Exercise 3.23a, page 157.
• Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), "The Recognition of Series Parallel Digraphs", SIAM Journal on Computing, 11 (2): 298–313, doi:10.1137/0211023.
External links
• Weisstein, Eric W. "Fence Poset". MathWorld.
| Wikipedia |
Zilber–Pink conjecture
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André–Oort, Manin–Mumford, and Mordell–Lang. For algebraic tori and semiabelian varieties it was proposed by Boris Zilber[1] and independently by Enrico Bombieri, David Masser, Umberto Zannier[2] in the early 2000's. For semiabelian varieties the conjecture implies the Mordell–Lang and Manin–Mumford conjectures. Richard Pink proposed (again independently) a more general conjecture for Shimura varieties which also implies the André–Oort conjecture.[3] In the case of algebraic tori, Zilber called it the Conjecture on Intersection with Tori (CIT). The general version is now known as the Zilber–Pink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties.
Statement
Atypical and unlikely intersections
The intersection of two algebraic varieties is called atypical if its dimension is larger than expected. More precisely, given three varieties $X,Y\subseteq U$, a component $Z$ of the intersection $X\cap Y$ is said to be atypical in $U$ if $\dim Z>\dim X+\dim Y-\dim U$. Since the expected dimension of $X\cap Y$ is $\dim X+\dim Y-\dim U$, atypical intersections are "atypically large" and are not expected to occur. When $\dim X+\dim Y-\dim U<0$, the varieties $X$ and $Y$ are not expected to intersect at all, so when they do, the intersection is said to be unlikely. For example, if in a 3-dimensional space two lines intersect, then it is an unlikely intersection, for two randomly chosen lines would almost never intersect.
Special varieties
Special varieties of a Shimura variety are certain arithmetically defined subvarieties. They are higher dimensional versions of special points. For example, in semiabelian varieties special points are torsion points and special varieties are translates of irreducible algebraic subgroups by torsion points. In the modular setting special points are the singular moduli and special varieties are irreducible components of varieties defined by modular equations.
Given a mixed Shimura variety $X$ and a subvariety $V\subseteq X$, an atypical subvariety of $V$ is an atypical component of an intersection $V\cap T$ where $T\subseteq X$ is a special subvariety.
The Zilber–Pink conjecture
Let $X$ be a mixed Shimura variety or a semiabelian variety defined over $\mathbb {C} $, and let $V\subseteq X$ be a subvariety. Then $V$ contains only finitely many maximal atypical subvarieties.[4]
The abelian and modular versions of the Zilber–Pink conjecture are special cases of the conjecture for Shimura varieties, while in general the semiabelian case is not. However, special subvarieties of semiabelian and Shimura varieties share many formal properties which makes the same formulation valid in both settings.
Partial results and special cases
While the Zilber–Pink conjecture is wide open, many special cases and weak versions have been proven.
If a variety $V\subseteq X$ contains a special variety $T$ then by definition $T$ is an atypical subvariety of $V$. Hence, the Zilber–Pink conjecture implies that $V$ contains only finitely many maximal special subvarieties. This is the Manin–Mumford conjecture in the semiabelian setting and the André–Oort conjecture in the Shimura setting. Both are now theorems; the former has been known for several decades,[5] while the latter was proven in full generality only recently.[6]
Many partial results have been proven on the Zilber–Pink conjecture.[7][8][9] An example in the modular setting is the result that any variety contains only finitely many maximal strongly atypical subvarieties, where a strongly atypical subvariety is an atypical subvariety with no constant coordinate.[10][11]
References
1. Zilber, Boris (2002), "Exponential sums equations and the Schanuel conjecture", J. London Math. Soc., 65 (2): 27–44, doi:10.1112/S0024610701002861.
2. Bombieri, Enrico; Masser, David; Zannier, Umberto (2007), Anomalous Subvarieties—Structure Theorems and Applications, International Mathematics Research Notices, vol. 2007.
3. Pink, Richard (2005). "A Combination of the Conjectures of Mordell–Lang and André–Oort". Geometric Methods in Algebra and Number Theory. Progress in Mathematics. Vol. 235. pp. 251–282. CiteSeerX 10.1.1.499.3023. doi:10.1007/0-8176-4417-2_11. ISBN 0-8176-4349-4.
4. Habegger, Philipp; Pila, Jonathan (2016), o-minimality and certain atypical intersections, Ann. Sci. Éc. Norm. Supér, vol. 49, pp. 813–858.
5. Raynaud, Michel (1983). "Sous-variétés d'une variété abélienne et points de torsion". In Artin, Michael; Tate, John (eds.). Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic. Progress in Mathematics (in French). Vol. 35. Birkhäuser-Boston. pp. 327–352. MR 0717600. Zbl 0581.14031.
6. Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob; Esnault, Hélène; Groechenig, Michael (2021-09-17). "Canonical Heights on Shimura Varieties and the André-Oort Conjecture". arXiv:2109.08788 [math.NT].
7. Habegger, Philipp; Pila, Jonathan (2012), Some unlikely intersections beyond André-Oort, Compositio Math., vol. 148, pp. 1–27.
8. Daw, Christopher; Orr, Martin (2021), Unlikely intersections with $E\times $CM curves in ${\mathcal {A}}_{2}$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), vol. 22, pp. 1705–1745.
9. Daw, Christopher; Orr, Martin (2022), Quantitative Reduction Theory and Unlikely Intersections, IMRN, vol. 2022, pp. 16138–16195.
10. Pila, Jonathan; Tsimerman, Jacob (2016), "Ax-Schanuel for the j-function", Duke Math. J., 165 (13): 2587–2605, arXiv:1412.8255, doi:10.1215/00127094-3620005, S2CID 118973278
11. Aslanyan, Vahagn (2021), "Weak Modular Zilber–Pink with Derivatives", Math. Ann., arXiv:1803.05895, doi:10.1007/s00208-021-02213-7, S2CID 119654268
Further reading
• Pila, Jonathan (2022). Point-Counting and the Zilber–Pink Conjecture. Cambridge University Press. ISBN 9781009170321.
• Zannier, Umberto (2012). Some Problems of Unlikely Intersections in Arithmetic and Geometry. Princeton: Princeton University Press. ISBN 978-0-691-15370-4.
| Wikipedia |
Zimmer's conjecture
Zimmer's conjecture is a statement in mathematics "which has to do with the circumstances under which geometric spaces exhibit certain kinds of symmetries."[1] It was named after the mathematician Robert Zimmer. The conjecture states that there can exist symmetries (specifically higher-rank lattices) in a higher dimension that cannot exist in lower dimensions.
In 2017, the conjecture was proven by Aaron Brown and Sebastián Hurtado-Salazar of the University of Chicago and David Fisher of Indiana University.[1][2][3]
References
1. Hartnett, Kevin (2018-10-23). "A Proof About Where Symmetries Can't Exist". Quanta Magazine. Retrieved 2018-11-02.
2. Brown, Aaron; Fisher, David; Hurtado, Sebastian (2017-10-07). "Zimmer's conjecture for actions of SL(𝑚,ℤ)". arXiv:1710.02735 [math.DS].
3. "New Methods for Zimmer's Conjecture". IPAM. Retrieved 2018-11-02.
| Wikipedia |
Zimmert set
In mathematics, a Zimmert set is a set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.
Definition
Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d). The Zimmert set Z(d) is the set of positive integers n such that 4n2 < -D-3 and n ≠ 2; D is a quadratic non-residue of all odd primes in d; n is odd if D is not congruent to 5 modulo 8. The cardinality of Z(d) may be denoted by z(d).
Property
For all but a finite number of d we have z(d) > 1: indeed this is true for all d > 10476.[1]
Application
Let Γd denote the Bianchi group PSL(2,Od), where Od is the ring of integers of. As a subgroup of PSL(2,C), there is an action of Γd on hyperbolic 3-space H3, with a fundamental domain. It is a theorem that there are only finitely many values of d for which Γd can contain an arithmetic subgroup G for which the quotient H3/G is a link complement. Zimmert sets are used to obtain results in this direction: z(d) is a lower bound for the rank of the largest free quotient of Γd[2] and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.[1]
References
1. Mason, A.W.; Odoni, R.W.K.; Stothers, W.W. (1992). "Almost all Bianchi groups have free, non-cyclic quotients". Math. Proc. Camb. Philos. Soc. 111 (1): 1–6. Bibcode:1992MPCPS.111....1M. doi:10.1017/S0305004100075101. S2CID 122325132. Zbl 0758.20009.
2. Zimmert, R. (1973). "Zur SL2 der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers". Inventiones Mathematicae. 19: 73–81. Bibcode:1973InMat..19...73Z. doi:10.1007/BF01418852. S2CID 121281237. Zbl 0254.10019.
• Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. Vol. 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.
| Wikipedia |
Zinovy Reichstein
Zinovy Reichstein (born 1961) is a Russian-born American mathematician. He is a professor at the University of British Columbia in Vancouver. He studies mainly algebra, algebraic geometry and algebraic groups. He introduced (with Joe P. Buhler) the concept of essential dimension.[1]
Zinovy Reichstein
Born1961
Alma materHarvard University
Known forEssential dimension
Scientific career
FieldsMathematics
InstitutionsUniversity of British Columbia
Doctoral advisorMichael Artin
Early life and education
In high school, Reichstein participated in the national mathematics olympiad in Russia and was the third highest scorer in 1977 and second highest scorer in 1978.
Because of the Antisemitism in the Soviet Union at the time, Reichstein was not accepted to Moscow University, even though he had passed the special math entrance exams. He attended a semester of college at Russian University of Transport instead.
His family then decided to emigrate, arriving in Vienna, Austria, in August 1979 and New York, United States in the fall of 1980. Reichstein worked as a delivery boy for a short period of time in New York. He was then accepted to and attended California Institute of Technology for his undergraduate studies.[2]
Reichstein received his PhD degree in 1988 from Harvard University under the supervision of Michael Artin. Parts of his thesis entitled "The Behavior of Stability under Equivariant Maps" were published in the journal Inventiones Mathematicae.[3]
Career
As of 2011, he is on the editorial board of the mathematics journal Transformation groups.[4]
Awards
• Winner of the 2013 Jeffery-Williams Prize awarded by the Canadian Mathematical Society[5]
• Fellow of the American Mathematical Society, 2012[6]
• Invited Speaker to the International Congress of Mathematicians (Hyderabad, India 2010)[7]
References
1. J. Buhler, Z. Reichstein (1997). "On the Essential Dimension of a Finite Group". Compositio Mathematica. 106 (2): 159–179. doi:10.1023/A:1000144403695.
2. Dietrich, JS. "To Do Mathematics: The Odyssey of a Soviet Emigre" (PDF). calteches.library.caltech.edu. Archived from the original (PDF) on 2010-08-06.
3. Reichstein, Zinovy (1989), "Stability and equivariant maps", Inventiones Mathematicae, 96 (2): 349–383, Bibcode:1989InMat..96..349R, doi:10.1007/BF01393967, S2CID 120929091
4. "Transformation groups (editorial board)". Springer.
5. UBC PROFESSOR GARNERS PRESTIGIOUS NATIONAL AWARD
6. List of Fellows of the American Mathematical Society, retrieved 2013-06-09.
7. Speakers of the International Congress of Mathematicians, retrieved 2011-05-24
External links
• Official website
• Zinovy Reichstein at the Mathematics Genealogy Project
Authority control
International
• VIAF
Academics
• CiNii
• DBLP
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Zipf–Mandelbrot law
In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it.
Zipf–Mandelbrot
Parameters $N\in \{1,2,3\ldots \}$ (integer)
$q\in [0;\infty )$ (real)
$s>0\,$ (real)
Support $k\in \{1,2,\ldots ,N\}$
PMF ${\frac {1/(k+q)^{s}}{H_{N,q,s}}}$
CDF ${\frac {H_{k,q,s}}{H_{N,q,s}}}$
Mean ${\frac {H_{N,q,s-1}}{H_{N,q,s}}}-q$
Mode $1\,$
Entropy ${\frac {s}{H_{N,q,s}}}\sum _{k=1}^{N}{\frac {\ln(k+q)}{(k+q)^{s}}}+\ln(H_{N,q,s})$
The probability mass function is given by:
$f(k;N,q,s)={\frac {1/(k+q)^{s}}{H_{N,q,s}}}$
where $H_{N,q,s}$ is given by:
$H_{N,q,s}=\sum _{i=1}^{N}{\frac {1}{(i+q)^{s}}}$
which may be thought of as a generalization of a harmonic number. In the formula, $k$ is the rank of the data, and $q$ and $s$ are parameters of the distribution. In the limit as $N$ approaches infinity, this becomes the Hurwitz zeta function $\zeta (s,q)$. For finite $N$ and $q=0$ the Zipf–Mandelbrot law becomes Zipf's law. For infinite $N$ and $q=0$ it becomes a Zeta distribution.
Applications
The distribution of words ranked by their frequency in a random text corpus is approximated by a power-law distribution, known as Zipf's law.
If one plots the frequency rank of words contained in a moderately sized corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Powers, 1998 and Gelbukh & Sidorov, 2001). Zipf's law implicitly assumes a fixed vocabulary size, but the Harmonic series with s=1 does not converge, while the Zipf–Mandelbrot generalization with s>1 does. Furthermore, there is evidence that the closed class of functional words that define a language obeys a Zipf–Mandelbrot distribution with different parameters from the open classes of contentive words that vary by topic, field and register.[1]
In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.[2]
Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandelbrot distributions.[3]
Notes
1. Powers, David M W (1998). "Applications and explanations of Zipf's law". New methods in language processing and computational natural language learning. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160.
2. Mouillot, D; Lepretre, A (2000). "Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity". Environmental Monitoring and Assessment. Springer. 63 (2): 279–295. doi:10.1023/A:1006297211561. S2CID 102285701. Retrieved 24 Dec 2008.
3. Manaris, B; Vaughan, D; Wagner, CS; Romero, J; Davis, RB. "Evolutionary Music and the Zipf–Mandelbrot Law: Developing Fitness Functions for Pleasant Music". Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003). 611.
References
• Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics". In B.B. Wolman and E. Nagel (ed.). Scientific psychology. Basic Books. Reprinted as
• Mandelbrot, Benoît (1968) [1965]. "Information Theory and Psycholinguistics". In R.C. Oldfield and J.C. Marchall (ed.). Language. Penguin Books.
• Powers, David M W (1998). "Applications and explanations of Zipf's law". New methods in language processing and computational natural language learning. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160.
• Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press.
• Van Droogenbroeck F.J., 'An essential rephrasing of the Zipf–Mandelbrot law to solve authorship attribution applications by Gaussian statistics' (2019)
External links
• Z. K. Silagadze: Citations and the Zipf–Mandelbrot's law
• NIST: Zipf's law
• W. Li's References on Zipf's law
• Gelbukh & Sidorov, 2001: Zipf and Heaps Laws’ Coefficients Depend on Language
• C++ Library for generating random Zipf–Mandelbrot deviates.
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
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| Wikipedia |
Zipf's law
Zipf's law (/zɪf/, German: [ts͡ɪpf]) is an empirical law that often holds, approximately, when a list of measured values is sorted in decreasing order. It states that the value of the nth entry is inversely proportional to n.
The best known instance of Zipf's law applies to the frequency table of words in a text or corpus of natural language:
${\text{word frequency}}\propto {\frac {1}{\text{word rank}}}$
Namely, it is usually found that the most common word occurs approximately twice as often as the next common one, three times as often as the third most common, and so on. For example, in the Brown Corpus of American English text, the word "the" is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's Law, the second-place word "of" accounts for slightly over 3.5% of words (36,411 occurrences), followed by "and" (28,852).[2] It is often used in the following form, called Zipf-Mandelbrot law:
${\text{frequency}}\propto {\frac {1}{({\text{rank}}+b)^{a}}}$
where $a,b$ are fitted parameters, with $a\approx 1$, and $b\approx 2.7$.[1]
This "law" is named after the American linguist George Kingsley Zipf,[3][4][5] and is still an important concept in quantitative linguistics. It has been found to apply to many other types of data studied in the physical and social sciences.
In mathematical statistics, the concept has been formalized as the Zipfian distribution: a family of related discrete probability distributions whose rank-frequency distribution is an inverse power law relation. They are related to Benford's law and the Pareto distribution.
Some sets of time-dependent empirical data deviate somewhat from Zipf's law. Such empirical distributions are said to be quasi-Zipfian.
History
In 1913, the German physicist Felix Auerbach observed an inverse proportionality between the population sizes of cities, and their ranks when sorted by decreasing order of that variable.[6]
Zipf's law has been discovered before Zipf,[lower-alpha 1] by the French stenographer Jean-Baptiste Estoup' Gammes Stenographiques (4th ed) in 1916,[7] with G. Dewey in 1923,[8] and with E. Condon in 1928.[9]
The same relation for frequencies of words in natural language texts was observed by George Zipf in 1932,[4] but he never claimed to have originated it. In fact, Zipf didn't like mathematics. In his 1932 publication,[10] the author speaks with disdain about mathematical involvement in linguistics, a. o. ibidem, p. 21: (…) let me say here for the sake of any mathematician who may plan to formulate the ensuing data more exactly, the ability of the highly intense positive to become the highly intense negative, in my opinion, introduces the devil into the formula in the form of √(-i). The only mathematical expression Zipf used looks like a.b2 = constant, which he "borrowed" from Alfred J. Lotka's 1926 publication.[11]
The same relationship was found to occur in many other contexts, and for other variables besides frequency.[1] For example, when corporations are ranked by decreasing size, their sizes are found to be inversely proportional to the rank.[12] The same relation is found for personal incomes (where it is called Pareto principle[13]), number of people watching the same TV channel,[14] notes in music,[15] cells transcriptomes[16][17] and more.
Formal definition
Zipf's law
Probability mass function
Zipf PMF for N = 10 on a log–log scale. The horizontal axis is the index k . (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Zipf CDF for N = 10. The horizontal axis is the index k . (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters $s\geq 0\,$ (real)
$N\in \{1,2,3\ldots \}$ (integer)
Support $k\in \{1,2,\ldots ,N\}$
PMF ${\frac {1/k^{s}}{H_{N,s}}}$ where HN,s is the Nth generalized harmonic number
CDF ${\frac {H_{k,s}}{H_{N,s}}}$
Mean ${\frac {H_{N,s-1}}{H_{N,s}}}$
Mode $1\,$
Variance ${\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}$
Entropy ${\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})$
MGF ${\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}$
CF ${\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}$
Formally, the Zipf distribution on N elements assigns to the element of rank k (counting from 1) the probability
$f(k;N)={\frac {1}{H_{N}}}\,{\frac {1}{k}}$
where HN is a normalization constant, the Nth harmonic number:
$H_{N}=\sum _{k=1}^{N}{\frac {1}{k}}\ .$
The distribution is sometimes generalized to an inverse power law with exponent s instead of 1.[18] Namely,
$f(k;s,N)={\frac {1}{H_{s,N}}}\,{\frac {1}{k^{s}}}$
where Hs,N is a generalized harmonic number
$H_{s,N}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}\ .$
The generalized Zipf distribution can be extended to infinitely many items (N = ∞) only if the exponent s exceeds 1. In that case, the normalization constant Hs,N becomes Riemann's zeta function,
$\zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty \ .$
If the exponent s is 1 or less, the normalization constant Hs,N diverges as N tends to infinity.
Empirical testing
Empirically, a data set can be tested to see whether Zipf's law applies by checking the goodness of fit of an empirical distribution to the hypothesized power law distribution with a Kolmogorov–Smirnov test, and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.[19]
Zipf's law can be visuallized by plotting the item frequency data on a log-log graph, with the axes being the logarithm of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent s to the extent that the plot approximates a linear (more precisely, affine) function with slope −s. For exponent s = 1, one can also plot the reciprocal of the frequency (mean interword interval) against rank, or the reciprocal of rank against frequency, and compare the result with the line through the origin with slope 1.[3]
Statistical explanations
Although Zipf's Law holds for most natural languages, even some non-natural ones like Esperanto,[20] the reason is still not well understood.[21] Recent reviews of generative processes for Zipf's law include.[22][23]
However, it may be partially explained by the statistical analysis of randomly generated texts. Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters (plus a space character), the "words" with different lengths follow the macro-trend of the Zipf's law (the more probable words are the shortest with equal probability).[24] In 1959, Vitold Belevitch observed that if any of a large class of well-behaved statistical distributions (not only the normal distribution) is expressed in terms of rank and expanded into a Taylor series, the first-order truncation of the series results in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law.[25][26]
The principle of least effort is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language want to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.[5][27]
A minimal explanation assumes that words are generated by monkeys typing randomly. If language is generated by a single monkey typing randomly, with fixed and nonzero probability of hitting each letter key or white space, then the words (letter strings separated by white spaces) produced by the monkey follows Zipf's law.[28]
Another possible cause for the Zipf distribution is a preferential attachment process, in which the value x of an item tends to grow at a rate proportional to x (intuitively, "the rich get richer" or "success breeds success"). Such a growth process results in the Yule–Simon distribution, which has been shown to fit word frequency versus rank in language[29] and population versus city rank[30] better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon.
A similar explanation is based on atlas models, systems of exchangeable positive-valued diffusion processes with drift and variance parameters that depend only on the rank of the process. It has been shown mathematically that Zipf's law holds for Atlas models that satisfy certain natural regularity conditions.[31][32] Quasi-Zipfian distributions can result from quasi-Atlas models.
Related laws
A generalization of Zipf's law is the Zipf–Mandelbrot law, proposed by Benoit Mandelbrot, whose frequencies are:
$f(k;N,q,s)={\frac {1}{C}}\,{\frac {(k+q)^{s}}{.}}\,$
The constant C is the Hurwitz zeta function evaluated at s.
Zipfian distributions can be obtained from Pareto distributions by an exchange of variables.[18]
The Zipf distribution is sometimes called the discrete Pareto distribution[33] because it is analogous to the continuous Pareto distribution in the same way that the discrete uniform distribution is analogous to the continuous uniform distribution.
The tail frequencies of the Yule–Simon distribution are approximately
$f(k;\rho )\approx {\frac {[{\text{constant}}]}{k^{\rho +1}}}$
for any choice of ρ > 0.
In the parabolic fractal distribution, the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.[34] Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.[35]
It has been argued that Benford's law is a special bounded case of Zipf's law,[34] with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.[36] The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with s = 1 satisfy Benford's law.
$n$ Benford's law: $P(n)=$
$\log _{10}(n+1)-\log _{10}(n)$
${\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}$
1 0.30103000
2 0.17609126 −0.7735840
3 0.12493874 −0.8463832
4 0.09691001 −0.8830605
5 0.07918125 −0.9054412
6 0.06694679 −0.9205788
7 0.05799195 −0.9315169
8 0.05115252 −0.9397966
9 0.04575749 −0.9462848
Occurrences
City sizes
Following Auerbach's 1913 observation, there has been substantial examination of Zipf's law for city sizes.[37] However, more recent empirical[38][39] and theoretical[40] studies have challenged the relevance of Zipf's law for cities.
Word frequencies in natural languages
In many texts in human languages, word frequencies approximately follow a Zipf distribution with exponent s close to 1: that is, the most common word occurs about n times the nth most common one.
The actual rank-frequency plot of a natural language text deviates in some extent from the ideal Zipf distribution, especially at the two ends of the range. The deviations may depend on the language, on the topic of the text, on the author, on whether the text was translated from another language, and on the spelling rules used. Some deviation is inevitable because of sampling error.
At the low-frequency end, where the rank approaches N, the plot takes a staircase shape, because each word can occur only an integer number of times.
• Zipf's law plots for several languages
• Texts in German (1669), Russian (1972), French (1865), Italian (1840), and Medieval English (1460).
• Cervantes' Don Quixote Part I (Spanish, 1605) and Assis's Dom Casmurro (Portuguese, 1899).
• Ge'ez (14th century), Arabic (~650 CE), Hebrew (500-800 CE), all with vowels.
• Lhasa Tibetan, Chinese, Vietnamese, all with separated syllables.
• Biblical texts: Pentateuch from the Latin Vulgate and Russian Synodal Bible, the four Gospels from the Byzantine Greek Majority version
• Cervantes's Don Quixote, Part I (1605) and Part II (1615).
• First five books of the Old Testament (the Torah) in Hebrew, with vowels.
• First five books of the Old Testament (the Pentateuch) in the Latin Vulgate version.
• First four books of the New Testament (the Gospels) in the Latin Vulgate version.
In some Romance languages, the frequencies of the dozen or so most frequent words deviate significantly from the ideal Zipf distribution, because of those words include articles inflected for grammatical gender and number.
In many East Asian languages, such as Chinese, Lhasa Tibetan, and Vietnamese, each "word" consists of a single syllable; a word of English being often translated to a compound of two such syllables. The rank-frequency table for those "words" deviates significantly from the ideal Zipf law, at both ends of the range.
Even in English, the deviations from the ideal Zipf's law become more apparent as one examines large collections of texts. Analysis of a corpus of 30,000 English texts showed that only about 15% of the texts in have a good fit to Zipf's law. Slight changes in the definition of Zipf's law can increase this percentage up to close to 50%.[41]
In these cases, the observed frequency-rank relation can be modeled more accurately as by separate Zipf–Mandelbrot laws distributions for different subsets or subtypes of words. This is the case for the frequency-rank plot of the first 10 million words of the English Wikipedia. In particular, the frequencies of the closed class of function words in English is better described with s lower than 1, while open-ended vocabulary growth with document size and corpus size require s greater than 1 for convergence of the Generalized Harmonic Series.[3]
When a text is encrypted in such a way that every occurrence of each distinct plaintext word is always mapped to the same encrypted word (as in the case of simple substitution ciphers, like the Caesar ciphers, or simple codebook ciphers), the frequency-rank distribution is not affected. On the other hand, if separate occurrences of the same word may be mapped to two or more different words (as happens with the Vigenère cipher), the Zipf distribution will typically have a flat part at the high-frequency end.
Applications
Zipf's law has been used for extraction of parallel fragments of texts out of comparable corpora.[42] Zipf's law has also been used in the search for extraterrestrial intelligence.[43][44]
The frequency-rank word distribution is often characteristic of the author and changes little over time. This feature has been used in the analysis of texts for authorship attribution.[45][46]
The word-like sign groups of the 15th-century codex Voynich Manuscript have been found to satisfy Zipf's law, suggesting that text is most likely not a hoax but rather written in an obscure language or cipher.[47][48]
See also
• 1% rule (Internet culture) – Hypothesis that more people will lurk in a virtual community than will participatePages displaying short descriptions of redirect targets
• Benford's law – Observation that in many real-life datasets, the leading digit is likely to be small
• Bradford's law – Pattern of references in science journals
• Brevity law – Linguistics law
• Demographic gravitation
• Frequency list – Bare list of a language's words in corpus linguisticsPages displaying short descriptions of redirect targets
• Gibrat's law – Economic principle
• Hapax legomenon – Word that only appears once in a given text or record
• Heaps' law – Heuristic for distinct words in a document
• King effect – Phenomenon in statistics where highest-ranked data points are outliers
• Long tail – Feature of some statistical distributions
• Lorenz curve – Graphical representation of the distribution of income or of wealth
• Lotka's law – An application of Zipf's law describing the frequency of publication by authors in any given field
• Menzerath's law – Linguistic law
• Pareto distribution – Probability distribution
• Pareto principle – Statistical principle about ratio of effects to causes, a.k.a. the "80–20 rule"
• Price's law – Historian of SciencePages displaying short descriptions of redirect targets
• Principle of least effort – Idea that agents prefer to do what's easiest
• Rank-size distribution – distribution of size by rankPages displaying wikidata descriptions as a fallback
• Stigler's law of eponymy – Observation that no scientific discovery is named after its discoverer
Notes
1. as Zipf acknowledged[5]: 546
References
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2. Fagan, Stephen; Gençay, Ramazan (2010), "An introduction to textual econometrics", in Ullah, Aman; Giles, David E. A. (eds.), Handbook of Empirical Economics and Finance, CRC Press, pp. 133–153, ISBN 9781420070361. P. 139: "For example, in the Brown Corpus, consisting of over one million words, half of the word volume consists of repeated uses of only 135 words."
3. Powers, David M W (1998). Applications and explanations of Zipf's law. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151–160.
4. George K. Zipf (1935): The Psychobiology of Language. Houghton-Mifflin.
5. George K. Zipf (1949). Human Behavior and the Principle of Least Effort. Cambridge, Massachusetts: Addison-Wesley. p. 1.
6. Auerbach F. (1913) Das Gesetz der Bevölkerungskonzentration. Petermann’s Geographische Mitteilungen 59, 74–76
7. Christopher D. Manning, Hinrich Schütze Foundations of Statistical Natural Language Processing, MIT Press (1999), ISBN 978-0-262-13360-9, p. 24
8. Dewey, Godfrey. Relativ frequency of English speech sounds. Harvard University Press, 1923.
9. Condon, EDWARD U. "Statistics of vocabulary." Science 67.1733 (1928): 300-300.
10. George K. Zipf (1932): Selected Studies on the Principle of Relative Frequency in Language. Harvard, MA: Harvard University Press.
11. Zipf, George Kingsley (1942). "The Unity of Nature, Least-Action, and Natural Social Science". Sociometry. 5 (1): 48–62. doi:10.2307/2784953. ISSN 0038-0431. JSTOR 2784953.
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Further reading
• Alexander Gelbukh and Grigori Sidorov (2001) "Zipf and Heaps Laws’ Coefficients Depend on Language". Proc. CICLing-2001, Conference on Intelligent Text Processing and Computational Linguistics, February 18–24, 2001, Mexico City. Lecture Notes in Computer Science N 2004, ISSN 0302-9743, ISBN 3-540-41687-0, Springer-Verlag: 332–335.
• Kali R. (2003) "The city as a giant component: a random graph approach to Zipf's law," Applied Economics Letters 10: 717–720(4)
• Shyklo A. (2017); Simple Explanation of Zipf's Mystery via New Rank-Share Distribution, Derived from Combinatorics of the Ranking Process, Available at SSRN: https://ssrn.com/abstract=2918642.
External links
Library resources about
Zipf's law
• Resources in your library
• Resources in other libraries
Wikimedia Commons has media related to Zipf's law.
• Strogatz, Steven (2009-05-29). "Guest Column: Math and the City". The New York Times. Archived from the original on 2015-09-27. Retrieved 2009-05-29.—An article on Zipf's law applied to city populations
• Seeing Around Corners (Artificial societies turn up Zipf's law)
• PlanetMath article on Zipf's law
• Distributions de type "fractal parabolique" dans la Nature (French, with English summary) Archived 2004-10-24 at the Wayback Machine
• An analysis of income distribution
• Zipf List of French words Archived 2007-06-23 at the Wayback Machine
• Zipf list for English, French, Spanish, Italian, Swedish, Icelandic, Latin, Portuguese and Finnish from Gutenberg Project and online calculator to rank words in texts Archived 2011-04-08 at the Wayback Machine
• Citations and the Zipf–Mandelbrot's law
• Zipf's Law examples and modelling (1985)
• Complex systems: Unzipping Zipf's law (2011)
• Benford’s law, Zipf’s law, and the Pareto distribution by Terence Tao.
• "Zipf law", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
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• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
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supported
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• Johnson's SU
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• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
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with support
whose type varies
• generalized chi-squared
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Mixed
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bivariate von Mises
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Singular
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| Wikipedia |
Zlil Sela
Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution[1] of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.[2]
Biographical data
Sela received his Ph.D. in 1991 from the Hebrew University of Jerusalem, where his doctoral advisor was Eliyahu Rips. Prior to his current appointment at the Hebrew University, he held an Associate Professor position at Columbia University in New York.[3] While at Columbia, Sela won the Sloan Fellowship from the Sloan Foundation.[3][4]
Sela gave an Invited Address at the 2002 International Congress of Mathematicians in Beijing.[2][5] He gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic,[6] and he delivered an AMS Invited Address at the October 2003 meeting of the American Mathematical Society[7] and the 2005 Tarski Lectures at the University of California at Berkeley.[8] He was also awarded the 2003 Erdős Prize from the Israel Mathematical Union.[9] Sela also received the 2008 Carol Karp Prize from the Association for Symbolic Logic for his work on the Tarski conjecture and on discovering and developing new connections between model theory and geometric group theory.[10][11]
Mathematical contributions
Sela's early important work was his solution[1] in mid-1990s of the isomorphism problem for torsion-free word-hyperbolic groups. The machinery of group actions on real trees, developed by Eliyahu Rips, played a key role in Sela's approach. The solution of the isomorphism problem also relied on the notion of canonical representatives for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper.[12] The machinery of the canonical representatives allowed Rips and Sela to prove[12] algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in free groups, where the Makanin–Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani[13] to the case of relatively hyperbolic groups and played a key role in the solution of the isomorphism problem for toral relatively hyperbolic groups.[14]
In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups,[15] motivated by the notion of a JSJ decomposition for 3-manifolds. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclic subgroups. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups[16] and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians.[17][18][19][20] Sela applied a combination of his JSJ-decomposition and real tree techniques to prove that torsion-free word-hyperbolic groups are Hopfian.[21] This result and Sela's approach were later generalized by others to finitely generated subgroups of hyperbolic groups[22] and to the setting of relatively hyperbolic groups.
Sela's most important work came in early 2000s when he produced a solution to a famous Tarski conjecture. Namely, in a long series of papers,[23][24][25][26][27][28][29] he proved that any two non-abelian finitely generated free groups have the same first-order theory. Sela's work relied on applying his earlier JSJ-decomposition and real tree techniques as well as developing new ideas and machinery of "algebraic geometry" over free groups.
Sela pushed this work further to study first-order theory of arbitrary torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first-order theory as) a given torsion-free word-hyperbolic group. In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word-hyperbolic group then G is word-hyperbolic as well.
Sela also proved that the first-order theory of a finitely generated free group is stable in the model-theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory.
An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich and Alexei Myasnikov.[30][31][32][33]
The work of Sela on first-order theory of free and word-hyperbolic groups substantially influenced the development of geometric group theory, in particular by stimulating the development and the study of the notion of limit groups and of relatively hyperbolic groups.[34]
Sela's classification theorem
Theorem. Two non-abelian torsion-free hyperbolic groups are elementarily equivalent if and only if their cores are isomorphic.[35]
Published work
• Sela, Zlil; Rips, Eliyahu (1995), "Canonical representatives and equations in hyperbolic groups", Inventiones Mathematicae, 120 (3): 489–512, Bibcode:1995InMat.120..489R, doi:10.1007/BF01241140, MR 1334482, S2CID 121404710
• Sela, Zlil (1995), "The isomorphism problem for hyperbolic groups", Annals of Mathematics, Second Series, 141 (2): 217–283, doi:10.2307/2118520, JSTOR 2118520, MR 1324134
• Sela, Zlil (1997), "Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II.", Geometric and Functional Analysis, 7 (3): 561–593, doi:10.1007/s000390050019, MR 1466338, S2CID 120486267
• Sela, Zlil; Rips, Eliyahu (1997), "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition", Annals of Mathematics, Second Series, 146 (1): 53–109, doi:10.2307/2951832, JSTOR 2951832, MR 1469317
• Sela, Zlil (1997), "Acylindrical accessibility for groups", Inventiones Mathematicae, 129 (3): 527–565, Bibcode:1997InMat.129..527S, doi:10.1007/s002220050172, S2CID 122548154 (Sela's theorem on acylindrical accessibility for groups)[36]
• Sela, Zlil (2001), "Diophantine geometry over groups. I. Makanin-Razborov diagrams" (PDF), Publications Mathématiques de l'IHÉS, 93 (1): 31–105, doi:10.1007/s10240-001-8188-y, MR 1863735, S2CID 51799226
• Sela, Zlil (2003), "Diophantine geometry over groups. II. Completions, closures and formal solutions", Israel Journal of Mathematics, 134 (1): 173–254, doi:10.1007/BF02787407, MR 1972179
• Sela, Zlil (2006), "Diophantine geometry over groups. VI. The elementary theory of a free group", Geometric and Functional Analysis, 16 (3): 707–730, doi:10.1007/s00039-006-0565-8, MR 2238945, S2CID 123197664
See also
• Geometric group theory
• Stable theory
• Free group
• Word-hyperbolic group
• Group isomorphism problem
• Real trees
• JSJ decomposition
References
1. Z. Sela. "The isomorphism problem for hyperbolic groups. I." Annals of Mathematics (2), vol. 141 (1995), no. 2, pp. 217–283.
2. Z. Sela. Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87 92, Higher Ed. Press, Beijing, 2002. ISBN 7-04-008690-5
3. Faculty Members Win Fellowships Columbia University Record, May 15, 1996, Vol. 21, No. 27.
4. Sloan Fellowships Awarded Notices of the American Mathematical Society, vol. 43 (1996), no. 7, pp. 781–782
5. Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345
6. The 2002 annual meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic, vol. 9 (2003), pp. 51–70
7. AMS Meeting at Binghamton, New York. Notices of the American Mathematical Society, vol. 50 (2003), no. 9, p. 1174
8. 2005 Tarski Lectures. Department of Mathematics, University of California at Berkeley. Accessed September 14, 2008.
9. Erdős Prize. Israel Mathematical Union. Accessed September 14, 2008
10. Karp Prize Recipients. Archived 2008-05-13 at the Wayback Machine Association for Symbolic Logic. Accessed September 13, 2008
11. ASL Karp and Sacks Prizes Awarded, Notices of the American Mathematical Society, vol. 56 (2009), no. 5, p. 638
12. Z. Sela, and E. Rips. Canonical representatives and equations in hyperbolic groups, Inventiones Mathematicae vol. 120 (1995), no. 3, pp. 489–512
13. François Dahmani. "Accidental parabolics and relatively hyperbolic groups." Israel Journal of Mathematics, vol. 153 (2006), pp. 93–127
14. François Dahmani, and Daniel Groves, "The isomorphism problem for toral relatively hyperbolic groups". Publications Mathématiques de l'IHÉS, vol. 107 (2008), pp. 211–290
15. Z. Sela. "Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II." Geometric and Functional Analysis, vol. 7 (1997), no. 3, pp. 561–593
16. E. Rips, and Z. Sela. "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition." Annals of Mathematics (2), vol. 146 (1997), no. 1, pp. 53–109
17. M. J. Dunwoody, and M. E. Sageev. "JSJ-splittings for finitely presented groups over slender groups." Inventiones Mathematicae, vol. 135 (1999), no. 1, pp. 25 44
18. P. Scott and G. A. Swarup. "Regular neighbourhoods and canonical decompositions for groups." Electronic Research Announcements of the American Mathematical Society, vol. 8 (2002), pp. 20–28
19. B. H. Bowditch. "Cut points and canonical splittings of hyperbolic groups." Acta Mathematica, vol. 180 (1998), no. 2, pp. 145–186
20. K. Fujiwara, and P. Papasoglu, "JSJ-decompositions of finitely presented groups and complexes of groups." Geometric and Functional Analysis, vol. 16 (2006), no. 1, pp. 70–125
21. Zlil Sela, "Endomorphisms of hyperbolic groups. I. The Hopf property." Topology, vol. 38 (1999), no. 2, pp. 301–321
22. Inna Bumagina, "The Hopf property for subgroups of hyperbolic groups." Geometriae Dedicata, vol. 106 (2004), pp. 211–230
23. Z. Sela. "Diophantine geometry over groups. I. Makanin-Razborov diagrams." Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 31–105
24. Z. Sela. Diophantine geometry over groups. II. Completions, closures and formal solutions. Israel Journal of Mathematics, vol. 134 (2003), pp. 173–254
25. Z. Sela. "Diophantine geometry over groups. III. Rigid and solid solutions." Israel Journal of Mathematics, vol. 147 (2005), pp. 1–73
26. Z. Sela. "Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence." Israel Journal of Mathematics, vol. 143 (2004), pp. 1–130
27. Z. Sela. "Diophantine geometry over groups. V1. Quantifier elimination. I." Israel Journal of Mathematics, vol. 150 (2005), pp. 1–197
28. Z. Sela. "Diophantine geometry over groups. V2. Quantifier elimination. II." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 537–706
29. Z. Sela. "Diophantine geometry over groups. VI. The elementary theory of a free group." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 707–730
30. O. Kharlampovich, and A. Myasnikov. "Tarski's problem about the elementary theory of free groups has a positive solution." Electronic Research Announcements of the American Mathematical Society, vol. 4 (1998), pp. 101–108
31. O. Kharlampovich, and A. Myasnikov. Implicit function theorem over free groups. Journal of Algebra, vol. 290 (2005), no. 1, pp. 1–203
32. O. Kharlampovich, and A. Myasnikov. "Algebraic geometry over free groups: lifting solutions into generic points." Groups, languages, algorithms, pp. 213–318, Contemporary Mathematics, vol. 378, American Mathematical Society, Providence, RI, 2005
33. O. Kharlampovich, and A. Myasnikov. "Elementary theory of free non-abelian groups." Journal of Algebra, vol. 302 (2006), no. 2, pp. 451–552
34. Frédéric Paulin. Sur la théorie élémentaire des groupes libres (d'après Sela). Astérisque No. 294 (2004), pp. 63–402
35. Guirardel, Vincent; Levitt, Gilbert; Salinos, Rizos (2020). "Towers and the first-order theory of hyperbolic groups". arXiv:2007.14148 [math.GR]. (See p. 8.)
36. Kapovich, Ilya; Weidmann, Richard (2002). "Acylindrical accessibility for groups acting on R-tree". arXiv:math/0210308.
External links
• Zlil Sela's webpage at the Hebrew University
• Zlil Sela at the Mathematics Genealogy Project
Authority control
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• VIAF
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| Wikipedia |
Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set $\{0,1,\dots ,n-1\}$ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
This quotient group, usually denoted $(\mathbb {Z} /n\mathbb {Z} )^{\times }$, is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: $|(\mathbb {Z} /n\mathbb {Z} )^{\times }|=\varphi (n).$ For prime n the group is cyclic, and in general the structure is easy to describe, but no simple general formula for finding generators is known.
Group axioms
It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n that are coprime to n satisfy the axioms for an abelian group.
Indeed, a is coprime to n if and only if gcd(a, n) = 1. Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n), hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined.
Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under multiplication.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity.
Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ 1 (mod n). It exists precisely when a is coprime to n, because in that case gcd(a, n) = 1 and by Bézout's lemma there are integers x and y satisfying ax + ny = 1. Notice that the equation ax + ny = 1 implies that x is coprime to n, so the multiplicative inverse belongs to the group.
Notation
The set of (congruence classes of) integers modulo n with the operations of addition and multiplication is a ring. It is denoted $\mathbb {Z} /n\mathbb {Z} $ or $\mathbb {Z} /(n)$ (the notation refers to taking the quotient of integers modulo the ideal $n\mathbb {Z} $ or $(n)$ consisting of the multiples of n). Outside of number theory the simpler notation $\mathbb {Z} _{n}$ is often used, though it can be confused with the p-adic integers when n is a prime number.
The multiplicative group of integers modulo n, which is the group of units in this ring, may be written as (depending on the author) $(\mathbb {Z} /n\mathbb {Z} )^{\times },$ $(\mathbb {Z} /n\mathbb {Z} )^{*},$ $\mathrm {U} (\mathbb {Z} /n\mathbb {Z} ),$ $\mathrm {E} (\mathbb {Z} /n\mathbb {Z} )$ (for German Einheit, which translates as unit), $\mathbb {Z} _{n}^{*}$, or similar notations. This article uses $(\mathbb {Z} /n\mathbb {Z} )^{\times }.$
The notation $\mathrm {C} _{n}$ refers to the cyclic group of order n. It is isomorphic to the group of integers modulo n under addition. Note that $\mathbb {Z} /n\mathbb {Z} $ or $\mathbb {Z} _{n}$ may also refer to the group under addition. For example, the multiplicative group $(\mathbb {Z} /p\mathbb {Z} )^{\times }$ for a prime p is cyclic and hence isomorphic to the additive group $\mathbb {Z} /(p-1)\mathbb {Z} $, but the isomorphism is not obvious.
Structure
The order of the multiplicative group of integers modulo n is the number of integers in $\{0,1,\dots ,n-1\}$ coprime to n. It is given by Euler's totient function: $|(\mathbb {Z} /n\mathbb {Z} )^{\times }|=\varphi (n)$ (sequence A000010 in the OEIS). For prime p, $\varphi (p)=p-1$.
Cyclic case
Main article: primitive root modulo n
The group $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is cyclic if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime and k > 0. For all other values of n the group is not cyclic.[1][2][3] This was first proved by Gauss.[4]
This means that for these n:
$(\mathbb {Z} /n\mathbb {Z} )^{\times }\cong \mathrm {C} _{\varphi (n)},$ where $\varphi (p^{k})=\varphi (2p^{k})=p^{k}-p^{k-1}.$
By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers $g^{0},g^{1},g^{2},\dots ,$ give all possible residues modulo n coprime to n (the first $\varphi (n)$ powers $g^{0},\dots ,g^{\varphi (n)-1}$ give each exactly once). A generator of $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is called a primitive root modulo n.[5] If there is any generator, then there are $\varphi (\varphi (n))$ of them.
Powers of 2
Modulo 1 any two integers are congruent, i.e., there is only one congruence class, [0], coprime to 1. Therefore, $(\mathbb {Z} /1\,\mathbb {Z} )^{\times }\cong \mathrm {C} _{1}$ is the trivial group with φ(1) = 1 element. Because of its trivial nature, the case of congruences modulo 1 is generally ignored and some authors choose not to include the case of n = 1 in theorem statements.
Modulo 2 there is only one coprime congruence class, [1], so $(\mathbb {Z} /2\mathbb {Z} )^{\times }\cong \mathrm {C} _{1}$ is the trivial group.
Modulo 4 there are two coprime congruence classes, [1] and [3], so $(\mathbb {Z} /4\mathbb {Z} )^{\times }\cong \mathrm {C} _{2},$ the cyclic group with two elements.
Modulo 8 there are four coprime congruence classes, [1], [3], [5] and [7]. The square of each of these is 1, so $(\mathbb {Z} /8\mathbb {Z} )^{\times }\cong \mathrm {C} _{2}\times \mathrm {C} _{2},$ the Klein four-group.
Modulo 16 there are eight coprime congruence classes [1], [3], [5], [7], [9], [11], [13] and [15]. $\{\pm 1,\pm 7\}\cong \mathrm {C} _{2}\times \mathrm {C} _{2},$ is the 2-torsion subgroup (i.e., the square of each element is 1), so $(\mathbb {Z} /16\mathbb {Z} )^{\times }$ is not cyclic. The powers of 3, $\{1,3,9,11\}$ are a subgroup of order 4, as are the powers of 5, $\{1,5,9,13\}.$ Thus $(\mathbb {Z} /16\mathbb {Z} )^{\times }\cong \mathrm {C} _{2}\times \mathrm {C} _{4}.$
The pattern shown by 8 and 16 holds[6] for higher powers 2k, k > 2: $\{\pm 1,2^{k-1}\pm 1\}\cong \mathrm {C} _{2}\times \mathrm {C} _{2},$ is the 2-torsion subgroup (so $(\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }$ is not cyclic) and the powers of 3 are a cyclic subgroup of order 2k − 2, so $(\mathbb {Z} /2^{k}\mathbb {Z} )^{\times }\cong \mathrm {C} _{2}\times \mathrm {C} _{2^{k-2}}.$
General composite numbers
By the fundamental theorem of finite abelian groups, the group $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is isomorphic to a direct product of cyclic groups of prime power orders.
More specifically, the Chinese remainder theorem[7] says that if $\;\;n=p_{1}^{k_{1}}p_{2}^{k_{2}}p_{3}^{k_{3}}\dots ,\;$ then the ring $\mathbb {Z} /n\mathbb {Z} $ is the direct product of the rings corresponding to each of its prime power factors:
$\mathbb {Z} /n\mathbb {Z} \cong \mathbb {Z} /{p_{1}^{k_{1}}}\mathbb {Z} \;\times \;\mathbb {Z} /{p_{2}^{k_{2}}}\mathbb {Z} \;\times \;\mathbb {Z} /{p_{3}^{k_{3}}}\mathbb {Z} \dots \;\;$
Similarly, the group of units $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is the direct product of the groups corresponding to each of the prime power factors:
$(\mathbb {Z} /n\mathbb {Z} )^{\times }\cong (\mathbb {Z} /{p_{1}^{k_{1}}}\mathbb {Z} )^{\times }\times (\mathbb {Z} /{p_{2}^{k_{2}}}\mathbb {Z} )^{\times }\times (\mathbb {Z} /{p_{3}^{k_{3}}}\mathbb {Z} )^{\times }\dots \;.$
For each odd prime power $p^{k}$ the corresponding factor $(\mathbb {Z} /{p^{k}}\mathbb {Z} )^{\times }$ is the cyclic group of order $\varphi (p^{k})=p^{k}-p^{k-1}$, which may further factor into cyclic groups of prime-power orders. For powers of 2 the factor $(\mathbb {Z} /{2^{k}}\mathbb {Z} )^{\times }$ is not cyclic unless k = 0, 1, 2, but factors into cyclic groups as described above.
The order of the group $\varphi (n)$ is the product of the orders of the cyclic groups in the direct product. The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function $\lambda (n)$ (sequence A002322 in the OEIS). In other words, $\lambda (n)$ is the smallest number such that for each a coprime to n, $a^{\lambda (n)}\equiv 1{\pmod {n}}$ holds. It divides $\varphi (n)$ and is equal to it if and only if the group is cyclic.
Subgroup of false witnesses
If n is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power n − 1, are congruent to 1 modulo n. (Because the residue 1 when raised to any power is congruent to 1 modulo n, the set of such elements is nonempty.)[8] One could say, because of Fermat's Little Theorem, that such residues are "false positives" or "false witnesses" for the primality of n. The number 2 is the residue most often used in this basic primality check, hence 341 = 11 × 31 is famous since 2340 is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the 300 element multiplicative group mod 341.
n = 9
The smallest example with a nontrivial subgroup of false witnesses is 9 = 3 × 3. There are 6 residues coprime to 9: 1, 2, 4, 5, 7, 8. Since 8 is congruent to −1 modulo 9, it follows that 88 is congruent to 1 modulo 9. So 1 and 8 are false positives for the "primality" of 9 (since 9 is not actually prime). These are in fact the only ones, so the subgroup {1,8} is the subgroup of false witnesses. The same argument shows that n − 1 is a "false witness" for any odd composite n.
n = 91
For n = 91 (= 7 × 13), there are $\varphi (91)=72$ residues coprime to 91, half of them (i.e., 36 of them) are false witnesses of 91, namely 1, 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, and 90, since for these values of x, x90 is congruent to 1 mod 91.
n = 561
n = 561 (= 3 × 11 × 17) is a Carmichael number, thus s560 is congruent to 1 modulo 561 for any integer s coprime to 561. The subgroup of false witnesses is, in this case, not proper; it is the entire group of multiplicative units modulo 561, which consists of 320 residues.
Examples
This table shows the cyclic decomposition of $(\mathbb {Z} /n\mathbb {Z} )^{\times }$ and a generating set for n ≤ 128. The decomposition and generating sets are not unique; for example,
$\displaystyle {\begin{aligned}(\mathbb {Z} /35\mathbb {Z} )^{\times }&\cong (\mathbb {Z} /5\mathbb {Z} )^{\times }\times (\mathbb {Z} /7\mathbb {Z} )^{\times }\cong \mathrm {C} _{4}\times \mathrm {C} _{6}\cong \mathrm {C} _{4}\times \mathrm {C} _{2}\times \mathrm {C} _{3}\cong \mathrm {C} _{2}\times \mathrm {C} _{12}\cong (\mathbb {Z} /4\mathbb {Z} )^{\times }\times (\mathbb {Z} /13\mathbb {Z} )^{\times }\\&\cong (\mathbb {Z} /52\mathbb {Z} )^{\times }\end{aligned}}$
(but $\not \cong \mathrm {C} _{24}\cong \mathrm {C} _{8}\times \mathrm {C} _{3}$). The table below lists the shortest decomposition (among those, the lexicographically first is chosen – this guarantees isomorphic groups are listed with the same decompositions). The generating set is also chosen to be as short as possible, and for n with primitive root, the smallest primitive root modulo n is listed.
For example, take $(\mathbb {Z} /20\mathbb {Z} )^{\times }$. Then $\varphi (20)=8$ means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); $\lambda (20)=4$ means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20). The set {3,19} generates the group, which means that every element of $(\mathbb {Z} /20\mathbb {Z} )^{\times }$ is of the form 3a × 19b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2).
Smallest primitive root mod n are (0 if no root exists)
0, 1, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 6, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, 0, 0, 0, 3, 0, 2, 7, 2, 0, 0, 3, 0, 0, 3, 0, ... (sequence A046145 in the OEIS)
Numbers of the elements in a minimal generating set of mod n are
0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 3, 1, 1, 1, 3, 2, 1, 2, 3, 1, 2, ... (sequence A046072 in the OEIS)
Group structure of (Z/nZ)×
$n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set $n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set $n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set $n\;$$(\mathbb {Z} /n\mathbb {Z} )^{\times }$$\varphi (n)$$\lambda (n)\;$Generating set
1 C1110 33 C2×C1020102, 10 65 C4×C1248122, 12 97 C9696965
2 C1111 34 C1616163 66 C2×C1020105, 7 98 C4242423
3 C2222 35 C2×C1224122, 6 67 C6666662 99 C2×C3060302, 5
4 C2223 36 C2×C61265, 19 68 C2×C1632163, 67 100 C2×C2040203, 99
5 C4442 37 C3636362 69 C2×C2244222, 68 101 C1001001002
6 C2225 38 C1818183 70 C2×C1224123, 69 102 C2×C1632165, 101
7 C6663 39 C2×C1224122, 38 71 C7070707 103 C1021021025
8 C2×C2423, 5 40 C2×C2×C41643, 11, 39 72 C2×C2×C62465, 17, 19 104 C2×C2×C1248123, 5, 103
9 C6662 41 C4040406 73 C7272725 105 C2×C2×C1248122, 29, 41
10 C4443 42 C2×C61265, 13 74 C3636365 106 C5252523
11 C1010102 43 C4242423 75 C2×C2040202, 74 107 C1061061062
12 C2×C2425, 7 44 C2×C1020103, 43 76 C2×C1836183, 37 108 C2×C1836185, 107
13 C1212122 45 C2×C1224122, 44 77 C2×C3060302, 76 109 C1081081086
14 C6663 46 C2222225 78 C2×C1224125, 7 110 C2×C2040203, 109
15 C2×C4842, 14 47 C4646465 79 C7878783 111 C2×C3672362, 110
16 C2×C4843, 15 48 C2×C2×C41645, 7, 47 80 C2×C4×C43243, 7, 79 112 C2×C2×C1248123, 5, 111
17 C1616163 49 C4242423 81 C5454542 113 C1121121123
18 C6665 50 C2020203 82 C4040407 114 C2×C1836185, 37
19 C1818182 51 C2×C1632165, 50 83 C8282822 115 C2×C4488442, 114
20 C2×C4843, 19 52 C2×C1224127, 51 84 C2×C2×C62465, 11, 13 116 C2×C2856283, 115
21 C2×C61262, 20 53 C5252522 85 C4×C1664162, 3 117 C6×C1272122, 17
22 C1010107 54 C1818185 86 C4242423 118 C58585811
23 C2222225 55 C2×C2040202, 21 87 C2×C2856282, 86 119 C2×C4896483, 118
24 C2×C2×C2825, 7, 13 56 C2×C2×C62463, 13, 29 88 C2×C2×C1040103, 5, 7 120 C2×C2×C2×C43247, 11, 19, 29
25 C2020202 57 C2×C1836182, 20 89 C8888883 121 C1101101102
26 C1212127 58 C2828283 90 C2×C1224127, 11 122 C6060607
27 C1818182 59 C5858582 91 C6×C1272122, 3 123 C2×C4080407, 83
28 C2×C61263, 13 60 C2×C2×C41647, 11, 19 92 C2×C2244223, 91 124 C2×C3060303, 61
29 C2828282 61 C6060602 93 C2×C30603011, 61 125 C1001001002
30 C2×C4847, 11 62 C3030303 94 C4646465 126 C6×C63665, 13
31 C3030303 63 C6×C63662, 5 95 C2×C3672362, 94 127 C1261261263
32 C2×C81683, 31 64 C2×C1632163, 63 96 C2×C2×C83285, 17, 31 128 C2×C3264323, 127
See also
• Lenstra elliptic curve factorization
Notes
1. Weisstein, Eric W. "Modulo Multiplication Group". MathWorld.
2. Primitive root, Encyclopedia of Mathematics
3. (Vinogradov 2003, pp. 105–121, § VI PRIMITIVE ROOTS AND INDICES)
4. (Gauss & Clarke 1986, arts. 52–56, 82–891) harv error: no target: CITEREFGaussClarke1986 (help)
5. (Vinogradov 2003, p. 106)
6. (Gauss & Clarke 1986, arts. 90–91) harv error: no target: CITEREFGaussClarke1986 (help)
7. Riesel covers all of this. (Riesel 1994, pp. 267–275)
8. Erdős, Paul; Pomerance, Carl (1986). "On the number of false witnesses for a composite number". Math. Comput. 46 (173): 259–279. doi:10.1090/s0025-5718-1986-0815848-x. Zbl 0586.10003.
References
The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
• Gauss, Carl Friedrich (1986), Disquisitiones Arithmeticae (English translation, Second, corrected edition), translated by Clarke, Arthur A., New York: Springer, ISBN 978-0-387-96254-2
• Gauss, Carl Friedrich (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (German translation, Second edition), translated by Maser, H., New York: Chelsea, ISBN 978-0-8284-0191-3
• Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 978-0-8176-3743-9
• Vinogradov, I. M. (2003), "§ VI PRIMITIVE ROOTS AND INDICES", Elements of Number Theory, Mineola, NY: Dover Publications, pp. 105–121, ISBN 978-0-486-49530-9
External links
• Weisstein, Eric W. "Modulo Multiplication Group". MathWorld.
• Weisstein, Eric W. "Primitive Root". MathWorld.
• Web-based tool to interactively compute group tables by John Jones
• OEIS sequence A033948 (Numbers that have a primitive root (the multiplicative group modulo n is cyclic))
• Numbers n such that the multiplicative group modulo n is the direct product of k cyclic groups:
• k = 2 OEIS sequence A272592 (2 cyclic groups)
• k = 3 OEIS sequence A272593 (3 cyclic groups)
• k = 4 OEIS sequence A272594 (4 cyclic groups)
• OEIS sequence A272590 (The smallest number m such that the multiplicative group modulo m is the direct product of n cyclic groups)
| Wikipedia |
Znám's problem
In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time.
The initial terms of Sylvester's sequence almost solve this problem, except that the last chosen term equals one plus the product of the others, rather than being a proper divisor. Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each $k\geq 5$. Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values.
The Znám problem is closely related to Egyptian fractions. It is known that there are only finitely many solutions for any fixed $k$. It is unknown whether there are any solutions to Znám's problem using only odd numbers, and there remain several other open questions.
The problem
Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. That is, given $k$, what sets of integers
$\{n_{1},\ldots ,n_{k}\}$
are there such that, for each $i$, $n_{i}$ divides but is not equal to
${\Bigl (}\prod _{j\neq i}^{n}n_{j}{\Bigr )}+1?$
A closely related problem concerns sets of integers in which each integer in the set is a divisor, but not necessarily a proper divisor, of one plus the product of the other integers in the set. This problem does not seem to have been named in the literature, and will be referred to as the improper Znám problem. Any solution to Znám's problem is also a solution to the improper Znám problem, but not necessarily vice versa.
History
Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972. Barbeau (1971) had posed the improper Znám problem for $k=3$, and Mordell (1973), independently of Znám, found all solutions to the improper problem for $k\leq 5$. Skula (1975) showed that Znám's problem is unsolvable for $k<5$, and credited J. Janák with finding the solution $\{2,3,11,23,315\}$ for $k=5$.[1]
Examples
Sylvester's sequence is an integer sequence in which each term is one plus the product of the previous terms. The first few terms of the sequence are
2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).
Stopping the sequence early produces a set like $\{2,3,7,43\}$ that almost meets the conditions of Znám's problem, except that the largest value equals one plus the product of the other terms, rather than being a proper divisor.[2] Thus, it is a solution to the improper Znám problem, but not a solution to Znám's problem as it is usually defined.
One solution to the proper Znám problem, for $k=5$, is $\{2,3,7,47,395\}$. A few calculations will show that
3 × 7 × 47 × 395+ 1 =389866, which is divisible by but unequal to 2,
2 × 7 × 47 × 395+ 1 =259911, which is divisible by but unequal to 3,
2 × 3 × 47 × 395+ 1 =111391, which is divisible by but unequal to 7,
2 × 3 × 7 × 395+ 1 =16591, which is divisible by but unequal to 47, and
2 × 3 × 7 × 47+ 1 =1975, which is divisible by but unequal to 395.
Connection to Egyptian fractions
Any solution to the improper Znám problem is equivalent (via division by the product of the values $x_{i}$) to a solution to the equation
$\sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=y,$
where $y$ as well as each $x_{i}$ must be an integer, and conversely any such solution corresponds to a solution to the improper Znám problem. However, all known solutions have $y=1$, so they satisfy the equation
$\sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.$
That is, they lead to an Egyptian fraction representation of the number one as a sum of unit fractions. Several of the cited papers on Znám's problem study also the solutions to this equation. Brenton & Hill (1988) describe an application of the equation in topology, to the classification of singularities on surfaces,[2] and Domaratzki et al. (2005) describe an application to the theory of nondeterministic finite automata.[3]
Number of solutions
The number of solutions to Znám's problem for any $k$ is finite, so it makes sense to count the total number of solutions for each $k$.[4] Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each $k\geq 5$. Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values.[5] The number of solutions for small values of $k$, starting with $k=5$, forms the sequence[6]
2, 5, 18, 96 (sequence A075441 in the OEIS).
Presently, a few solutions are known for $k=9$ and $k=10$, but it is unclear how many solutions remain undiscovered for those values of $k$. However, there are infinitely many solutions if $k$ is not fixed: Cao & Jing (1998) showed that there are at least 39 solutions for each $k\geq 12$, improving earlier results proving the existence of fewer solutions;[7] Sun & Cao (1988) conjecture that the number of solutions for each value of $k$ grows monotonically with $k$.[8]
It is unknown whether there are any solutions to Znám's problem using only odd numbers. With one exception, all known solutions start with 2. If all numbers in a solution to Znám's problem or the improper Znám problem are prime, their product is a primary pseudoperfect number;[9] it is unknown whether infinitely many solutions of this type exist.
References
Notes
1. Barbeau 1971; Mordell 1973; Skula 1975
2. Brenton & Hill 1988.
3. Domaratzki et al. 2005.
4. Janák & Skula 1978.
5. Sun 1983.
6. Brenton & Vasiliu 2002.
7. Cao, Liu & Zhang 1987 Sun & Cao 1988
8. Sun & Cao 1988.
9. Butske, Jaje & Mayernik 2000.
Sources
• Barbeau, G. E. J. (1971), "Problem 179", Canadian Mathematical Bulletin, 14 (1): 129.
• Brenton, Lawrence; Hill, Richard (1988), "On the Diophantine equation $ 1=\sum 1/n_{i}+1/\prod n_{i}$ and a class of homologically trivial complex surface singularities", Pacific Journal of Mathematics, 133 (1): 41–67, doi:10.2140/pjm.1988.133.41, MR 0936356.
• Brenton, Lawrence; Vasiliu, Ana (2002), "Znám's problem", Mathematics Magazine, 75 (1): 3–11, doi:10.2307/3219178, JSTOR 3219178.
• Butske, William; Jaje, Lynda M.; Mayernik, Daniel R. (2000), "On the equation $ \sum _{p|N}{\frac {1}{p}}+{\frac {1}{N}}=1$, pseudoperfect numbers, and perfectly weighted graphs", Mathematics of Computation, 69: 407–420, doi:10.1090/S0025-5718-99-01088-1, MR 1648363.
• Cao, Zhen Fu; Jing, Cheng Ming (1998), "On the number of solutions of Znám's problem", J. Harbin Inst. Tech., 30 (1): 46–49, MR 1651784.
• Cao, Zhen Fu; Liu, Rui; Zhang, Liang Rui (1987), "On the equation $ \sum _{j=1}^{s}(1/x_{j})+(1/(x_{1}\cdots x_{s}))=1$ and Znám's problem", Journal of Number Theory, 27 (2): 206–211, doi:10.1016/0022-314X(87)90062-X, MR 0909837.
• Domaratzki, Michael; Ellul, Keith; Shallit, Jeffrey; Wang, Ming-Wei (2005), "Non-uniqueness and radius of cyclic unary NFAs", International Journal of Foundations of Computer Science, 16 (5): 883–896, doi:10.1142/S0129054105003352, MR 2174328.
• Janák, Jaroslav; Skula, Ladislav (1978), "On the integers $ x_{i}$ for which $ x_{i}|x_{1}\cdots x_{i-1}x_{i+1}\cdots x_{n}+1$", Math. Slovaca, 28 (3): 305–310, MR 0534998.
• Mordell, L. J. (1973), "Systems of congruences", Canadian Mathematical Bulletin, 16 (3): 457–462, doi:10.4153/CMB-1973-077-3, MR 0332650.
• Skula, Ladislav (1975), "On a problem of Znám", Acta Fac. Rerum Natur. Univ. Comenian. Math. (Russian, Slovak summary), 32: 87–90, MR 0539862.
• Sun, Qi (1983), "On a problem of Š. Znám", Sichuan Daxue Xuebao (4): 9–12, MR 0750288.
• Sun, Qi; Cao, Zhen Fu (1988), "On the equation $ \sum _{j=1}^{s}1/x_{j}+1/x_{1}\cdots x_{s}=n$ and the number of solutions of Znám's problem", Northeastern Mathematics Journal, 4 (1): 43–48, MR 0970644.
External links
• Primefan, Solutions to Znám's Problem
• Weisstein, Eric W., "Znám's Problem", MathWorld
| Wikipedia |
Zoel García de Galdeano
Zoel García de Galdeano y Yanguas (5 July 1846 – 28 March 1924) was a Spanish mathematician. He was considered by Julio Rey Pastor as "The apostle of modern mathematics".[1]
Zoel García de Galdeano
Born(1846-07-05)5 July 1846
Pamplona, Spain
Died28 March 1924(1924-03-28) (aged 77)
Zaragoza, Spain
Nationality Spanish
Alma materUniversity of Zaragoza
Scientific career
FieldsMathematics
Biography
His father was a military man, and was killed in war action, so his maternal grandfather, the historian José Yanguas y Miranda (1782-1863), took care of Zoel. To continue his studies, in 1863, Zoel moved to Zaragoza, where he received the title of professor and expert surveyor. In 1869 he graduated as Bachelor. Later he began his studies of Philosophy and Letters, and Sciences at the University of Zaragoza. In 1871, he graduated from these two specialties.[2]
Between 1872 and 1879, Zoel served as professor of mathematics at various schools and institutes in Spain. While he worked in the city of Toledo, he began to write mathematical works that introduced the modern concepts of the European Mathematical in Spain.
In 1889 he obtained the professorship of Analytic geometry at the University of Zaragoza, and in 1896, he was appointed to the professorship of Infinitesimal calculus. He worked at this university until his retirement in 1918.
In 1891, Zoel created El Progreso Matemático, the first strictly mathematical journal published in Spain.[3] He was the principal editor in the two periods in which the journal was published (1891 – 1895 and 1899 – 1900). He was also the first contemporary Spanish mathematician to regularly participate in international congresses of mathematics.
He died in Zaragoza on 28 March 1924.
Notes
1. "García de Galdeano, Zoel". Gran Enciclopedia Aragonesa (in Spanish).
2. Ausejo Martínez, Elena (2010). "Zoel García de Galdeano y Yanguas (Pamplona, 1846 - Zaragoza, 1924)". Números (in Spanish). 73: 5–22. Retrieved 23 June 2016.
3. Hormigón, Mariano (1981). "El Progreso Matemático (1891-1900): Un estudio sobre la primera revista matemática española". Llull: Revista de la Sociedad Española de Historia de las Ciencias y de las Técnicas (in Spanish). Llull. 4 (6): 87–115. Retrieved 23 June 2016.
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| Wikipedia |
Zofia Szmydt
Zofia Szmydt (29 July 1923 – 26 November 2010) was a Polish mathematician working in the areas of differential equations, potential theory and the theory of distributions. She was a winner of the Stefan Banach Prize for mathematics in 1956.
Zofia Szmydt
Born(1923-07-29)July 29, 1923
Warsaw
DiedNovember 26, 2010(2010-11-26) (aged 87)
CitizenshipPolish
Alma materJagiellonian University
AwardsStefan Banach Prize
Scientific career
FieldsDifferential equations
InstitutionsJagiellonian University,
University of Warsaw
ThesisO całkach pierwszych równania różniczkowego
Doctoral advisorTadeusz Ważewski
Life
Zofia Szmydt was born in Warsaw on 29 July 1923. Her mother, Zofia Szmydtowa (née Gąsiorowska), was a historian and philologist.[1]
Szmydt studied at the University of Warsaw in clandestine classes during the Second World War. Following the Warsaw Uprising, she and her family were deported to Krakow.[1]
In 1946, Szmydt graduated in mathematics from the Jagiellonian University. She defended her doctoral thesis in 1949, written under the direction of Tadeusz Ważewski.[2]
Szmydt died on 27 November 2010.[3]
Career
Until 1952, Szmydt worked at the Jagiellonian University. She was a member of the Mathematical Institute of the Polish Academy of Sciences between 1949 and 1971. In 1971, she joined the University of Warsaw where she became a professor in 1984. She retired in 1993.
Contributions
In her 1951 paper, Sur l’allure asymptotique des intégrales des équations différentielles ordinaires, Szmydt applied the topological method by Ważewski to generalizations of Perron's classic results on the asymptotics of systems of solutions of ordinary differential equations.[4]
Szmydt's work on hyperbolic differential equations Sur un problème concernant un systèmes d’équations différentielles hyperboliques d’ordre arbitraire à deux variables indépendantes (1957) proposed a generalised solution for the functional differential equation, which subsumed the Darboux, Cauchy, Picard and Goursat problems as special cases.[5] This was in later literature referred to as the Szmydt problem.[6]
Szmydt's textbook Fourier Transformation and Linear Differential Equations (1971) was the first on the topic to be published in the Polish language.[7] Her motivation was to present the basics of the theory of partial differential equations with a particular emphasis on distributions in limit problems of the classical equations (the heat equation, Schrödinger equation, and the Laplace and Poisson equations).[8]
In Paley–Wiener theorems for the Mellin transformations (1990), Szmydt gave a full characterization of the space of multipliers for Mellin's distribution in terms of the Mellin transform (equivalent to the Paley–Wiener theorem) and established relationships between Schwartz and Mellin distribution spaces.[9]
Honours
In 1956, Szmydt won the Stefan Banach prize of the Polish Academy of Sciences for her research into topological methods in nonlinear ordinary differential equations.[2] In 1973, she awarded the Commander's Cross of the Order of Polonia Restituta for her services to mathematical education.[7]
Selected works
Books
• Topological Imbedding of Laplace Distributions in Laplace Hyperfunctions. Polish Academy of Sciences. 1998. (with Bogdan Ziemian)
• The Mellin Transformation and Fuchsian Type Partial Differential Equations. Springer. 1992. ISBN 978-0792316831. (with Bogdan Ziemian)
• Fourier Transformation and Linear Differential Equations. Springer. 1977. ISBN 978-90-277-0622-5.
Articles
• "Paley–Wiener theorems for the Mellin transformations". Ann. Polon. Math. 51. 1990.
• "Sur un problème concernant un systèmes d'équations différentielles hyperboliques d'ordre arbitraire à deux variables indépendantes". Bull. Acad. Polon. Sci. III (5). 1957.
• "Sur l'allure asymptotique des intégrales des équations différentielles ordinaires". Ann. Soc. Polon. Math. 24 (2). 1951.
References
1. Łysik 2015, p. 283.
2. Kenney 2017, p. 76.
3. Łysik 2015, p. 285.
4. Łysik 2015, p. 287.
5. Karpowicz 2014, p. 866.
6. Łysik 2015, p. 288.
7. Łysik 2015, p. 284.
8. Łysik 2015, p. 290.
9. Łysik 2015, p. 291.
Bibliography
• Łysik, Grzegorz (2015). "Zofia Szmydt (1923–2010)". Wiadomości Matematyczne. 51 (2).
• Karpowicz, Adrian (2014). "The Existence of a Unique Solution of the Hyperbolic Functional Differential Equation" (PDF). Demonstratio Mathematica. XLVII (4).
• Kenney, Emelie Agnes (2017). "Making Her Mark on a Century of Turmoil and Triumph: A Tribute to Polish Women in Mathematics". In Janet L. Beery; Sarah J. Greenwald; Jacqueline A. Jensen-Vallin; Maura B. Mast (eds.). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Springer. ISBN 978-3-319-66694-5.
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| Wikipedia |
Zoll surface
In mathematics, particularly in differential geometry, a Zoll surface, named after Otto Zoll, is a surface homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on S2 obviously has this property, it also has an infinite-dimensional family of geometrically distinct deformations that are still Zoll surfaces. In particular, most Zoll surfaces do not have constant curvature.
Zoll, a student of David Hilbert, discovered the first non-trivial examples.
See also
• Funk transform: The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.
References
• Besse, Arthur L. (1978), Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 93, Springer, Berlin, doi:10.1007/978-3-642-61876-5
• Funk, Paul (1913), "Über Flächen mit lauter geschlossenen geodätischen Linien", Mathematische Annalen, 74: 278–300, doi:10.1007/BF01456044
• Guillemin, Victor (1976), "The Radon transform on Zoll surfaces", Advances in Mathematics, 22 (1): 85–119, doi:10.1016/0001-8708(76)90139-0
• LeBrun, Claude; Mason, L.J. (July 2002), "Zoll manifolds and complex surfaces", Journal of Differential Geometry, 61 (3): 453–535, doi:10.4310/jdg/1090351530
• Zoll, Otto (March 1903). "Über Flächen mit Scharen geschlossener geodätischer Linien". Mathematische Annalen (in German). 57 (1): 108–133. doi:10.1007/bf01449019.
External links
• Tannery's pear, an example of Zoll surface where all closed geodesics (up to the meridians) are shaped like a curved-figure eight.
| Wikipedia |
Zolotarev's lemma
In number theory, Zolotarev's lemma states that the Legendre symbol
$\left({\frac {a}{p}}\right)$
for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation:
$\left({\frac {a}{p}}\right)=\varepsilon (\pi _{a})$
where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a.
For example, take a = 2 and p = 7. The nonzero squares mod 7 are 1, 2, and 4, so (2|7) = 1 and (6|7) = −1. Multiplication by 2 on the nonzero numbers mod 7 has the cycle decomposition (1,2,4)(3,6,5), so the sign of this permutation is 1, which is (2|7). Multiplication by 6 on the nonzero numbers mod 7 has cycle decomposition (1,6)(2,5)(3,4), whose sign is −1, which is (6|7).
Proof
In general, for any finite group G of order n, it is straightforward to determine the signature of the permutation πg made by left-multiplication by the element g of G. The permutation πg will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for πg to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup <g> generated by g should have odd index.
We will apply this to the group of nonzero numbers mod p, which is a cyclic group of order p − 1. The jth power of a primitive root modulo p will have index the greatest common divisor
i = (j, p − 1).
The condition for a nonzero number mod p to be a quadratic non-residue is to be an odd power of a primitive root. The lemma therefore comes down to saying that i is odd when j is odd, which is true a fortiori, and j is odd when i is odd, which is true because p − 1 is even (p is odd).
Another proof
Zolotarev's lemma can be deduced easily from Gauss's lemma and vice versa. The example
$\left({\frac {3}{11}}\right)$,
i.e. the Legendre symbol (a/p) with a = 3 and p = 11, will illustrate how the proof goes. Start with the set {1, 2, . . . , p − 1} arranged as a matrix of two rows such that the sum of the two elements in any column is zero mod p, say:
1 2 3 4 5
10 9 8 7 6
Apply the permutation $U:x\mapsto ax{\pmod {p}}$:
3 6 9 1 4
8 5 2 10 7
The columns still have the property that the sum of two elements in one column is zero mod p. Now apply a permutation V which swaps any pairs in which the upper member was originally a lower member:
3 5 2 1 4
8 6 9 10 7
Finally, apply a permutation W which gets back the original matrix:
1 2 3 4 5
10 9 8 7 6
We have W−1 = VU. Zolotarev's lemma says (a/p) = 1 if and only if the permutation U is even. Gauss's lemma says (a/p) = 1 iff V is even. But W is even, so the two lemmas are equivalent for the given (but arbitrary) a and p.
Jacobi symbol
This interpretation of the Legendre symbol as the sign of a permutation can be extended to the Jacobi symbol
$\left({\frac {a}{n}}\right),$
where a and n are relatively prime integers with odd n > 0: a is invertible mod n, so multiplication by a on Z/nZ is a permutation and a generalization of Zolotarev's lemma is that the Jacobi symbol above is the sign of this permutation.
For example, multiplication by 2 on Z/21Z has cycle decomposition (0)(1,2,4,8,16,11)(3,6,12)(5,10,20,19,17,13)(7,14)(9,18,15), so the sign of this permutation is (1)(−1)(1)(−1)(−1)(1) = −1 and the Jacobi symbol (2|21) is −1. (Note that multiplication by 2 on the units mod 21 is a product of two 6-cycles, so its sign is 1. Thus it's important to use all integers mod n and not just the units mod n to define the right permutation.)
When n = p is an odd prime and a is not divisible by p, multiplication by a fixes 0 mod p, so the sign of multiplication by a on all numbers mod p and on the units mod p have the same sign. But for composite n that is not the case, as we see in the example above.
History
This lemma was introduced by Yegor Ivanovich Zolotarev in an 1872 proof of quadratic reciprocity.
See also: Gauss's lemma
References
• Zolotareff G. (1872). "Nouvelle démonstration de la loi de réciprocité de Legendre" (PDF). Nouvelles Annales de Mathématiques. 2e série. 11: 354–362.
External links
• PlanetMath article on Zolotarev's lemma; includes his proof of quadratic reciprocity
| Wikipedia |
Zonal polynomial
In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials.
They appear as zonal spherical functions of the Gelfand pairs $(S_{2n},H_{n})$ (here, $H_{n}$ is the hyperoctahedral group) and $(Gl_{n}(\mathbb {R} ),O_{n})$, which means that they describe canonical basis of the double class algebras $\mathbb {C} [H_{n}\backslash S_{2n}/H_{n}]$ and $\mathbb {C} [O_{d}(\mathbb {R} )\backslash M_{d}(\mathbb {R} )/O_{d}(\mathbb {R} )]$.
They are applied in multivariate statistics.
The zonal polynomials are the $\alpha =2$ case of the C normalization of the Jack function.
References
• Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.
| Wikipedia |
Zonal spherical function
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.
Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra. For special linear groups, they were independently discovered by Israel Gelfand and Mark Naimark. For complex groups, the theory simplifies significantly, because G is the complexification of K, and the formulas are related to analytic continuations of the Weyl character formula on K. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group G also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra of G on L2(G/K), as differential operators on the symmetric space G/K. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for singular ordinary differential equations: they were obtained in full generality in the 1960s in terms of Harish-Chandra's c-function.
The name "zonal spherical function" comes from the case when G is SO(3,R) acting on a 2-sphere and K is the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis.
Definitions
See also: Hecke algebra of a locally compact group
Let G be a locally compact unimodular topological group and K a compact subgroup and let H1 = L2(G/K). Thus, H1 admits a unitary representation π of G by left translation. This is a subrepresentation of the regular representation, since if H= L2(G) with left and right regular representations λ and ρ of G and P is the orthogonal projection
$P=\int _{K}\rho (k)\,dk$
from H to H1 then H1 can naturally be identified with PH with the action of G given by the restriction of λ.
On the other hand, by von Neumann's commutation theorem[1]
$\lambda (G)^{\prime }=\rho (G)^{\prime \prime },$
where S' denotes the commutant of a set of operators S, so that
$\pi (G)^{\prime }=P\rho (G)^{\prime \prime }P.$
Thus the commutant of π is generated as a von Neumann algebra by operators
$P\rho (f)P=\int _{G}f(g)(P\rho (g)P)\,dg$
where f is a continuous function of compact support on G.[lower-alpha 1]
However Pρ(f) P is just the restriction of ρ(F) to H1, where
$F(g)=\int _{K}\int _{K}f(kgk^{\prime })\,dk\,dk^{\prime }$
is the K-biinvariant continuous function of compact support obtained by averaging f by K on both sides.
Thus the commutant of π is generated by the restriction of the operators ρ(F) with F in Cc(K\G/K), the K-biinvariant continuous functions of compact support on G.
These functions form a * algebra under convolution with involution
$F^{*}(g)={\overline {F(g^{-1})}},$
often called the Hecke algebra for the pair (G, K).
Let A(K\G/K) denote the C* algebra generated by the operators ρ(F) on H1.
The pair (G, K) is said to be a Gelfand pair[2] if one, and hence all, of the following algebras are commutative:
• $\pi (G)^{\prime }$
• $C_{c}(K\backslash G/K)$
• $A(K\backslash G/K).$
Since A(K\G/K) is a commutative C* algebra, by the Gelfand–Naimark theorem it has the form C0(X), where X is the locally compact space of norm continuous * homomorphisms of A(K\G/K) into C.
A concrete realization of the * homomorphisms in X as K-biinvariant uniformly bounded functions on G is obtained as follows.[2][3][4][5][6]
Because of the estimate
$\|\pi (F)\|\leq \int _{G}|F(g)|\,dg\equiv \|F\|_{1},$
the representation π of Cc(K\G/K) in A(K\G/K) extends by continuity to L1(K\G/K), the * algebra of K-biinvariant integrable functions. The image forms a dense * subalgebra of A(K\G/K). The restriction of a * homomorphism χ continuous for the operator norm is also continuous for the norm ||·||1. Since the Banach space dual of L1 is L∞, it follows that
$\chi (\pi (F))=\int _{G}F(g)h(g)\,dg,$
for some unique uniformly bounded K-biinvariant function h on G. These functions h are exactly the zonal spherical functions for the pair (G, K).
Properties
A zonal spherical function h has the following properties:[2]
1. h is uniformly continuous on G
2. $h(x)h(y)=\int _{K}h(xky)\,dk\,\,(x,y\in G).$
3. h(1) =1 (normalisation)
4. h is a positive definite function on G
5. f * h is proportional to h for all f in Cc(K\G/K).
These are easy consequences of the fact that the bounded linear functional χ defined by h is a homomorphism. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection with unitary representations. For semisimple Lie groups, there is a further characterization as eigenfunctions of invariant differential operators on G/K (see below).
In fact, as a special case of the Gelfand–Naimark–Segal construction, there is one-one correspondence between irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions h given by
$h(g)=(\sigma (g)v,v).$
Such irreducible representations are often described as having class one. They are precisely the irreducible representations required to decompose the induced representation π on H1. Each representation σ extends uniquely by continuity to A(K\G/K), so that each zonal spherical function satisfies
$\left|\int _{G}f(g)h(g)\,dg\right|\leq \|\pi (f)\|$
for f in A(K\G/K). Moreover, since the commutant π(G)' is commutative, there is a unique probability measure μ on the space of * homomorphisms X such that
$\int _{G}|f(g)|^{2}\,dg=\int _{X}|\chi (\pi (f))|^{2}\,d\mu (\chi ).$
μ is called the Plancherel measure. Since π(G)' is the centre of the von Neumann algebra generated by G, it also gives the measure associated with the direct integral decomposition of H1 in terms of the irreducible representations σχ.
Gelfand pairs
See also: Gelfand pair
If G is a connected Lie group, then, thanks to the work of Cartan, Malcev, Iwasawa and Chevalley, G has a maximal compact subgroup, unique up to conjugation.[7][8] In this case K is connected and the quotient G/K is diffeomorphic to a Euclidean space. When G is in addition semisimple, this can be seen directly using the Cartan decomposition associated to the symmetric space G/K, a generalisation of the polar decomposition of invertible matrices. Indeed, if τ is the associated period two automorphism of G with fixed point subgroup K, then
$G=P\cdot K,$
where
$P=\{g\in G|\tau (g)=g^{-1}\}.$
Under the exponential map, P is diffeomorphic to the -1 eigenspace of τ in the Lie algebra of G. Since τ preserves K, it induces an automorphism of the Hecke algebra Cc(K\G/K). On the other hand, if F lies in Cc(K\G/K), then
F(τg) = F(g−1),
so that τ induces an anti-automorphism, because inversion does. Hence, when G is semisimple,
• the Hecke algebra is commutative
• (G,K) is a Gelfand pair.
More generally the same argument gives the following criterion of Gelfand for (G,K) to be a Gelfand pair:[9]
• G is a unimodular locally compact group;
• K is a compact subgroup arising as the fixed points of a period two automorphism τ of G;
• G =K·P (not necessarily a direct product), where P is defined as above.
The two most important examples covered by this are when:
• G is a compact connected semisimple Lie group with τ a period two automorphism;[10][11]
• G is a semidirect product $A\rtimes K$, with A a locally compact Abelian group without 2-torsion and τ(a· k)= k·a−1 for a in A and k in K.
The three cases cover the three types of symmetric spaces G/K:[5]
1. Non-compact type, when K is a maximal compact subgroup of a non-compact real semisimple Lie group G;
2. Compact type, when K is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group G;
3. Euclidean type, when A is a finite-dimensional Euclidean space with an orthogonal action of K.
Cartan–Helgason theorem
Let G be a compact semisimple connected and simply connected Lie group and τ a period two automorphism of a G with fixed point subgroup K = Gτ. In this case K is a connected compact Lie group.[5] In addition let T be a maximal torus of G invariant under τ, such that T $\cap $ P is a maximal torus in P, and set[12]
$S=K\cap T=T^{\tau }.$
S is the direct product of a torus and an elementary abelian 2-group.
In 1929 Élie Cartan found a rule to determine the decomposition of L2(G/K) into the direct sum of finite-dimensional irreducible representations of G, which was proved rigorously only in 1970 by Sigurdur Helgason. Because the commutant of G on L2(G/K) is commutative, each irreducible representation appears with multiplicity one. By Frobenius reciprocity for compact groups, the irreducible representations V that occur are precisely those admitting a non-zero vector fixed by K.
From the representation theory of compact semisimple groups, irreducible representations of G are classified by their highest weight. This is specified by a homomorphism of the maximal torus T into T.
The Cartan–Helgason theorem[13][14] states that
the irreducible representations of G admitting a non-zero vector fixed by K are precisely those with highest weights corresponding to homomorphisms trivial on S.
The corresponding irreducible representations are called spherical representations.
The theorem can be proved[5] using the Iwasawa decomposition:
${\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {a}}\oplus {\mathfrak {n}},$
where ${\mathfrak {g}}$, ${\mathfrak {k}}$, ${\mathfrak {a}}$ are the complexifications of the Lie algebras of G, K, A = T $\cap $ P and
${\mathfrak {n}}=\bigoplus {\mathfrak {g}}_{\alpha },$
summed over all eigenspaces for T in ${\mathfrak {g}}$ corresponding to positive roots α not fixed by τ.
Let V be a spherical representation with highest weight vector v0 and K-fixed vector vK. Since v0 is an eigenvector of the solvable Lie algebra ${\mathfrak {a}}\oplus {\mathfrak {n}}$, the Poincaré–Birkhoff–Witt theorem implies that the K-module generated by v0 is the whole of V. If Q is the orthogonal projection onto the fixed points of K in V obtained by averaging over G with respect to Haar measure, it follows that
$\displaystyle {v_{K}=cQv_{0}}$
for some non-zero constant c. Because vK is fixed by S and v0 is an eigenvector for S, the subgroup S must actually fix v0, an equivalent form of the triviality condition on S.
Conversely if v0 is fixed by S, then it can be shown[15] that the matrix coefficient
$\displaystyle {f(g)=(gv_{0},v_{0})}$
is non-negative on K. Since f(1) > 0, it follows that (Qv0, v0) > 0 and hence that Qv0 is a non-zero vector fixed by K.
Harish-Chandra's formula
If G is a non-compact semisimple Lie group, its maximal compact subgroup K acts by conjugation on the component P in the Cartan decomposition. If A is a maximal Abelian subgroup of G contained in P, then A is diffeomorphic to its Lie algebra under the exponential map and, as a further generalisation of the polar decomposition of matrices, every element of P is conjugate under K to an element of A, so that[16]
G =KAK.
There is also an associated Iwasawa decomposition
G =KAN,
where N is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by A. Thus S=AN is a closed solvable subgroup of G, the semidirect product of N by A, and G = KS.
If α in Hom(A,T) is a character of A, then α extends to a character of S, by defining it to be trivial on N. There is a corresponding unitary induced representation σ of G on L2(G/S) = L2(K),[17] a so-called (spherical) principal series representation.
This representation can be described explicitly as follows. Unlike G and K, the solvable Lie group S is not unimodular. Let dx denote left invariant Haar measure on S and ΔS the modular function of S. Then[5]
$\int _{G}f(g)\,dg=\int _{S}\int _{K}f(x\cdot k)\,dx\,dk=\int _{S}\int _{K}f(k\cdot x)\Delta _{S}(x)\,dx\,dk.$
The principal series representation σ is realised on L2(K) as[18]
$(\sigma (g)\xi )(k)=\alpha ^{\prime }(g^{-1}k)^{-1}\,\xi (U(g^{-1}k)),$
where
$g=U(g)\cdot X(g)$
is the Iwasawa decomposition of g with U(g) in K and X(g) in S and
$\alpha ^{\prime }(kx)=\Delta _{S}(x)^{1/2}\alpha (x)$
for k in K and x in S.
The representation σ is irreducible, so that if v denotes the constant function 1 on K, fixed by K,
$\varphi _{\alpha }(g)=(\sigma (g)v,v)$
defines a zonal spherical function of G.
Computing the inner product above leads to Harish-Chandra's formula for the zonal spherical function
$\varphi _{\alpha }(g)=\int _{K}\alpha ^{\prime }(gk)^{-1}\,dk$
as an integral over K.
Harish-Chandra proved that these zonal spherical functions exhaust the characters of the C* algebra generated by the Cc(K \ G / K) acting by right convolution on L2(G / K). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·s, where s is in the Weyl group of A
$W(A)=N_{K}(A)/C_{K}(A),$
the quotient of the normaliser of A in K by its centraliser, a finite reflection group.
It can also be verified directly[2] that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of G-invariant differential operators on G/K and their simultaneous eigenfunctions (see below).[4][5] In the case of complex semisimple groups, Harish-Chandra and Felix Berezin realised independently that the formula simplified considerably and could be proved more directly.[5][19][20][21][22]
The remaining positive-definite zonal spherical functions are given by Harish-Chandra's formula with α in Hom(A,C*) instead of Hom(A,T). Only certain α are permitted and the corresponding irreducible representations arise as analytic continuations of the spherical principal series. This so-called "complementary series" was first studied by Bargmann (1947) for G = SL(2,R) and by Harish-Chandra (1947) and Gelfand & Naimark (1947) for G = SL(2,C). Subsequently in the 1960s, the construction of a complementary series by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, Elias Stein and Bertram Kostant.[23][24][25] Since these irreducible representations are not tempered, they are not usually required for harmonic analysis on G (or G / K).
Eigenfunctions
Harish-Chandra proved[4][5] that zonal spherical functions can be characterised as those normalised positive definite K-invariant functions on G/K that are eigenfunctions of D(G/K), the algebra of invariant differential operators on G. This algebra acts on G/K and commutes with the natural action of G by left translation. It can be identified with the subalgebra of the universal enveloping algebra of G fixed under the adjoint action of K. As for the commutant of G on L2(G/K) and the corresponding Hecke algebra, this algebra of operators is commutative; indeed it is a subalgebra of the algebra of mesurable operators affiliated with the commutant π(G)', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of W(A)-invariant polynomials on the Lie algebra of A, which itself is a polynomial ring by the Chevalley–Shephard–Todd theorem on polynomial invariants of finite reflection groups. The simplest invariant differential operator on G/K is the Laplacian operator; up to a sign this operator is just the image under π of the Casimir operator in the centre of the universal enveloping algebra of G.
Thus a normalised positive definite K-biinvariant function f on G is a zonal spherical function if and only if for each D in D(G/K) there is a constant λD such that
$\displaystyle \pi (D)f=\lambda _{D}f,$
i.e. f is a simultaneous eigenfunction of the operators π(D).
If ψ is a zonal spherical function, then, regarded as a function on G/K, it is an eigenfunction of the Laplacian there, an elliptic differential operator with real analytic coefficients. By analytic elliptic regularity, ψ is a real analytic function on G/K, and hence G.
Harish-Chandra used these facts about the structure of the invariant operators to prove that his formula gave all zonal spherical functions for real semisimple Lie groups.[26][27][28] Indeed, the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula.
Example: SL(2,C)
See also: SL(2,C); Representations of the Lorentz group; and Spectral theory of ordinary differential equations
The group G = SL(2,C) is the complexification of the compact Lie group K = SU(2) and the double cover of the Lorentz group. The infinite-dimensional representations of the Lorentz group were first studied by Dirac in 1945, who considered the discrete series representations, which he termed expansors. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial component of the Laplacian operator.[5]
Indeed, any unimodular complex 2×2 matrix g admits a unique polar decomposition g = pv with v unitary and p positive. In turn p = uau*, with u unitary and a a diagonal matrix with positive entries. Thus g = uaw with w = u* v, so that any K-biinvariant function on G corresponds to a function of the diagonal matrix
$a={\begin{pmatrix}e^{r/2}&0\\0&e^{-r/2}\end{pmatrix}},$
invariant under the Weyl group. Identifying G/K with hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinate r, the Laplacian is given by[29]
$L=-\partial _{r}^{2}-2\coth r\partial _{r}.$
Setting f(r) = sinh (r)·ψ(r), it follows that f is an odd function of r and an eigenfunction of $\partial _{r}^{2}$.
Hence
$\varphi (r)={\sin(\ell r) \over \ell \sinh r}$
where $\ell $ is real.
There is a similar elementary treatment for the generalized Lorentz groups SO(N,1) in Takahashi (1963) and Faraut & Korányi (1994) (recall that SO0(3,1) = SL(2,C) / ±I).
Complex case
If G is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup K. If ${\mathfrak {g}}$ and ${\mathfrak {k}}$ are their Lie algebras, then
${\mathfrak {g}}={\mathfrak {k}}\oplus i{\mathfrak {k}}.$
Let T be a maximal torus in K with Lie algebra ${\mathfrak {t}}$. Then
$A=\exp i{\mathfrak {t}},\,\,P=\exp i{\mathfrak {k}}.$
Let
$W=N_{K}(T)/T$
be the Weyl group of T in K. Recall characters in Hom(T,T) are called weights and can be identified with elements of the weight lattice Λ in Hom(${\mathfrak {t}}$, R) = ${\mathfrak {t}}^{*}$. There is a natural ordering on weights and every finite-dimensional irreducible representation (π, V) of K has a unique highest weight λ. The weights of the adjoint representation of K on ${\mathfrak {k}}\ominus {\mathfrak {t}}$ are called roots and ρ is used to denote half the sum of the positive roots α, Weyl's character formula asserts that for z = exp X in T
$\displaystyle \chi _{\lambda }(e^{X})\equiv {\rm {Tr}}\,\pi (z)=A_{\lambda +\rho }(e^{X})/A_{\rho }(e^{X}),$
where, for μ in ${\mathfrak {t}}^{*}$, Aμ denotes the antisymmetrisation
$\displaystyle A_{\mu }(e^{X})=\sum _{s\in W}\varepsilon (s)e^{i\mu (sX)},$
and ε denotes the sign character of the finite reflection group W.
Weyl's denominator formula expresses the denominator Aρ as a product:
$\displaystyle A_{\rho }(e^{X})=e^{i\rho (X)}\prod _{\alpha >0}(1-e^{-i\alpha (X)}),$
where the product is over the positive roots.
Weyl's dimension formula asserts that
$\displaystyle \chi _{\lambda }(1)\equiv {\rm {dim}}\,V={\prod _{\alpha >0}(\lambda +\rho ,\alpha ) \over \prod _{\alpha >0}(\rho ,\alpha )}.$
where the inner product on ${\mathfrak {t}}^{*}$ is that associated with the Killing form on ${\mathfrak {k}}$.
Now
• every irreducible representation of K extends holomorphically to the complexification G
• every irreducible character χλ(k) of K extends holomorphically to the complexification of K and ${\mathfrak {t}}^{*}$.
• for every λ in Hom(A,T) = $i{\mathfrak {t}}^{*}$, there is a zonal spherical function φλ.
The Berezin–Harish–Chandra formula[5] asserts that for X in $i{\mathfrak {t}}$
$\varphi _{\lambda }(e^{X})={\chi _{\lambda }(e^{X}) \over \chi _{\lambda }(1)}.$
In other words:
• the zonal spherical functions on a complex semisimple Lie group are given by analytic continuation of the formula for the normalised characters.
One of the simplest proofs[30] of this formula involves the radial component on A of the Laplacian on G, a proof formally parallel to Helgason's reworking of Freudenthal's classical proof of the Weyl character formula, using the radial component on T of the Laplacian on K.[31]
In the latter case the class functions on K can be identified with W-invariant functions on T. The radial component of ΔK on T is just the expression for the restriction of ΔK to W-invariant functions on T, where it is given by the formula
$\displaystyle \Delta _{K}=h^{-1}\circ \Delta _{T}\circ h+\|\rho \|^{2},$
where
$\displaystyle h(e^{X})=A_{\rho }(e^{X})$
for X in ${\mathfrak {t}}$. If χ is a character with highest weight λ, it follows that φ = h·χ satisfies
$\Delta _{T}\varphi =(\|\lambda +\rho \|^{2}-\|\rho \|^{2})\varphi .$
Thus for every weight μ with non-zero Fourier coefficient in φ,
$\displaystyle \|\lambda +\rho \|^{2}=\|\mu +\rho \|^{2}.$
The classical argument of Freudenthal shows that μ + ρ must have the form s(λ + ρ) for some s in W, so the character formula follows from the antisymmetry of φ.
Similarly K-biinvariant functions on G can be identified with W(A)-invariant functions on A. The radial component of ΔG on A is just the expression for the restriction of ΔG to W(A)-invariant functions on A. It is given by the formula
$\displaystyle \Delta _{G}=H^{-1}\circ \Delta _{A}\circ H-\|\rho \|^{2},$
where
$\displaystyle H(e^{X})=A_{\rho }(e^{X})$
for X in $i{\mathfrak {t}}$.
The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function
$\displaystyle f=H\cdot \varphi ,$
which is an eigenfunction of the Laplacian ΔA. Since K is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to simple roots, its complexification G is generated by the corresponding homomorphic images of SL(2,C). The formula for zonal spherical functions of SL(2,C) implies that f is a periodic function on $i{\mathfrak {t}}$ with respect to some sublattice. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula.
Example: SL(2,R)
See also: SL(2,R); Representation theory of SL2(R); and Spectral theory of ordinary differential equations
The theory of zonal spherical functions for SL(2,R) originated in the work of Mehler in 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed the Mehler–Fock transform. It was already put on a firm footing in 1910 by Hermann Weyl's important work on the spectral theory of ordinary differential equations. The radial part of the Laplacian in this case leads to a hypergeometric differential equation, the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general semisimple Lie groups. Following the work of Dirac on the discrete series representations of SL(2,R), the general theory of unitary irreducible representations of SL(2,R) was developed independently by Bargmann, Harish-Chandra and Gelfand–Naimark. The irreducible representations of class one, or equivalently the theory of zonal spherical functions, form an important special case of this theory.
The group G = SL(2,R) is a double cover of the 3-dimensional Lorentz group SO(2,1), the symmetry group of the hyperbolic plane with its Poincaré metric. It acts by Möbius transformations. The upper half-plane can be identified with the unit disc by the Cayley transform. Under this identification G becomes identified with the group SU(1,1), also acting by Möbius transformations. Because the action is transitive, both spaces can be identified with G/K, where K = SO(2). The metric is invariant under G and the associated Laplacian is G-invariant, coinciding with the image of the Casimir operator. In the upper half-plane model the Laplacian is given by the formula[5][6]
$\displaystyle \Delta =-4y^{2}(\partial _{x}^{2}+\partial _{y}^{2}).$
If s is a complex number and z = x + i y with y > 0, the function
$\displaystyle f_{s}(z)=y^{s}=\exp({s}\cdot \log y),$
is an eigenfunction of Δ:
$\displaystyle \Delta f_{s}=4s(1-s)f_{s}.$
Since Δ commutes with G, any left translate of fs is also an eigenfunction with the same eigenvalue. In particular, averaging over K, the function
$\varphi _{s}(z)=\int _{K}f_{s}(k\cdot z)\,dk$
is a K-invariant eigenfunction of Δ on G/K. When
$\displaystyle s={1 \over 2}+i\tau ,$
with τ real, these functions give all the zonal spherical functions on G. As with Harish-Chandra's more general formula for semisimple Lie groups, φs is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by K in the principal series. Various arguments are available to prove that there are no others. One of the simplest classical Lie algebraic arguments[5][6][32][33][34] is to note that, since Δ is an elliptic operator with analytic coefficients, by analytic elliptic regularity any eigenfunction is necessarily real analytic. Hence, if the zonal spherical function corresponds to the matrix coefficient for a vector v and representation σ, the vector v is an analytic vector for G and
$\displaystyle (\sigma (e^{X})v,v)=\sum _{n=0}^{\infty }(\sigma (X)^{n}v,v)/n!$
for X in $i{\mathfrak {t}}$. The infinitesimal form of the irreducible unitary representations with a vector fixed by K were worked out classically by Bargmann.[32][33] They correspond precisely to the principal series of SL(2,R). It follows that the zonal spherical function corresponds to a principal series representation.
Another classical argument[35] proceeds by showing that on radial functions the Laplacian has the form
$\displaystyle \Delta =-\partial _{r}^{2}-\coth(r)\cdot \partial _{r},$
so that, as a function of r, the zonal spherical function φ(r) must satisfy the ordinary differential equation
$\displaystyle \varphi ^{\prime \prime }+\coth r\,\varphi ^{\prime }=\alpha \,\varphi $
for some constant α. The change of variables t = sinh r transforms this equation into the hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by[2][36]
$\varphi (r)=P_{\rho }(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }(\cosh r+\sinh r\,\cos \theta )^{\rho }\,d\theta ,$
where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function on G.
There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL(2,R), including the Plancherel formula, from the corresponding results for SL(2,C), which are simple consequences of the Plancherel formula and Fourier inversion formula for R. This "method of descent" works more generally, allowing results for a real semisimple Lie group to be derived by descent from the corresponding results for its complexification.[37][38]
Further directions
• The theory of zonal functions that are not necessarily positive-definite. These are given by the same formulas as above, but without restrictions on the complex parameter s or ρ. They correspond to non-unitary representations.[5]
• Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions.[39] This is an important special case of his Plancherel theorem for real semisimple Lie groups.
• The structure of the Hecke algebra. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between Cc∞(K \ G / K ) and Cc∞(A)W, the subalgebra invariant under the Weyl group.[3] This is straightforward to establish for SL(2,R).[6]
• Spherical functions for Euclidean motion groups and compact Lie groups.[5]
• Spherical functions for p-adic Lie groups. These were studied in depth by Satake and Macdonald.[40][41] Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the Langlands program.
See also
• Plancherel theorem for spherical functions
• Hecke algebra of a locally compact group
• Representations of Lie groups
• Non-commutative harmonic analysis
• Tempered representation
• Positive definite function on a group
• Symmetric space
• Gelfand pair
Notes
1. If σ is a unitary representation of G, then $\sigma (f)=\int _{G}f(g)\sigma (g)\,dg$.
Citations
1. Dixmier 1996, Algèbres hilbertiennes.
2. Dieudonné 1978.
3. Godement 1952.
4. Helgason 2001.
5. Helgason 1984.
6. Lang 1985.
7. Cartier 1954–1955.
8. Hochschild 1965.
9. Dieudonné 1978, pp. 55–57.
10. Dieudonné 1977.
11. Helgason 1978, p. 249.
12. Helgason 1978, pp. 257–264.
13. Helgason 1984, pp. 534–538.
14. Goodman & Wallach 1998, pp. 549–550.
15. Goodman & Wallach 1998, p. 550.
16. Helgason 1978, Chapter IX.
17. Harish-Chandra 1954a, p. 251.
18. Wallach 1973.
19. Berezin 1956a.
20. Berezin 1956b.
21. Harish-Chandra 1954b.
22. Harish-Chandra 1954c.
23. Kunze & Stein 1961.
24. Stein 1970.
25. Kostant 1969.
26. Harish-Chandra 1958.
27. Helgason 2001, pages 418–422, 427-434
28. Helgason 1984, p. 418.
29. Davies 1990.
30. Helgason 1984, pp. 432–433.
31. Helgason 1984, pp. 501–502.
32. Bargmann 1947.
33. Howe & Tan 1992.
34. Wallach 1988.
35. Helgason 2001, p. 405.
36. Bateman & Erdélyi 1953, p. 156.
37. Flensted-Jensen 1978.
38. Helgason 1984, pp. 489–491.
39. Helgason 1984, pp. 434–458.
40. Satake 1963.
41. Macdonald 1971.
Sources
• Bargmann, V. (1947), "Irreducible Unitary Representations of the Lorentz Group", Annals of Mathematics, 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129
• Barnett, Adam; Smart, Nigel P. (2003), "Mental Poker Revisited", Cryptography and Coding, Lecture Notes in Computer Science, vol. 2898, pp. 370–383, doi:10.1007/978-3-540-40974-8_29, ISBN 978-3-540-20663-7
• Bateman, Harry; Erdélyi, Arthur (1953), Higher transcendental functions, Vol. I (PDF), McGraw–Hill, ISBN 0-07-019546-3
• Berezin, F. A. (1956a), "Операторы Лапласа на полупростых группах" [Laplace operators on semisimple groups], Doklady Akademii Nauk SSSR, 107: 9–12.
• Berezin, F. A. (1956b), "Representation of complex semisimple Lie groups in Banach space", Doklady Akademii Nauk SSSR, 110: 897–900
• Brychkov, Yu A.; Prudnikov, A.P. (2001) [1994], "Spherical functions", Encyclopedia of Mathematics, EMS Press
• Cartier, Pierre (1954–1955), Structure topologique des groupes de Lie généraux, Exposé No. 22 (PDF), Séminaire "Sophus Lie", vol. 1.
• Davies, E. B. (1990), Heat Kernels and Spectral Theory, Cambridge University Press, ISBN 0-521-40997-7
• Dieudonné, Jean (1977), Treatise on Analysis, Vol. V, Academic Press
• Dieudonné, Jean (1978), Treatise on Analysis, Vol. VI, Academic Press, ISBN 0-12-215506-8
• Dirac, P. A. M. (1945), "Unitary representations of the Lorentz group", Proceedings of the Royal Society A, 183 (994): 284–295, Bibcode:1945RSPSA.183..284D, doi:10.1098/rspa.1945.0003
• Dixmier, Jacques (1996), Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Les Grands Classiques Gauthier-Villars., Éditions Jacques Gabay, ISBN 2-87647-012-8
• Faraut, Jacques; Korányi, Adam (1994), Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, ISBN 0-19-853477-9, MR 1446489, Chapter XIV.
• Flensted-Jensen, Mogens (1978), "Spherical functions of a real semisimple Lie group. A method of reduction to the complex case", J. Funct. Anal., 30: 106–146, doi:10.1016/0022-1236(78)90058-7
• Gelfand, I.M.; Naimark, M.A. (1947), "Unitary representations of the Lorentz group", Izv. Akad. Nauk SSSR, Ser. Mat., 37: 411–504
• Gelfand, I.M.; Naimark, M.A. (1948), "An analogue of Plancherel's theorem for the complex unimodular group", Doklady Akademii Nauk SSSR, 63: 609–612
• Gelfand, I.M.; Naimark, M.A. (1952), "Unitary representations of the unimodular group containing the identity representation of the unitary subgroup", Trudy Moscov. Mat. Obšč., 1: 423–475
• Godement, Roger (1952), "A theory of spherical functions. I", Transactions of the American Mathematical Society, 73 (3): 496–556, doi:10.2307/1990805, JSTOR 1990805
• Goodman, Roe; Wallach, Nolan (1998), Representations and Invariants of the Classical Groups, Cambridge University Press, ISBN 0-521-66348-2
• Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proceedings of the Royal Society A, 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047
• Harish-Chandra (1954a), "Representations of Semisimple Lie Groups. II", Trans. Amer. Math. Soc., 76 (1): 26–65, doi:10.2307/1990743, JSTOR 1990743, PMC 1063373, PMID 16578367
• Harish-Chandra (1954b), "Representations of Semisimple Lie Groups. III", Trans. Amer. Math. Soc., 76 (2): 234–253, doi:10.2307/1990767, JSTOR 1990767 (Simplification of formula for complex semisimple Lie groups)
• Harish-Chandra (1954c), "The Plancherel formula for complex semisimple Lie groups", Trans. Amer. Math. Soc., 76 (3): 485–528, doi:10.2307/1990793, JSTOR 1990793, PMC 1063477, PMID 16589034 (Second proof of formula for complex semisimple Lie groups)
• Harish-Chandra (1958), "Spherical functions on a semisimple Lie group I, II", Amer. J. Math., 80 (2): 241–310, 553–613, doi:10.2307/2372786, JSTOR 2372786, PMC 528464, PMID 16590028 (Determination of Plancherel measure)
• Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5
• Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3 – via Internet Archive
• Helgason, Sigurdur (2001), Differential geometry and symmetric spaces (reprint of 1962 edition), American Mathematical Society, ISBN 0-8218-2735-9
• Hochschild, Gerhard P. (1965), The structure of Lie groups, Holden–Day
• Howe, Roger; Tan, Eng-chye (1992), Non-abelian Harmonic Analysis: Applications of SL(2,R), Universitext, Springer-Verlag, ISBN 0-387-97768-6
• Kostant, Bertram (1969), "On the existence and irreducibility of certain series of representations", Bull. Amer. Math. Soc., 75 (4): 627–642, doi:10.1090/S0002-9904-1969-12235-4
• Kunze, Raymond A.; Stein, Elias M. (1961), "Analytic continuation of the principal series", Bull. Amer. Math. Soc., 67 (6): 593–596, doi:10.1090/S0002-9904-1961-10705-2
• Lang, Serge (1985), SL(2,R), Graduate Texts in Mathematics, vol. 105, Springer-Verlag, ISBN 0-387-96198-4
• Macdonald, Ian G. (1971), Spherical Functions on a Group of p-adic Type, Publ. Ramanujan Institute, vol. 2, University of Madras
• Satake, I. (1963), "Theory of spherical functions on reductive algebraic groups over p-adic fields", Publ. Math. IHÉS, 18: 5–70, doi:10.1007/bf02684781, S2CID 4666554
• Stein, Elias M. (1970), "Analytic continuation of group representations", Advances in Mathematics, 4 (2): 172–207, doi:10.1016/0001-8708(70)90022-8
• Takahashi, R. (1963), "Sur les représentations unitaires des groupes de Lorentz généralisés", Bull. Soc. Math. France, 91: 289–433, doi:10.24033/bsmf.1598
• Wallach, Nolan (1973), Harmonic Analysis on Homogeneous Spaces, Marcel Decker, ISBN 0-8247-6010-7
• Wallach, Nolan (1988), Real Reductive Groups I, Academic Press, ISBN 0-12-732960-9 – via Internet Archive
External links
• Casselman, William, Notes on spherical functions (PDF)
| Wikipedia |
Zonal spherical harmonics
In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.
On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates by
$Z^{(\ell )}(\theta ,\phi )=P_{\ell }(\cos \theta )$
where Pℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by $Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )$, where x is a point on the sphere representing the fixed axis, and y is the variable of the function. This can be obtained by rotation of the basic zonal harmonic $Z^{(\ell )}(\theta ,\phi ).$
In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define $Z_{\mathbf {x} }^{(\ell )}$ to be the dual representation of the linear functional
$P\mapsto P(\mathbf {x} )$
in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:
$Y(\mathbf {x} )=\int _{S^{n-1}}Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )Y(\mathbf {y} )\,d\Omega (y)$
for all Y ∈ Hℓ. The integral is taken with respect to the invariant probability measure.
Relationship with harmonic potentials
The zonal harmonics appear naturally as coefficients of the Poisson kernel for the unit ball in Rn: for x and y unit vectors,
${\frac {1}{\omega _{n-1}}}{\frac {1-r^{2}}{|\mathbf {x} -r\mathbf {y} |^{n}}}=\sum _{k=0}^{\infty }r^{k}Z_{\mathbf {x} }^{(k)}(\mathbf {y} ),$
where $\omega _{n-1}$ is the surface area of the (n-1)-dimensional sphere. They are also related to the Newton kernel via
${\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }c_{n,k}{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{n+k-2}}}Z_{\mathbf {x} /|\mathbf {x} |}^{(k)}(\mathbf {y} /|\mathbf {y} |)$
where x,y ∈ Rn and the constants cn,k are given by
$c_{n,k}={\frac {1}{\omega _{n-1}}}{\frac {2k+n-2}{(n-2)}}.$
The coefficients of the Taylor series of the Newton kernel (with suitable normalization) are precisely the ultraspherical polynomials. Thus, the zonal spherical harmonics can be expressed as follows. If α = (n−2)/2, then
$Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )={\frac {n+2\ell -2}{n-2}}C_{\ell }^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} )$
where cn,ℓ are the constants above and $C_{\ell }^{(\alpha )}$ is the ultraspherical polynomial of degree ℓ.
Properties
• The zonal spherical harmonics are rotationally invariant, meaning that
$Z_{R\mathbf {x} }^{(\ell )}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )$
for every orthogonal transformation R. Conversely, any function f(x,y) on Sn−1×Sn−1 that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree ℓ zonal harmonic.
• If Y1, ..., Yd is an orthonormal basis of Hℓ, then
$Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )=\sum _{k=1}^{d}Y_{k}(\mathbf {x} ){\overline {Y_{k}(\mathbf {y} )}}.$
• Evaluating at x = y gives
$Z_{\mathbf {x} }^{(\ell )}(\mathbf {x} )=\omega _{n-1}^{-1}\dim \mathbf {H} _{\ell }.$
References
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-08078-9.
| Wikipedia |
Zone theorem
In geometry, the zone theorem is a result that establishes the complexity of the zone of a line in an arrangement of lines.
Definition
A line arrangement, denoted as $A(L)$, is a subdivision of the plane, induced by a set of lines $L$, into cells ($2$-dimensional faces), edges ($1$-dimensional faces) and vertices ($0$-dimensional faces). Given a set of $n$ lines $L$, the line arrangement $A(L)$, and a line $l$ (not belonging to $L$), the zone of $l$ is the set of faces intersected by $l$. The complexity of a zone is the total number of edges in its boundary, expressed as a function of $n$. The zone theorem states that said complexity is $O(n)$.
History
This result was published for the first time in 1985;[1] Chazelle et al. gave the upper bound of $10n+2$ for the complexity of the zone of a line in an arrangement. In 1991,[2] this bound was improved to $\lfloor 9.5n\rfloor -1$, and it was also shown that this is the best possible upper bound up to a small additive factor. Then, in 2011,[3] Rom Pinchasi proved that the complexity of the zone of a line in an arrangement is at most $\lfloor 9.5n\rfloor -3$, and this is a tight bound.
Some paradigms used in the different proofs of the theorem are induction,[1] sweep technique,[2][4] tree construction,[5] and Davenport-Schinzel sequences.[6][7]
Generalizations
Although the most popular version is for arrangements of lines in the plane, there exist some generalizations of the zone theorem. For instance, in dimension $d$, considering arrangements of hyperplanes, the complexity of the zone of a hyperplane $h$ is the number of facets ($d-1$ - dimensional faces) bounding the set of cells ($d$-dimensional faces) intersected by $h$. Analogously, the $d$-dimensional zone theorem states that the complexity of the zone of a hyperplane is $O(n^{d-1})$.[7] There are considerably fewer proofs for the theorem for dimension $d\geq 3$. For the $3$-dimensional case, there are proofs based on sweep techniques and for higher dimensions is used Euler’s relation:[8]
$\sum _{i=0}^{d}(-1)^{i}F_{i}\geq 0.$
Another generalization is considering arrangements of pseudolines (and pseudohyperplanes in dimension $d$) instead of lines (and hyperplanes). Some proofs for the theorem work well in this case since they do not use the straightness of the lines substantially through their arguments.[7]
Motivation
The primary motivation to study the zone complexity in arrangements arises from looking for efficient algorithms to construct arrangements. A classical algorithm is the incremental construction, which can be roughly described as adding the lines one after the other and storing all faces generated by each in an appropriate data structure (the usual structure for arrangements is the doubly connected edge list (DCEL)). Here, the consequence of the zone theorem is that the entire construction of any arrangement of $n$ lines can be done in time $O(n^{2})$, since the insertion of each line takes time $O(n)$.
Notes
1. Chazelle, Guibas & Lee (1985)
2. Bern et al. (1991)
3. Pinchasi (2011)
4. Edelsbrunner, O'Rourke & Seidel (1986)
5. Edelsbrunner & Guibas (1989)
6. Edelsbrunner et al. (1992)
7. Edelsbrunner, Seidel & Sharir (1991)
8. Saxena (2021)
References
• Agarwal, P. K.; Sharir, M. (2000), "Arrangements and their applications" (PDF), in Sack, J.-R.; Urrutia, J. (eds.), Handbook of Computational Geometry, Elsevier, pp. 49–119.
• Agarwal, P. K.; Sharir, M. (2002), "Pseudo-line arrangements: duality, algorithms, and applications", Proc. 13th ACM-SIAM Symposium on Discrete Algorithms (SODA '02), San Francisco: Society for Industrial and Applied Mathematics, pp. 800–809.
• Aharoni, Y.; Halperin, D.; Hanniel, I.; Har-Peled, S.; Linhart, C. (1999), "On-line zone construction in arrangements of lines in the plane", in Vitter, Jeffrey S.; Zaroliagis, Christos D. (eds.), Algorithm Engineering: 3rd International Workshop, WAE'99, London, UK, July 19–21, 1999, Proceedings, Lecture Notes in Computer Science, vol. 1668, Springer-Verlag, pp. 139–153, CiteSeerX 10.1.1.35.7681, doi:10.1007/3-540-48318-7_13, ISBN 978-3-540-66427-7.
• Bern, M. W.; Eppstein, D.; Plassman, P. E.; Yao, F. F. (1991), "Horizon theorems for lines and polygons", in Goodman, J. E.; Pollack, R.; Steiger, W. (eds.), Discrete and Computational Geometry: Papers from the DIMACS Special Year, DIMACS Ser. Discrete Math. and Theoretical Computer Science (6 ed.), Amer. Math. Soc., pp. 45–66, MR 1143288.
• Chazelle, B.; Guibas, L. J.; Lee, D. T. (1985), "The power of geometric duality", BIT Numerical Mathematics, 25 (1): 76–90, doi:10.1007/BF01934990, S2CID 122411548.
• Edelsbrunner, H. (1987), Algorithms in Combinatorial Geometry, EATCS Monographs in Theoretical Computer Science, Springer-Verlag, ISBN 978-3-540-13722-1.
• Edelsbrunner, H.; O'Rourke, J.; Seidel, R. (1986), "Constructing arrangements of lines and hyperplanes with applications", SIAM Journal on Computing, 15 (2): 341–363, doi:10.1137/0215024.
• Edelsbrunner, H.; Guibas, L. J. (1989), "Topologically sweeping an arrangement", Journal of Computer and System Sciences, 38 (1): 165–194, doi:10.1016/0022-0000(89)90038-X.
• Edelsbrunner, H.; Guibas, L. J.; Pach, J.; Pollack, R.; Seidel, R.; Sharir, M. (1992), "Arrangements of curves in the plane—topology, combinatorics, and algorithms", Theoretical Computer Science, 92 (2): 319–336, doi:10.1016/0304-3975(92)90319-B.
• Edelsbrunner, H.; Seidel, R.; Sharir, M. (1991), "On the zone theorem for hyperplane arrangements", New Results and New Trends in Computer Science, Graz, Austria: Springer Science & Business Media, 555: 108, doi:10.1016/0304-3975(92)90319-B
• Grünbaum, B. (1972), Arrangements and Spreads, Regional Conference Series in Mathematics, vol. 10, Providence, R.I.: American Mathematical Society.
• Pinchasi, R. (2011), "The zone theorem revisited", Manuscript
• Saxena, S. (2021), "Zone theorem for arrangements in dimension three", Information Processing Letters, 172: 106161, arXiv:2006.01428, doi:10.1016/j.ipl.2021.106161, S2CID 219179345
| Wikipedia |
Zonohedron
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
Zonohedra that tile space
The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron (including the cube), hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron.
Zonohedra from Minkowski sums
Let $\{v_{0},v_{1},\dots \}$ be a collection of three-dimensional vectors. With each vector $v_{i}$ we may associate a line segment $ \{x_{i}v_{i}\mid 0\leq x_{i}\leq 1\}$. The Minkowski sum $ \ \sum _{i}x_{i}v_{i}\mid 0\leq x_{i}\leq 1\}$ forms a zonohedron, and all zonohedra that contain the origin have this form. The vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, giving zonotopes.
Each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, and by setting all vector lengths equal, we may form an equilateral version of any combinatorial type of zonohedron.
By choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. For instance, generators equally spaced around the equator of a sphere, together with another pair of generators through the poles of the sphere, form zonohedra in the form of prism over regular $2k$-gons: the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc. Generators parallel to the edges of an octahedron form a truncated octahedron, and generators parallel to the long diagonals of a cube form a rhombic dodecahedron.[1]
The Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Thus, the Minkowski sum of a cube and a truncated octahedron forms the truncated cuboctahedron, while the Minkowski sum of the cube and the rhombic dodecahedron forms the truncated rhombic dodecahedron. Both of these zonohedra are simple (three faces meet at each vertex), as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, and rhombic dodecahedron.[1]
Zonohedra from arrangements
The Gauss map of any convex polyhedron maps each face of the polygon to a point on the unit sphere, and maps each edge of the polygon separating a pair of faces to a great circle arc connecting the corresponding two points. In the case of a zonohedron, the edges surrounding each face can be grouped into pairs of parallel edges, and when translated via the Gauss map any such pair becomes a pair of contiguous segments on the same great circle. Thus, the edges of the zonohedron can be grouped into zones of parallel edges, which correspond to the segments of a common great circle on the Gauss map, and the 1-skeleton of the zonohedron can be viewed as the planar dual graph to an arrangement of great circles on the sphere. Conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles.
Any simple zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane. There are three known infinite families of simplicial arrangements, one of which leads to the prisms when converted to zonohedra, and the other two of which correspond to additional infinite families of simple zonohedra. There are also many sporadic examples that do not fit into these three families.[2]
It follows from the correspondence between zonohedra and arrangements, and from the Sylvester–Gallai theorem which (in its projective dual form) proves the existence of crossings of only two lines in any arrangement, that every zonohedron has at least one pair of opposite parallelogram faces. (Squares, rectangles, and rhombuses count for this purpose as special cases of parallelograms.) More strongly, every zonohedron has at least six parallelogram faces, and every zonohedron has a number of parallelogram faces that is linear in its number of generators.[3]
Types of zonohedra
Any prism over a regular polygon with an even number of sides forms a zonohedron. These prisms can be formed so that all faces are regular: two opposite faces are equal to the regular polygon from which the prism was formed, and these are connected by a sequence of square faces. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc.
In addition to this infinite family of regular-faced zonohedra, there are three Archimedean solids, all omnitruncations of the regular forms:
• The truncated octahedron, with 6 square and 8 hexagonal faces. (Omnitruncated tetrahedron)
• The truncated cuboctahedron, with 12 squares, 8 hexagons, and 6 octagons. (Omnitruncated cube)
• The truncated icosidodecahedron, with 30 squares, 20 hexagons and 12 decagons. (Omnitruncated dodecahedron)
In addition, certain Catalan solids (duals of Archimedean solids) are again zonohedra:
• Kepler's rhombic dodecahedron is the dual of the cuboctahedron.
• The rhombic triacontahedron is the dual of the icosidodecahedron.
Others with congruent rhombic faces:
• Bilinski's rhombic dodecahedron.
• Rhombic icosahedron
• Rhombohedron
There are infinitely many zonohedra with rhombic faces that are not all congruent to each other. They include:
• Rhombic enneacontahedron
zonohedron image number of
generators
regular face face
transitive
edge
transitive
vertex
transitive
Parallelohedron
(space-filling)
simple
Cube
4.4.4
3 Yes Yes Yes Yes Yes Yes
Hexagonal prism
4.4.6
4 Yes No No Yes Yes Yes
2n-prism (n > 3)
4.4.2n
n + 1 Yes No No Yes No Yes
Truncated octahedron
4.6.6
6 Yes No No Yes Yes Yes
Truncated cuboctahedron
4.6.8
9 Yes No No Yes No Yes
Truncated icosidodecahedron
4.6.10
15 Yes No No Yes No Yes
Parallelepiped 3 No Yes No No Yes Yes
Rhombic dodecahedron
V3.4.3.4
4 No Yes Yes No Yes No
Bilinski dodecahedron 4 No No No No Yes No
Rhombic icosahedron 5 No No No No No No
Rhombic triacontahedron
V3.5.3.5
6 No Yes Yes No No No
Rhombo-hexagonal dodecahedron 5 No No No No Yes No
Truncated rhombic dodecahedron 7 No No No No No Yes
Dissection of zonohedra
Although it is not generally true that any polyhedron has a dissection into any other polyhedron of the same volume (see Hilbert's third problem), it is known that any two zonohedra of equal volumes can be dissected into each other.
Zonohedrification
Zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron.[4][5]
First the vertices of any seed polyhedron are considered vectors from the polyhedron center. These vectors create the zonohedron which we call the zonohedrification of the original polyhedron. If the seed polyhedron has central symmetry, opposite points define the same direction, so the number of zones in the zonohedron is half the number of vertices of the seed. For any two vertices of the original polyhedron, there are two opposite planes of the zonohedrification which each have two edges parallel to the vertex vectors.
Examples
SymmetryDihedralOctahedralicosahedral
Seed
8 vertex
V4.4.6
6 vertex
{3,4}
8 vertex
{4,3}
12 vertex
3.4.3.4
14 vertex
V3.4.3.4
12 vertex
{3,5}
20 vertex
{5,3}
30 vertex
3.5.3.5
32 vertex
V3.5.3.5
Zonohedron
4 zone
4.4.6
3 zone
{4,3}
4 zone
Rhomb.12
6 zone
4.6.6
7 zone
Ch.cube
6 zone
Rhomb.30
10 zone
Rhomb.90
15 zone
4.6.10
16 zone
Rhomb.90
Zonotopes
The Minkowski sum of line segments in any dimension forms a type of polytope called a zonotope. Equivalently, a zonotope $Z$ generated by vectors $v_{1},...,v_{k}\in \mathbb {R} ^{n}$ is given by $Z=\{a_{1}v_{1}+\cdots +a_{k}v_{k}|\;\forall (j)a_{j}\in [0,1]\}$. Note that in the special case where $k\leq n$, the zonotope $Z$ is a (possibly degenerate) parallelotope.
The facets of any zonotope are themselves zonotopes of one lower dimension; for instance, the faces of zonohedra are zonogons. Examples of four-dimensional zonotopes include the tesseract (Minkowski sums of d mutually perpendicular equal length line segments), the omnitruncated 5-cell, and the truncated 24-cell. Every permutohedron is a zonotope.
Zonotopes and Matroids
Fix a zonotope $Z$ defined from the set of vectors $V=\{v_{1},\dots ,v_{n}\}\subset \mathbb {R} ^{d}$ and let $M$ be the $d\times n$ matrix whose columns are the $v_{i}$. Then the vector matroid ${\underline {\mathcal {M}}}$ on the columns of $M$ encodes a wealth of information about $Z$, that is, many properties of $Z$ are purely combinatorial in nature.
For example, pairs of opposite facets of $Z$ are naturally indexed by the cocircuits of ${\mathcal {M}}$ and if we consider the oriented matroid ${\mathcal {M}}$ represented by ${M}$, then we obtain a bijection between facets of $Z$ and signed cocircuits of ${\mathcal {M}}$ which extends to a poset anti-isomorphism between the face lattice of $Z$ and the covectors of ${\mathcal {M}}$ ordered by component-wise extension of $0\prec +,-$. In particular, if $M$ and $N$ are two matrices that differ by a projective transformation then their respective zonotopes are combinatorially equivalent. The converse of the previous statement does not hold: the segment $[0,2]\subset \mathbb {R} $ is a zonotope and is generated by both $\{2\mathbf {e} _{1}\}$ and by $\{\mathbf {e} _{1},\mathbf {e} _{1}\}$ whose corresponding matrices, $[2]$ and $[1~1]$, do not differ by a projective transformation.
Tilings
Tiling properties of the zonotope $Z$ are also closely related to the oriented matroid ${\mathcal {M}}$ associated to it. First we consider the space-tiling property. The zonotope $Z$ is said to tile $\mathbb {R} ^{d}$ if there is a set of vectors $\Lambda \subset \mathbb {R} ^{d}$ such that the union of all translates $Z+\lambda $ ($\lambda \in \Lambda $) is $\mathbb {R} ^{d}$ and any two translates intersect in a (possibly empty) face of each. Such a zonotope is called a space-tiling zonotope. The following classification of space-tiling zonotopes is due to McMullen:[6] The zonotope $Z$ generated by the vectors $V$ tiles space if and only if the corresponding oriented matroid is regular. So the seemingly geometric condition of being a space-tiling zonotope actually depends only on the combinatorial structure of the generating vectors.
Another family of tilings associated to the zonotope $Z$ are the zonotopal tilings of $Z$. A collection of zonotopes is a zonotopal tiling of $Z$ if it a polyhedral complex with support $Z$, that is, if the union of all zonotopes in the collection is $Z$ and any two intersect in a common (possibly empty) face of each. Many of the images of zonohedra on this page can be viewed as zonotopal tilings of a 2-dimensional zonotope by simply considering them as planar objects (as opposed to planar representations of three dimensional objects). The Bohne-Dress Theorem states that there is a bijection between zonotopal tilings of the zonotope $Z$ and single-element lifts of the oriented matroid ${\mathcal {M}}$ associated to $Z$.[7][8]
Volume
Zonohedra, and n-dimensional zonotopes in general, are noteworthy for admitting a simple analytic formula for their volume.[9]
Let $Z(S)$ be the zonotope $Z=\{a_{1}v_{1}+\cdots +a_{k}v_{k}|\;\forall (j)a_{j}\in [0,1]\}$ generated by a set of vectors $S=\{v_{1},\dots ,v_{k}\in \mathbb {R} ^{n}\}$. Then the n-dimensional volume of $Z(S)$ is given by $\sum _{T\subset S\;:\;|T|=n}|\det(Z(T))|$.
The determinant in this formula makes sense because (as noted above) when the set $T$ has cardinality equal to the dimension $n$ of the ambient space, the zonotope is a parallelotope.
Note that when $k<n$, this formula simply states that the zonotope has n-volume zero.
References
1. Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21.
2. Grünbaum, Branko (2009). "A catalogue of simplicial arrangements in the real projective plane". Ars Mathematica Contemporanea. 2 (1): 1–25. doi:10.26493/1855-3974.88.e12. hdl:1773/2269. MR 2485643.
3. Shephard, G. C. (1968). "Twenty problems on convex polyhedra, part I". The Mathematical Gazette. 52 (380): 136–156. doi:10.2307/3612678. JSTOR 3612678. MR 0231278. S2CID 250442107.
4. "Zonohedrification".
5. Zonohedrification, George W. Hart, The Mathematica Journal, 1999, Volume: 7, Issue: 3, pp. 374-389
6. McMullen, Peter (1975). "Space tiling zonotopes". Mathematika. 22 (2): 202–211. doi:10.1112/S0025579300006082.
7. J. Bohne, Eine kombinatorische Analyse zonotopaler Raumaufteilungen, Dissertation, Bielefeld 1992; Preprint 92-041, SFB 343, Universität Bielefeld 1992, 100 pages.
8. Richter-Gebert, J., & Ziegler, G. M. (1994). Zonotopal tilings and the Bohne-Dress theorem. Contemporary Mathematics, 178, 211-211.
9. McMullen, Peter (1984-05-01). "Volumes of Projections of unit Cubes". Bulletin of the London Mathematical Society. 16 (3): 278–280. doi:10.1112/blms/16.3.278. ISSN 0024-6093.
• Coxeter, H. S. M (1962). "The Classification of Zonohedra by Means of Projective Diagrams". J. Math. Pures Appl. 41: 137–156. Reprinted in Coxeter, H. S. M (1999). The Beauty of Geometry. Mineola, NY: Dover. pp. 54–74. ISBN 0-486-40919-8.
• Fedorov, E. S. (1893). "Elemente der Gestaltenlehre". Zeitschrift für Krystallographie und Mineralogie. 21: 671–694.
• Rolf Schneider, Chapter 3.5 "Zonoids and other classes of convex bodies" in Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
• Shephard, G. C. (1974). "Space-filling zonotopes". Mathematika. 21 (2): 261–269. doi:10.1112/S0025579300008652.
• Taylor, Jean E. (1992). "Zonohedra and generalized zonohedra". American Mathematical Monthly. 99 (2): 108–111. doi:10.2307/2324178. JSTOR 2324178.
• Beck, M.; Robins, S. (2007). Computing the continuous discretely. Springer Science+ Business Media, LLC.
External links
• Weisstein, Eric W. "Zonohedron". MathWorld.
• Eppstein, David. "The Geometry Junkyard: Zonohedra and Zonotopes".
• Hart, George W. "Virtual Polyhedra: Zonohedra".
• Weisstein, Eric W. "Primary Parallelohedron". MathWorld.
• Bulatov, Vladimir. "Zonohedral Polyhedra Completion".
• Centore, Paul. "Chap. 2 of The Geometry of Colour" (PDF).
| Wikipedia |
Zonogon
In geometry, a zonogon is a centrally-symmetric, convex polygon.[1] Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.
Examples
A regular polygon is a zonogon if and only if it has an even number of sides.[2] Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the rectangles, the rhombi, and the parallelograms.
Tiling and equidissection
The four-sided and six-sided zonogons are parallelogons, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form.[3]
Every $2n$-sided zonogon can be tiled by ${\tbinom {n}{2}}$ parallelograms.[4] (For equilateral zonogons, a $2n$-sided one can be tiled by ${\tbinom {n}{2}}$ rhombi.) In this tiling, there is parallelogram for each pair of slopes of sides in the $2n$-sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling.[5] For instance, the regular octagon can be tiled by two squares and four 45° rhombi.[6]
In a generalization of Monsky's theorem, Paul Monsky (1990) proved that no zonogon has an equidissection into an odd number of equal-area triangles.[7][8]
Other properties
In an $n$-sided zonogon, at most $2n-3$ pairs of vertices can be at unit distance from each other. There exist $n$-sided zonogons with $2n-O({\sqrt {n}})$ unit-distance pairs.[9]
Related shapes
Zonogons are the two-dimensional analogues of three-dimensional zonohedra and higher-dimensional zonotopes. As such, each zonogon can be generated as the Minkowski sum of a collection of line segments in the plane.[1] If no two of the generating line segments are parallel, there will be one pair of parallel edges for each line segment. Every face of a zonohedron is a zonogon, and every zonogon is the face of at least one zonohedron, the prism over that zonogon. Additionally, every planar cross-section through the center of a centrally-symmetric polyhedron (such as a zonohedron) is a zonogon.
References
1. Boltyanski, Vladimir; Martini, Horst; Soltan, P. S. (2012), Excursions into Combinatorial Geometry, Springer, p. 319, ISBN 9783642592379
2. Young, John Wesley; Schwartz, Albert John (1915), Plane Geometry, H. Holt, p. 121, If a regular polygon has an even number of sides, its center is a center of symmetry of the polygon
3. Alexandrov, A. D. (2005), Convex Polyhedra, Springer, p. 351, ISBN 9783540231585
4. Beck, József (2014), Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting, Springer, p. 28, ISBN 9783319107417
5. Andreescu, Titu; Feng, Zuming (2000), Mathematical Olympiads 1998-1999: Problems and Solutions from Around the World, Cambridge University Press, p. 125, ISBN 9780883858035
6. Frederickson, Greg N. (1997), Dissections: Plane and Fancy, Cambridge University Press, Cambridge, p. 10, doi:10.1017/CBO9780511574917, ISBN 978-0-521-57197-5, MR 1735254
7. Monsky, Paul (1990), "A conjecture of Stein on plane dissections", Mathematische Zeitschrift, 205 (4): 583–592, doi:10.1007/BF02571264, MR 1082876, S2CID 122009844
8. Stein, Sherman; Szabó, Sandor (1994), Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs, vol. 25, Cambridge University Press, p. 130, ISBN 9780883850282
9. Ábrego, Bernardo M.; Fernández-Merchant, Silvia (2002), "The unit distance problem for centrally symmetric convex polygons", Discrete & Computational Geometry, 28 (4): 467–473, doi:10.1007/s00454-002-2882-5, MR 1949894
| Wikipedia |
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935.[2] It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis,[3] Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.[4]
Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF (Zermelo–Fraenkel set theory without the axiom of choice) any one of the three is sufficient to prove the other two.[5] An earlier formulation of Zorn's lemma is Hausdorff's maximum principle which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set.[6]
Motivation
To prove the existence of a mathematical object that can be viewed as a maximal element in some partially ordered set in some way, one can try proving the existence of such an object by assuming there is no maximal element and using transfinite induction and the assumptions of the situation to get a contradiction. Zorn's lemma tidies up the conditions a situation needs to satisfy in order for such an argument to work and enables mathematicians to not have to repeat the transfinite induction argument by hand each time, but just check the conditions of Zorn's lemma.
If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn’s lemma may well be able to help you.
— William Timothy Gowers, "How to use Zorn’s lemma"[7]
Statement of the lemma
Preliminary notions:
• A set P equipped with a binary relation ≤ that is reflexive (x ≤ x for every x), antisymmetric (if both x ≤ y and y ≤ x hold, then x = y), and transitive (if x ≤ y and y ≤ z then x ≤ z) is said to be (partially) ordered by ≤. Given two elements x and y of P with x ≤ y, y is said to be greater than or equal to x. The word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be comparable under the order relation, that is, in a partially ordered set P with order relation ≤ there may be elements x and y with neither x ≤ y nor y ≤ x. An ordered set in which every pair of elements is comparable is called totally ordered.
• Every subset S of a partially ordered set P can itself be seen as partially ordered by restricting the order relation inherited from P to S. A subset S of a partially ordered set P is called a chain (in P) if it is totally ordered in the inherited order.
• An element m of a partially ordered set P with order relation ≤ is maximal (with respect to ≤) if there is no other element of P greater than m, that is, if there is no s in P with s ≠ m and m ≤ s. Depending on the order relation, a partially ordered set may have any number of maximal elements. However, a totally ordered set can have at most one maximal element.
• Given a subset S of a partially ordered set P, an element u of P is an upper bound of S if it is greater than or equal to every element of S. Here, S is not required to be a chain, and u is required to be comparable to every element of S but need not itself be an element of S.
Zorn's lemma can then be stated as:
Zorn's lemma — Suppose a partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element.
Variants of this formulation are sometimes used, such as requiring that the set P and the chains be non-empty.[8]
Zorn's lemma (for non-empty sets) — Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P. Then the set P contains at least one maximal element.
Although this formulation appears to be formally weaker (since it places on P the additional condition of being non-empty, but obtains the same conclusion about P), in fact the two formulations are equivalent. To verify this, suppose first that P satisfies the condition that every chain in P has an upper bound in P. Then the empty subset of P is a chain, as it satisfies the definition vacuously; so the hypothesis implies that this subset must have an upper bound in P, and this upper bound shows that P is in fact non-empty. Conversely, if P is assumed to be non-empty and satisfies the hypothesis that every non-empty chain has an upper bound in P, then P also satisfies the condition that every chain has an upper bound, as an arbitrary element of P serves as an upper bound for the empty chain (that is, the empty subset viewed as a chain).
The difference may seem subtle, but in many proofs that invoke Zorn's lemma one takes unions of some sort to produce an upper bound, and so the case of the empty chain may be overlooked; that is, the verification that all chains have upper bounds may have to deal with empty and non-empty chains separately. So many authors prefer to verify the non-emptiness of the set P rather than deal with the empty chain in the general argument.[9]
Example applications
Every vector space has a basis
Zorn's lemma can be used to show that every vector space V has a basis.[10]
If V = {0}, then the empty set is a basis for V. Now, suppose that V ≠ {0}. Let P be the set consisting of all linearly independent subsets of V. Since V is not the zero vector space, there exists a nonzero element v of V, so P contains the linearly independent subset {v}. Furthermore, P is partially ordered by set inclusion (see inclusion order). Finding a maximal linearly independent subset of V is the same as finding a maximal element in P.
To apply Zorn's lemma, take a chain T in P (that is, T is a subset of P that is totally ordered). If T is the empty set, then {v} is an upper bound for T in P. Suppose then that T is non-empty. We need to show that T has an upper bound, that is, there exists a linearly independent subset B of V containing all the members of T.
Take B to be the union of all the sets in T. We wish to show that B is an upper bound for T in P. To do this, it suffices to show that B is a linearly independent subset of V.
Suppose otherwise, that B is not linearly independent. Then there exists vectors v1, v2, ..., vk ∈ B and scalars a1, a2, ..., ak, not all zero, such that
$a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k}=\mathbf {0} .$
Since B is the union of all the sets in T, there are some sets S1, S2, ..., Sk ∈ T such that vi ∈ Si for every i = 1, 2, ..., k. As T is totally ordered, one of the sets S1, S2, ..., Sk must contain the others, so there is some set Si that contains all of v1, v2, ..., vk. This tells us there is a linearly dependent set of vectors in Si, contradicting that Si is linearly independent (because it is a member of P).
The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in P, in other words a maximal linearly independent subset B of V.
Finally, we show that B is indeed a basis of V. It suffices to show that B is a spanning set of V. Suppose for the sake of contradiction that B is not spanning. Then there exists some v ∈ V not covered by the span of B. This says that B ∪ {v} is a linearly independent subset of V that is larger than B, contradicting the maximality of B. Therefore, B is a spanning set of V, and thus, a basis of V.
Every nontrivial ring with unity contains a maximal ideal
Zorn's lemma can be used to show that every nontrivial ring R with unity contains a maximal ideal.
Let P be the set consisting of all proper ideals in R (that is, all ideals in R except R itself). Since R is non-trivial, the set P contains the trivial ideal {0}. Furthermore, P is partially ordered by set inclusion. Finding a maximal ideal in R is the same as finding a maximal element in P.
To apply Zorn's lemma, take a chain T in P. If T is empty, then the trivial ideal {0} is an upper bound for T in P. Assume then that T is non-empty. It is necessary to show that T has an upper bound, that is, there exists an ideal I ⊆ R containing all the members of T but still smaller than R (otherwise it would not be a proper ideal, so it is not in P).
Take I to be the union of all the ideals in T. We wish to show that I is an upper bound for T in P. We will first show that I is an ideal of R. For I to be an ideal, it must satisfy three conditions:
1. I is a nonempty subset of R,
2. For every x, y ∈ I, the sum x + y is in I,
3. For every r ∈ R and every x ∈ I, the product rx is in I.
#1 - I is a nonempty subset of R.
Because T contains at least one element, and that element contains at least 0, the union I contains at least 0 and is not empty. Every element of T is a subset of R, so the union I only consists of elements in R.
#2 - For every x, y ∈ I, the sum x + y is in I.
Suppose x and y are elements of I. Then there exist two ideals J, K ∈ T such that x is an element of J and y is an element of K. Since T is totally ordered, we know that J ⊆ K or K ⊆ J. Without loss of generality, assume the first case. Both x and y are members of the ideal K, therefore their sum x + y is a member of K, which shows that x + y is a member of I.
#3 - For every r ∈ R and every x ∈ I, the product rx is in I.
Suppose x is an element of I. Then there exists an ideal J ∈ T such that x is in J. If r ∈ R, then rx is an element of J and hence an element of I. Thus, I is an ideal in R.
Now, we show that I is a proper ideal. An ideal is equal to R if and only if it contains 1. (It is clear that if it is R then it contains 1; on the other hand, if it contains 1 and r is an arbitrary element of R, then r1 = r is an element of the ideal, and so the ideal is equal to R.) So, if I were equal to R, then it would contain 1, and that means one of the members of T would contain 1 and would thus be equal to R – but R is explicitly excluded from P.
The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in P, in other words a maximal ideal in R.
Proof sketch
A sketch of the proof of Zorn's lemma follows, assuming the axiom of choice. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such that every totally ordered subset has an upper bound, and that for every element in P there is another element bigger than it. For every totally ordered subset T we may then define a bigger element b(T), because T has an upper bound, and that upper bound has a bigger element. To actually define the function b, we need to employ the axiom of choice.
Using the function b, we are going to define elements a0 < a1 < a2 < a3 < ... < aω < aω+1 <…, in P. This uncountably infinite sequence is really long: the indices are not just the natural numbers, but all ordinals. In fact, the sequence is too long for the set P; there are too many ordinals (a proper class), more than there are elements in any set (in other words, given any set of ordinals, there exists a larger ordinal), and the set P will be exhausted before long and then we will run into the desired contradiction.
The ai are defined by transfinite recursion: we pick a0 in P arbitrary (this is possible, since P contains an upper bound for the empty set and is thus not empty) and for any other ordinal w we set aw = b({av : v < w}). Because the av are totally ordered, this is a well-founded definition.
The above proof can be formulated without explicitly referring to ordinals by considering the initial segments {av : v < w} as subsets of P. Such sets can be easily characterized as well-ordered chains S ⊆ P where each x ∈ S satisfies x = b({y ∈ S : y < x}). Contradiction is reached by noting that we can always find a "next" initial segment either by taking the union of all such S (corresponding to the limit ordinal case) or by appending b(S) to the "last" S (corresponding to the successor ordinal case).[11]
This proof shows that actually a slightly stronger version of Zorn's lemma is true:
Lemma — If P is a poset in which every well-ordered subset has an upper bound, and if x is any element of P, then P has a maximal element greater than or equal to x. That is, there is a maximal element which is comparable to x.
History
The Hausdorff maximal principle is an early statement similar to Zorn's lemma.
Kazimierz Kuratowski proved in 1922[12] a version of the lemma close to its modern formulation (it applies to sets ordered by inclusion and closed under unions of well-ordered chains). Essentially the same formulation (weakened by using arbitrary chains, not just well-ordered) was independently given by Max Zorn in 1935,[13] who proposed it as a new axiom of set theory replacing the well-ordering theorem, exhibited some of its applications in algebra, and promised to show its equivalence with the axiom of choice in another paper, which never appeared.
The name "Zorn's lemma" appears to be due to John Tukey, who used it in his book Convergence and Uniformity in Topology in 1940. Bourbaki's Théorie des Ensembles of 1939 refers to a similar maximal principle as "le théorème de Zorn".[14] The name "Kuratowski–Zorn lemma" prevails in Poland and Russia.
Equivalent forms of Zorn's lemma
See also: Axiom of choice § Equivalents
Zorn's lemma is equivalent (in ZF) to three main results:
1. Hausdorff maximal principle
2. Axiom of choice
3. Well-ordering theorem.
A well-known joke alluding to this equivalency (which may defy human intuition) is attributed to Jerry Bona: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"[15]
Zorn's lemma is also equivalent to the strong completeness theorem of first-order logic.[16]
Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,
1. Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem
2. Every vector space has a basis, a result from linear algebra (to which it is equivalent[17]). In particular, the real numbers, as a vector space over the rational numbers, possess a Hamel basis.
3. Every commutative unital ring has a maximal ideal, a result from ring theory known as Krull's theorem, to which Zorn's lemma is equivalent[18]
4. Tychonoff's theorem in topology (to which it is also equivalent[19])
5. Every proper filter is contained in an ultrafilter, a result that yields completeness theorem of first-order logic[20]
In this sense, we see how Zorn's lemma can be seen as a powerful tool, applicable to many areas of mathematics.
Analogs under weakenings of the axiom of choice
See also: Axiom of dependent choice
A weakened form of Zorn's lemma can be proven from ZF + DC (Zermelo–Fraenkel set theory with the axiom of choice replaced by the axiom of dependent choice). Zorn's lemma can be expressed straightforwardly by observing that the set having no maximal element would be equivalent to stating that the set's ordering relation would be entire, which would allow us to apply the axiom of dependent choice to construct a countable chain. As a result, any partially ordered set with exclusively finite chains must have a maximal element.[21]
More generally, strengthening the axiom of dependent choice to higher ordinals allows us to generalize the statement in the previous paragraph to higher cardinalities.[21] In the limit where we allow arbitrarily large ordinals, we recover the proof of the full Zorn's lemma using the axiom of choice in the preceding section.
In popular culture
The 1970 film Zorns Lemma is named after the lemma.
The lemma was referenced on The Simpsons in the episode "Bart's New Friend".[22]
See also
• Antichain – Subset of incomparable elements
• Bourbaki–Witt theorem
• Chain-complete partial order – a partially ordered set in which every chain has a least upper bound
• Szpilrajn extension theorem – Mathematical result on order relations
• Tarski finiteness – Mathematical set containing a finite number of elementsPages displaying short descriptions of redirect targets
• Teichmüller–Tukey lemma (sometimes named Tukey's lemma)
Notes
1. Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Springer, p. 23
2. Moore 2013, p. 168
3. Wilansky, Albert (1964). Functional Analysis. New York: Blaisdell. pp. 16–17.
4. Jech 2008, ch. 2, §2 Some applications of the Axiom of Choice in mathematics
5. Jech 2008, p. 9
6. Moore 2013, p. 168
7. William Timothy Gowers (12 August 2008). "How to use Zorn's lemma".
8. For example, Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (Revised 3rd ed.). Springer-Verlag. p. 880. ISBN 978-0-387-95385-4., Dummit, David S.; Foote, Richard M. (1998). Abstract Algebra (2nd ed.). Prentice Hall. p. 875. ISBN 978-0-13-569302-5., and Bergman, George M (2015). An Invitation to General Algebra and Universal Constructions. Universitext (2nd ed.). Springer-Verlag. p. 162. ISBN 978-3-319-11477-4..
9. Bergman, George M (2015). An Invitation to General Algebra and Universal Constructions. Universitext (Second ed.). Springer-Verlag. p. 164. ISBN 978-3-319-11477-4.
10. Smits, Tim. "A Proof that every Vector Space has a Basis" (PDF). Retrieved 14 August 2022.
11. Lewin, Jonathan W. (1991). "A simple proof of Zorn's lemma". The American Mathematical Monthly. 98 (4): 353–354. doi:10.1080/00029890.1991.12000768.
12. Kuratowski, Casimir (1922). "Une méthode d'élimination des nombres transfinis des raisonnements mathématiques" [A method of disposing of transfinite numbers of mathematical reasoning] (PDF). Fundamenta Mathematicae (in French). 3: 76–108. doi:10.4064/fm-3-1-76-108. Retrieved 24 April 2013.
13. Zorn, Max (1935). "A remark on method in transfinite algebra". Bulletin of the American Mathematical Society. 41 (10): 667–670. doi:10.1090/S0002-9904-1935-06166-X.
14. Campbell 1978, p. 82.
15. Krantz, Steven G. (2002), "The Axiom of Choice", Handbook of Logic and Proof Techniques for Computer Science, Springer, pp. 121–126, doi:10.1007/978-1-4612-0115-1_9, ISBN 978-1-4612-6619-8.
16. J.L. Bell & A.B. Slomson (1969). Models and Ultraproducts. North Holland Publishing Company. Chapter 5, Theorem 4.3, page 103.
17. Blass, Andreas (1984). "Existence of bases implies the Axiom of Choice". Axiomatic Set Theory. pp. 31–33. doi:10.1090/conm/031/763890. ISBN 9780821850268. {{cite book}}: |journal= ignored (help)
18. Hodges, W. (1979). "Krull implies Zorn". Journal of the London Mathematical Society. s2-19 (2): 285–287. doi:10.1112/jlms/s2-19.2.285.
19. Kelley, John L. (1950). "The Tychonoff product theorem implies the axiom of choice". Fundamenta Mathematicae. 37: 75–76. doi:10.4064/fm-37-1-75-76.
20. J.L. Bell & A.B. Slomson (1969). Models and Ultraproducts. North Holland Publishing Company.
21. Wolk, Elliot S. (1983), "On the principle of dependent choices and some forms of Zorn's lemma", Canadian Mathematical Bulletin, 26 (3): 365–367, doi:10.4153/CMB-1983-062-5
22. "Zorn's Lemma | The Simpsons and their Mathematical Secrets".
References
• Campbell, Paul J. (February 1978). "The Origin of 'Zorn's Lemma'". Historia Mathematica. 5 (1): 77–89. doi:10.1016/0315-0860(78)90136-2.
• Ciesielski, Krzysztof (1997). Set Theory for the Working Mathematician. Cambridge University Press. ISBN 978-0-521-59465-3.
• Jech, Thomas (2008) [1973]. The Axiom of Choice. Mineola, New York: Dover Publications. ISBN 978-0-486-46624-8.
• Moore, Gregory H. (2013) [1982]. Zermelo's axiom of choice: Its origins, development & influence. Dover Publications. ISBN 978-0-486-48841-7.
External links
• "Zorn lemma", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Zorn's Lemma at ProvenMath contains a formal proof down to the finest detail of the equivalence of the axiom of choice and Zorn's Lemma.
• Zorn's Lemma at Metamath is another formal proof. (Unicode version for recent browsers.)
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| Wikipedia |
Zorn ring
In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent (Kaplansky 1968, pages 19, 25). Kaplansky (1951) named them after Max August Zorn, who studied a similar condition in (Zorn 1941).
For associative rings, the definition of Zorn ring can be restated as follows: the Jacobson radical J(R) is a nil ideal and every right ideal of R which is not contained in J(R) contains a nonzero idempotent. Replacing "right ideal" with "left ideal" yields an equivalent definition. Left or right Artinian rings, left or right perfect rings, semiprimary rings and von Neumann regular rings are all examples of associative Zorn rings.
References
• Kaplansky, Irving (1951), "Semi-simple alternative rings", Portugaliae Mathematica, 10 (1): 37–50, MR 0041835
• Kaplansky, I. (1968), Rings of Operators, New York: W. A. Benjamin, Inc.
• Tuganbaev, A. A. (2002), "Semiregular, weakly regular, and π-regular rings", J. Math. Sci. (New York), 109 (3): 1509–1588, doi:10.1023/A:1013929008743, MR 1871186, S2CID 189870092
• Zorn, Max (1941), "Alternative rings and related questions I: existence of the radical", Annals of Mathematics, Second Series, 42 (3): 676–686, doi:10.2307/1969256, JSTOR 1969256, MR 0005098
| Wikipedia |
Zorya Shapiro
Zorya Yakovlevna Shapiro (Russian: Зоря Яковлевна Шапиро; 7 December 1914 – 4 July 2013) was a Soviet mathematician, educator and translator. She is known for her contributions to representation theory and functional analysis in her collaboration with Israel Gelfand, and the Shapiro-Lobatinski condition in elliptical boundary value problems.
Zorya Shapiro
Born(1914-12-07)December 7, 1914
DiedJuly 4, 2013(2013-07-04) (aged 98)
River Forest, Illinois
CitizenshipSoviet
Alma materMSU Faculty of Mechanics and Mathematics
Known forShapiro-Lopatinski condition in elliptic boundary value problems
SpouseIsrael Gelfand
Scientific career
Fieldsrepresentation theory
Thesis (1938)
Life
Zorya Shapiro attended the Moscow State University Faculty of Mechanics and Mathematics from where she received her undergraduate and doctoral degrees by 1938.[1] She was active in the military department of the university, especially in aviation, learning to fly and land aeroplanes.[2]
She started her teaching career at the Faculty, shortly after Zoya Kishkina (1917–1989) and Natalya Eisenstadt (1912–1985), and very quickly became recognized for her courses in analysis.[1]
Shapiro married Israel Gelfand in 1942. They had 3 sons, one of whom died in childhood.[3] Shapiro and Gelfand later divorced.[4]
In the 1980s, Shapiro lived in the same house as Akiva Yaglom.[5] In 1991 Shapiro moved to River Forest, Illinois to live with her younger son. She died there on 4 July 2013.
Career
Shapiro published several works on representation theory. A contribution (with Gelfand) in integral geometry was to find inversion formulae for the reconstruction of the value of a function on a manifold in terms of integrals over a family of submanifolds, a result with applicability in non-linear differential equations, tomography, multi-dimensional complex analysis and other domains.[6] Another work was on the representations of rotation groups of 3-dimensional spaces.[7]
Shapiro is best known for her elucidation of the conditions for well-defined solutions to the elliptical boundary value problem on Sobolev spaces.[8]
Selected publications
Articles
• "О существовании квазиконформных отображений". Доклады АН СССР. 30 (8). 1941.
• "Об эллиптических системах уравнений с частными производными". Доклады АН СССР. XLVI (4): 146–149. 1945.
• "Первая краевая задача для эллиптической системы дифференциальных уравнений" (PDF). Математический сборник. 28(70) (1): 55–78. 1951.
• "Представления группы вращений трёхмерного пространства и их применения". УМН. 7 (1(47)): 3–117. 1952. (with I.M. Gelfand)
• "Об общих краевых задачах для уравнений эллиптического типа" (PDF). Известия АН СССР. 17 (6): 539–565. 1953.
• "Однородные функции и их приложения" (PDF). Успехи математических наук. 10 (3(65)): 3–70. 1955. (with I.M. Gelfand)
• "Об одном классе обобщённых функций" (PDF). Успехи математических наук. 13 (3(81)): 205–212. 1958.
• "Интегральная геометрия на многообразии k-мерных плоскостей". Доклады АН СССР. 168 (6): 1236–1238. 1966. (with I.M. Gelfand, M.I. Graev)
• "Интегральная геометрия на k-мерных плоскостях" (PDF). Функциональный анализ и его приложения. 1 (1): 15–31. 1967. (with I.M. Gelfand, M.I. Graev)
• "Дифференциальные формы и интегральная геометрия" (PDF). Функциональный анализ и его приложения. 3 (2): 24–40. 1969. (with I.M. Gelfand, M.I. Graev)
• "Интегральная геометрия в проективном пространстве". Функциональный анализ и его приложения. 4 (1): 14–32. 1970. (with I.M. Gelfand, M.I. Graev)
• "Локальная задача интегральной геометрии в пространстве кривых" (PDF). Функциональный анализ и его приложения. 13 (2): 11–31. 1979. (with I.M. Gelfand, S.G. Gindikin)
Books
• Representations of the rotation and Lorentz groups and their applications. Macmillan. 1963. (with I.M. Gelfand, R.A. Minlos)
From French
• Jean Leray (1961). Дифференциальное и интегральное исчисления на комплексном аналитическом многообразии. Moscow: Foreign Literature.
From English
• Stanislaw Ulam (1964). Collection of Mathematical Problems [Нерешённые математические задачи]. Moscow: Nauka.
• Robert Finn (1989). Equilibrium Capillary Surfaces [Равновесные капиллярные поверхности: Математическая теория]. Moscow: Mir.
References
1. Vladimir Tikhomirov (2010). "Прогулки с И.М. Гельфандом". Семь Исскуств. 11 (12).
2. Vladimir Tikhomirov (2011). Ровесники Октября. p. 6. ISBN 9785040175239.
3. "Israel Gelfand". The Daily Telegraph. 26 October 2009. Retrieved 28 October 2018.
4. Thomas Maugh (2 November 2009). "Mathematics genius had it all figured out". The Sydney Morning Herald. Retrieved 28 October 2018.
5. "Akiva M. Yaglom, Dec. 2, 1988; Part 2". Retrieved 28 October 2018.
6. David Kazhdan (2003). "Works of I. Gelfand on the theory of representations" (PDF). An International Conference on "The Unity of Mathematics": 5.
7. A.A. Yushkevich (5 September 2017). "Колмогоров на моем пути в математику". Колмогоров в воспоминаниях учеников. p. 425. ISBN 978-5-457-91890-0.
8. Katsiaryna Krupchyk; Jukka Tuomela (2006). "The Shapiro–Lopatinskij Condition for Elliptic Boundary Value Problems". LMS Journal of Computation and Mathematics. 9: 287–329. doi:10.1112/S1461157000001285.
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| Wikipedia |
Zoé Chatzidakis
Zoé Maria Chatzidakis is a mathematician who works as a director of research at the École Normale Supérieure in Paris, France.[1] Her research concerns model theory and difference algebra. She was invited to give the Tarski Lectures in 2020, though the lectures were postponed due to the COVID-19 pandemic.[2]
Zoé Chatzidakis
Alma materYale university
AwardsLeconte Prize (2013) Tarski Lectures (2020)
Scientific career
FieldsMathematics, Model theory, Algebra
InstitutionsÉcole normale supérieure (Paris)
ThesisModel Theory of Profinite Groups (1984)
Doctoral advisorAngus John Macintyre
Education and employment
Chatzidakis earned her Ph.D. in 1984 from Yale University, under the supervision of Angus Macintyre, with a dissertation on the model theory of profinite groups.[3] She is Senior researcher and team director in Algebra and Geometry in the Département de mathématiques et applications de l'École Normale Supérieure.[4][5]
Honors and awards
She was the 2013 winner of the Leconte Prize,[6] and was an invited speaker at the International Congress of Mathematicians in 2014.[7] She was named MSRI Chern Professor for Fall 2020.[8]
References
1. Member directory, ENS/DMA, retrieved 2016-07-02.
2. "The Tarski Lectures | Department of Mathematics at University of California Berkeley". math.berkeley.edu. Retrieved 2021-11-02. Update on March 10th 2020: The event has been postponed to next year
3. Zoé Chatzidakis at the Mathematics Genealogy Project.
4. "Mathematics at Ecole Normale Supérieure - Algebra and Geometry". www.math.ens.fr. Retrieved 2021-06-07.
5. "Gestion membre". www.math.ens.fr. Retrieved 2021-06-07.
6. Leconte Prize citation, French Academy of Sciences, retrieved 2016-07-02.
7. ICM Plenary and Invited Speakers since 1897, International Mathematical Union, retrieved 2016-07-02.
8. MSRI. "Mathematical Sciences Research Institute". www.msri.org. Retrieved 2021-06-07.
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• Home page
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| Wikipedia |
Zsolt Baranyai
Zsolt Baranyai (June 23, 1948 in Budapest – April 6, 1978) was a Hungarian mathematician known for his work in combinatorics.
He graduated from Fazekas High School where he was a classmate of László Lovász, Miklós Laczkovich, and Lajos Pósa. He studied mathematics at Eötvös Loránd University and went on to become a lecturer in the Analysis Department. He earned his Ph.D. in 1975 and was posthumously awarded the Candidate degree of the Hungarian Academy of Sciences in 1978.
Baranyai is best known for his theorem on the decompositions of complete hypergraphs, which solved a long-standing open problem.[1] In addition to his mathematical pursuits, Baranyai was also a professional musician who played the recorder. He died while touring Hungary with the Bakfark Consort in a car accident after a concert.[2]
References
1. van Lint & Wilson (2001). Chapter 38, "Baranyai's theorem", pp. 536–541.
2. van Lint, J. H.; Wilson, R. M. (2001), A Course in Combinatorics, Cambridge University Press, p. 540, ISBN 9780521006019.
External links
• A Panorama of Hungarian Mathematics in the Twentieth Century, p. 567.
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Zubov's method
Zubov's method is a technique for computing the basin of attraction for a set of ordinary differential equations (a dynamical system). The domain of attraction is the set $\{x:\,v(x)<1\}$, where $v(x)$ is the solution to a partial differential equation known as the Zubov equation.[1] Zubov's method can be used in a number of ways.
Statement
Zubov's theorem states that:
If $x'=f(x),t\in \mathbb {R} $ is an ordinary differential equation in $\mathbb {R} ^{n}$ with $f(0)=0$, a set $A$ containing 0 in its interior is the domain of attraction of zero if and only if there exist continuous functions $v,h$ such that:
• $v(0)=h(0)=0$, $0<v(x)<1$ for $x\in A\setminus \{0\}$, $h>0$ on $\mathbb {R} ^{n}\setminus \{0\}$
• for every $\gamma _{2}>0$ there exist $\gamma _{1}>0,\alpha _{1}>0$ such that $v(x)>\gamma _{1},h(x)>\alpha _{1}$ , if $||x||>\gamma _{2}$
• $v(x_{n})\rightarrow 1$ for $x_{n}\rightarrow \partial A$ or $||x_{n}||\rightarrow \infty $
• $\nabla v(x)\cdot f(x)=-h(x)(1-v(x)){\sqrt {1+||f(x)||^{2}}}$
If f is continuously differentiable, then the differential equation has at most one continuously differentiable solution satisfying $v(0)=0$.
References
1. Vladimir Ivanovich Zubov, Methods of A.M. Lyapunov and their application, Izdatel'stvo Leningradskogo Universiteta, 1961. (Translated by the United States Atomic Energy Commission, 1964.) ASIN B0007F2CDQ.
| Wikipedia |
Zuckerman functor
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related.
This article is about the Zuckerman induction functor, which is not the same as the (Zuckerman) translation functor.
Notation and terminology
• G is a connected reductive real affine algebraic group (for simplicity; the theory works for more general groups), and g is the Lie algebra of G. K is a maximal compact subgroup of G.
• L is a Levi subgroup of G, the centralizer of a compact connected abelian subgroup, and *l is the Lie algebra of L.
• A representation of K is called K-finite if every vector is contained in a finite-dimensional representation of K. Denote by WK the subspace of K-finite vectors of a representation W of K.
• A (g,K)-module is a vector space with compatible actions of g and K, on which the action of K is K-finite.
• R(g,K) is the Hecke algebra of G of all distributions on G with support in K that are left and right K finite. This is a ring which does not have an identity but has an approximate identity, and the approximately unital R(g,K)- modules are the same as (g,K) modules.
Definition
The Zuckerman functor Γ is defined by
$\Gamma _{g,L\cap K}^{g,K}(W)=\hom _{R(g,L\cap K)}(R(g,K),W)_{K}$
and the Bernstein functor Π is defined by
$\Pi _{g,L\cap K}^{g,K}(W)=R(g,K)\otimes _{R(g,L\cap K)}W.$
References
• David A. Vogan, Representations of real reductive Lie groups, ISBN 3-7643-3037-6
• Anthony W. Knapp, David A. Vogan, Cohomological induction and unitary representations, ISBN 0-691-03756-6 prefacereview by Dan BarbaschMR1330919
• David A. Vogan, Unitary Representations of Reductive Lie Groups. (AM-118) (Annals of Mathematics Studies) ISBN 0-691-08482-3
• Gregg J. Zuckerman, Construction of representations via derived functors, unpublished lecture series at the Institute for Advanced Study, 1978.
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Translation functor
In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by Zuckerman (1977) and Jantzen (1979). Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.
Definition
By the Harish-Chandra isomorphism, the characters of the center Z of the universal enveloping algebra of a complex reductive Lie algebra can be identified with the points of L⊗C/W, where L is the weight lattice and W is the Weyl group. If λ is a point of L⊗C/W then write χλ for the corresponding character of Z.
A representation of the Lie algebra is said to have central character χλ if every vector v is a generalized eigenvector of the center Z with eigenvalue χλ; in other words if z∈Z and v∈V then (z − χλ(z))n(v)=0 for some n.
The translation functor ψμ
λ
takes representations V with central character χλ to representations with central character χμ. It is constructed in two steps:
• First take the tensor product of V with an irreducible finite dimensional representation with extremal weight λ−μ (if one exists).
• Then take the generalized eigenspace of this with eigenvalue χμ.
References
• Jantzen, Jens Carsten (1979), Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069521, ISBN 978-3-540-09558-3, MR 0552943
• Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, doi:10.1515/9781400883936, ISBN 978-0-691-03756-1, MR 1330919
• Zuckerman, Gregg (1977), "Tensor products of finite and infinite dimensional representations of semisimple Lie groups", Ann. Math., 2, 106 (2): 295–308, doi:10.2307/1971097, JSTOR 1971097, MR 0457636
| Wikipedia |
Wadim Zudilin
Wadim Zudilin (Вадим Валентинович Зудилин) is a Russian mathematician and number theorist who is active in studying hypergeometric functions and zeta constants. He studied under Yuri V. Nesterenko and worked at Moscow State University, the Steklov Institute of Mathematics, the Max Planck Institute for Mathematics and the University of Newcastle, Australia. He now works at the Radboud University Nijmegen, the Netherlands.[1]
He has reproved Apéry's theorem that ζ(3) is irrational, and expanded it. Zudilin proved that at least one of the four numbers ζ(5), ζ(7), ζ(9), or ζ(11) is irrational.[2] For that accomplishment he won the Distinguished Award of the Hardy-Ramanujan Society in 2001.
With Doron Zeilberger, Zudilin[3] improved upper bound of irrationality measure for π, which as of November 2022 is the current best estimate.
References
1. "Wadim Zudilin appointed Professor of Pure Mathematics".
2. W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/RM2001v056n04ABEH000427. S2CID 250734661.
3. Zeilberger, Doron; Zudilin, Wadim (2020-01-07). "The irrationality measure of π is at most 7.103205334137…". Moscow Journal of Combinatorics and Number Theory. 9 (4): 407–419. arXiv:1912.06345. doi:10.2140/moscow.2020.9.407. S2CID 209370638.
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• Wadim Zudilin's homepage
• Wadim Zudilin's research profile
• Wadim Zudilin's list of published works
• Wadim Zudilin at the Mathematics Genealogy Project
| Wikipedia |
Zvezdelina Stankova
Zvezdelina Entcheva Stankova (Bulgarian: Звезделина Енчева Станкова; born 15 September 1969) is an American mathematician who is a professor of mathematics at Mills College and a teaching professor at the University of California, Berkeley, the founder of the Berkeley Math Circle, and an expert in the combinatorial enumeration of permutations with forbidden patterns.[1]
Zvezdelina Entcheva Stankova
Stankova in 2012
Born (1969-09-15) 15 September 1969
Ruse, People's Republic of Bulgaria
NationalityAmerican
Alma materBryn Mawr College
Harvard University
Known forSkew-merged permutation
Studying permutations with forbidden subsequences
Establishing math circles
AwardsAlice T. Schafer Prize (1992)
Deborah and Franklin Tepper Haimo Award (2011)
Scientific career
FieldsMathematics
InstitutionsMills College
University of California, Berkeley
Doctoral advisorJoe Harris
Websitehttps://math.berkeley.edu/~stankova/
Biography
Stankova was born in Ruse, Bulgaria.[2] She began attending the Ruse math circle as a fifth grader in Bulgaria, the same year she learned to solve the Rubik's Cube[3] and began winning regional mathematics competitions.[1] She later wrote of this experience that "if I was not a member of Ruse SMC I would not be able to make such profound achievements in mathematics".[4] She became a student at an elite English-language high school, and competed on the Bulgarian team in the International Mathematical Olympiads in 1987 and 1988, earning silver medals both times.[2][5] She entered Sofia University but in 1989, as the Iron Curtain was falling, became one of 15 Bulgarian students selected to travel to the US to complete their studies.[2]
Stankova studied at Bryn Mawr College, completing bachelor's and master's degrees there in 1992,[6] with Rhonda Hughes as a faculty mentor.[7] While an undergraduate, she participated in a summer research program with Joseph Gallian at the University of Minnesota Duluth, which began her interest in permutation patterns.[8] Next, she went to Harvard University for her doctoral studies, and earned a Ph.D. there in 1997; her dissertation, entitled Moduli of Trigonal Curves, was supervised by Joe Harris.[9]
She worked at the University of California, Berkeley as Morrey Assistant Professor of Mathematics[10] before joining the Mills College faculty in 1999,[6] and continues to teach one course per year as a visiting professor at Berkeley.[11][12] She also serves on the advisory board of the Proof School in San Francisco.[1]
Contributions
In the theory of permutation patterns, Stankova is known for proving that the permutations with the forbidden pattern 1342 are equinumerous with the permutations with forbidden pattern 2413, an important step in the enumeration of permutations avoiding a pattern of length 4.[8][13]
In 1998 she became the founder and director of the Berkeley Math Circle, an after-school mathematics enrichment program that Stankova modeled after her early experiences learning mathematics in Bulgaria.[3][7][14][15] The Berkeley circle was only the second math circle in the US (after one in Boston); following its success, over 100 other circles have been created,[3] and Stankova has assisted in the formation of many of them.[11]
Also in 1998, she founded the Bay Area Mathematical Olympiad.[10] For six years, she served as a coach of the US International Mathematical Olympiad team.[7][16]
Since 2013, she has featured in several videos on the mathematics-themed YouTube channel "Numberphile".[17]
Publications
With Tom Rike, she is co-editor of two books about her work with the Berkeley Math Circle, A Decade of the Berkeley Math Circle: The American Experience (Vol. I, 2008, Vol. II, 2014).[18]
Awards and honors
In 1992, Stankova won the Alice T. Schafer Prize of the Association for Women in Mathematics for her undergraduate research in permutation patterns.[8][11] In 2004 she became one of two inaugural winners of the Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member.[19] In 2011 Stankova won the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching, given by the Mathematical Association of America, "for her outstanding work in teaching, mentoring, and inspiring students at all levels, and in leading the development of Math Circles, and promoting participation in mathematics competitions".[7][11][16] From 2009 to 2012 she was the Frederick A. Rice Professor of Mathematics at Mills.[20]
References
1. "Zvezdelina Stankova". Proof School. Archived from the original on 2018-12-09. Retrieved 2016-02-03.
2. Vigoda, Ralph (October 23, 1991), "Bulgarian Math Whiz Wows 'Em: Zvezdelina Stankova Already Is Joining The Ranks Of Top Mathematicians. "She's A Genius," Says Bryn Mawr's Head Of Math", The Philadelphia Inquirer, archived from the original on December 22, 2015.
3. Weld, Sarah (April 2014), "Proving Their Passion: The Berkeley Math Circle gives math kids a place to find solutions—together", The East Bay Monthly
4. Ruse Students Mathematical Circle (PDF), Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, retrieved 2016-02-02.
5. Participant record: Zvezdelina Stankova, International Mathematical Olympiad, retrieved 2016-02-02.
6. "Zvezdelina Stankova", Mathematics Faculty and Staff, Mills College, retrieved 2016-02-01.
7. Bakke, Katherine (April 6, 2011), "Zvezdelina Stankova '92 Carries on a Bryn Mawr Tradition: Excellent Teaching in Mathematics", Meaningful Contributions, Bryn Mawr College.
8. Third Annual Alice T. Schafer Prize, Association for Women in Mathematics, July 1992, retrieved 2016-02-02.
9. Zvezdelina Stankova at the Mathematics Genealogy Project
10. Keith, Tamara (February 10, 1999), UC Berkeley mathematicians feed the minds of young local math whizzes, University of California, Berkeley.
11. "Haimo Award Citation: Zvezdelina Stankova" (PDF), January 2011 Prizes and Awards, American Mathematical Society, p. 4, January 7, 2011.
12. Stankova's home page at Berkeley, retrieved 2016-02-02.
13. Bona, Miklos (2012), Combinatorics of Permutations, Discrete Mathematics and Its Applications (2nd ed.), CRC Press, pp. 154–155, ISBN 9781439850527.
14. Vandervelde, Sam (2009), Circle in a Box, MSRI mathematical circles library, vol. 2, Mathematical Sciences Research Institute and American Mathematical Society, pp. 4, 34, ISBN 9780821847527.
15. Bloom, Melanie (October 21, 2013), Moms Everyday: Making Math Fun, 10/11.
16. Melendez, Lyanne (February 25, 2011), Mills College professor wins highest math award, ABC7 News.
17. Numberphile (2013-12-19), Pebbling a Chessboard - Numberphile, retrieved 2016-08-21
18. MSRI Mathematical Circles Library, National Association of Math Circles, retrieved 2016-02-01.
19. Henry L. Alder Award, Mathematical Association of America, retrieved 2018-06-08
20. Curriculum vitae: Zvezdelina Stankova (PDF), retrieved 2016-02-01
External links
• Zvezdelina Stankova in the Oberwolfach photo collection
• Moduli of Trigonal Curves Paper based on PhD thesis.
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Zvi Arad
Zvi Arad (Hebrew: צבי ארד,16 April 1942, in Petah Tikva, Mandatory Palestine – 4 February 2018, in Petah Tikva, Israel) was an Israeli mathematician, acting president of Bar-Ilan University, and president of Netanya Academic College.[1][2]
Zvi Arad
Zvi Arad (2008)
Born
צבי ארד
(1942-04-16)April 16, 1942
Petah Tikva, Mandatory Palestine
DiedFebruary 4, 2018(2018-02-04) (aged 75)
Petah Tikva, Israel
NationalityIsraeli
Occupationmathematician
Known for
• Acting President of Bar-Ilan University
• President of Netanya Academic College
Biography
Zvi Arad began his academic studies in the Mathematics Department of Bar-Ilan University. He received his first degree in 1964 and after army service went on to complete a second and third degree in the Mathematics Department of Tel Aviv University.
Academic career
In 1968 Arad joined the academic staff at Bar-Ilan University as an assistant and in 1983 was appointed a full professor. During the years 1978/9 he held the position of visiting scientist at the University of Chicago, and from 1982 to 1983 held the position of visiting professor at the University of Toronto.
Arad held a variety of senior academic posts at Bar-Ilan University. He served as chairman of the Mathematics and Computer Science Department, dean of the Faculty of Natural Sciences and Mathematics, rector and president of the university (succeeding Ernest Krausz, and followed by Shlomo Eckstein).[3] Together with Professor Bernard Pinchuk he founded Gelbart Institute, an international research institute named after Abe Gelbart, and the Emmy Noether Institute (Minerva Center). Together with colleagues he established a journal, the Israel Mathematics Conference Proceedings, distributed by the American Mathematical Society (AMS). From 1984–1985 he served as a member of the Council for Higher Education of the State of Israel. In 1982 he was elected a member of Russia's Academy of Natural Sciences.
From 1994 he served on the editorial board of the Algebra Colloquium, a journal of the Chinese Academy of Sciences published by Springer-Verlag. He also serves on the editorial board of various international publications: South East Asian Bulletin of Mathematics of the Asian Mathematical Society, the IMCP of Contemporary Mathematics published by the American Mathematical Society, and the publication Cubo Matemática Educacional, Temuco, Chile.
He initiated numerous agreements of cooperation with universities and institutions throughout the world including academic institutes in the former Soviet Union, universities and research centers in America, Canada, Germany, the United Kingdom, Italy, Russia, China, South Africa, etc.
He was a member of Israel's first official delegation to the former USSR, under the leadership of President Ezer Weizman. In an official address, President Mikhail Gorbachev mentioned Professor Arad's contributions towards the establishment of scientific communications between Israel and the former USSR. In an effort to advance cooperation in research he has headed delegations of scientists to Russia, China, and East Germany.
Haaretz newspaper (January 21, 1998) described him as one of the pioneers of higher education reform in Israel. The Encyclopaedia Hebraica lists Zvi Arad as "fulfilling a key role in the development and advancement of Bar-Ilan University and in the establishment of the University's regional colleges in Safed, Ashkelon and the Jordan Valley)." For this achievement he was awarded a certificate of honor by the mayor of each city. The establishment of these colleges began in 1985 and went on to affect the whole of Israel. These colleges advanced the Galilee and Southern Israel and brought higher education to the peripheries of Israel.
Netanya Academic College
In 1994, at the request of the mayors of the city of Netanya, Yoel Elroi and Zvi Poleg, Arad established the Netanya Academic College. He served as president of the college for 24 years.[4] A partner in the initiation and establishment of the college was Miriam Feirberg, who at that time served as head of the Education Department of the City of Netanya. Today the college is an accredited institute of higher education that grant first and second academic degrees in a variety of fields.
Published works
Together with his colleague Professor Marcel Herzog, Arad wrote Products of Conjugacy Classes, published by Springer-Verlag. The book facilitated the basis of the establishment of mathematical theory and today forms part of the branch of abstract algebra known as Table Algebras, and is attached to central branches in mathematics: Graph theory, algebra combinations, and theory presentation. Arad coauthored two other books on the subject of table algebra. In 2000 his book was published in the series American Mathematical Society Memoirs and in January 2002 another book on table algebras was published in the international publication, Springer. Arad was the editor of Contemporary Mathematics, Volume 402.
See also
• Education in Israel
References
1. For 11-year-old Math Prodigy, 2 Plus 2 Equals 'Super-gifted'
2. "Bar-Ilan Presidents | Bar Ilan University". .biu.ac.il. Retrieved 2020-02-15.
3. "Bar-Ilan Presidents | Bar Ilan University". .biu.ac.il. Retrieved 2020-02-18.
4. "Professor Zvi Arad passes away". Israel National News. 4 February 2018. Retrieved 19 February 2018.
External links
• Profile Netanya Academic College
• Zvi Arad Dun's 100 (in Hebrew)
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Zvi Galil
Zvi Galil (Hebrew: צבי גליל; born June 26, 1947) is an Israeli-American computer scientist and mathematician. Galil served as the president of Tel Aviv University from 2007 through 2009. From 2010 to 2019, he was the dean of the Georgia Institute of Technology College of Computing.[3] His research interests include the design and analysis of algorithms, computational complexity and cryptography. He has been credited with coining the terms stringology and sparsification.[4][5] He has published over 200 scientific papers[6] and is listed as an ISI highly cited researcher.[7]
Zvi Galil
Galil in 2010
Born (1947-06-26) June 26, 1947[1]
Tel Aviv, Mandatory Palestine[1]
Alma mater
• Tel Aviv University (BSc, MSc)
• Cornell University (PhD)
Awards
• ACM Fellow
• NAE Member
• American Academy of Arts and Sciences Fellow
Scientific career
Fields
• Computer science
• Mathematics
Institutions
• IBM
• Tel Aviv University
• Columbia University
• Georgia Institute of Technology
Doctoral advisorJohn Hopcroft[2]
Doctoral students
• Mordechai Ben-Ari
• Moti Yung
• David Eppstein
• Giuseppe F. Italiano
• Matthew K. Franklin[2]
• Jonathan Katz
• Stuart Haber
Early life and education
Zvi Galil was born in Tel Aviv in Mandatory Palestine in 1947. He completed both his B.Sc. (1970) and his M.Sc. (1971) in applied mathematics, both summa cum laude, at Tel Aviv University before earning his Ph.D. in computer science at Cornell in 1975 under the supervision of John Hopcroft.[2] He then spent a year working as a post-doctorate researcher at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.[8]
Career
From 1976 until 1995 he worked in the computer science department of Tel Aviv University, serving as its chair from 1979 to 1982. In 1982 he joined the faculty of Columbia University, serving as the chair of the computer science department from 1989-1994.[1][8] From 1995-2007, he served as the dean of the Fu Foundation School of Engineering & Applied Science.[9] In this position, he oversaw the naming of the school in honor of Chinese businessman Z. Y. Fu after a large donation was given in his name.[10] At Columbia, he was appointed the Julian Clarence Levi Professor of Mathematical Methods and Computer Science in 1987, and the Morris and Alma A. Schapiro Dean of Engineering in 1995.[1]
Galil served as the president of Tel Aviv University starting in 2007 (following Itamar Rabinovich),[11] but resigned and returned to the faculty in 2009, and was succeeded by Joseph Klafter.[12][13] He was named as the dean of Georgia Tech's college of computing on April 9, 2010.[3] At Georgia Tech, together with Udacity founder Sebastian Thrun, Galil conceived of the college of computing's online Master of Science in computer science (OMSCS) program, and he led the faculty creation of the program.[14] OMSCS went on to become the largest online master’s program in computer science in the United States.[15] OMSCS has been featured in hundreds of articles, including a 2013 front-page article in The New York Times and 2021 interviews in The Wall Street Journal and Forbes.[14][16][17] Inside Higher Education noted that OMSCS "suggests that institutions can successfully deliver high-quality, low-cost degrees to students at scale".[18] The Chronicle of Higher Education noted that OMSCS "may have the best chance of changing how much students pay for a traditional degree".[19] Galil stepped down as dean and returned to a regular faculty position in June 2019.[20][21] He now serves as the Frederick G. Storey Chair in Computing and Executive Advisor to Online Programs at Georgia Tech.
Professional service
In 1982, Galil founded the Columbia University Theory Day and organized the event for the first 15 years. It still exists as the New York Area Theory Day.[22] From 1983 to 1987, Galil served as the chairman of ACM SIGACT, an organization that promotes research in theoretical computer science.[23] He served as managing editor of SIAM Journal on Computing from 1991 to 1997 and editor in chief of Journal of Algorithms from 1988 to 2003.
Research
Galil's research is in the areas of algorithms (particularly string and graph algorithms) complexity, and cryptography. He has also conducted research in experimental design with Jack Kiefer.
Galil's real-time algorithms are the fastest possible for string matching and palindrome recognition, and they work even on the most basic computer model, the multi-tape Turing machine. More generally, he formulated a "predictability" condition that allows any complying online algorithm to be converted to a real-time algorithm.[24][25] With Joel Seiferas, Galil improved the time-optimal algorithms to be space optimal (logarithmic space) as well.[26]
Galil worked with Dany Breslauer to design a linear-work, O(loglogn) parallel algorithm for string matching,[27] and they later proved it to have the best possible time complexity among linear work algorithms.[28] With other computer scientists, he designed a constant-time linear-work randomized search algorithm to be used when the pattern preprocessing is given.[29]
With his students, Galil designed more than a dozen currently-fastest algorithms for exact or approximate, sequential or parallel, and one- or multi-dimensional string matching.
Galil worked with other computer scientists to develop several currently-fastest graph algorithms. Examples include: maximum weighted matching;[30] trivalent graph isomorphism;[31] and minimum weight spanning trees.[32]
With his students, Galil devised a technique he called "sparsification"[33] and a method he called "sparse dynamic programming".[34] The first was used to speed up dynamic graph algorithms. The second was used to speed up the computations of various edit distances between strings.
In 1979, together with Ofer Gabber, Galil solved the previously open problem of constructing a family of expander graphs with an explicit expansion ratio,[35] useful in the design of fast graph algorithms.
Awards and honors
In 1995, Galil was inducted as a Fellow of the Association for Computing Machinery for "fundamental contributions to the design and analysis of algorithms and outstanding service to the theoretical computer science community,"[36] and in 2004, he was elected to the National Academy of Engineering for "contributions to the design and analysis of algorithms and for leadership in computer science and engineering."[37][38] In 2005, he was selected as a Fellow of the American Academy of Arts and Sciences.[39] In 2008, Columbia University established the Zvi Galil award for student life.[40] In 2009, the Columbia Society of Graduates awarded him the Great Teacher Award.[41] In 2012, The University of Waterloo awarded Galil with an honorary Doctor of Mathematics degree for his "fundamental contributions in the areas of graph algorithms and string matching."[42] In 2020, Academic Influence included Galil in the list of the 10 most influential computer scientists of the last decade, and the advisory board of the College of Computing at Georgia Tech raised over $2 million from over 130 donors to establish an endowed chair named after Galil.[43][44]
References
1. Eppstein, David; Italiano, Giuseppe F. (March 1999). "PREFACE: Festschrift for Zvi Galil". Journal of Complexity. 15 (1): 1–3. doi:10.1006/jcom.1998.0492.
2. Zvi Galil at the Mathematics Genealogy Project
3. "Institute names next College of Computing Dean" (Press release). Georgia Institute of Technology. 2010-04-09. Retrieved 2010-04-09.
4. "Introduction to Stringology". The Prague Stringology Club. Czech Technical University in Prague. Retrieved May 14, 2012.
5. Zvi, Galil; David Eppstein; Giuseppe F. Italiano; Amnon Nissenzweig (September 1997). "Sparsification - a technique for speeding up dynamic graph algorithms". Journal of the ACM. 44 (5): 669–696. doi:10.1145/265910.265914. S2CID 340999.
6. "Zvi Galil". The DBLP Computer Science Bibliography. Digital Bibliography & Library Project. Retrieved 2016-03-24.
7. "ISI Highly Cited Researchers Version 1.1: Zvi Galil". ISI Web of Knowledge. Retrieved 2011-06-27.
8. "Zvi Galil Named Dean of Columbia's Engineering School" (Press release). Columbia University. July 14, 1995. Retrieved 2019-06-05.
9. McCaughey, Robert (2014). A Lever Long Enough: A History of Columbia's School of Engineering and Applied Science since 1864. Columbia University Press. p. 240. ISBN 9780231166881.
10. Arenson, Karen W. (1997-10-01). "Chinese Tycoon Gives Columbia $26 Million". The New York Times. Retrieved 2010-04-20.
11. "Computer expert nominated for TAU presidency". The Jerusalem Post. November 5, 2006.
12. Basch_Interactive (1980-01-01). "Presidents of Tel Aviv University | Tel Aviv University | Tel Aviv University". English.tau.ac.il. Retrieved 2020-02-18.
13. Ilani, Ofri; Kashti, Or (2009-07-02). "Tel Aviv University president quits / Sources: Galil was forced out of office". Haaretz. Retrieved 2011-06-27.
14. Lewin, Tamar (2013-08-18). "Master's Degree Is New Frontier of Study Online". The New York Times. ISSN 0362-4331. Retrieved 2023-01-02.
15. Galil, Zvi. "OMSCS: The Revolution Will Be Digitized". cacm.acm.org. Retrieved 2020-07-27.
16. Varadarajan, Tunku (2021-04-02). "Opinion | The Man Who Made Online College Work". Wall Street Journal. ISSN 0099-9660. Retrieved 2021-11-01.
17. Nietzel, Michael T. "Georgia Tech's Online MS In Computer Science Continues To Thrive. Why That's Important For The Future of MOOCs". Forbes. Retrieved 2022-03-25.
18. "Analysis shows Georgia Tech's online master's in computer science expanded access | Inside Higher Ed". www.insidehighered.com. 20 March 2018. Retrieved 2022-03-25.
19. "What Georgia Tech's Online Degree in Computer Science Means for Low-Cost Programs". www.chronicle.com. 6 November 2014. Retrieved 2022-03-25.
20. "College's Skyrocketing Stature, Global Impact Highlight Galil's Legacy". Georgia Tech College of Computing. April 16, 2019. Retrieved 2019-06-05.
21. "Georgia Tech Alumni Magazine, Vol. 95 No. 3, Fall 2019". Issuu. Retrieved 2020-04-21.
22. "New York Area Theory Day". www.cs.columbia.edu. Retrieved 2020-06-03.
23. "Front matter". ACM SIGACT News. 19 (1). Fall 1987.
24. Galil, Zvi (1981-01-01). "String Matching in Real Time". Journal of the ACM. 28 (1): 134–149. doi:10.1145/322234.322244. ISSN 0004-5411. S2CID 9164969.
25. Galil, Zvi (1978-04-01). "Palindrome recognition in real time by a multitape turing machine". Journal of Computer and System Sciences. 16 (2): 140–157. doi:10.1016/0022-0000(78)90042-9. ISSN 0022-0000.
26. Galil, Zvi; Seiferas, Joel (1983-06-01). "Time-space-optimal string matching". Journal of Computer and System Sciences. 26 (3): 280–294. doi:10.1016/0022-0000(83)90002-8. ISSN 0022-0000.
27. Breslauer, Dany; Galil, Zvi (1990-12-01). "An Optimal $O(\log\log n)$ Time Parallel String Matching Algorithm". SIAM Journal on Computing. 19 (6): 1051–1058. doi:10.1137/0219072. ISSN 0097-5397.
28. Breslauer, Dany; Galil, Zvi (1992-10-01). "A Lower Bound for Parallel String Matching". SIAM Journal on Computing. 21 (5): 856–862. doi:10.1137/0221050. ISSN 0097-5397.
29. Crochemore, Maxime; Galil, Zvi; Gasieniec, Leszek; Park, Kunsoo; Rytter, Wojciech (1997-08-01). "Constant-Time Randomized Parallel String Matching". SIAM Journal on Computing. 26 (4): 950–960. doi:10.1137/S009753979528007X. ISSN 0097-5397.
30. Galil, Zvi; Micali, Silvio; Gabow, Harold (1986-02-01). "An $O(EV\log V)$ Algorithm for Finding a Maximal Weighted Matching in General Graphs". SIAM Journal on Computing. 15 (1): 120–130. doi:10.1137/0215009. ISSN 0097-5397. S2CID 12854446.
31. Galil, Zvi; Hoffmann, Christoph M.; Luks, Eugene M.; Schnorr, Claus P.; Weber, Andreas (1987-07-01). "An O(n3log n) deterministic and an O(n3) Las Vegs isomorphism test for trivalent graphs". Journal of the ACM. 34 (3): 513–531. doi:10.1145/28869.28870. ISSN 0004-5411. S2CID 18031646.
32. Gabow, Harold N.; Galil, Zvi; Spencer, Thomas; Tarjan, Robert E. (1986-06-01). "Efficient algorithms for finding minimum spanning trees in undirected and directed graphs". Combinatorica. 6 (2): 109–122. doi:10.1007/BF02579168. ISSN 1439-6912. S2CID 35618095.
33. Eppstein, David; Galil, Zvi; Italiano, Giuseppe F.; Nissenzweig, Amnon (1997-09-01). "Sparsification—a technique for speeding up dynamic graph algorithms". Journal of the ACM. 44 (5): 669–696. doi:10.1145/265910.265914. ISSN 0004-5411. S2CID 340999.
34. Eppstein, David; Galil, Zvi; Giancarlo, Raffaele; Italiano, Giuseppe F. (1992-07-01). "Sparse dynamic programming I: linear cost functions". Journal of the ACM. 39 (3): 519–545. doi:10.1145/146637.146650. ISSN 0004-5411. S2CID 17060840.
35. Gabber, Ofer; Galil, Zvi (1981-06-01). "Explicit constructions of linear-sized superconcentrators". Journal of Computer and System Sciences. 22 (3): 407–420. doi:10.1016/0022-0000(81)90040-4. ISSN 0022-0000.
36. ACM Fellow Award / Zvi Galil
37. "Dr. Zvi Galil". NAE Members. National Academy of Engineering. Retrieved May 11, 2012.
38. "Zvi Galil Elected to National Academy of Engineering". Columbia News. Columbia University. Retrieved May 11, 2012.
39. Academy Elects 225th Class of Fellows and Foreign Honorary Members, American Association for the Advancement of Science, April 26, 2005
40. "Zvi Galil Award". Columbia College. Retrieved 2019-06-05.
41. "Quigley, Galil To Receive Great Teacher Awards". Columbia College Today. September 2009. Retrieved 2019-06-05.
42. Smyth, Pamela. "University of Waterloo to award eight honorary degrees at spring convocation". Waterloo Communications. University of Waterloo. Retrieved May 11, 2012.
43. Larson, Erik J.; PhD. "Top Influential Computer Scientists Today". academicinfluence.com. Retrieved 2021-05-05.
44. "New Endowed Chair Honors Inclusion and Diversity". College of Computing. 2021-06-02. Retrieved 2021-06-09.
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| Wikipedia |
Zvika Brakerski
Zvika Brakerski is an Israeli mathematician, known for his work on homomorphic encryption, particularly in developing the foundations of the second generation FHE schema, for which he was awarded the 2022 Gödel Prize.[1][2] Brakerski is an Associate Professor in the Department of Computer Science and Applied Mathematics at the Weizmann Institute of Science.
Zvika Brakerski
OccupationAssociate Professor
Known forhomomorphic encryption
AwardsGödel Prize
Academic background
Doctoral advisorShafi Goldwasser
Other advisorsDan Boneh
Academic work
Disciplinecryptography
Research
In 2012 Brakerski published a paper at the Annual Cryptology Conference "Fully homomorphic encryption without modulus switching from classical GapSVP Authors",[3] this formed the basis of the Brakerski-Gentry-Vaikuntanathan (BGV)[4] - for which they were jointly awarded the Gödel Prize - and BFV Fully Homomorphic Encryption (FHE) schema. The two dominant second-generation FHE schema.
References
1. "ACM SIGACT - Gödel Prize". sigact.org. Archived from the original on 2022-11-24. Retrieved 2022-11-24.
2. "School of Engineering second quarter 2022 awards". MIT News | Massachusetts Institute of Technology. Archived from the original on 2022-11-24. Retrieved 2022-11-24.
3. Brakerski, Zvika (2012). "Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP". In Safavi-Naini, Reihaneh; Canetti, Ran (eds.). Advances in Cryptology – CRYPTO 2012. Lecture Notes in Computer Science. Vol. 7417. Berlin, Heidelberg: Springer. pp. 868–886. doi:10.1007/978-3-642-32009-5_50. ISBN 978-3-642-32009-5.
4. Brakerski, Zvika; Gentry, Craig; Vaikuntanathan, Vinod (2011). "Fully Homomorphic Encryption without Bootstrapping". Cryptology ePrint Archive.
| Wikipedia |
Thomas von Randow
Thomas von Randow (26 December 1921 Breslau, Schlesien – 29 July 2009 Hamburg) was a German mathematician and journalist who published mathematical and logical puzzles under the pseudonym Zweistein in the "Logelei" column in Die Zeit. (After 2005 his column and pseudonym were continued by Bernhard Seckinger and Immanuel Halupczok.)
Publications
Many of his logic puzzles were published in the following books:
• 99 Logeleien von Zweistein. Christian Wegner, Hamburg 1968
• Neue Logeleien von Zweistein. Hoffmann und Campe, Hamburg 1976
• Logeleien für Kenner. Hoffmann und Campe, Hamburg 1975
• 88 neue Logeleien. Nymphenburger, München 1983
• 87 neue Logeleien. Rasch und Röhring, Hamburg 1985
• Weitere Logeleien von Zweistein. Deutscher Taschenbuchverlag (dtv), München 1985, ISBN 3-485-00446-4
• Zweisteins Zahlenmagie. Mathematisches und Mystisches über einen abstrakten Gebrauchsgegenstand. Von Eins bis Dreizehn. Illustrationen von Gerhard Gepp. Christian Brandstätter, Wien 1993, ISBN 3854474814
• Zweisteins Zahlen-Logeleien. Insel, Frankfurt am Main und Leipzig 1993, ISBN 3-458-33210-3
References
• Interview in Die Zeit, 15 November 2005
• Thomas von Randow – Visionär seines Fachs. Obituary in Die Zeit, 32/2009
• Thomas von Randow at the Mathematics Genealogy Project
External links
• Logelei puzzle by Zweistein in Die Zeit
• Collection of logical puzzles by Zweistein (in German)
• Index to articles by Thomas von Randow in Die Zeit
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| Wikipedia |
Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century.[1][2][3][4][5] Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986.[1][2][3][4]
Antoni Zygmund
Antoni Zygmund
Born(1900-12-25)December 25, 1900
Warsaw, Congress Poland, Russian Empire
DiedMay 30, 1992(1992-05-30) (aged 91)
Chicago, Illinois, United States
NationalityPolish
CitizenshipPolish, American
Alma materUniversity of Warsaw (Ph.D., 1923)
Known forSingular integral operators
Calderón–Zygmund lemma
Marcinkiewicz–Zygmund inequality
Paley–Zygmund inequality
Calderón–Zygmund kernel
AwardsLeroy P. Steele Prize (1979)
National Medal of Science (1986)
Scientific career
FieldsMathematics
InstitutionsUniversity of Chicago
Stefan Batory University
Doctoral advisorAleksander Rajchman
Stefan Mazurkiewicz
Doctoral studentsAlberto Calderón
Elias M. Stein
Paul Cohen
Biography
Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago.
He was a member of several scientific societies. From 1930 until 1952 he was a member of the Warsaw Scientific Society (TNW), from 1946 of the Polish Academy of Learning (PAU), from 1959 of the Polish Academy of Sciences (PAN), and from 1961 of the National Academy of Sciences in the United States. In 1986 he received the National Medal of Science.
In 1935 Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume Trigonometric Series. It was described by Robert A. Fefferman as "one of the most influential books in the history of mathematical analysis" and "an extraordinarily comprehensive and masterful presentation of a ... vast field".[6] Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst. The theory of trigonometric series had remained the largest component of Zygmund's mathematical investigations.[5]
His work has had a pervasive influence in many fields of mathematics, mostly in mathematical analysis, and particularly in harmonic analysis. Among the most significant were the results he obtained with Calderón on singular integral operators.[7][6] George G. Lorentz called it Zygmund's crowning achievement, one that "stands somewhat apart from the rest of Zygmund's work".[5]
Zygmund's students included Alberto Calderón, Paul Cohen, Nathan Fine, Józef Marcinkiewicz, Victor L. Shapiro, Guido Weiss, Elias M. Stein and Mischa Cotlar. He died in Chicago.
Mathematical objects named after Zygmund
• Calderón–Zygmund lemma
• Marcinkiewicz–Zygmund inequality
• Paley–Zygmund inequality
• Calderón–Zygmund kernel
Books
• Trigonometric Series (Cambridge University Press 1959, 2002)
• Intégrales singulières (Springer-Verlag, 1971)
• Trigonometric Interpolation (University of Chicago, 1950)
• Measure and Integral: An Introduction to Real Analysis, With Richard L. Wheeden (Marcel Dekker, 1977)
• Analytic Functions, with Stanislaw Saks (Elsevier Science Ltd, 1971)
See also
• Calderón–Zygmund lemma
• Zygmunt Janiszewski
• Marcinkiewicz–Zygmund inequality
• Paley–Zygmund inequality
• List of Poles
• Centipede mathematics
References
1. Noble, Holcomb B. (1998-04-20). "Alberto Calderon, 77, Pioneer Of Mathematical Analysis". The New York Times. ISSN 0362-4331. Retrieved 2019-06-23.
2. Warnick, Mark S. (19 April 1998). "ALBERTO CALDERON, MATH GENIUS". chicagotribune.com. Retrieved 2019-06-23.
3. "Antoni Zygmund (1900-1992)". www-history.mcs.st-and.ac.uk. Retrieved 2019-06-23.
4. "PROFESSOR ALBERTO CALDERON, 77, DIES". Washington Post. ISSN 0190-8286. Retrieved 2019-06-22.
5. Lorentz, G. G. (1993). "Antoni Zygmund and His Work" (PDF). Journal of Approximation Theory. 75: 1–7. doi:10.1006/jath.1993.1084.
6. The 2nd edition of Zygmund's Trigonometric Series (Cambridge University Press, 1959) consists of 2 separate volumes. The 3rd edition (Cambridge University Press, 2002, ISBN 0 521 89053 5) consists of the two volumes combined with a foreword by Robert A. Fefferman. The nine pages in Fefferman's foreword (biographic and other information concerning Zygmund) are not numbered.
7. Carbery, Tony (17 July 1992). "Obituary: Professor Antoni Zygmund". The Independent. Archived from the original on 2022-05-07.
Further reading
• Kazimierz Kuratowski, A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford, Pergamon Press, 1980, ISBN 0-08-023046-6.
• Gray, Jeremy (1970–1980). "Zygmund, Antoni". Dictionary of Scientific Biography. Vol. 25. New York: Charles Scribner's Sons. pp. 414–416. ISBN 978-0-684-10114-9.
External links
• Antoni Zygmund at the Mathematics Genealogy Project
• Mount Holyoke biography
• O'Connor, John J.; Robertson, Edmund F., "Antoni Zygmund", MacTutor History of Mathematics Archive, University of St Andrews
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1975
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1988
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1992
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1993
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1994
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1995
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1996
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1997
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1998
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1999
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2001
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2002
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2012
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2013
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2014
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1987
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1988
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1989
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1997
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1998
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1999
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2013
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1965
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1975
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1979
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1986
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Esther M. Conwell
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Sandra Faber
Sylvester James Gates
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Sean C. Solomon
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| Wikipedia |
Zygmunt Janiszewski
Zygmunt Janiszewski (12 July 1888 – 3 January 1920)[1] was a Polish mathematician.
Zygmunt Janiszewski
Born(1888-07-12)12 July 1888
Warsaw, Vistula Land, Russian Empire
Died3 January 1920(1920-01-03) (aged 31)
Lwów, Poland
Resting placeLychakiv cemetery, Lwów
Alma materUniversity of Paris
Known forJaniszewski's theorem
Brouwer–Janiszewski–Knaster continuum
Scientific career
FieldsMathematics
InstitutionsUniversity of Warsaw
Doctoral advisorHenri Lebesgue
Doctoral studentsKazimierz Kuratowski
Early life and education
He was born to mother Julia Szulc-Chojnicka and father, Czeslaw Janiszewski who was a graduate of the University of Warsaw and served as the director of the Société du Crédit Municipal in Warsaw.
Janiszewski left Poland to study mathematics in Zurich, Munich and Göttingen, where he was taught by some of the most prominent mathematicians of the time, such as Heinrich Burkhardt, David Hilbert, Hermann Minkowski and Ernst Zermelo.[2] He then went to Paris and in 1911 received his doctorate in topology under the supervision of Henri Lebesgue. His thesis was titled Sur les continus irréductibles entre deux points (On the Irreducible Continuous Curves Between Two Points).[2] In 1913, he published a seminal work in the field of topology of surface entitled On Cutting the Plane by Continua.
Career
Janiszewski taught at the University of Lwów and was professor at the University of Warsaw. At the outbreak of World War I he was a soldier in the Polish Legions of Józef Piłsudski, and took part in operations around Volyn.[3] Along with other officers, he refused to swear an oath of allegiance to the Austrian government. He subsequently left the Legions and went into hiding under an assumed identity, Zygmunt Wicherkiewicz, in Boiska, near Zwoleń.[2] From Boiska he moved on to Ewin, near Włoszczowa, where he directed a shelter for homeless children.[2]
In 1917, he published an article O potrzebach matematyki w Polsce in the Nauka Polska journal, thus initiating the Polish School of Mathematics.[4] He also founded the journal Fundamenta Mathematicae.[5] Janiszewski proposed the name of the journal in 1919, though the first issue was published in 1920, after his death.
Janiszewski devoted the family property that he had inherited from his father to charity and education. He also donated all the prize money that he received from mathematical awards and competitions to the education and development of young Polish students.
Death
Z. Janiszewski's fiancée was Janina Kelles-Krauz, daughter of Kazimierz Kelles-Krauz. The date of the wedding was set, but due to Z. Janiszewski's death, it did not take place[6]. His life was cut short by the influenza pandemic of 1918–19,[5] which took his life at Lwów on 3 January 1920 at the age of 31. He willed his body for medical research, and his cranium for craniological study, desiring to be "useful after his death".
Samuel Dickstein wrote a commemorative address after Janiszewski's death, honoring his humility, kindness and dedication to his work:
Enthusiasm and strong will characterized Janiszewski not only in his scientific work, but in his life generally. His active participation in the Legions, his refusal to take an oath which was not compatible with his patriotic conscience, his work in the field of education, when at a most difficult time he entered that field as an enlightened and wise worker, free of any prejudice and partiality and ardently keen only to propagate light and truth - these facts prove that in the heart of a mathematician seemingly detached from active life there glowed the purest emotions of affection and self-denial. If we also mention that, having very moderate needs himself, he dispensed all the means at his disposal to educate young talents, and that he bequeathed the property that he had inherited from his parents for educational purposes, and in particular for the education of outstanding individuals, then we may indeed exclaim from the bottom of our hearts that the memory of that life, devoted to the cause and interrupted so early, lives on in its results and deeds and will remain treasured and living for us, the witnesses of his work, and for generations to come.[7]
While Janiszewski best remembered for his many contributions to topological mathematics in the early 20th century, for the founding of Fundamenta Mathematicae, and for his enthusiasm for teaching young minds, his loyalty to his homeland during World War I perhaps gives the greatest insight into his psyche. The orphans' shelter that he set up during the war doubtlessly saved many lives, and is perhaps his greatest contribution to the world.
On 3 January 2020, the 100th anniversary of his death, a researcher from Australia travelled to Lviv and met with the director of Lychakiv Cemetery. Restoration of the grave was arranged, and the stone was restored. Janiszewski is buried in field 58, plot 82 of Lychakiv Cemetery.
See also
• List of Polish mathematicians
Notes
1. "Janiszewski Zygmunt". Astro-Databank. Retrieved 13 November 2021.
2. "Zygmunt Janiszewski (1888 - 1920)". mathshistory.st-andrews.ac.uk. Retrieved 2020-02-11.
3. Domoradzki, Stanislaw; Stawiska, Malgorzata (6 April 2018). "Polish mathematicians and mathematics in World War I. Part II. Russian Empire". Studia Historiae Scientiarum (published 2019). 18: 55–92. arXiv:1804.02448. doi:10.4467/2543702XSHS.19.004.11010.
4. Iłowiecki, Maciej (1981). Dzieje nauki polskiej. Warsaw: Interpress. pp. 251–256.
5. "Placing World War I in the History of Mathematics". HAL archives-ouvertes (Sorbonne). 8 July 2014. Retrieved 11 February 2020.
6. Wilczkowska-Grabowska, Magdalena (1986). "Memories of Janina Kelles-Krauz. Librarian - Monthly of the Association of Polish Librarians, Number 1986 7-8, pages 37-39., 1986". {{cite journal}}: Cite journal requires |journal= (help)
7. Kuratowski 1980, pp. 162–163
References
• Kuratowski, Kazimierz (1980), A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford: Pergamon Press, pp. 158–163, ISBN 0-08-023046-6 et passim.
External links
• Zygmunt Janiszewski at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Zygmunt Janiszewski", MacTutor History of Mathematics Archive, University of St Andrews
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| Wikipedia |
Vertical bar
The vertical bar, |, is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally the word "or"), vbar, and others.[1]
" ‖ " redirects here. For the use of a similar-looking character in African languages, see lateral clicks. For the parallelism symbol see parallel (geometry) and parallel (operator).
|
Vertical bar
In UnicodeU+007C | VERTICAL LINE (|, |, |)
Related
See alsoU+00A6 ¦ BROKEN BAR (¦)
U+2016 ‖ DOUBLE VERTICAL LINE (‖, ‖)
U+2223 ∣ DIVIDES
Usage
Mathematics
The vertical bar is used as a mathematical symbol in numerous ways:
• absolute value: $|x|$, read "the absolute value of x"[2]
• cardinality: $|S|$, read "the cardinality of the set S" or "the length of a string S".
• conditional probability: $P(X|Y)$, read "the probability of X given Y"
• determinant: $|A|$, read "the determinant of the matrix A".[2] When the matrix entries are written out, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses of the matrix, as in ${\begin{vmatrix}a&b\\c&d\end{vmatrix}}$.
• distance: $P|ab$, denoting the shortest distance between point $P$ to line $ab$, so line $P|ab$ is perpendicular to line $ab$
• divisibility: $a\mid b$, read "a divides b" or "a is a factor of b", though Unicode also provides special 'divides' and 'does not divide' symbols (U+2223 and U+2224: ∣, ∤)[2]
• function evaluation: $f(x)|_{x=4}$, read "f of x, evaluated at x equals 4" (see subscripts at Wikibooks)
• order: $|G|$, read "the order of the group G", or $|g|$, "the order of the element $g\in G$"
• restriction: $f|_{A}$, denoting the restriction of the function $f$, with a domain that is a superset of $A$, to just $A$
• set-builder notation: $\{x|x<2\}$, read "the set of x such that x is less than two". Often, a colon ':' is used instead of a vertical bar
• the Sheffer stroke in logic: $a|b$, read "a nand b"
• subtraction: $f(x)\vert _{a}^{b}$, read "f(x) from a to b", denoting $f(b)-f(a)$. Used in the context of a definite integral with variable x.
• A vertical bar can be used to separate variables from fixed parameters in a function, for example $f(x|\mu ,\sigma )$, or in the notation for elliptic integrals.
The double vertical bar, $\|$, is also employed in mathematics.
• parallelism: $AB\parallel CD$, read "the line $AB$ is parallel to the line $CD$"
• norm: $\|A\|$, read "the norm (length, size, magnitude etc.) of the matrix $A$". The norm of a one-dimensional vector is the absolute value and single bars are used.[3]
• Propositional truncation (a type former that truncates a type down to a mere proposition in homotopy type theory): for any $a:A$ (read "term $a$ of type $A$") we have $|a|:\left\|A\right\|$[4] (here $|a|$ reads "image of $a:A$ in $\left\|A\right\|$" and $|a|:\left\|A\right\|$ reads "propositional truncation of $A$")[5]
In LaTeX mathematical mode, the ASCII vertical bar produces a vertical line, and \| creates a double vertical line (a | b \| c is set as $a|b\|c$). This has different spacing from \mid and \parallel, which are relational operators: a \mid b \parallel c is set as $a\mid b\parallel c$. See below about LaTeX in text mode.
Physics
The vertical bar is used in bra–ket notation in quantum physics. Examples:
• $|\psi \rangle $: the quantum physical state $\psi $
• $\langle \psi |$: the dual state corresponding to the state above
• $\langle \psi |\rho \rangle $: the inner product of states $\psi $ and $\rho $
• Supergroups in physics are denoted G(N|M), which reads "G, M vertical bar N"; here G denotes any supergroup, M denotes the bosonic dimensions, and N denotes the Grassmann dimensions.[6]
Pipe
A pipe is an inter-process communication mechanism originating in Unix, which directs the output (standard out and, optionally, standard error) of one process to the input (standard in) of another. In this way, a series of commands can be "piped" together, giving users the ability to quickly perform complex multi-stage processing from the command line or as part of a Unix shell script ("bash file"). In most Unix shells (command interpreters), this is represented by the vertical bar character. For example:
grep -i 'blair' filename.log | more
where the output from the grep process (all lines containing 'blair') is piped to the more process (which allows a command line user to read through results one page at a time).
The same "pipe" feature is also found in later versions of DOS and Microsoft Windows.
This usage has led to the character itself being called "pipe".
Disjunction
In many programming languages, the vertical bar is used to designate the logic operation or, either bitwise or or logical or.
Specifically, in C and other languages following C syntax conventions, such as C++, Perl, Java and C#, a | b denotes a bitwise or; whereas a double vertical bar a || b denotes a (short-circuited) logical or. Since the character was originally not available in all code pages and keyboard layouts, ANSI C can transcribe it in form of the trigraph ??!, which, outside string literals, is equivalent to the | character.
In regular expression syntax, the vertical bar again indicates logical or (alternation). For example: the Unix command grep -E 'fu|bar' matches lines containing 'fu' or 'bar'.
Concatenation
The double vertical bar operator "||" denotes string concatenation in PL/I, standard ANSI SQL, and theoretical computer science (particularly cryptography).
Delimiter
Although not as common as commas or tabs, the vertical bar can be used as a delimiter in a flat file. Examples of a pipe-delimited standard data format are LEDES 1998B and HL7. It is frequently used because vertical bars are typically uncommon in the data itself.
Similarly, the vertical bar may see use as a delimiter for regular expression operations (e.g. in sed). This is useful when the regular expression contains instances of the more common forward slash (/) delimiter; using a vertical bar eliminates the need to escape all instances of the forward slash. However, this makes the bar unusable as the regular expression "alternative" operator.
Backus–Naur form
In Backus–Naur form, an expression consists of sequences of symbols and/or sequences separated by '|', indicating a choice, the whole being a possible substitution for the symbol on the left.
<personal-name> ::= <name> | <initial>
Concurrency operator
In calculi of communicating processes (like pi-calculus), the vertical bar is used to indicate that processes execute in parallel.
APL
The pipe in APL is the modulo or residue function between two operands and the absolute value function next to one operand.
List comprehensions
The vertical bar is used for list comprehensions in some functional languages, e.g. Haskell and Erlang. Compare set-builder notation.
Text markup
The vertical bar is used as a special character in lightweight markup languages, notably MediaWiki's Wikitext (in the templates and internal links).
In LaTeX text mode, the vertical bar produces an em dash (—). The \textbar command can be used to produce a vertical bar.
Phonetics and orthography
In the Khoisan languages and the International Phonetic Alphabet, the vertical bar is used to write the dental click (ǀ). A double vertical bar is used to write the alveolar lateral click (ǁ). Since these are technically letters, they have their own Unicode code points in the Latin Extended-B range: U+01C0 for the single bar and U+01C1 for the double bar.
Some Northwest and Northeast Caucasian languages written in the Cyrillic script have a vertical bar called palochka (Russian: палочка, lit. 'little stick'), indicating the preceding consonant is an ejective.
Longer single and double vertical bars are used to mark prosodic boundaries in the IPA.
Literature
Punctuation
In medieval European manuscripts, a single vertical bar was a common variant of the virgula ⟨/⟩ used as a period, scratch comma,[7] and caesura mark.[7]
In Sanskrit and other Indian languages, a single vertical mark, a danda, has a similar function as a period (full stop). Two bars || (a 'double danda') is the equivalent of a pilcrow in marking the end of a stanza, paragraph or section. The danda has its own Unicode code point, U+0964.
Poetry
A double vertical bar ⟨||⟩ or ⟨ǁ⟩ is the standard caesura mark in English literary criticism and analysis. It marks the strong break or caesura common to many forms of poetry, particularly Old English verse. It is also traditionally used to mark the division between lines of verse printed as prose (the style preferred by Oxford University Press), though it is now often replaced by the forward slash.
Notation
In the Geneva Bible and early printings of the King James Version, a double vertical bar is used to mark margin notes that contain an alternative translation from the original text. These margin notes always begin with the conjunction "Or". In later printings of the King James Version, the double vertical bar is irregularly used to mark any comment in the margins.
Music scoring
In music, when writing chord sheets, single vertical bars associated with a colon (|: A / / / :|) represents the beginning and end of a section (e.g. Intro, Interlude, Verse, Chorus) of music. Single bars can also represent the beginning and end of measures (|: A / / / | D / / / | E / / / :|). A double vertical bar associated with a colon can represent the repeat of a given section (||: A / / / :|| - play twice).
Encoding
Solid vertical bar vs broken bar
Many early video terminals and dot-matrix printers rendered the vertical bar character as the allograph broken bar ¦. This may have been to distinguish the character from the lower-case 'L' and the upper-case 'I' on these limited-resolution devices, and to make a vertical line of them look more like a horizontal line of dashes. It was also (briefly) part of the ASCII standard.
An initial draft for a 7-bit character set that was published by the X3.2 subcommittee for Coded Character Sets and Data Format on June 8, 1961, was the first to include the vertical bar in a standard set. The bar was intended to be used as the representation for the logical OR symbol.[8] A subsequent draft on May 12, 1966, places the vertical bar in column 7 alongside regional entry codepoints, and formed the basis for the original draft proposal used by the International Standards Organisation.[8] This draft received opposition from the IBM user group SHARE, with its chairman, H. W. Nelson, writing a letter to the American Standards Association titled "The Proposed revised American Standard Code for Information Interchange does NOT meet the needs of computer programmers!"; in this letter, he argues that no characters within the international subset designated at columns 2-5 of the character set would be able to adequately represent logical OR and logical NOT in languages such as IBM's PL/I universally on all platforms.[9] As a compromise, a requirement was introduced where the exclamation mark (!) and circumflex (^) would display as logical OR (|) and logical NOT (¬) respectively in use cases such as programming, while outside of these use cases they would represent their original typographic symbols:
It may be desirable to employ distinctive styling to facilitate their use for specific purposes as, for example, to stylize the graphics in code positions 2/1 and 5/14 to those frequently associated with logical OR (|) and logical NOT (¬) respectively.
— X3.2 document X3.2/475[10]
The original vertical bar encoded at 0x7C in the original May 12, 1966 draft was then broken as ¦, so it could not be confused with the unbroken logical OR. In the 1967 revision of ASCII, along with the equivalent ISO 464 code published the same year, the code point was defined to be a broken vertical bar, and the exclamation mark character was allowed to be rendered as a solid vertical bar.[11][12] However, the 1977 revision (ANSI X.3-1977) undid the changes made in the 1967 revision, enforcing that the circumflex could no longer be stylised as a logical NOT symbol, the exclamation mark likewise no longer allowing stylisation as a vertical bar, and defining the code point originally set to the broken bar as a solid vertical bar instead;[11] the same changes were also reverted in ISO 646-1973 published four years prior.
Some variants of EBCDIC included both versions of the character as different code points. The broad implementation of the extended ASCII ISO/IEC 8859 series in the 1990s also made a distinction between the two forms. This was preserved in Unicode as a separate character at U+00A6 BROKEN BAR (the term "parted rule" is used sometimes in Unicode documentation). Some fonts draw the characters the same (both are solid vertical bars, or both are broken vertical bars).[13] The broken bar does not appear to have any clearly identified uses distinct from those of the vertical bar.[14] In non-computing use — for example in mathematics, physics and general typography — the broken bar is not an acceptable substitute for the vertical bar.
Many keyboards with US or US-International layout display the broken bar on a keycap even though the solid vertical bar character is produced in modern operating systems. This includes many German QWERTZ keyboards. This is a legacy of keyboards manufactured during the 1980s and 1990s for IBM PC compatible computers featuring the broken bar, as such computers used IBM's 8-bit Code page 437 character set based on ASCII, which continued to display the glyph for the broken bar at codepoint 7C on displays from MDA (1981) to VGA (1987) despite the changes made to ASCII in 1977. The UK/Ireland keyboard has both symbols engraved: the broken bar is given as an alternate graphic on the "grave" (backtick) key; the solid bar is on the backslash key.
The broken bar character can be typed (depending on the layout) as AltGr+` or AltGr+6 or AltGr+⇧ Shift+Right \ on Windows and Compose!^ on Linux. It can be inserted into HTML as ¦
In some dictionaries, the broken bar is used to mark stress that may be either primary or secondary. That is, [¦ba] covers the pronunciations [ˈba] and [ˌba].[15]
Unicode code points
These glyphs are encoded in Unicode as follows:
• U+007C | VERTICAL LINE (|, |, |) (single vertical line)
• U+00A6 ¦ BROKEN BAR (¦) (single broken line)
• U+2016 ‖ DOUBLE VERTICAL LINE (‖, ‖) (double vertical line ( $\|$ ): used in pairs to indicate norm)
• U+FF5C ｜ FULLWIDTH VERTICAL LINE (Fullwidth form)
• U+2225 ∥ PARALLEL TO (∥, ∥, ∥, ∥, ∥)
• U+01C0 ǀ LATIN LETTER DENTAL CLICK
• U+01C1 ǁ LATIN LETTER LATERAL CLICK
• U+2223 ∣ DIVIDES (∣, ∣, ∣, ∣)
• U+2502 │ BOX DRAWINGS LIGHT VERTICAL (│) (and various other box drawing characters in the range U+2500 to U+257F)
• U+0964 । DEVANAGARI DANDA
• U+0965 ॥ DEVANAGARI DOUBLE DANDA
Code pages and other historical encodings
Code pages, ASCII, ISO/IEC, EBCDIC, Shift-JIS, etc. Vertical bar (|) Broken bar (¦)
ASCII,
CP437, CP667, CP720, CP737, CP790, CP819, CP852, CP855, CP860, CP861, CP862, CP865, CP866, CP867, CP869, CP872, CP895, CP932, CP991
124 (7Ch) none
CP775 167 (A7h)
CP850, CP857, CP858 221 (DDh)
CP863 160 (A0h)
CP864 219 (DBh)
ISO/IEC 8859-1, -7, -8, -9, -13,
CP1250, CP1251, CP1252, CP1253, CP1254, CP1255, CP1256, CP1257, CP1258
166 (A6h)
ISO/IEC 8859-2, -3, -4, -5, -6, -10, -11, -14, -15, -16 none
EBCDIC CCSID 37 79 (4Fh) 106 (6Ah)
EBCDIC CCSID 500 187 (BBh)
JIS X 0208, JIS X 0213 Men-ku-ten 1-01-35 (7-bit: 2143h; Shift JIS: 8162h; EUC: A1C3h)[lower-alpha 1] none
See also
Look up vertical bar in Wiktionary, the free dictionary.
• Bar (diacritic) – Diacritic used in some languages
• Triple bar – Symbol with multiple meanings
Notes
1. The Shift JIS and EUC encoded forms also include the ASCII vertical bar in its usual encoding (see halfwidth and fullwidth forms). The same applies when the 7-bit form is used as part of ISO-2022-JP (allowing switching to and from ASCII).
References
1. Raymond, Eric S. "ASCII". The Jargon File.
2. Weisstein, Eric W. "Single Bar". mathworld.wolfram.com. Retrieved 2020-08-24.
3. Weisstein, Eric W. "Matrix Norm". mathworld.wolfram.com. Retrieved 2020-08-24.
4. Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics (GitHub version) (PDF). Institute for Advanced Study. p. 108. Archived from the original (PDF) on 2017-07-07. Retrieved 2017-07-01.
5. Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics (print version). Institute for Advanced Study. p. 450.
6. Larus Thorlacius, Thordur Jonsson (eds.), M-Theory and Quantum Geometry, Springer, 2012, p. 263.
7. "virgula, n.", Oxford English Dictionary, 1st ed., Oxford: Oxford University Press, 1917.
8. Fischer, Eric (2012). The Evolution of Character Codes, 1874-1968 (Thesis). Penn State University. CiteSeerX 10.1.1.96.678. Retrieved July 10, 2020.
9. H. W. Nelson, letter to Thomas B. Steel, June 8, 1966, Honeywell Inc. X3.2 Standards Subcommittee Records, 1961-1969 (CBI 67), Charles Babbage Institute, University of Minnesota, Minneapolis, box 1, folder 23.
10. X3.2 document X3.2/475, December 13, 1966, Honeywell Inc. X3.2 Standards Subcommittee Records, 1961-1969 (CBI 67), Charles Babbage Institute, University of Minnesota, Minneapolis, box 1, folder 22.
11. Salste, Tuomas (January 2016). "7-bit character sets: Revisions of ASCII". Aivosto Oy. urn:nbn:fi-fe201201011004. Archived from the original on 2016-06-13. Retrieved 2016-06-13.
12. Korpela, Jukka. "Character histories - notes on some Ascii code positions". Archived from the original on 2020-03-11. Retrieved 2020-05-31.
13. Jim Price (2010-05-24). "ASCII Chart: IBM PC Extended ASCII Display Characters". Retrieved 2012-02-23.
14. Jukka "Yucca" Korpela (2006-09-20). "Detailed descriptions of the characters". Retrieved 2012-02-23.
15. For example, "Balearic". Merriam-Webster Dictionary..
Common punctuation marks and other typographical marks or symbols
• space
• , comma
• : colon
• ; semicolon
• ‐ hyphen
• ’ ' apostrophe
• ′ ″ ‴ prime
• . full stop
• & ampersand
• @ at sign
• ^ caret
• / slash
• \ backslash
• … ellipsis
• * asterisk
• ⁂ asterism
• * * * dinkus
• - hyphen-minus
• ‒ – — dash
• = ⸗ double hyphen
• ? question mark
• ! exclamation mark
• ‽ interrobang
• ¡ ¿ inverted ! and ?
• ⸮ irony punctuation
• # number sign
• № numero sign
• º ª ordinal indicator
• % percent sign
• ‰ per mille
• ‱ basis point
• ° degree symbol
• ⌀ diameter sign
• + − plus and minus signs
• × multiplication sign
• ÷ division sign
• ~ tilde
• ± plus–minus sign
• ∓ minus-plus sign
• _ underscore
• ⁀ tie
• | ¦ ‖ vertical bar
• • bullet
• · interpunct
• © copyright symbol
• © copyleft
• ℗ sound recording copyright
• ® registered trademark
• SM service mark symbol
• TM trademark symbol
• ‘ ’ “ ” ' ' " " quotation mark
• ‹ › « » guillemet
• ( ) [ ] { } ⟨ ⟩ bracket
• ” 〃 ditto mark
• † ‡ dagger
• ❧ hedera/floral heart
• ☞ manicule
• ◊ lozenge
• ¶ ⸿ pilcrow (paragraph mark)
• ※ reference mark
• § section mark
• Version of this table as a sortable list
• Currency symbols
• Diacritics (accents)
• Logic symbols
• Math symbols
• Whitespace
• Chinese punctuation
• Hebrew punctuation
• Japanese punctuation
• Korean punctuation
| Wikipedia |
Hexadecimal
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and "A"–"F" (or alternatively "a"–"f") to represent values from ten to fifteen.
"Sexadecimal" redirects here. For base 60, see Sexagesimal.
Part of a series on
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By radix/base
Common radices/bases
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• (table)
Non-standard radices/bases
• Bijective (1)
• Signed-digit (balanced ternary)
• Mixed (factorial)
• Negative
• Complex (2i)
• Non-integer (φ)
• Asymmetric
Sign-value notation
Non-alphabetic
• Aegean
• Attic
• Aztec
• Brahmi
• Chuvash
• Egyptian
• Etruscan
• Kharosthi
• Prehistoric counting
• Proto-cuneiform
• Roman
• Tally marks
Alphabetic
• Abjad
• Armenian
• Alphasyllabic
• Akṣarapallī
• Āryabhaṭa
• Kaṭapayādi
• Coptic
• Cyrillic
• Geʽez
• Georgian
• Glagolitic
• Greek
• Hebrew
List of numeral systems
Software developers and system designers widely use hexadecimal numbers because they provide a human-friendly representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble).[1] For example, an 8-bit byte can have values ranging from 00000000 to 11111111 in binary form, which can be conveniently represented as 00 to FF in hexadecimal.
In mathematics, a subscript is typically used to specify the base. For example, the decimal value 38,967 would be expressed in hexadecimal as 983716. In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this value as 0x9837.
Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.
Representation
Written representation
In most current use cases, the letters A–F or a–f represent the values 10–15, while the numerals 0–9 are used to represent their decimal values.
There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Seven-segment displays use mixed-case AbCdEF to make digits that can be distinguished from each other.
There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following hex dump, each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.
00000000 57 69 6b 69 70 65 64 69 61 2c 20 74 68 65 20 66
00000010 72 65 65 20 65 6e 63 79 63 6c 6f 70 65 64 69 61
00000020 20 74 68 61 74 20 61 6e 79 6f 6e 65 20 63 61 6e
00000030 20 65 64 69 74 0a
Distinguishing from decimal
In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 15910 is decimal 159; 15916 is hexadecimal 159, which equals 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h.
Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.[2] Hexadecimal representations are written there in a typewriter typeface: 5A3
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
• Unix (and related) shells, AT&T assembly language and likewise the C programming language (and its syntactic descendants such as C++, C#, Go, D, Java, JavaScript, Python and Windows PowerShell) use the prefix 0x for numeric constants represented in hex: 0x5A3. Character and string constants may express character codes in hexadecimal with the prefix \x followed by two hex digits: '\x1B' represents the Esc control character; "\x1B[0m\x1B[25;1H" is a string containing 11 characters with two embedded Esc characters.[3] To output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used.
• In XML and XHTML, characters can be expressed as hexadecimal numeric character references using the notation ode;, for instance ’ represents the character U+2019 (the right single quotation mark). If there is no x the number is decimal (thus ’ is the same character).[4]
• In Intel-derived assembly languages and Modula-2,[5] hexadecimal is denoted with a suffixed H or h: FFh or 05A3H. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh instead of FFh. Some other implementations (such as NASM) allow C-style numbers (0x42).
• Other assembly languages (6502, Motorola), Pascal, Delphi, some versions of BASIC (Commodore), GameMaker Language, Godot and Forth use $ as a prefix: $5A3.
• Some assembly languages (Microchip) use the notation H'ABCD' (for ABCD16). Similarly, Fortran 95 uses Z'ABCD'.
• Ada and VHDL enclose hexadecimal numerals in based "numeric quotes": 16#5A3#. For bit vector constants VHDL uses the notation x"5A3".[6]
• Verilog represents hexadecimal constants in the form 8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant.
• The Smalltalk language uses the prefix 16r: 16r5A3
• PostScript and the Bourne shell and its derivatives denote hex with prefix 16#: 16#5A3.
• Common Lisp uses the prefixes #x and #16r. Setting the variables *read-base*[7] and *print-base*[8] to 16 can also be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.
• MSX BASIC,[9] QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H: &H5A3
• BBC BASIC and Locomotive BASIC use & for hex.[10]
• TI-89 and 92 series uses a 0h prefix: 0h5A3
• ALGOL 68 uses the prefix 16r to denote hexadecimal numbers: 16r5a3. Binary, quaternary (base-4) and octal numbers can be specified similarly.
• The most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (IBM i) running the traditional OS's (zOS, zVSE, zVM, TPF, IBM i) is X'5A3', and is used in Assembler, PL/I, COBOL, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.
Syntax that is always Hex
Sometimes the numbers are known to be Hex.
• In URIs (including URLs), character codes are written as hexadecimal pairs prefixed with %: http://www.example.com/name%20with%20spaces where %20 is the code for the space (blank) character, ASCII code point 20 in hex, 32 in decimal.
• In the Unicode standard, a character value is represented with U+ followed by the hex value, e.g. U+20AC is the Euro sign (€).
• Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits (two each for the red, green and blue components, in that order) prefixed with #: white, for example, is represented as #FFFFFF.[11] CSS also allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange: ).
• In MIME (e-mail extensions) quoted-printable encoding, character codes are written as hexadecimal pairs prefixed with =: Espa=F1a is "España" (F1 is the code for ñ in the ISO/IEC 8859-1 character set).[12])
• PostScript binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC...
• Any IPv6 address can be written as eight groups of four hexadecimal digits (sometimes called hextets), where each group is separated by a colon (:). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334 or abbreviated by removing zeros as 2001:db8:85a3::8a2e:370:7334 (IPv4 addresses are usually written in decimal).
• Globally unique identifiers are written as thirty-two hexadecimal digits, often in unequal hyphen-separated groupings, for example 3F2504E0-4F89-41D3-9A0C-0305E82C3301.
Other symbols for 10–15 and mostly different symbol sets
The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.
• During the 1950s, some installations, such as Bendix-14, favored using the digits 0 through 5 with an overline to denote the values 10–15 as 0, 1, 2, 3, 4 and 5.
• The SWAC (1950)[13] and Bendix G-15 (1956)[14][13] computers used the lowercase letters u, v, w, x, y and z for the values 10 to 15.
• The ORDVAC and ILLIAC I (1952) computers (and some derived designs, e.g. BRLESC) used the uppercase letters K, S, N, J, F and L for the values 10 to 15.[15][13]
• The Librascope LGP-30 (1956) used the letters F, G, J, K, Q and W for the values 10 to 15.[16][13]
• On the PERM (1956) computer, hexadecimal numbers were written as letters O for zero, A to N and P for 1 to 15. Many machine instructions had mnemonic hex-codes (A=add, M=multiply, L=load, F=fixed-point etc.); programs were written without instruction names.[17]
• The Honeywell Datamatic D-1000 (1957) used the lowercase letters b, c, d, e, f, and g whereas the Elbit 100 (1967) used the uppercase letters B, C, D, E, F and G for the values 10 to 15.[13]
• The Monrobot XI (1960) used the letters S, T, U, V, W and X for the values 10 to 15.[13]
• The NEC parametron computer NEAC 1103 (1960) used the letters D, G, H, J, K (and possibly V) for values 10–15.[18]
• The Pacific Data Systems 1020 (1964) used the letters L, C, A, S, M and D for the values 10 to 15.[13]
• New numeric symbols and names were introduced in the Bibi-binary notation by Boby Lapointe in 1968.
• Bruce Alan Martin of Brookhaven National Laboratory considered the choice of A–F "ridiculous". In a 1968 letter to the editor of the CACM, he proposed an entirely new set of symbols based on the bit locations.[19]
• Ronald O. Whitaker of Rowco Engineering Co., in 1972, proposed a triangular font that allows "direct binary reading" in order to "permit both input and output from computers without respect to encoding matrices."[20][21]
• Some seven-segment display decoder chips (i.e., 74LS47) show unexpected output due to logic designed only to produce 0–9 correctly.[22]
Verbal and digital representations
Since there were no traditional numerals to represent the quantities from ten to fifteen, alphabetic letters were re-employed as a substitute. Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. Some people read hexadecimal numbers digit by digit, like a phone number, or using the NATO phonetic alphabet, the Joint Army/Navy Phonetic Alphabet, or a similar ad-hoc system. In the wake of the adoption of hexadecimal among IBM System/360 programmers, Magnuson (1968)[23] suggested a pronunciation guide that gave short names to the letters of hexadecimal – for instance, "A" was pronounced "ann", B "bet", C "chris", etc.[23] Another naming-system was published online by Rogers (2007)[24] that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below. Yet another naming system was elaborated by Babb (2015), based on a joke in Silicon Valley.[25]
Others have proposed using the verbal Morse Code conventions to express four-bit hexadecimal digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), dah-dit-dit-dah (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hexadecimal digit for decimal 15.
Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 102310 on ten fingers.[26] Another system for counting up to FF16 (25510) is illustrated on the right.
Magnuson (1968)[23]
naming method
NumberPronunciation
Aann
Bbet
Cchris
Ddot
Eernest
Ffrost
1Aannteen
A0annty
5Bfifty-bet
A01Cannty christeen
1AD0annteen dotty
3A7Dthirty-ann seventy-dot
Rogers (2007)[24]
naming method
NumberPronunciation
Aten
Beleven
Ctwelve
Ddraze
Eeptwin
Ffim
10tex
11oneteek
1Ffimteek
50fiftek
C0twelftek
100hundrek
1000thousek
3Ethirtek-eptwin
E1eptek-one
C4Atwelve-hundrek-fourtek-ten
1743one-thousek-seven-
-hundrek-fourtek-three
Signs
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −4210 and so on.
Hexadecimal can also be used to express the exact bit patterns used in the processor, so a sequence of hexadecimal digits may represent a signed or even a floating-point value. This way, the negative number −4210 can be written as FFFF FFD6 in a 32-bit CPU register (in two's-complement), as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard).
Hexadecimal exponential notation
Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. P notation uses the letter P (or p, for "power"), whereas E (or e) serves a similar purpose in decimal E notation. The number after the P is decimal and represents the binary exponent. Increasing the exponent by 1 multiplies by 2, not 16: 20p0 = 10p1 = 8p2 = 4p3 = 2p4 = 1.0p5. Usually, the number is normalized so that the hexadecimal digits start with 1. (zero is usually 0 with no P).
Example: 1.3DEp42 represents 1.3DE16 × 24210.
P notation is required by the IEEE 754-2008 binary floating-point standard, and can be used for floating-point literals in the C99 edition of the C programming language.[27] Using the %a or %A conversion specifiers, this notation can be produced by implementations of the printf family of functions following the C99 specification[28] and Single Unix Specification (IEEE Std 1003.1) POSIX standard.[29]
Conversion
Binary conversion
Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (410). This example converts 11112 to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:
• 00012 = 110
• 00102 = 210
• 01002 = 410
• 10002 = 810
Therefore:
11112= 810 + 410 + 210 + 110
= 1510
With little practice, mapping 11112 to F16 in one step becomes easy: see table in written representation. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.[30]
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
(01011110101101010010)2= 26214410 + 6553610 + 3276810 + 1638410 + 819210 + 204810 + 51210 + 25610 + 6410 + 1610 + 210
= 38792210
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
(01011110101101010010)2=0101 1110 1011 0101 00102
=5EB5216
=5EB5216
The conversion from hexadecimal to binary is equally direct.[30]
Other simple conversions
Although quaternary (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits and each quaternary digit corresponds to a pair of binary digits. In the above example 5 E B 5 216 = 11 32 23 11 024.
The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.
Division-remainder in source base
As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans only decimal and for most computers only binary (which can be converted by far more efficient methods) can be easily handled with this method.
Let d be the number to represent in hexadecimal, and the series hihi−1...h2h1 be the hexadecimal digits representing the number.
1. i ← 1
2. hi ← d mod 16
3. d ← (d − hi) / 16
4. If d = 0 (return series hi) else increment i and go to step 2
"16" may be replaced with any other base that may be desired.
The following is a JavaScript implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work with bitwise operators.
function toHex(d) {
var r = d % 16;
if (d - r == 0) {
return toChar(r);
}
return toHex((d - r) / 16) + toChar(r);
}
function toChar(n) {
const alpha = "0123456789ABCDEF";
return alpha.charAt(n);
}
Conversion through addition and multiplication
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value — before carrying out multiplication and addition to get the final representation. For example, to convert the number B3AD to decimal, one can split the hexadecimal number into its digits: B (1110), 3 (310), A (1010) and D (1310), and then get the final result by multiplying each decimal representation by 16p (p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:
B3AD = (11 × 163) + (3 × 162) + (10 × 161) + (13 × 160)
which is 45997 in base 10.
Tools for conversion
Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hexadecimal.
In Microsoft Windows, the Calculator utility can be set to Programmer mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal) and 2 (binary), the bases most commonly used by programmers. In Programmer Mode, the on-screen numeric keypad includes the hexadecimal digits A through F, which are active when "Hex" is selected. In hex mode, however, the Windows Calculator supports only integers.
Elementary arithmetic
Elementary operations such as addition, subtraction, multiplication and division can be carried out indirectly through conversion to an alternate numeral system, such as the commonly-used decimal system or the binary system where each hex digit corresponds to four binary digits.
Alternatively, one can also perform elementary operations directly within the hex system itself — by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm.
Real numbers
Rational numbers
As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although repeating expansions are common since sixteen (1016) has only a single prime factor; two.
For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1. Because the radix 16 is a perfect square (42), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.19 in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.062510 (one-sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560.
n Decimal
Prime factors of: base, b = 10: 2, 5;
b − 1 = 9: 3
Hexadecimal
Prime factors of: base, b = 1610 = 10: 2; b − 1 = 1510 = F: 3, 5
Reciprocal Prime factors Positional representation
(decimal)
Positional representation
(hexadecimal)
Prime factors Reciprocal
2 1/2 2 0.5 0.8 2 1/2
3 1/3 3 0.3333... = 0.3 0.5555... = 0.5 3 1/3
4 1/4 2 0.25 0.4 2 1/4
5 1/5 5 0.2 0.3 5 1/5
6 1/6 2, 3 0.16 0.2A 2, 3 1/6
7 1/7 7 0.142857 0.249 7 1/7
8 1/8 2 0.125 0.2 2 1/8
9 1/9 3 0.1 0.1C7 3 1/9
10 1/10 2, 5 0.1 0.19 2, 5 1/A
11 1/11 11 0.09 0.1745D B 1/B
12 1/12 2, 3 0.083 0.15 2, 3 1/C
13 1/13 13 0.076923 0.13B D 1/D
14 1/14 2, 7 0.0714285 0.1249 2, 7 1/E
15 1/15 3, 5 0.06 0.1 3, 5 1/F
16 1/16 2 0.0625 0.1 2 1/10
17 1/17 17 0.0588235294117647 0.0F 11 1/11
18 1/18 2, 3 0.05 0.0E38 2, 3 1/12
19 1/19 19 0.052631578947368421 0.0D79435E5 13 1/13
20 1/20 2, 5 0.05 0.0C 2, 5 1/14
21 1/21 3, 7 0.047619 0.0C3 3, 7 1/15
22 1/22 2, 11 0.045 0.0BA2E8 2, B 1/16
23 1/23 23 0.0434782608695652173913 0.0B21642C859 17 1/17
24 1/24 2, 3 0.0416 0.0A 2, 3 1/18
25 1/25 5 0.04 0.0A3D7 5 1/19
26 1/26 2, 13 0.0384615 0.09D8 2, D 1/1A
27 1/27 3 0.037 0.097B425ED 3 1/1B
28 1/28 2, 7 0.03571428 0.0924 2, 7 1/1C
29 1/29 29 0.0344827586206896551724137931 0.08D3DCB 1D 1/1D
30 1/30 2, 3, 5 0.03 0.08 2, 3, 5 1/1E
31 1/31 31 0.032258064516129 0.08421 1F 1/1F
32 1/32 2 0.03125 0.08 2 1/20
33 1/33 3, 11 0.03 0.07C1F 3, B 1/21
34 1/34 2, 17 0.02941176470588235 0.078 2, 11 1/22
35 1/35 5, 7 0.0285714 0.075 5, 7 1/23
36 1/36 2, 3 0.027 0.071C 2, 3 1/24
Irrational numbers
The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.
Number Positional representation
Decimal Hexadecimal
√2 (the length of the diagonal of a unit square) 1.414213562373095048... 1.6A09E667F3BCD...
√3 (the length of the diagonal of a unit cube) 1.732050807568877293... 1.BB67AE8584CAA...
√5 (the length of the diagonal of a 1×2 rectangle) 2.236067977499789696... 2.3C6EF372FE95...
φ (phi, the golden ratio = (1+√5)/2) 1.618033988749894848... 1.9E3779B97F4A...
π (pi, the ratio of circumference to diameter of a circle) 3.141592653589793238462643
383279502884197169399375105...
3.243F6A8885A308D313198A2E0
3707344A4093822299F31D008...
e (the base of the natural logarithm) 2.718281828459045235... 2.B7E151628AED2A6B...
τ (the Thue–Morse constant) 0.412454033640107597... 0.6996 9669 9669 6996...
γ (the limiting difference between the harmonic series and the natural logarithm) 0.577215664901532860... 0.93C467E37DB0C7A4D1B...
Powers
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
2xValueValue (Decimal)
2011
2122
2244
2388
2410hex16dec
2520hex32dec
2640hex64dec
2780hex128dec
28100hex256dec
29200hex512dec
2A (210dec)400hex1024dec
2B (211dec)800hex2048dec
2C (212dec)1000hex4096dec
2D (213dec)2000hex8192dec
2E (214dec)4000hex16,384dec
2F (215dec)8000hex32,768dec
210 (216dec)10000hex65,536dec
Cultural history
The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hexadecimal calculations such as additions and subtractions.[31]
As with the duodecimal system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals.[32] Some proposals unify standard measures so that they are multiples of 16.[33][34] An early such proposal was put forward by John W. Nystrom in Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base, published in 1862.[35] Nystrom among other things suggested hexadecimal time, which subdivides a day by 16, so that there are 16 "hours" (or "10 tims", pronounced tontim) in a day.[36]
Look up hexadecimal in Wiktionary, the free dictionary.
The word hexadecimal is first recorded in 1952.[37] It is macaronic in the sense that it combines Greek ἕξ (hex) "six" with Latinate -decimal. The all-Latin alternative sexadecimal (compare the word sexagesimal for base 60) is older, and sees at least occasional use from the late 19th century.[38] It is still in use in the 1950s in Bendix documentation. Schwartzman (1994) argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex.[39] Many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal (e.g. French hexadécimal, Italian esadecimale, Romanian hexazecimal, Serbian хексадецимални, etc.) but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic sextándakerfi, Russian шестнадцатеричной etc.)
Terminology and notation did not become settled until the end of the 1960s. Donald Knuth in 1969 argued that the etymologically correct term would be senidenary, or possibly sedenary, a Latinate term intended to convey "grouped by 16" modelled on binary, ternary and quaternary etc. According to Knuth's argument, the correct terms for decimal and octal arithmetic would be denary and octonary, respectively.[40] Alfred B. Taylor used senidenary in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".[41][42]
The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of the publication of the Fortran IV manual for IBM System/360, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants.[43] As noted above, alternative notations were used by NEC (1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martin in his letter to the editor of the CACM complains that
With the ridiculous choice of letters A, B, C, D, E, F as hexadecimal number symbols adding to already troublesome problems of distinguishing octal (or hex) numbers from decimal numbers (or variable names), the time is overripe for reconsideration of our number symbols. This should have been done before poor choices gelled into a de facto standard!
Martin's argument was that use of numerals 0 to 9 in nondecimal numbers "imply to us a base-ten place-value scheme": "Why not use entirely new symbols (and names) for the seven or fifteen nonzero digits needed in octal or hex. Even use of the letters A through P would be an improvement, but entirely new symbols could reflect the binary nature of the system".[19] He also argued that "re-using alphabetic letters for numerical digits represents a gigantic backward step from the invention of distinct, non-alphabetic glyphs for numerals sixteen centuries ago" (as Brahmi numerals, and later in a Hindu–Arabic numeral system), and that the recent ASCII standards (ASA X3.4-1963 and USAS X3.4-1968) "should have preserved six code table positions following the ten decimal digits -- rather than needlessly filling these with punctuation characters" (":;<=>?") that might have been placed elsewhere among the 128 available positions.
Base16 (transfer encoding)
Base16 (as a proper name without a space) can also refer to a binary to text encoding belonging to the same family as Base32, Base58, and Base64.
In this case, data is broken into 4-bit sequences, and each value (between 0 and 15 inclusively) is encoded using one of 16 symbols from the ASCII character set. Although any 16 symbols from the ASCII character set can be used, in practice the ASCII digits '0'–'9' and the letters 'A'–'F' (or the lowercase 'a'–'f') are always chosen in order to align with standard written notation for hexadecimal numbers.
There are several advantages of Base16 encoding:
• Most programming languages already have facilities to parse ASCII-encoded hexadecimal
• Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64 respectively
• The symbols 0–9 and A–F are universal in hexadecimal notation, so it is easily understood at a glance without needing to rely on a symbol lookup table
• Many CPU architectures have dedicated instructions that allow access to a half-byte (otherwise known as a "nibble"), making it more efficient in hardware than Base32 and Base64
The main disadvantages of Base16 encoding are:
• Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte. In contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively.
• Possible added complexity of having to accept both uppercase and lowercase letters
Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the W3C standard for URL percent encoding, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers.
See also
• Base32, Base64 (content encoding schemes)
• Hexadecimal time
• IBM hexadecimal floating-point
• Hex editor
• Hex dump
• Bailey–Borwein–Plouffe formula (BBP)
• Hexspeak
• P notation
References
1. "The hexadecimal system". Ionos Digital Guide. Archived from the original on 2022-08-26. Retrieved 2022-08-26.
2. Knuth, Donald Ervin (1986). The TeXbook. Duane Bibby. Reading, Mass. ISBN 0-201-13447-0. OCLC 12973034. Archived from the original on 2022-01-16. Retrieved 2022-03-15.{{cite book}}: CS1 maint: location missing publisher (link)
3. The string "\x1B[0m\x1B[25;1H" specifies the character sequence Esc [ 0 m Esc [ 2 5 ; 1 H Nul. These are the escape sequences used on an ANSI terminal that reset the character set and color, and then move the cursor to line 25.
4. "The Unicode Standard, Version 7" (PDF). Unicode. Archived (PDF) from the original on 2016-03-03. Retrieved 2018-10-28.
5. "Modula-2 – Vocabulary and representation". Modula −2. Archived from the original on 2015-12-13. Retrieved 2015-11-01.
6. "An Introduction to VHDL Data Types". FPGA Tutorial. 2020-05-10. Archived from the original on 2020-08-23. Retrieved 2020-08-21.
7. "*read-base* variable in Common Lisp". CLHS. Archived from the original on 2016-02-03. Retrieved 2015-01-10.
8. "*print-base* variable in Common Lisp". CLHS. Archived from the original on 2014-12-26. Retrieved 2015-01-10.
9. MSX is Coming — Part 2: Inside MSX Archived 2010-11-24 at the Wayback Machine Compute!, issue 56, January 1985, p. 52
10. BBC BASIC programs are not fully portable to Microsoft BASIC (without modification) since the latter takes & to prefix octal values. (Microsoft BASIC primarily uses &O to prefix octal, and it uses &H to prefix hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.
11. "Hexadecimal web colors explained". Archived from the original on 2006-04-22. Retrieved 2006-01-11.
12. "ISO-8859-1 (ISO Latin 1) Character Encoding". www.ic.unicamp.br. Archived from the original on 2019-06-29. Retrieved 2019-06-26.
13. Savard, John J. G. (2018) [2005]. "Computer Arithmetic". quadibloc. The Early Days of Hexadecimal. Archived from the original on 2018-07-16. Retrieved 2018-07-16.
14. "2.1.3 Sexadecimal notation". G15D Programmer's Reference Manual (PDF). Los Angeles, CA, USA: Bendix Computer, Division of Bendix Aviation Corporation. p. 4. Archived (PDF) from the original on 2017-06-01. Retrieved 2017-06-01. This base is used because a group of four bits can represent any one of sixteen different numbers (zero to fifteen). By assigning a symbol to each of these combinations we arrive at a notation called sexadecimal (usually hex in conversation because nobody wants to abbreviate sex). The symbols in the sexadecimal language are the ten decimal digits and, on the G-15 typewriter, the letters u, v, w, x, y and z. These are arbitrary markings; other computers may use different alphabet characters for these last six digits.
15. Gill, S.; Neagher, R. E.; Muller, D. E.; Nash, J. P.; Robertson, J. E.; Shapin, T.; Whesler, D. J. (1956-09-01). Nash, J. P. (ed.). "ILLIAC Programming – A Guide to the Preparation of Problems For Solution by the University of Illinois Digital Computer" (PDF). bitsavers.org (Fourth printing. Revised and corrected ed.). Urbana, Illinois, USA: Digital Computer Laboratory, Graduate College, University of Illinois. pp. 3–2. Archived (PDF) from the original on 2017-05-31. Retrieved 2014-12-18.
16. ROYAL PRECISION Electronic Computer LGP – 30 PROGRAMMING MANUAL. Port Chester, New York: Royal McBee Corporation. April 1957. Archived from the original on 2017-05-31. Retrieved 2017-05-31. (NB. This somewhat odd sequence was from the next six sequential numeric keyboard codes in the LGP-30's 6-bit character code.)
17. Manthey, Steffen; Leibrandt, Klaus (2002-07-02). "Die PERM und ALGOL" (PDF) (in German). Archived (PDF) from the original on 2018-10-03. Retrieved 2018-05-19.
18. NEC Parametron Digital Computer Type NEAC-1103 (PDF). Tokyo, Japan: Nippon Electric Company Ltd. 1960. Cat. No. 3405-C. Archived (PDF) from the original on 2017-05-31. Retrieved 2017-05-31.
19. Martin, Bruce Alan (October 1968). "Letters to the editor: On binary notation". Communications of the ACM. Associated Universities Inc. 11 (10): 658. doi:10.1145/364096.364107. S2CID 28248410.
20. Whitaker, Ronald O. (January 1972). Written at Indianapolis, Indiana, USA. "More on man/machine" (PDF). Letters. Datamation. Vol. 18, no. 1. Barrington, Illinois, USA: Technical Publishing Company. p. 103. Archived (PDF) from the original on 2022-12-05. Retrieved 2022-12-24. (1 page)
21. Whitaker, Ronald O. (1976-08-10) [1975-02-24]. "Combined display and range selector for use with digital instruments employing the binary numbering system" (PDF). Indianapolis, Indiana, USA. US Patent 3974444A. Archived (PDF) from the original on 2022-12-24. Retrieved 2022-12-24. (7 pages)
22. "SN5446A, '47A, '48, SN54LS47, 'LS48, 'LS49, SN7446A, '47A, '48, SN74LS47, 'LS48, 'LS49 BCD-to-Seven-Segment Decoders/Drivers". Dallas, Texas, USA: Texas Instruments Incorporated. March 1988 [1974]. SDLS111. Archived (PDF) from the original on 2021-10-20. Retrieved 2021-09-15. (29 pages)
23. Magnuson, Robert A. (January 1968). "A hexadecimal pronunciation guide". Datamation. Vol. 14, no. 1. p. 45.
24. Rogers, S.R. (2007). "Hexadecimal number words". Intuitor. Archived from the original on 2019-09-17. Retrieved 2019-08-26.
25. Babb, Tim (2015). "How to pronounce hexadecimal". Bzarg. Archived from the original on 2020-11-11. Retrieved 2021-01-01.
26. Clarke, Arthur; Pohl, Frederik (2008). The Last Theorem. Ballantine. p. 91. ISBN 978-0007289981.
27. "ISO/IEC 9899:1999 – Programming languages – C". ISO. Iso.org. 2011-12-08. Archived from the original on 2016-10-10. Retrieved 2014-04-08.
28. "Rationale for International Standard – Programming Languages – C" (PDF). Open Standards. 5.10. April 2003. pp. 52, 153–154, 159. Archived (PDF) from the original on 2016-06-06. Retrieved 2010-10-17.
29. The IEEE and The Open Group (2013) [2001]. "dprintf, fprintf, printf, snprintf, sprintf – print formatted output". The Open Group Base Specifications (Issue 7, IEEE Std 1003.1, 2013 ed.). Archived from the original on 2016-06-21. Retrieved 2016-06-21.
30. Mano, M. Morris; Ciletti, Michael D. (2013). Digital Design – With an Introduction to the Verilog HDL (Fifth ed.). Pearson Education. pp. 6, 8–10. ISBN 978-0-13-277420-8.
31. "算盤 Hexadecimal Addition & Subtraction on a Chinese Abacus". totton.idirect.com. Archived from the original on 2019-07-06. Retrieved 2019-06-26.
32. "Base 4^2 Hexadecimal Symbol Proposal". Hauptmech. Archived from the original on 2021-10-20. Retrieved 2008-09-04.
33. "Intuitor Hex Headquarters". Intuitor. Archived from the original on 2010-09-04. Retrieved 2018-10-28.
34. Niemietz, Ricardo Cancho (2003-10-21). "A proposal for addition of the six Hexadecimal digits (A-F) to Unicode". DKUUG Standardizing. Archived from the original on 2011-06-04. Retrieved 2018-10-28.
35. Nystrom, John William (1862). Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base. Philadelphia: Lippincott.
36. Nystrom (1862), p. 33: "In expressing time, angle of a circle, or points on the compass, the unit tim should be noted as integer, and parts thereof as tonal fractions, as 5·86 tims is five times and metonby [*"sutim and metonby" John Nystrom accidentally gives part of the number in decimal names; in Nystrom's pronunciation scheme, 5=su, 8=me, 6=by, c.f. unifoundry.com Archived 2021-05-19 at the Wayback Machine ]."
37. C. E. Fröberg, Hexadecimal Conversion Tables, Lund (1952).
38. The Century Dictionary of 1895 has sexadecimal in the more general sense of "relating to sixteen". An early explicit use of sexadecimal in the sense of "using base 16" is found also in 1895, in the Journal of the American Geographical Society of New York, vols. 27–28, p. 197.
39. Schwartzman, Steven (1994). The Words of Mathematics: An etymological dictionary of mathematical terms used in English. The Mathematical Association of America. p. 105. ISBN 0-88385-511-9. s.v. hexadecimal
40. Knuth, Donald. (1969). The Art of Computer Programming, Volume 2. ISBN 0-201-03802-1. (Chapter 17.)
41. Alfred B. Taylor, Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, 15 September 1859. See pages and 33 and 41.
42. Alfred B. Taylor, "Octonary numeration and its application to a system of weights and measures", Proc Amer. Phil. Soc. Vol XXIV Archived 2016-06-24 at the Wayback Machine, Philadelphia, 1887; pages 296–366. See pages 317 and 322.
43. IBM System/360 FORTRAN IV Language Archived 2021-05-19 at the Wayback Machine (1966), p. 13.
| Wikipedia |
Easter
Easter,[nb 1] also called Pascha[nb 2] (Aramaic, Greek, Latin) or Resurrection Sunday,[nb 3] is a Christian festival and cultural holiday commemorating the resurrection of Jesus from the dead, described in the New Testament as having occurred on the third day of his burial following his crucifixion by the Romans at Calvary c. 30 AD.[10][11] It is the culmination of the Passion of Jesus Christ, preceded by Lent (or Great Lent), a 40-day period of fasting, prayer, and penance.
Easter
Icon of the Resurrection depicting Christ having destroyed the gates of hell and removing Adam and Eve from the grave. Christ is flanked by saints, and Satan, depicted as an old man, is bound and chained.
Observed byChristians
SignificanceCelebrates the resurrection of Jesus
CelebrationsChurch services, festive family meals, Easter egg decoration, and gift-giving
ObservancesPrayer, all-night vigil, sunrise service
DateVariable, determined by the Computus
2022 date
• April 17 (Western)
• April 24 (Eastern)
2023 date
• April 9 (Western)
• April 16 (Eastern)
2024 date
• March 31 (Western)
• May 5 (Eastern)
2025 date
• April 20 (Western)
• April 20 (Eastern)
Related toPassover, Septuagesima, Sexagesima, Quinquagesima, Shrove Tuesday, Ash Wednesday, Clean Monday, Lent, Great Lent, Palm Sunday, Holy Week, Maundy Thursday, Good Friday, and Holy Saturday which lead up to Easter; and Divine Mercy Sunday, Ascension, Pentecost, Trinity Sunday, Corpus Christi and Feast of the Sacred Heart which follow it.
Easter-observing Christians commonly refer to the week before Easter as Holy Week, which in Western Christianity begins on Palm Sunday (marking the entrance of Jesus in Jerusalem), includes Spy Wednesday (on which the betrayal of Jesus is mourned),[12] and contains the days of the Easter Triduum including Maundy Thursday, commemorating the Maundy and Last Supper,[13][14] as well as Good Friday, commemorating the crucifixion and death of Jesus.[15] In Eastern Christianity, the same days and events are commemorated with the names of days all starting with "Holy" or "Holy and Great"; and Easter itself might be called "Great and Holy Pascha", "Easter Sunday", "Pascha" or "Sunday of Pascha". In Western Christianity, Eastertide, or the Easter Season, begins on Easter Sunday and lasts seven weeks, ending with the coming of the 50th day, Pentecost Sunday. In Eastern Christianity, the Paschal season ends with Pentecost as well, but the leave-taking of the Great Feast of Pascha is on the 39th day, the day before the Feast of the Ascension.
Easter and its related holidays are moveable feasts, not falling on a fixed date; its date is computed based on a lunisolar calendar (solar year plus Moon phase) similar to the Hebrew calendar. The First Council of Nicaea (325) established only two rules, namely independence from the Hebrew calendar and worldwide uniformity. No details for the computation were specified; these were worked out in practice, a process that took centuries and generated a number of controversies. It has come to be the first Sunday after the ecclesiastical full moon that occurs on or soonest after 21 March.[16] Even if calculated on the basis of the Gregorian calendar, the date of that full moon sometimes differs from that of the astronomical first full moon after the March equinox.[17]
The English term is derived from the Saxon spring festival Ēostre;[18] Easter is linked to the Jewish Passover by its name (Hebrew: פֶּסַח pesach, Aramaic: פָּסחָא pascha are the basis of the term Pascha), by its origin (according to the synoptic Gospels, both the crucifixion and the resurrection took place during the week of Passover)[19][20] and by much of its symbolism, as well as by its position in the calendar. In most European languages, both the Christian Easter and the Jewish Passover are called by the same name; and in the older English versions of the Bible, as well, the term Easter was used to translate Passover.[21]
Easter traditions vary across the Christian world, and include sunrise services or late-night vigils, exclamations and exchanges of Paschal greetings, flowering the cross,[22] the wearing of Easter bonnets by women, clipping the church,[23] and the decoration and the communal breaking of Easter eggs (a symbol of the empty tomb).[24][25][26] The Easter lily, a symbol of the resurrection in Western Christianity,[27][28] traditionally decorates the chancel area of churches on this day and for the rest of Eastertide.[29] Additional customs that have become associated with Easter and are observed by both Christians and some non-Christians include Easter parades, communal dancing (Eastern Europe), the Easter Bunny and egg hunting.[30][31][32][33][34] There are also traditional Easter foods that vary by region and culture.
Etymology
The modern English term Easter, cognate with modern Dutch ooster and German Ostern, developed from an Old English word that usually appears in the form Ēastrun, Ēastron, or Ēastran; but also as Ēastru, Ēastro; and Ēastre or Ēostre.[nb 4] Bede provides the only documentary source for the etymology of the word, in his eighth-century The Reckoning of Time. He wrote that Ēosturmōnaþ (Old English for 'Month of Ēostre', translated in Bede's time as "Paschal month") was an English month, corresponding to April, which he says "was once called after a goddess of theirs named Ēostre, in whose honour feasts were celebrated in that month".[35]
In Latin and Greek, the Christian celebration was, and still is, called Pascha (Greek: Πάσχα), a word derived from Aramaic פסחא (Paskha), cognate to the Hebrew פֶּסַח (Pesach). The word originally denoted the Jewish festival known in English as Passover, commemorating the Jewish Exodus from slavery in Egypt.[36][37] As early as the 50s of the 1st century, Paul the Apostle, writing from Ephesus to the Christians in Corinth,[38] applied the term to Christ, and it is unlikely that the Ephesian and Corinthian Christians were the first to hear Exodus 12 interpreted as speaking about the death of Jesus, not just about the Jewish Passover ritual.[39] In most languages, Germanic languages such as English being exceptions, the feast is known by names derived from the Greek and Latin Pascha.[7][40] Pascha is also a name by which Jesus himself is remembered in the Orthodox Church, especially in connection with his resurrection and with the season of its celebration.[41] Others call the holiday "Resurrection Sunday" or "Resurrection Day", after the Greek Ἀνάστασις, Anastasis, 'Resurrection' day.[8][9][42][43]
Theological significance
Easter celebrates Jesus' supernatural resurrection from the dead, which is one of the chief tenets of the Christian faith.[44] Paul writes that, for those who trust in Jesus's death and resurrection, "death is swallowed up in victory." The First Epistle of Peter declares that God has given believers "a new birth into a living hope through the resurrection of Jesus Christ from the dead". Christian theology holds that, through faith in the working of God, those who follow Jesus are spiritually resurrected with him so that they may walk in a new way of life and receive eternal salvation, and can hope to be physically resurrected to dwell with him in the Kingdom of Heaven.[45]
Easter is linked to Passover and the Exodus from Egypt recorded in the Old Testament through the Last Supper, sufferings, and crucifixion of Jesus that preceded the resurrection.[40] According to the three Synoptic Gospels, Jesus gave the Passover meal a new meaning, as in the upper room during the Last Supper he prepared himself and his disciples for his death.[40] He identified the bread and cup of wine as his body, soon to be sacrificed, and his blood, soon to be shed. The Apostle Paul states, in his First Epistle to the Corinthians, "Get rid of the old yeast that you may be a new batch without yeast—as you really are. For Christ, our Passover lamb, has been sacrificed." This refers to the requirement in Jewish law that Jews eliminate all chametz, or leavening, from their homes in advance of Passover, and to the allegory of Jesus as the Paschal lamb.[46][47]
Early Christianity
As the Gospels assert that both the crucifixion and resurrection of Jesus during the week of Passover, the first Christians timed the observance of the annual celebration of the resurrections in relation to Passover.[48] Direct evidence for a more fully formed Christian festival of Pascha (Easter) begins to appear in the mid-2nd century. Perhaps the earliest extant primary source referring to Easter is a mid-2nd-century Paschal homily attributed to Melito of Sardis, which characterizes the celebration as a well-established one.[49] Evidence for another kind of annually recurring Christian festival, those commemorating the martyrs, began to appear at about the same time as the above homily.[50]
While martyrs' days (usually the individual dates of martyrdom) were celebrated on fixed dates in the local solar calendar, the date of Easter was fixed by means of the local Jewish[51] lunisolar calendar. This is consistent with the celebration of Easter having entered Christianity during its earliest, Jewish, period, but does not leave the question free of doubt.[52]
The ecclesiastical historian Socrates Scholasticus attributes the observance of Easter by the church to the perpetuation of pre-Christian custom, "just as many other customs have been established", stating that neither Jesus nor his Apostles enjoined the keeping of this or any other festival. Although he describes the details of the Easter celebration as deriving from local custom, he insists the feast itself is universally observed.[53]
Date
Easter and the holidays that are related to it are moveable feasts, in that they do not fall on a fixed date in the Gregorian or Julian calendars (both of which follow the cycle of the sun and the seasons). Instead, the date for Easter is determined on a lunisolar calendar similar to the Hebrew calendar. The First Council of Nicaea (325) established two rules, independence of the Jewish calendar and worldwide uniformity, which were the only rules for Easter explicitly laid down by the Council. No details for the computation were specified; these were worked out in practice, a process that took centuries and generated a number of controversies. (See also Computus and Reform of the date of Easter.) In particular, the Council did not decree that Easter must fall on Sunday, but this was already the practice almost everywhere.[55]
In Western Christianity, using the Gregorian calendar, Easter always falls on a Sunday between 22 March and 25 April,[56] within about seven days after the astronomical full moon.[57] The following day, Easter Monday, is a legal holiday in many countries with predominantly Christian traditions.[58]
Eastern Orthodox Christians base Paschal date calculations on the Julian calendar. Because of the thirteen-day difference between the calendars between 1900 and 2099, 21 March corresponds, during the 21st century, to 3 April in the Gregorian calendar. Since the Julian calendar is no longer used as the civil calendar of the countries where Eastern Christian traditions predominate, Easter varies between 4 April and 8 May in the Gregorian calendar. Because the Julian "full moon" is always several days after the astronomical full moon, the Eastern Easter is also often later, relative to the visible lunar phases, than Western Easter.[59]
Among the Oriental Orthodox, some churches have changed from the Julian to the Gregorian calendar and the date for Easter, as for other fixed and moveable feasts, is the same as in the Western church.[60]
Computations
In 725, Bede succinctly wrote, "The Sunday following the full Moon which falls on or after the equinox will give the lawful Easter."[61] However, this does not precisely reflect the ecclesiastical rules. The full moon referred to (called the Paschal full moon) is not an astronomical full moon, but the 14th day of a lunar month. Another difference is that the astronomical equinox is a natural astronomical phenomenon, which can fall on 19, 20 or 21 March,[62] while the ecclesiastical date is fixed by convention on 21 March.[63]
In addition, the lunar tables of the Julian calendar are currently five days behind those of the Gregorian calendar. Therefore, the Julian computation of the Paschal full moon is a full five days later than the astronomical full moon. The result of this combination of solar and lunar discrepancies is divergence in the date of Easter in most years (see table).[64]
Easter is determined on the basis of lunisolar cycles. The lunar year consists of 30-day and 29-day lunar months, generally alternating, with an embolismic month added periodically to bring the lunar cycle into line with the solar cycle. In each solar year (1 January to 31 December inclusive), the lunar month beginning with an ecclesiastical new moon falling in the 29-day period from 8 March to 5 April inclusive is designated as the paschal lunar month for that year.[65]
Easter is the third Sunday in the paschal lunar month, or, in other words, the Sunday after the paschal lunar month's 14th day. The 14th of the paschal lunar month is designated by convention as the Paschal full moon, although the 14th of the lunar month may differ from the date of the astronomical full moon by up to two days.[65] Since the ecclesiastical new moon falls on a date from 8 March to 5 April inclusive, the paschal full moon (the 14th of that lunar month) must fall on a date from 22 March to 18 April inclusive.[64]
The Gregorian calculation of Easter was based on a method devised by the Calabrian doctor Aloysius Lilius (or Lilio) for adjusting the epacts of the Moon,[66] and has been adopted by almost all Western Christians and by Western countries which celebrate national holidays at Easter. For the British Empire and colonies, a determination of the date of Easter Sunday using Golden Numbers and Sunday letters was defined by the Calendar (New Style) Act 1750 with its Annexe. This was designed to match exactly the Gregorian calculation.[67]
Controversies over the date
The precise date of Easter has at times been a matter of contention. By the later 2nd century, it was widely accepted that the celebration of the holiday was a practice of the disciples and an undisputed tradition. The Quartodeciman controversy, the first of several Easter controversies, arose concerning the date on which the holiday should be celebrated.[68]
The term "Quartodeciman" refers to the practice of ending the Lenten fast on Nisan 14 of the Hebrew calendar, "the LORD's passover".[69] According to the church historian Eusebius, the Quartodeciman Polycarp (bishop of Smyrna, by tradition a disciple of John the Apostle) debated the question with Anicetus (bishop of Rome). The Roman province of Asia was Quartodeciman, while the Roman and Alexandrian churches continued the fast until the Sunday following (the Sunday of Unleavened Bread), wishing to associate Easter with Sunday. Neither Polycarp nor Anicetus persuaded the other, but they did not consider the matter schismatic either, parting in peace and leaving the question unsettled.[70]
Controversy arose when Victor, bishop of Rome a generation after Anicetus, attempted to excommunicate Polycrates of Ephesus and all other bishops of Asia for their Quartodecimanism. According to Eusebius, a number of synods were convened to deal with the controversy, which he regarded as all ruling in support of Easter on Sunday.[71] Polycrates (c. 190), however, wrote to Victor defending the antiquity of Asian Quartodecimanism. Victor's attempted excommunication was apparently rescinded, and the two sides reconciled upon the intervention of bishop Irenaeus and others, who reminded Victor of the tolerant precedent of Anicetus.[72][73]
Quartodecimanism seems to have lingered into the 4th century, when Socrates of Constantinople recorded that some Quartodecimans were deprived of their churches by John Chrysostom[74] and that some were harassed by Nestorius.[75]
It is not known how long the Nisan 14 practice continued. But both those who followed the Nisan 14 custom, and those who set Easter to the following Sunday, had in common the custom of consulting their Jewish neighbors to learn when the month of Nisan would fall, and setting their festival accordingly. By the later 3rd century, however, some Christians began to express dissatisfaction with the custom of relying on the Jewish community to determine the date of Easter. The chief complaint was that the Jewish communities sometimes erred in setting Passover to fall before the Northern Hemisphere spring equinox.[76][77] The Sardica paschal table[78] confirms these complaints, for it indicates that the Jews of some eastern Mediterranean city (possibly Antioch) fixed Nisan 14 on dates well before the spring equinox on multiple occasions.[79]
Because of this dissatisfaction with reliance on the Jewish calendar, some Christians began to experiment with independent computations.[nb 5] Others, however, believed that the customary practice of consulting Jews should continue, even if the Jewish computations were in error.[82]
First Council of Nicaea (325 AD)
This controversy between those who advocated independent computations, and those who wished to continue the custom of relying on the Jewish calendar, was formally resolved by the First Council of Nicaea in 325, which endorsed changing to an independent computation by the Christian community in order to celebrate in common. This effectively required the abandonment of the old custom of consulting the Jewish community in those places where it was still used. Epiphanius of Salamis wrote in the mid-4th century:
[T]he emperor [...] convened a council of 318 bishops [...] in the city of Nicaea [...] They passed certain ecclesiastical canons at the council besides, and at the same time decreed in regard to the Passover [i.e., Easter] that there must be one unanimous concord on the celebration of God's holy and supremely excellent day. For it was variously observed by people; some kept it early, some between [the disputed dates], but others late. And in a word, there was a great deal of controversy at that time.[83]
Canons[84] and sermons[85] condemning the custom of computing Easter's date based on the Jewish calendar indicate that this custom (called "protopaschite" by historians) did not die out at once, but persisted for a time after the Council of Nicaea.[86]
Dionysius Exiguus, and others following him, maintained that the 318 bishops assembled at Nicaea had specified a particular method of determining the date of Easter; subsequent scholarship has refuted this tradition.[87] In any case, in the years following the council, the computational system that was worked out by the church of Alexandria came to be normative. The Alexandrian system, however, was not immediately adopted throughout Christian Europe. Following Augustalis' treatise De ratione Paschae (On the Measurement of Easter), Rome retired the earlier 8-year cycle in favor of Augustalis' 84-year lunisolar calendar cycle, which it used until 457. It then switched to Victorius of Aquitaine's adaptation of the Alexandrian system.[88][89]
Because this Victorian cycle differed from the unmodified Alexandrian cycle in the dates of some of the Paschal full moons, and because it tried to respect the Roman custom of fixing Easter to the Sunday in the week of the 16th to the 22nd of the lunar month (rather than the 15th to the 21st as at Alexandria), by providing alternative "Latin" and "Greek" dates in some years, occasional differences in the date of Easter as fixed by Alexandrian rules continued.[88][89] The Alexandrian rules were adopted in the West following the tables of Dionysius Exiguus in 525.[90]
Early Christians in Britain and Ireland also used an 84-year cycle. From the 5th century onward this cycle set its equinox to 25 March and fixed Easter to the Sunday falling in the 14th to the 20th of the lunar month inclusive.[91][92] This 84-year cycle was replaced by the Alexandrian method in the course of the 7th and 8th centuries. Churches in western continental Europe used a late Roman method until the late 8th century during the reign of Charlemagne, when they finally adopted the Alexandrian method. Since 1582, when the Roman Catholic Church adopted the Gregorian calendar while most of Europe used the Julian calendar, the date on which Easter is celebrated has again differed.[93]
The Greek island of Syros, whose population is divided almost equally between Catholics and Orthodox, is one of the few places where the two Churches share a common date for Easter, with the Catholics accepting the Orthodox date—a practice helping considerably in maintaining good relations between the two communities.[94] Conversely, Orthodox Christians in Finland celebrate Easter according to the Western Christian date.[95]
Proposed reforms of the date
In the 20th and 21st centuries, some individuals and institutions have propounded changing the method of calculating the date for Easter, the most prominent proposal being the Sunday after the second Saturday in April. Despite having some support, proposals to reform the date have not been implemented.[96] An Orthodox congress of Eastern Orthodox bishops, which included representatives mostly from the Patriarch of Constantinople and the Serbian Patriarch, met in Constantinople in 1923, where the bishops agreed to the Revised Julian calendar.[97]
The original form of this calendar would have determined Easter using precise astronomical calculations based on the meridian of Jerusalem.[98][99] However, all the Eastern Orthodox countries that subsequently adopted the Revised Julian calendar adopted only that part of the revised calendar that applied to festivals falling on fixed dates in the Julian calendar. The revised Easter computation that had been part of the original 1923 agreement was never permanently implemented in any Orthodox diocese.[97]
In the United Kingdom, Parliament passed the Easter Act 1928 to change the date of Easter to be the first Sunday after the second Saturday in April (or, in other words, the Sunday in the period from 9 to 15 April). However, the legislation has not been implemented, although it remains on the Statute book and could be implemented, subject to approval by the various Christian churches.[100]
At a summit in Aleppo, Syria, in 1997, the World Council of Churches (WCC) proposed a reform in the calculation of Easter which would have replaced the present divergent practices of calculating Easter with modern scientific knowledge taking into account actual astronomical instances of the spring equinox and full moon based on the meridian of Jerusalem, while also following the tradition of Easter being on the Sunday following the full moon.[101] The recommended World Council of Churches changes would have sidestepped the calendar issues and eliminated the difference in date between the Eastern and Western churches. The reform was proposed for implementation starting in 2001, and despite repeated calls for reform, it was not ultimately adopted by any member body.[102][103]
In January 2016, the Anglican Communion, Coptic Orthodox Church, Greek Orthodox Church, and Roman Catholic Church again considered agreeing on a common, universal date for Easter, while also simplifying the calculation of that date, with either the second or third Sunday in April being popular choices.[104]
In November 2022, the Patriarch of Constantinople said that conversations between the Roman Catholic Church and the Orthodox Churches had begun to determine a common date for the celebration of Easter. The agreement is expected to be reached for the 1700th anniversary of the Council of Nicaea in 2025.[105]
Table of the dates of Easter by Gregorian and Julian calendars
The WCC presented comparative data of the relationships:
Table of dates of Easter 2001–2025 (in Gregorian dates)[106]
Year Full Moon Jewish Passover [note 1] Astronomical Easter [note 2] Gregorian Easter Julian Easter
2001 8 April 15 April
2002 28 March 31 March5 May
2003 16 April17 April 20 April27 April
2004 5 April6 April 11 April
2005 25 March24 April 27 March1 May
2006 13 April 16 April23 April
2007 2 April3 April 8 April
2008 21 March20 April 23 March27 April
2009 9 April 12 April19 April
2010 30 March 4 April
2011 18 April19 April 24 April
2012 6 April7 April 8 April15 April
2013 27 March26 March 31 March5 May
2014 15 April 20 April
2015 4 April 5 April12 April
2016 23 March23 April 27 March1 May
2017 11 April 16 April
2018 31 March 1 April8 April
2019 21 March20 April 24 March21 April28 April
2020 8 April9 April 12 April19 April
2021 28 March 4 April2 May
2022 16 April 17 April24 April
2023 6 April 9 April16 April
2024 25 March23 April 31 March5 May
2025 13 April 20 April
1. Jewish Passover is on Nisan 15 of its calendar. It commences at sunset preceding the date indicated (as does Easter in many traditions).
2. Astronomical Easter is the first Sunday after the astronomical full moon after the astronomical March equinox as measured at the meridian of Jerusalem according to this WCC proposal.
Position in the church year
Western Christianity
In most branches of Western Christianity, Easter is preceded by Lent, a period of penitence that begins on Ash Wednesday, lasts 40 days (not counting Sundays), and is often marked with fasting. The week before Easter, known as Holy Week, is an important time for observers to commemorate the final week of Jesus' life on earth.[107] The Sunday before Easter is Palm Sunday, with the Wednesday before Easter being known as Spy Wednesday (or Holy Wednesday). The last three days before Easter are Maundy Thursday, Good Friday and Holy Saturday (sometimes referred to as Silent Saturday).[108]
Palm Sunday, Maundy Thursday and Good Friday respectively commemorate Jesus's entry in Jerusalem, the Last Supper and the crucifixion. Maundy Thursday, Good Friday, and Holy Saturday are sometimes referred to as the Easter Triduum (Latin for "Three Days"). Many churches begin celebrating Easter late in the evening of Holy Saturday at a service called the Easter Vigil.[109]
The week beginning with Easter Sunday is called Easter Week or the Octave of Easter, and each day is prefaced with "Easter", e.g. Easter Monday (a public holiday in many countries), Easter Tuesday (a much less widespread public holiday), etc. Easter Saturday is therefore the Saturday after Easter Sunday. The day before Easter is properly called Holy Saturday. Eastertide, or Paschaltide, the season of Easter, begins on Easter Sunday and lasts until the day of Pentecost, seven weeks later.[110][111][112]
Eastern Christianity
In Eastern Christianity, the spiritual preparation for Easter/Pascha begins with Great Lent, which starts on Clean Monday and lasts for 40 continuous days (including Sundays). Great Lent ends on a Friday, and the next day is Lazarus Saturday. The Vespers which begins Lazarus Saturday officially brings Great Lent to a close, although the fast continues through the following week.[113][114]
The Paschal Vigil begins with the Midnight Office, which is the last service of the Lenten Triodion and is timed so that it ends a little before midnight on Holy Saturday night. At the stroke of midnight the Paschal celebration itself begins, consisting of Paschal Matins, Paschal Hours, and Paschal Divine Liturgy.[115]
The liturgical season from Easter to the Sunday of All Saints (the Sunday after Pentecost) is known as the Pentecostarion (the "50 days"). The week which begins on Easter Sunday is called Bright Week, during which there is no fasting, even on Wednesday and Friday. The Afterfeast of Easter lasts 39 days, with its Apodosis (leave-taking) on the day before the Feast of the Ascension. Pentecost Sunday is the 50th day from Easter (counted inclusively).[116] In the Pentecostarion published by Apostoliki Diakonia of the Church of Greece, the Great Feast Pentecost is noted in the synaxarion portion of Matins to be the 8th Sunday of Pascha. However, the Paschal greeting of "Christ is risen!" is no longer exchanged among the faithful after the Apodosis of Pascha.[117][118]
Liturgical observance
Western Christianity
The Easter festival is kept in many different ways among Western Christians. The traditional, liturgical observation of Easter, as practised among Roman Catholics, Lutherans,[121] and some Anglicans begins on the night of Holy Saturday with the Easter Vigil which follows an ancient liturgy involving symbols of light, candles and water and numerous readings form the Old and New Testament.[122]
Services continue on Easter Sunday and in a number of countries on Easter Monday. In parishes of the Moravian Church, as well as some other denominations such as the Methodist Churches, there is a tradition of Easter Sunrise Services[123] often starting in cemeteries[124] in remembrance of the biblical narrative in the Gospels, or other places in the open where the sunrise is visible.[125]
In some traditions, Easter services typically begin with the Paschal greeting: "Christ is risen!" The response is: "He is risen indeed. Alleluia!"[126]
Eastern Christianity
Eastern Orthodox, Eastern Catholics and Byzantine Rite Lutherans have a similar emphasis on Easter in their calendars, and many of their liturgical customs are very similar.[127]
Preparation for Easter begins with the season of Great Lent, which begins on Clean Monday.[128] While the end of Lent is Lazarus Saturday, fasting does not end until Easter Sunday.[129] The Orthodox service begins late Saturday evening, observing the Jewish tradition that evening is the start of liturgical holy days.[129]
The church is darkened, then the priest lights a candle at midnight, representing the resurrection of Jesus Christ. Altar servers light additional candles, with a procession which moves three times around the church to represent the three days in the tomb.[129] The service continues early into Sunday morning, with a feast to end the fasting. An additional service is held later that day on Easter Sunday.[129]
Non-observing Christian groups
Many Puritans saw traditional feasts of the established Anglican Church, such as All Saints' Day and Easter, as abominations because the Bible does not mention them.[130][131] Conservative Reformed denominations such as the Free Presbyterian Church of Scotland and the Reformed Presbyterian Church of North America likewise reject the celebration of Easter as a violation of the regulative principle of worship and what they see as its non-Scriptural origin.[132][133]
Members of the Religious Society of Friends (Quakers), as part of their historic testimony against times and seasons, do not celebrate or observe Easter or any traditional feast days of the established Church, believing instead that "every day is the Lord's Day," and that elevation of one day above others suggests that it is acceptable to do un-Christian acts on other days.[134][135] During the 17th and 18th centuries, Quakers were persecuted for this non-observance of Holy Days.[136]
Groups such as the Restored Church of God reject the celebration of Easter, seeing it as originating in a pagan spring festival adopted by the Roman Catholic Church.[137][138][139]
Jehovah's Witnesses maintain a similar view, observing a yearly commemorative service of the Last Supper and the subsequent execution of Christ on the evening of Nisan 14 (as they calculate the dates derived from the lunar Hebrew calendar). It is commonly referred to by many Witnesses as simply "The Memorial". Jehovah's Witnesses believe that such verses as Luke 22:19–20 and 1 Corinthians 11:26 constitute a commandment to remember the death of Christ though not the resurrection.[140][141]
Easter celebrations around the world
In countries where Christianity is a state religion, or those with large Christian populations, Easter is often a public holiday.[142] As Easter always falls on a Sunday, many countries in the world also recognize Easter Monday as a public holiday.[143] Some retail stores, shopping malls, and restaurants are closed on Easter Sunday. Good Friday, which occurs two days before Easter Sunday, is also a public holiday in many countries, as well as in 12 U.S. states. Even in states where Good Friday is not a holiday, many financial institutions, stock markets, and public schools are closed – the few banks that are normally open on regular Sundays are closed on Easter.[144]
In the Nordic countries Good Friday, Easter Sunday, and Easter Monday are public holidays,[145] and Good Friday and Easter Monday are bank holidays.[146] In Denmark, Iceland and Norway Maundy Thursday is also a public holiday. It is a holiday for most workers, except those operating some shopping malls which keep open for a half-day. Many businesses give their employees almost a week off, called Easter break.[147] Schools are closed between Palm Sunday and Easter Monday. According to a 2014 poll, 6 of 10 Norwegians travel during Easter, often to a countryside cottage; 3 of 10 said their typical Easter included skiing.[148]
In the Netherlands both Easter Sunday and Easter Monday are national holidays. Like first and second Christmas Day, they are both considered Sundays, which results in a first and a second Easter Sunday, after which the week continues to a Tuesday.[149]
In Greece Good Friday and Saturday as well as Easter Sunday and Monday are traditionally observed public holidays. It is custom for employees of the public sector to receive Easter bonuses as a gift from the state.[150]
In Commonwealth nations Easter Day is rarely a public holiday, as is the case for celebrations which fall on a Sunday. In the United Kingdom both Good Friday and Easter Monday are bank holidays, except for Scotland, where only Good Friday is a bank holiday.[151] In Canada, Easter Monday is a statutory holiday for federal employees. In the Canadian province of Quebec, either Good Friday or Easter Monday are statutory holidays (although most companies give both).[152]
In Australia, Easter is associated with harvest time.[153] Good Friday and Easter Monday are public holidays across all states and territories. "Easter Saturday" (the Saturday before Easter Sunday) is a public holiday in every state except Tasmania and Western Australia, while Easter Sunday itself is a public holiday only in New South Wales. Easter Tuesday is additionally a conditional public holiday in Tasmania, varying between award, and was also a public holiday in Victoria until 1994.[154]
In the United States, because Easter falls on a Sunday, which is already a non-working day for federal and state employees, it has not been designated as a federal or state holiday.[155] Easter parades are held in many American cities, involving festive strolling processions.[30]
Traditional customs
The egg is an ancient symbol of new life and rebirth.[156] In Christianity it became associated with Jesus's crucifixion and resurrection.[157] The custom of the Easter egg originated in the early Christian community of Mesopotamia, who stained eggs red in memory of the blood of Christ, shed at his crucifixion.[158][159] As such, for Christians, the Easter egg is a symbol of the empty tomb.[25][26] The oldest tradition is to use dyed chicken eggs.
In the Eastern Orthodox Church Easter eggs are blessed by a priest[160] both in families' baskets together with other foods forbidden during Great Lent and alone for distribution or in church or elsewhere.
• Traditional red Easter eggs for blessing by a priest
• A priest blessing baskets with Easter eggs and other foods forbidden during Great Lent
• A priest distributing blessed Easter eggs after blessing the Soyuz rocket
Easter eggs are a widely popular symbol of new life among the Eastern Orthodox but also in folk traditions in Slavic countries and elsewhere. A batik-like decorating process known as pisanka produces intricate, brilliantly colored eggs. The celebrated House of Fabergé workshops created exquisite jewelled Easter eggs for the Russian Imperial family from 1885 to 1916.[161]
Modern customs
A modern custom in the Western world is to substitute decorated chocolate, or plastic eggs filled with candy such as jellybeans; as many people give up sweets as their Lenten sacrifice, individuals enjoy them at Easter after having abstained from them during the preceding forty days of Lent.[162]
• Easter eggs, a symbol of the empty tomb, are a popular cultural symbol of Easter.[24]
• Marshmallow rabbits, candy eggs and other treats in an Easter basket
• An Easter egg decorated with the Easter Bunny
Manufacturing their first Easter egg in 1875, British chocolate company Cadbury sponsors the annual Easter egg hunt which takes place in over 250 National Trust locations in the United Kingdom.[163][164] On Easter Monday, the President of the United States holds an annual Easter egg roll on the White House lawn for young children.[165]
Easter Bunny
In some traditions, the children put out their empty baskets for the Easter bunny to fill while they sleep. They wake to find their baskets filled with candy eggs and other treats.[166][31] A custom originating in Germany,[166] the Easter Bunny is a popular legendary anthropomorphic Easter gift-giving character analogous to Santa Claus in American culture. Many children around the world follow the tradition of coloring hard-boiled eggs and giving baskets of candy.[31] Historically, foxes, cranes and storks were also sometimes named as the mystical creatures.[166] Since the rabbit is a pest in Australia, the Easter Bilby is available as an alternative.[167]
Music
• Marc-Antoine Charpentier:[168][169]
• Messe pour le samedi de Pâques, for soloists, chorus and continuo, H.8 (1690).
• Prose pour le jour de Pâques, for 3 voices and continuo, H.13 (1670)
• Chant joyeux du temps de Pâques, for soloists, chorus, 2 treble viols, and continuo, H.339 (1685).
• O filii à 3 voix pareilles, for 3 voices, 2 flutes, and continuo, H.312 (1670).
• Pour Pâques, for 2 voices, 2 flutes, and continuo, H.308 (1670).
• O filii pour les voix, violons, flûtes et orgue, for soloists, chorus, flutes, strings, and continuo, H.356 (1685?).
• Louis-Nicolas Clérambault: Motet pour le Saint jour de Pâques, in F major, opus 73
• André Campra: Au Christ triomphant, cantata for Easter
• Dieterich Buxtehude: Cantatas BuxWV 15 and BuxWV 62
• Carl Heinrich Graun: Easter Oratorio
• Henrich Biber: Missa Christi resurgentis C.3 (1674)
• Michael Praetorius: Easter Mass
• Johann Sebastian Bach: Christ lag in Todesbanden, BWV 4; Der Himmel lacht! Die Erde jubilieret, BWV 31; Oster-Oratorium, BWV 249.
• Georg Philipp Telemann: more than 100 cantatas for Eastertide.
• Jacques-Nicolas Lemmens: Sonata n° 2 "O Filii", Sonata n° 3 "Pascale", for organ.
• Charles Gounod: Messe solennelle de Pâques (1883).
• Nikolai Rimsky-Korsakov: La Grande Pâque russe, symphonic overture (1888).
• Sergueï Vassilievitch Rachmaninov: Suite pour deux pianos n°1 – Pâques, op. 5, n° 4 (1893).
See also
• Divine Mercy Sunday
• Life of Jesus in the New Testament
• List of Easter films
• List of Easter hymns
• List of Easter television episodes
• Movable Eastern Christian Observances
• Regina Caeli
• Category:Film portrayals of Jesus' death and resurrection
Footnotes
1. Traditional names for the feast in English are "Easter Day", as in the Book of Common Prayer; "Easter Sunday", used by James Ussher (The Whole Works of the Most Rev. James Ussher, Volume 4[1]) and Samuel Pepys (The Diary of Samuel Pepys, Volume 2[2]), as well as the single word "Easter" in books printed in 1575,[3] 1584,[4] and 1586.[5]
2. In the Eastern Orthodox Church, the Greek word Pascha is used for the celebration; in English, the analogous word is Pasch.[6][7]
3. The term "Resurrection Sunday" is used particularly by Christian communities in the Middle East.[8][9]
4. Old English pronunciation: [ˈæːɑstre, ˈeːostre]
5. Eusebius reports that Dionysius, Bishop of Alexandria, proposed an 8-year Easter cycle, and quotes a letter from Anatolius, Bishop of Laodicea, that refers to a 19-year cycle.[80] An 8-year cycle has been found inscribed on a statue unearthed in Rome in the 17th century, and since dated to the 3rd century.[81]
References
1. Ussher, James; Elrington, Charles Richard (1631). The Whole Works of the Most Rev. James Ussher – James Ussher, Charles Richard Elrington – Google Books. Archived from the original on 1 August 2020. Retrieved 28 March 2023.
2. Pepys, Samuel (1665). The Diary of Samuel Pepys M.A. F.R.S. Archived from the original on 9 April 2023. Retrieved 7 April 2023.
3. Foxe, John (1575). A Sermon of Christ Crucified, Preached at Paules Crosse the Fridaie Before ... Archived from the original on 9 April 2023. Retrieved 20 June 2015.
4. Caradoc (St. of Llancarfan) (1584). The Historie of Cambria. Archived from the original on 9 April 2023. Retrieved 20 June 2015.
5. (de Granada), Luis (1586). "A Memoriall of a Christian Life: Wherein are Treated All Such Thinges, as ..." Archived from the original on 9 April 2023. Retrieved 20 June 2015.
6. Ferguson, Everett (2009). Baptism in the Early Church: History, Theology, and Liturgy in the First Five Centuries. Wm. B. Eerdmans Publishing. p. 351. ISBN 978-0802827487. Archived from the original on 1 August 2020. Retrieved 23 April 2014. The practices are usually interpreted in terms of baptism at the pasch (Easter), for which compare Tertullian, but the text does not specify this season, only that it was done on Sunday, and the instructions may apply to whenever the baptism was to be performed.
7. Davies, Norman (1998). Europe: A History. HarperCollins. p. 201. ISBN 978-0060974688. In most European languages Easter is called by some variant of the late Latin word Pascha, which in turn derives from the Hebrew pesach, meaning passover.
8. Gamman, Andrew; Bindon, Caroline (2014). Stations for Lent and Easter. Kereru Publishing Limited. p. 7. ISBN 978-0473276812. Easter Day, also known as Resurrection Sunday, marks the high point of the Christian year. It is the day that we celebrate the resurrection of Jesus Christ from the dead.
9. Boda, Mark J.; Smith, Gordon T. (2006). Repentance in Christian Theology. Liturgical Press. p. 316. ISBN 978-0814651759. Archived from the original on 4 August 2020. Retrieved 19 April 2014. Orthodox, Catholic, and all Reformed churches in the Middle East celebrate Easter according to the Eastern calendar, calling this holy day "Resurrection Sunday," not Easter.
10. Trawicky, Bernard; Gregory, Ruth Wilhelme (2000). Anniversaries and Holidays. American Library Association. ISBN 978-0838906958. Archived from the original on 12 October 2017. Retrieved 17 October 2020. Easter is the central celebration of the Christian liturgical year. It is the oldest and most important Christian feast, celebrating the Resurrection of Jesus Christ. The date of Easter determines the dates of all movable feasts except those of Advent.
11. Aveni, Anthony (2004). "The Easter/Passover Season: Connecting Time's Broken Circle", The Book of the Year: A Brief History of Our Seasonal Holidays. Oxford University Press. pp. 64–78. ISBN 0-19-517154-3. Archived from the original on 8 February 2021. Retrieved 17 October 2020.
12. Cooper, J.HB. (23 October 2013). Dictionary of Christianity. Routledge. p. 124. ISBN 9781134265466. Holy Week. The last week in LENT. It begins on PALM SUNDAY; the fourth day is called SPY WEDNESDAY; the fifth is MAUNDY THURSDAY or HOLY THURSDAY; the sixth is Good Friday; and the last 'Holy Saturday', or the 'Great Sabbath'.
13. Peter C. Bower (2003). The Companion to the Book of Common Worship. Geneva Press. ISBN 978-0664502324. Archived from the original on 8 June 2021. Retrieved 11 April 2009. Maundy Thursday (or le mandé; Thursday of the Mandatum, Latin, commandment). The name is taken from the first few words sung at the ceremony of the washing of the feet, "I give you a new commandment" (John 13:34); also from the commandment of Christ that we should imitate His loving humility in the washing of the feet (John 13:14–17). The term mandatum (maundy), therefore, was applied to the rite of foot-washing on this day.
14. Ramshaw, Gail (2004). Three Day Feast: Maundy Thursday, Good Friday, and Easter. Augsburg Fortress. ISBN 978-1451408164. Archived from the original on 5 November 2021. Retrieved 11 April 2009. In the liturgies of the Three Days, the service for Maundy Thursday includes both, telling the story of Jesus' last supper and enacting the footwashing.
15. Stuart, Leonard (1909). New century reference library of the world's most important knowledge: complete, thorough, practical, Volume 3. Syndicate Pub. Co. Archived from the original on 5 November 2021. Retrieved 11 April 2009. Holy Week, or Passion Week, the week which immediately precedes Easter, and is devoted especially to commemorating the passion of our Lord. The Days more especially solemnized during it are Holy Wednesday, Maundy Thursday, Good Friday, and Holy Saturday.
16. "Frequently asked questions about the date of Easter". Archived from the original on 22 April 2011. Retrieved 22 April 2009.
17. Woodman, Clarence E. (1923). "Clarence E. Woodman, "Easter and the Ecclesiastical Calendar" in Journal of the Royal Astronomical Society of Canada". Journal of the Royal Astronomical Society of Canada. 17: 141. Bibcode:1923JRASC..17..141W. Archived from the original on 12 May 2019. Retrieved 12 May 2019.
18. Gamber, Jenifer (September 2014). My Faith, My Life, Revised Edition: A Teen's Guide to the Episcopal Church. Church Publishing. p. 96. ISBN 978-0-8192-2962-5. The word "Easter" comes from the Anglo-Saxon spring festival called Eostre. Easter replaced the pagan festival of Eostre.
19. "5 April 2007: Mass of the Lord's Supper | BENEDICT XVI". www.vatican.va. Archived from the original on 5 April 2021. Retrieved 1 April 2021.
20. Reno, R. R. (14 April 2017). "The Profound Connection Between Easter and Passover". The Wall Street Journal. ISSN 0099-9660. Archived from the original on 17 December 2021. Retrieved 1 April 2021.
21. Weiser, Francis X. (1958). Handbook of Christian Feasts and Customs. New York: Harcourt, Brace and Company. p. 214. ISBN 0-15-138435-5.
22. Whitehouse, Bonnie Smith (15 November 2022). Seasons of Wonder: Making the Ordinary Sacred Through Projects, Prayers, Reflections, and Rituals: A 52-week devotional. Crown Publishing Group. p. 95-96. ISBN 978-0-593-44332-3.
23. Simpson, Jacqueline; Roud, Steve (2003). "clipping the church". Oxford Reference. Oxford University Press. doi:10.1093/acref/9780198607663.001.0001. ISBN 9780198607663. Archived from the original on 12 April 2020. Retrieved 31 March 2013.
24. Jordan, Anne (2000). Christianity. Nelson Thornes. ISBN 978-0748753208. Archived from the original on 8 February 2021. Retrieved 7 April 2012. Easter eggs are used as a Christian symbol to represent the empty tomb. The outside of the egg looks dead but inside there is new life, which is going to break out. The Easter egg is a reminder that Jesus will rise from His tomb and bring new life. Eastern Orthodox Christians dye boiled eggs red to represent the blood of Christ shed for the sins of the world.
25. The Guardian, Volume 29. H. Harbaugh. 1878. Archived from the original on 4 August 2020. Retrieved 7 April 2012. Just so, on that first Easter morning, Jesus came to life and walked out of the tomb, and left it, as it were, an empty shell. Just so, too, when the Christian dies, the body is left in the grave, an empty shell, but the soul takes wings and flies away to be with God. Thus you see that though an egg seems to be as dead as a stone, yet it really has life in it; and also it is like Christ's dead body, which was raised to life again. This is the reason we use eggs on Easter. (In olden times they used to color the eggs red, so as to show the kind of death by which Christ died, – a bloody death.)
26. Gordon Geddes, Jane Griffiths (2002). Christian belief and practice. Heinemann. ISBN 978-0435306915. Archived from the original on 29 July 2020. Retrieved 7 April 2012. Red eggs are given to Orthodox Christians after the Easter Liturgy. They crack their eggs against each other's. The cracking of the eggs symbolizes a wish to break away from the bonds of sin and misery and enter the new life issuing from Christ's resurrection.
27. Collins, Cynthia (19 April 2014). "Easter Lily Tradition and History". The Guardian. Archived from the original on 17 August 2020. Retrieved 20 April 2014. The Easter Lily is symbolic of the resurrection of Jesus Christ. Churches of all denominations, large and small, are filled with floral arrangements of these white flowers with their trumpet-like shape on Easter morning.
28. Schell, Stanley (1916). Easter Celebrations. Werner & Company. p. 84. We associate the lily with Easter, as pre-eminently the symbol of the Resurrection.
29. Luther League Review: 1936–1937. Luther League of America. 1936. Archived from the original on 3 August 2020. Retrieved 20 June 2015.
30. Duchak, Alicia (2002). An A–Z of Modern America. Rutledge. p. 372. ISBN 978-0415187558. Archived from the original on 27 December 2021. Retrieved 17 October 2020.
31. Sifferlin, Alexandra (21 February 2020) [2015]. "What's the Origin of the Easter Bunny?". Time. Archived from the original on 22 October 2021. Retrieved 4 April 2021.
32. Black, Vicki K. (2004). The Church Standard, Volume 74. Church Publishing, Inc. ISBN 978-0819225757. Archived from the original on 4 August 2020. Retrieved 7 April 2012. In parts of Europe, the eggs were dyed red and were then cracked together when people exchanged Easter greetings. Many congregations today continue to have Easter egg hunts for the children after the services on Easter Day.
33. The Church Standard, Volume 74. Walter N. Hering. 1897. Archived from the original on 30 August 2020. Retrieved 7 April 2012. When the custom was carried over into Christian practice the Easter eggs were usually sent to the priests to be blessed and sprinkled with holy water. In later times the coloring and decorating of eggs was introduced, and in a royal roll of the time of Edward I., which is preserved in the Tower of London, there is an entry of 18d. for 400 eggs, to be used for Easter gifts.
34. Brown, Eleanor Cooper (2010). From Preparation to Passion. ISBN 978-1609577650. Archived from the original on 4 August 2020. Retrieved 7 April 2012. So what preparations do most Christians and non-Christians make? Shopping for new clothing often signifies the belief that Spring has arrived, and it is a time of renewal. Preparations for the Easter Egg Hunts and the Easter Ham for the Sunday dinner are high on the list too.
35. Wallis, Faith (1999). Bede: The Reckoning of Time. Liverpool University Press. p. 54. ISBN 0853236933.
36. "History of Easter". The History Channel website. A&E Television Networks. Archived from the original on 31 May 2013. Retrieved 9 March 2013.
37. Karl Gerlach (1998). The Antenicene Pascha: A Rhetorical History. Peeters Publishers. p. xviii. ISBN 978-9042905702. Archived from the original on 8 August 2021. Retrieved 9 January 2020. The second century equivalent of easter and the paschal Triduum was called by both Greek and Latin writers "Pascha (πάσχα)", a Greek transliteration of the Aramaic form of the Hebrew פֶּסַח, the Passover feast of Ex. 12.
38. 1 Corinthians 5:7
39. Karl Gerlach (1998). The Antenicene Pascha: A Rhetorical History. Peters Publishers. p. 21. ISBN 978-9042905702. Archived from the original on 28 December 2021. Retrieved 17 October 2020. For while it is from Ephesus that Paul writes, "Christ our Pascha has been sacrificed for us," Ephesian Christians were not likely the first to hear that Ex 12 did not speak about the rituals of Pesach, but the death of Jesus of Nazareth.
40. Vicki K. Black (2004). Welcome to the Church Year: An Introduction to the Seasons of the Episcopal Church. Church Publishing, Inc. ISBN 978-0819219664. Archived from the original on 8 August 2021. Retrieved 9 January 2020. Easter is still called by its older Greek name, Pascha, which means "Passover", and it is this meaning as the Christian Passover-the celebration of Jesus's triumph over death and entrance into resurrected life-that is the heart of Easter in the church. For the early church, Jesus Christ was the fulfillment of the Jewish Passover feast: through Jesus, we have been freed from slavery of sin and granted to the Promised Land of everlasting life.
41. Orthros of Holy Pascha, Stichera: "Today the sacred Pascha is revealed to us. The new and holy Pascha, the mystical Pascha. The all-venerable Pascha. The Pascha which is Christ the Redeemer. The spotless Pascha. The great Pascha. The Pascha of the faithful. The Pascha which has opened unto us the gates of Paradise. The Pascha which sanctifies all faithful."
42. "Easter or Resurrection day?". Simply Catholic. 17 January 2019. Archived from the original on 8 June 2021. Retrieved 4 April 2021.
43. "Easter: 5 facts you need to know about resurrection sunday". Christian Post. 1 April 2018. Archived from the original on 22 November 2021. Retrieved 4 April 2021.
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Category:Easter
• Greek words (Wiktionary): Πάσχα (Easter) vs. πάσχα (Passover) vs. πάσχω (to suffer)
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• Orthodox Paschal Calculator Julian Easter and associated festivals in Gregorian calendar 1583–4099
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| Wikipedia |
2E6 (mathematics)
In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E6, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L) (thinking of the group as a subgroup of E6(L) fixed by an outer involution).
Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959).
Over finite fields
The group 2E6(q2) has order q36 (q12 − 1) (q9 + 1) (q8 − 1) (q6 − 1) (q5 + 1) (q2 − 1) /(3,q + 1).[1] This is similar to the order q36 (q12 − 1) (q9 − 1) (q8 − 1) (q6 − 1) (q5 − 1) (q2 − 1) /(3,q − 1) of E6(q).
Its Schur multiplier has order (3, q + 1) except for q=2, i. e. 2E6(22), when it has order 12 and is a product of cyclic groups of orders 2,2,3. One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.
The outer automorphism group has order (3, q + 1) · f where q2 = pf.
Over the real numbers
Over the real numbers, 2E6 is the quasisplit form of E6, and is one of the five real forms of E6 classified by Élie Cartan. Its maximal compact subgroup is of type F4.
Remarks
1. Reading example: If q2=22 in 2E6(q2) then q=2 in the order formula q36 (q12 − 1) (q9 + 1) (q8 − 1) (q6 − 1) (q5 + 1) (q2 − 1) /(3,q + 1). However, the group 2E6(22) is sometimes also written 2E6(2) (e. g. in Wilson's Atlas).
References
• Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50683-6, MR 0407163
• Steinberg, Robert (1959), "Variations on a theme of Chevalley", Pacific Journal of Mathematics, 9: 875–891, doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191
• Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335, archived from the original on 2012-09-10
• Tits, Jacques (1958), Les "formes réelles" des groupes de type E6, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162, vol. 15, Paris: Secrétariat math'ematique, MR 0106247
• Robert Wilson: Atlas of Finite Group Representations: Sporadic groups
| Wikipedia |
1/4 + 1/16 + 1/64 + 1/256 + ⋯
In mathematics, the infinite series 1/4 + 1/16 + 1/64 + 1/256 + ⋯ is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.[1] As it is a geometric series with first term 1/4 and common ratio 1/4, its sum is
$\sum _{n=1}^{\infty }{\frac {1}{4^{n}}}={\frac {\frac {1}{4}}{1-{\frac {1}{4}}}}={\frac {1}{3}}.$
Visual demonstrations
The series 1/4 + 1/16 + 1/64 + 1/256 + ⋯ lends itself to some particularly simple visual demonstrations because a square and a triangle both divide into four similar pieces, each of which contains 1/4 the area of the original.
In the figure on the left,[2][3] if the large square is taken to have area 1, then the largest black square has area 1/2 × 1/2 = 1/4. Likewise, the second largest black square has area 1/16, and the third largest black square has area 1/64. The area taken up by all of the black squares together is therefore 1/4 + 1/16 + 1/64 + ⋯, and this is also the area taken up by the gray squares and the white squares. Since these three areas cover the unit square, the figure demonstrates that
$3\left({\frac {1}{4}}+{\frac {1}{4^{2}}}+{\frac {1}{4^{3}}}+{\frac {1}{4^{4}}}+\cdots \right)=1.$
Archimedes' own illustration, adapted at top,[4] was slightly different, being closer to the equation
$\sum _{n=1}^{\infty }{\frac {3}{4^{n}}}={\frac {3}{4}}+{\frac {3}{4^{2}}}+{\frac {3}{4^{3}}}+{\frac {3}{4^{4}}}+\cdots =1.$
See below for details on Archimedes' interpretation.
The same geometric strategy also works for triangles, as in the figure on the right:[2][5][6] if the large triangle has area 1, then the largest black triangle has area 1/4, and so on. The figure as a whole has a self-similarity between the large triangle and its upper sub-triangle. A related construction making the figure similar to all three of its corner pieces produces the Sierpiński triangle.[7]
Proof by Archimedes
Archimedes encounters the series in his work Quadrature of the Parabola. He is finding the area inside a parabola by the method of exhaustion, and he gets a series of triangles; each stage of the construction adds an area 1/4 times the area of the previous stage. His desired result is that the total area is 4/3 times the area of the first stage. To get there, he takes a break from parabolas to introduce an algebraic lemma:
Proposition 23. Given a series of areas A, B, C, D, ... , Z, of which A is the greatest, and each is equal to four times the next in order, then[8]
$A+B+C+D+\cdots +Z+{\frac {1}{3}}Z={\frac {4}{3}}A.$
Archimedes proves the proposition by first calculating
${\begin{array}{rcl}\displaystyle B+C+\cdots +Z+{\frac {B}{3}}+{\frac {C}{3}}+\cdots +{\frac {Z}{3}}&=&\displaystyle {\frac {4B}{3}}+{\frac {4C}{3}}+\cdots +{\frac {4Z}{3}}\\[1em]&=&\displaystyle {\frac {1}{3}}(A+B+\cdots +Y).\end{array}}$
On the other hand,
${\frac {B}{3}}+{\frac {C}{3}}+\cdots +{\frac {Y}{3}}={\frac {1}{3}}(B+C+\cdots +Y).$
Subtracting this equation from the previous equation yields
$B+C+\cdots +Z+{\frac {Z}{3}}={\frac {1}{3}}A$
and adding A to both sides gives the desired result.[9]
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1/4 + 1/16 + ⋯ are:
$1+{\frac {1}{4}}+{\frac {1}{4^{2}}}+\cdots +{\frac {1}{4^{n}}}={\frac {1-\left({\frac {1}{4}}\right)^{n+1}}{1-{\frac {1}{4}}}}.$
This form can be proved by multiplying both sides by 1 − 1/4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs. The same strategy works for any finite geometric series.
The limit
Archimedes' Proposition 24 applies the finite (but indeterminate) sum in Proposition 23 to the area inside a parabola by a double reductio ad absurdum. He does not quite[10] take the limit of the above partial sums, but in modern calculus this step is easy enough:
$\lim _{n\to \infty }{\frac {1-\left({\frac {1}{4}}\right)^{n+1}}{1-{\frac {1}{4}}}}={\frac {1}{1-{\frac {1}{4}}}}={\frac {4}{3}}.$
Since the sum of an infinite series is defined as the limit of its partial sums,
$1+{\frac {1}{4}}+{\frac {1}{4^{2}}}+{\frac {1}{4^{3}}}+\cdots ={\frac {4}{3}}.$
Notes
1. Shawyer and Watson p. 3.
2. Nelsen and Alsina p. 74.
3. Ajose and Nelson. p. 230
4. Heath p. 250
5. Stein p. 46.
6. Mabry. p. 63
7. Nelson and Alsina p. 56
8. This is a quotation from Heath's English translation (p. 249).
9. This presentation is a shortened version of Heath p. 250.
10. Modern authors differ on how appropriate it is to say that Archimedes summed the infinite series. For example, Shawyer and Watson (p. 3) simply say he did; Swain and Dence say that "Archimedes applied an indirect limiting process"; and Stein (p. 45) stops short with the finite sums.
References
• Ajose, Sunday and Roger Nelsen (June 1994). "Proof without Words: Geometric Series". Mathematics Magazine. 67 (3): 230. doi:10.2307/2690617. JSTOR 2690617.
• Heath, T. L. (1953) [1897]. The Works of Archimedes. Cambridge UP. Page images at Casselman, Bill. "Archimedes' quadrature of the parabola". Archived from the original on 2012-03-20. Retrieved 2007-03-22. HTML with figures and commentary at Otero, Daniel E. (2002). "Archimedes of Syracuse". Archived from the original on 7 March 2007. Retrieved 2007-03-22.
• Mabry, Rick (February 1999). "Proof without Words: ${\frac {1}{4}}$ + $({\frac {1}{4}})^{2}$ + $({\frac {1}{4}})^{3}$ + ⋯ = ${\frac {1}{3}}$". Mathematics Magazine. 72 (1): 63. doi:10.1080/0025570X.1999.11996702. JSTOR 2691318.
• Nelsen, Roger B.; Alsina, Claudi (2006). Math Made Visual: Creating Images for Understanding Mathematics. MAA. ISBN 0-88385-746-4.
• Shawyer, Bruce; Watson, Bruce (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN 0-19-853585-6.
• Stein, Sherman K. (1999). Archimedes: What Did He Do Besides Cry Eureka?. MAA. ISBN 0-88385-718-9.
• Swain, Gordon; Dence, Thomas (April 1998). "Archimedes' Quadrature of the Parabola Revisited". Mathematics Magazine. 71 (2): 123–30. doi:10.2307/2691014. JSTOR 2691014.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
1/2 − 1/4 + 1/8 − 1/16 + ⋯
In mathematics, the infinite series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely.
It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is
$\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2^{n}}}={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {\frac {1}{2}}{1-(-{\frac {1}{2}})}}={\frac {1}{3}}.$
Hackenbush and the surreals
A slight rearrangement of the series reads
$1-{\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots ={\frac {1}{3}}.$
The series has the form of a positive integer plus a series containing every negative power of two with either a positive or negative sign, so it can be translated into the infinite blue-red Hackenbush string that represents the surreal number 1/3:
LRRLRLR... = 1/3.[1]
A slightly simpler Hackenbush string eliminates the repeated R:
LRLRLRL... = 2/3.[2]
In terms of the Hackenbush game structure, this equation means that the board depicted on the right has a value of 0; whichever player moves second has a winning strategy.
Related series
• The statement that 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is absolutely convergent means that the series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is convergent. In fact, the latter series converges to 1, and it proves that one of the binary expansions of 1 is 0.111....
• Pairing up the terms of the series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ results in another geometric series with the same sum, 1/4 + 1/16 + 1/64 + 1/256 + ⋯. This series is one of the first to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.[3]
• The Euler transform of the divergent series 1 − 2 + 4 − 8 + ⋯ is 1/2 − 1/4 + 1/8 − 1/16 + ⋯. Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to 1/3.[4]
Notes
1. Berkelamp et al. p. 79
2. Berkelamp et al. pp. 307–308
3. Shawyer and Watson p. 3
4. Korevaar p. 325
References
• Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (1982). Winning Ways for your Mathematical Plays. Academic Press. ISBN 0-12-091101-9.
• Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.
• Shawyer, Bruce; Watson, Bruce (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN 0-19-853585-6.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
Circulant graph
In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph,[1] but this term has other meanings.
For the square matrices, see Circulant matrix.
Equivalent definitions
Circulant graphs can be described in several equivalent ways:[2]
• The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has a graph automorphism, which is a cyclic permutation of its vertices.
• The graph has an adjacency matrix that is a circulant matrix.
• The n vertices of the graph can be numbered from 0 to n − 1 in such a way that, if some two vertices numbered x and (x + d) mod n are adjacent, then every two vertices numbered z and (z + d) mod n are adjacent.
• The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing.
• The graph is a Cayley graph of a cyclic group.[3]
Examples
Every cycle graph is a circulant graph, as is every crown graph with 2 modulo 4 vertices.
The Paley graphs of order n (where n is a prime number congruent to 1 modulo 4) is a graph in which the vertices are the numbers from 0 to n − 1 and two vertices are adjacent if their difference is a quadratic residue modulo n. Since the presence or absence of an edge depends only on the difference modulo n of two vertex numbers, any Paley graph is a circulant graph.
Every Möbius ladder is a circulant graph, as is every complete graph. A complete bipartite graph is a circulant graph if it has the same number of vertices on both sides of its bipartition.
If two numbers m and n are relatively prime, then the m × n rook's graph (a graph that has a vertex for each square of an m × n chessboard and an edge for each two squares that a chess rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group Cmn $\simeq $ Cm×Cn. More generally, in this case, the tensor product of graphs between any m- and n-vertex circulants is itself a circulant.[2]
Many of the known lower bounds on Ramsey numbers come from examples of circulant graphs that have small maximum cliques and small maximum independent sets.[1]
A specific example
The circulant graph $C_{n}^{s_{1},\ldots ,s_{k}}$ with jumps $s_{1},\ldots ,s_{k}$ is defined as the graph with $n$ nodes labeled $0,1,\ldots ,n-1$ where each node i is adjacent to 2k nodes $i\pm s_{1},\ldots ,i\pm s_{k}\mod n$.
• The graph $C_{n}^{s_{1},\ldots ,s_{k}}$ is connected if and only if $\gcd(n,s_{1},\ldots ,s_{k})=1$.
• If $1\leq s_{1}<\cdots <s_{k}$ are fixed integers then the number of spanning trees $t(C_{n}^{s_{1},\ldots ,s_{k}})=na_{n}^{2}$ where $a_{n}$ satisfies a recurrence relation of order $2^{s_{k}-1}$.
• In particular, $t(C_{n}^{1,2})=nF_{n}^{2}$ where $F_{n}$ is the n-th Fibonacci number.
Self-complementary circulants
A self-complementary graph is a graph in which replacing every edge by a non-edge and vice versa produces an isomorphic graph. For instance, a five-vertex cycle graph is self-complementary, and is also a circulant graph. More generally every Paley graph of prime order is a self-complementary circulant graph.[4] Horst Sachs showed that, if a number n has the property that every prime factor of n is congruent to 1 modulo 4, then there exists a self-complementary circulant with n vertices. He conjectured that this condition is also necessary: that no other values of n allow a self-complementary circulant to exist.[2][4] The conjecture was proven some 40 years later, by Vilfred.[2]
Ádám's conjecture
Define a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to n − 1 in such a way that, if some two vertices numbered x and y are adjacent, then every two vertices numbered z and (z − x + y) mod n are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix.
Let a be an integer that is relatively prime to n, and let b be any integer. Then the linear function that takes a number x to ax + b transforms a circulant numbering to another circulant numbering. András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if G and H are isomorphic circulant graphs, with different numberings, then there is a linear map that transforms the numbering for G into the numbering for H. However, Ádám's conjecture is now known to be false. A counterexample is given by graphs G and H with 16 vertices each; a vertex x in G is connected to the six neighbors x ± 1, x ± 2, and x ± 7 modulo 16, while in H the six neighbors are x ± 2, x ± 3, and x ± 5 modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.[2]
Toida's conjecture refines Ádám's conjecture by considering only a special class of circulant graphs, in which all of the differences between adjacent graph vertices are relatively prime to the number of vertices. According to this refined conjecture, these special circulant graphs should have the property that all of their symmetries come from symmetries of the underlying additive group of numbers modulo n. It was proven by two groups in 2001 and 2002.[5][6]
Algorithmic questions
There is a polynomial-time recognition algorithm for circulant graphs, and the isomorphism problem for circulant graphs can be solved in polynomial time.[7][8]
References
1. Small Ramsey Numbers, Stanisław P. Radziszowski, Electronic J. Combinatorics, dynamic survey 1, updated 2014.
2. Vilfred, V. (2004), "On circulant graphs", in Balakrishnan, R.; Sethuraman, G.; Wilson, Robin J. (eds.), Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001), Alpha Science, pp. 34–36.
3. Alspach, Brian (1997), "Isomorphism and Cayley graphs on abelian groups", Graph symmetry (Montreal, PQ, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 497, Dordrecht: Kluwer Acad. Publ., pp. 1–22, MR 1468786.
4. Sachs, Horst (1962). "Über selbstkomplementäre Graphen". Publicationes Mathematicae Debrecen. 9: 270–288. MR 0151953..
5. Muzychuk, Mikhail; Klin, Mikhail; Pöschel, Reinhard (2001), "The isomorphism problem for circulant graphs via Schur ring theory", Codes and association schemes (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 56, Providence, Rhode Island: American Mathematical Society, pp. 241–264, MR 1816402
6. Dobson, Edward; Morris, Joy (2002), "Toida's conjecture is true", Electronic Journal of Combinatorics, 9 (1): R35:1–R35:14, MR 1928787
7. Muzychuk, Mikhail (2004). "A Solution of the Isomorphism Problem for Circulant Graphs". Proc. London Math. Soc. 88: 1–41. doi:10.1112/s0024611503014412. MR 2018956.
8. Evdokimov, Sergei; Ponomarenko, Ilia (2004). "Recognition and verification of an isomorphism of circulant graphs in polynomial time". St. Petersburg Math. J. 15: 813–835. doi:10.1090/s1061-0022-04-00833-7. MR 2044629.
External links
• Weisstein, Eric W. "Circulant Graph". MathWorld.
| Wikipedia |
Ágnes Szendrei
Ágnes Szendrei is a Hungarian-American mathematician whose research concerns clones, the congruence lattice problem, and other topics in universal algebra. She is a professor of mathematics at the University of Colorado Boulder,[1] and the author of the well-cited book Clones in Universal Algebra (1986).[2] In May 2022[3][4], Dr. Szendrei was elected as an external member of the Hungarian Academy of Sciences[5]; such external memberships are for Hungarian scientists who live outside of Hungary and who have made exceptional contributions to scientific research.
Szendrei earned a doctorate from the Hungarian Academy of Sciences in 1982, and a habilitation in 1993.[6] Her 1982 dissertation was Clones of Linear Operations and Semi-Affine Algebras, supervised by Béla Csákány.[7] She was on the faculty of the University of Szeged from 1982 until 2003, when she moved to the University of Colorado.[6]
Szendrei is a Humboldt Fellow. She won the Kató Rényi Award for undergraduate research in 1975, the Géza Grünwald Commemorative Prize for young researchers of the János Bolyai Mathematical Society in 1978, and the Golden Ring of the Republic in 1979. She was the 1992 winner of the Paul Erdős Prize of the Hungarian Academy of Sciences, and the 2000 winner of the Academy's Farkas Bolyai Award.[6]
References
1. Agnes Szendrei, University of Colorado Boulder, 29 September 2016, retrieved 2019-10-11
2. Berman, Joel (1987), "Review of Clones in Universal Algebra", Mathematical Reviews, MR 0859550
3. University of Colorado Boulder Department of Mathematics News and Events, 28 September 2016, retrieved 2023-03-16
4. Hungarian Academy of Sciences announcement of new external members elected 2022, 13 May 2022, retrieved 2023-03-16
5. Department of Mathematical Sciences of the Hungarian Academy of Sciences, retrieved 2023-03-16
6. Curriculum vitae (PDF), January 31, 2019, retrieved 2019-10-11
7. Ágnes Szendrei at the Mathematics Genealogy Project
External links
• Home page
• Ágnes Szendrei publications indexed by Google Scholar
Authority control: Academics
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Árpád Varecza
Árpád Varecza (6 September 1941 – 26 September 2005), was a Hungarian mathematician, former lecturer at the College of Nyíregyháza, head of the Institute of Mathematics and Informatics, and deputy director general of the institution for three years.
Árpád Varecza
Born(1941-09-06)6 September 1941
Vác, Hungary
Died26 September 2005(2005-09-26) (aged 64)
Nyíregyháza, Hungary
NationalityHungarian
Alma materEötvös Loránd University, Budapest
Scientific career
FieldsMathematics
InstitutionsCollege of Nyíregyháza
Biography
He was born on September 6, 1941 in Vác. He graduated from the teacher training college in Szeged in 1963, then graduated from ELTE with a degree in mathematics, physics and technology. His first jobs were tied to his birthplace. He taught at the Primary School in Váchartyán and Verőce, then at the Géza Király Secondary School and Vocational Secondary School in Vác. He was admitted to the Department of Mathematics of the Teacher Training College in Nyíregyháza in 1969 as an assistant lecturer, in 1971 he was appointed an assistant professor, in 1977 he was appointed an associate professor and in 1983 he was appointed a college teacher. Between 1977 and 1980 he was an aspirant at the Mathematical Research Institute of the Hungarian Academy of Sciences. In 1975 he received his Ph.D. at Kossuth Lajos University, and in 1982 he defended his Ph.D. Following the era marked by the name of Gyula Bereznai, he was head of the Department of Mathematics in 1984, then Head of the Institute of Mathematics and Informatics in 2000, and served as Deputy Director General of the institution for three years.
Work
He specialized in combinatorics, including "sorting algorithms". He obtained his candidate's degree in his dissertation on "Optimal sorting algorithms". Honorary Heir President of the János Bolyai Mathematical Society, President of the Department of Mathematics, Physics and Astronomy of the DAB, Chairman of the College's Scientific Committee, Member of the MM Intensive Further Education Council and of the MM Computer Science Advisory Board, He was the editor of the Mathematics series. Under his guidance, in 1985, four colleagues earned their Ph.D.
Books
• Mathematics competitions for teacher training colleges[1]
• A tanárképző főiskolák Péter Rózsa matematikai versenyei[2]
Notes
1. Bereznai Gyula - Dr.Varecza Árpád - Dr.Rozgonyi Tibor: Tanárképző főiskolák matematika versenyei (1952-1970:ISBN 9789631736274; 1971-1979:ISBN 9789631761597; 1980-1985:ISBN 9789631816068)
Tanárképző főiskolák országos matematika versenyei: (ISBN 9789631736267)
2. Rozgonyi Tibor, Varecza Árpád: A tanárképző főiskolák Péter Rózsa matematikai versenyei; Typotex, 2003. - 160 p. ISBN 9639326798
External links
• Varecza Árpád
• On the smallest and largest elements
• In memoriam
Authority control
International
• ISNI
• VIAF
National
• Germany
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