4,211 results
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2. Addendum to the paper on partially stable algebras
- Author
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A. Adrian Albert
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Lemma (logic) ,Addendum ,Expression (computer science) ,Mathematics - Abstract
I regret to announce that there is a serious error in my paper in these Transactions, volume 84, pp. 430-443. The error was discovered by Louis Kokoris who found that on line 8 of page 434 the expression given as 4[g(bz)](az) should have been 4[g(az)](bz). As a consequence the computation of P(z, g, az, b) yields nothing, the proof of formula (30) is not valid, and the important Lemma 9 is not proved. Thus the paper does not give a proof of its major result stated as Theorem 1. Nevertheless, the theorems of the paper are all correct and we shall provide a revision of the proof here. This revised proof has been checked by Louis Kokoris to whom the author wishes to express his great thanks. We observe first that the equation
- Published
- 1958
3. Remarks concerning the paper of W. L. Ayres on the regular points of a continuum
- Author
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Karl Menger
- Subjects
Set (abstract data type) ,Discrete mathematics ,Kernel (set theory) ,Applied Mathematics ,General Mathematics ,Order (group theory) ,Point (geometry) ,Continuum (set theory) ,Mathematics - Abstract
The reading of Ayres' interesting paper suggested to me the following remarks: 1. The order of a subset of a set S in a point p4 cannot surpass the order of S in p. Hence if S2 denotes the set of all points of S of order 2, then S2 has in each point of S the order 2, the order 1, or the order 0, where the terms "order 0" and "0-dimensional" are used synonymously. SI(M), S2(1), S" may denote the set of all points of S in which S2 has the order 0, 1, 2, respectively. The points of order 2 of S are also called the ordinary points of S, and the set S2 of all ordinarv points of S may be called the ordinary part of S. The set S" of all ordinary points of the ordinary part of S may be designated the ordinary kernel of S. WVe have
- Published
- 1931
4. Invariant means and fixed points: A sequel to Mitchell’s paper
- Author
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L. N. Argabright
- Subjects
Discrete mathematics ,Combinatorics ,Uniform norm ,Invariant polynomial ,Applied Mathematics ,General Mathematics ,Banach space ,Convex set ,Fixed-point theorem ,Fixed point ,Fixed-point property ,Topological vector space ,Mathematics - Abstract
The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.
- Published
- 1968
5. On Fermat’s last theorem (thirteenth paper)
- Author
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Taro Morishima
- Subjects
Fermat's Last Theorem ,Applied Mathematics ,General Mathematics ,Calculus ,Mathematics ,Wiles' proof of Fermat's Last Theorem - Published
- 1952
6. Rectifications to the papers: Sets of uniqueness and sets of multiplicity, I and II
- Author
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R. Salem
- Subjects
Algebra ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Multiplicity (mathematics) ,Uniqueness ,Mathematics - Published
- 1948
7. Remarks on a paper of Hermann
- Author
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Shlomo Sternberg and Victor Guillemin
- Subjects
Applied Mathematics ,General Mathematics ,Classics ,Mathematics - Published
- 1968
8. A note on the preceding paper: 'On the location of the roots of certain types of polynomials' [Trans. Amer. Math. Soc. 24 (1922), no. 3, 163–180; 1501220] by J. L. Walsh
- Author
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D. R. Curtiss
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Mathematics - Published
- 1922
9. Remarks on the preceding paper of James A. Clarkson: 'Uniformly convex spaces' [Trans. Amer. Math. Soc. 40 (1936), no. 3; MR1501880]
- Author
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Nelson Dunford and Anthony P. Morse
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Uniformly convex space ,Mathematics - Published
- 1936
10. On the second derivatives of an extremal-integral with an application to a problem with variable end points (supplementary paper)
- Author
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Arnold Dresden
- Subjects
Applied Mathematics ,General Mathematics ,Calculus ,Applied mathematics ,Variable (mathematics) ,Mathematics ,Second derivative - Published
- 1923
11. Corrections to the paper: 'Integration in general analysis' [Trans. Amer. Math. Soc. 37 (1935), no. 3, 441–453; 1501796]
- Author
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Nelson Dunford
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Mathematics - Published
- 1935
12. Errata in my paper: 'On a special class of polynomials' [Trans. Amer. Math. Soc. 35 (1933), no. 3, 559–584; 1501703]
- Author
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Oystein Ore
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Calculus ,Special class ,Mathematics - Published
- 1934
13. Correction to the paper: 'A problem concerning orthogonal polynomials' [Trans. Amer. Math. Soc. 37 (1935), no. 1, 196–206; 1501782]
- Author
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G. Szegö
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Orthogonal polynomials ,Mathematics - Published
- 1936
14. Correction to a paper: 'The Moore-Kline problem' [Trans. Amer. Math. Soc. 34 (1932), no. 3, 705–721; 1501658]
- Author
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Leo Zippin
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Mathematics - Published
- 1933
15. Note on a paper by Mandelbrojt and MacLane
- Author
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Jacqueline Ferrand
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Domain (ring theory) ,Mathematics::Differential Geometry ,Function (mathematics) ,Mathematical proof ,Mathematics - Abstract
Let A8 be the domain in the s-plane (s =o+it) defined by -gl(a) -Ah. The proofs of Theorems 1, I1, and III are the same, with the new function S(oI5.
- Published
- 1947
16. A correction to the paper: 'On effective sets of points in relation to integral functions' [Trans. Amer. Math. Soc. 42 (1937), no. 3, 358–365; 1501926]
- Author
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V. Ganapathy Iyer
- Subjects
Algebra ,Relation (database) ,Applied Mathematics ,General Mathematics ,Calculus ,Mathematics - Published
- 1938
17. Correction to the paper: 'The multinomial solid and the chi test' [Trans. Amer. Math. Soc. 31 (1929), no. 1, 133–144; 1501472]
- Author
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Burton H. Camp
- Subjects
Applied Mathematics ,General Mathematics ,Statistics ,Econometrics ,Chi-square test ,Multinomial distribution ,Mathematics - Published
- 1938
18. Correction to a paper on the Whitehead-Huntington postulates: 'The complete existential theory of the Whitehead-Huntington set of postulates for the algebra of logic' [Trans. Amer. Math. Soc. 35 (1933), no. 4, 940–948; 1501726]
- Author
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A. H. Diamond
- Subjects
Set (abstract data type) ,Algebra ,Applied Mathematics ,General Mathematics ,Algebra over a field ,Existential theory ,Mathematics - Published
- 1934
19. The geometry of diagonal groups
- Author
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Peter J. Cameron, Cheryl E. Praeger, Csaba Schneider, R. A. Bailey, University of St Andrews. Pure Mathematics, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra, and University of St Andrews. Statistics
- Subjects
Mathematics(all) ,South china ,Primitive permutation group ,General Mathematics ,Diagonal group ,T-NDAS ,Library science ,Group Theory (math.GR) ,O'Nan-Scott Theorem ,01 natural sciences ,Hospitality ,FOS: Mathematics ,NCAD ,Mathematics - Combinatorics ,QA Mathematics ,0101 mathematics ,Diagonal semilattice ,QA ,Cartesian lattice ,Mathematics ,business.industry ,20B05 ,Applied Mathematics ,010102 general mathematics ,Latin square ,Semilattice ,Latin cube ,010101 applied mathematics ,Hamming graph ,Research council ,Diagonal graph ,Combinatorics (math.CO) ,business ,Mathematics - Group Theory ,Partition - Abstract
Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3). Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m
- Published
- 2022
20. Integration of modules – II: Exponentials
- Author
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Matthew Westaway and Dmitriy Rumynin
- Subjects
Applied Mathematics ,General Mathematics ,Restricted representation ,Representation (systemics) ,Group Theory (math.GR) ,Mathematics - Rings and Algebras ,Representation theory ,Exponential function ,Algebra ,Rings and Algebras (math.RA) ,Algebraic group ,Lie algebra ,FOS: Mathematics ,20G05 (primary), 17B45 (secondary) ,Representation Theory (math.RT) ,QA ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Group theory ,Mathematics - Abstract
We continue our exploration of various approaches to integration of representations from a Lie algebra $\mbox{Lie} (G)$ to an algebraic group $G$ in positive characteristic. In the present paper we concentrate on an approach exploiting exponentials. This approach works well for over-restricted representations, introduced in this paper, and takes no note of $G$-stability., Accepted by Transactions of the AMS. This paper is split off the earlier versions (1, 2 and 3) of arXiv:1708.06620. Some of the statements in these versions of arXiv:1708.06620 contain mistakes corrected here. Version 2 of this paper: close to the accepted version by the journal, minor improvements, compared to Version 1
- Published
- 2021
21. An improvement on Furstenberg’s intersection problem
- Author
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Han Yu
- Subjects
Combinatorics ,Intersection ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,Dimension (graph theory) ,Zero (complex analysis) ,0101 mathematics ,Invariant (mathematics) ,Dynamical system (definition) ,01 natural sciences ,Mathematics - Abstract
In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim A 2 + dim A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .
- Published
- 2021
22. Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks
- Author
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Armando Martino, Stefano Francaviglia, Francaviglia, Stefano, and Martino, Armando
- Subjects
Outer space, conjugacy problem, automorphisms of free groups, graphs ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Group Theory (math.GR) ,Train track map ,Automorphism ,Lipschitz continuity ,01 natural sciences ,Convexity ,Free product ,Metric (mathematics) ,FOS: Mathematics ,20E06, 20E36, 20E08 ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Group Theory ,Mathematics - Abstract
This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems., 50 pages. Originally part of arXiv:1703.09945 . We decided to split that paper following the recommendations of a referee. Updated subsequent to acceptance by Transactions of the American Mathematical Society
- Published
- 2021
23. The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces
- Author
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Paolo Mantero
- Subjects
Monomial ,Pure mathematics ,Mathematics::Commutative Algebra ,Betti number ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Algebraic geometry ,Star (graph theory) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Representation theory ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,Young tableau ,0101 mathematics ,Commutative algebra ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)
- Published
- 2020
24. Extremal growth of Betti numbers and trivial vanishing of (co)homology
- Author
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Jonathan Montaño and Justin Lyle
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Commutative Algebra ,Betti number ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Local ring ,Homology (mathematics) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,13D07, 13D02, 13C14, 13H10, 13D40 ,01 natural sciences ,Injective function ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke., to appear in Trans. Amer. Math. Soc
- Published
- 2020
25. Corrigendum to 'Strongly self-absorbing 𝐶*-dynamical systems'
- Author
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Gábor Szabó
- Subjects
Classical mechanics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,Mathematics - Abstract
We correct a mistake that appeared in the first section of the original article, which appeared in Tran. Amer. Math. Soc. 370 (2018), 99–130. Namely, Corollary 1.16 was false as stated and was subsequently used in later proofs in the paper. In this note it is argued that all the relevant statements after Corollary 1.16 can be saved with at most minor modifications. In particular, all the main results of the original paper remain valid as stated, but some intermediate claims are slightly modified or proved more directly without Corollary 1.16.
- Published
- 2020
26. Flow equivalence of G-SFTs
- Author
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Toke Meier Carlsen, Søren Eilers, and Mike Boyle
- Subjects
Pure mathematics ,Finite group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Dynamical Systems (math.DS) ,01 natural sciences ,Matrix (mathematics) ,Group action ,Flow (mathematics) ,FOS: Mathematics ,Equivariant map ,Mathematics - Dynamical Systems ,0101 mathematics ,Connection (algebraic framework) ,Equivalence (measure theory) ,Group ring ,Mathematics - Abstract
In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata., The paper has been augmented considerably and the second version is now 81 pages long. This version has been accepted for publication in Transactions of the American Mathematical Society
- Published
- 2020
27. Constructing hyperelliptic curves with surjective Galois representations
- Author
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Samuele Anni, Vladimir Dokchitser, Institut de Mathématiques de Marseille (I2M), and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
11F80 (Primary), 12F12, 11G10, 11G30 (Secondary) ,Symplectic group ,Mathematics - Number Theory ,Degree (graph theory) ,Inverse Galois problem ,Galois representations ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Image (category theory) ,Galois module ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,hyperelliptic curves ,inverse Galois problem ,Integer ,Abelian varieties ,Goldbach’s conjecture ,FOS: Mathematics ,Number Theory (math.NT) ,Monic polynomial ,Mathematics ,Symplectic geometry - Abstract
In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n (this hypothesis appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N and an explicit monic polynomial $f_0(x)\in \mathbb{Z}[x]$ of degree n, such that the Jacobian $J$ of every curve of the form $y^2=f(x)$ has $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and $Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, whenever $f(x)\in\mathbb{Z}[x]$ is monic with $f(x)\equiv f_0(x) \bmod{N}$ and with no roots of multiplicity greater than $2$ in $\overline{\mathbb{F}}_p$ for any p not dividing N., Comment: 24 pages, minor corrections
- Published
- 2019
28. Ultrametric properties for valuation spaces of normal surface singularities
- Author
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Evelia R. García Barroso, Patrick Popescu-Pampu, Pedro Daniel González Pérez, and Matteo Ruggiero
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Block (permutation group theory) ,14B05, 14J17, 32S25 ,Intersection number ,Function (mathematics) ,01 natural sciences ,Linear subspace ,Combinatorics ,Mathematics - Algebraic Geometry ,Tree (descriptive set theory) ,Singularity ,FOS: Mathematics ,0101 mathematics ,Normal surface ,Algebraic Geometry (math.AG) ,Ultrametric space ,Mathematics - Abstract
Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric., Comment: 50 pages, 14 figures. Final version
- Published
- 2019
29. The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation
- Author
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Łukasz Kubat, Eric Jespers, Arne Van Antwerpen, Mathematics, Algebra, and Faculty of Sciences and Bioengineering Sciences
- Subjects
Monoid ,Semidirect product ,Yang–Baxter equation ,Applied Mathematics ,General Mathematics ,Prime ideal ,010102 general mathematics ,Subalgebra ,Semiprime ,Normal extension ,Mathematics - Rings and Algebras ,Jacobson radical ,01 natural sciences ,Algebra ,Rings and Algebras (math.RA) ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of $M(X,r)$ as a regular submonoid in the semidirect product $A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module finite normal extension of a commutative affine subalgebra. In particular, $K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension bounded by $|X|$. We also characterize, in ring-theoretical terms of $K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $M(X,r)$. These results allow us to control the prime spectrum of the algebra $K[M(X,r)]$ and to describe the Jacobson radical and prime radical of $K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra $K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times \mathrm{M}_{v_m}(K[G_m])$., A subtle mistake in the proof of Theorem 4.4 has been corrected (will appear in a corrigendum et addendum, TAMS). In the latter paper we also strengthen some of the results by removing the "square free'' condition in Section 5 and in this paper we also prove new homological equivalences in Theorem 4.4
- Published
- 2019
30. Good coverings of Alexandrov spaces
- Author
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Takao Yamaguchi and Ayato Mitsuishi
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Stability (learning theory) ,Fibration ,Metric Geometry (math.MG) ,Type (model theory) ,Curvature ,Space (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,53C20, 53C23 ,Mathematics - Metric Geometry ,Mathematics::Category Theory ,Bounded function ,FOS: Mathematics ,Mathematics::Differential Geometry ,Isomorphism ,0101 mathematics ,Mathematics - Abstract
In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper., Minor change basically on the proof of Theorem 1.2 in Section 5
- Published
- 2019
31. On the local time process of a skew Brownian motion
- Author
-
Andrei N. Borodin and Paavo Salminen
- Subjects
Discontinuity (linguistics) ,Distribution (mathematics) ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,Local time ,Mathematical analysis ,Skew ,Measure (mathematics) ,Brownian motion ,Exponential function ,Mathematics - Abstract
We derive a Ray–Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both of these cases. The Ray–Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper.
- Published
- 2019
32. An 𝐿^{𝑝} theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions
- Author
-
Jennifer Chayes, Yufei Zhao, Henry Cohn, and Christian Borgs
- Subjects
Random graph ,Discrete mathematics ,Dense graph ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,01 natural sciences ,Power law ,Limit theory ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Equivalence (formal languages) ,Mathematics - Probability ,Mathematics - Abstract
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper., Comment: 44 pages
- Published
- 2019
33. On symmetric linear diffusions
- Author
-
Liping Li and Jiangang Ying
- Subjects
Discrete mathematics ,Representation theorem ,Dirichlet form ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Disjoint sets ,01 natural sciences ,Dirichlet distribution ,010104 statistics & probability ,symbols.namesake ,Closure (mathematics) ,symbols ,Interval (graph theory) ,Countable set ,0101 mathematics ,Mathematics - Abstract
The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.
- Published
- 2018
34. On period relations for automorphic 𝐿-functions I
- Author
-
Fabian Januszewski
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Statistics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Period (music) ,Mathematics - Abstract
This paper is the first in a series of two dedicated to the study of period relations of the type L ( 1 2 + k , Π ) ∈ ( 2 π i ) d ⋅ k Ω ( − 1 ) k \bf Q ( Π ) , 1 2 + k critical , \begin{equation*} L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}\textrm {\bf Q}(\Pi ),\quad \frac {1}{2}+k\;\text {critical}, \end{equation*} for certain automorphic representations Π \Pi of a reductive group G . G. In this paper we discuss the case G = G L ( n + 1 ) × G L ( n ) . G=\mathrm {GL}(n+1)\times \mathrm {GL}(n). The case G = G L ( 2 n ) G=\mathrm {GL}(2n) is discussed in part two. Our method is representation theoretic and relies on the author’s recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation Π \Pi under consideration. The new period relations we prove are in accordance with Deligne’s Conjecture on special values of L L -functions, and the author expects this method to apply to other cases as well.
- Published
- 2018
35. Wave front sets of reductive Lie group representations II
- Author
-
Benjamin Harris
- Subjects
Wavefront ,Induced representation ,Applied Mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Wave front set ,Lie group ,(g,K)-module ,01 natural sciences ,Algebra ,Representation of a Lie group ,0103 physical sciences ,FOS: Mathematics ,Tempered representation ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups., Accepted to Transactions of the American Mathematical Society
- Published
- 2017
36. Gromov hyperbolicity, the Kobayashi metric, and $\mathbb {C}$-convex sets
- Author
-
Andrew Zimmer
- Subjects
Pure mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Regular polygon ,Boundary (topology) ,Codimension ,01 natural sciences ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,Affine transformation ,Ball (mathematics) ,0101 mathematics ,Mathematics - Abstract
In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.
- Published
- 2017
37. Tame circle actions
- Author
-
Jordan Watts and Susan Tolman
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Kähler manifold ,Fixed point ,01 natural sciences ,Mathematics - Symplectic Geometry ,0103 physical sciences ,Symplectic category ,Slice theorem ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,53D20 (Primary) 53D05, 53B35 (Secondary) ,0101 mathematics ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic manifold ,Symplectic geometry ,Mathematics - Abstract
In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages
- Published
- 2017
38. Differentiability of the conjugacy in the Hartman-Grobman Theorem
- Author
-
Weinian Zhang, Kening Lu, and Wenmeng Zhang
- Subjects
Bump function ,0209 industrial biotechnology ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Invariant manifold ,02 engineering and technology ,01 natural sciences ,Hartman–Grobman theorem ,020901 industrial engineering & automation ,Conjugacy class ,Differentiable function ,0101 mathematics ,Mathematics - Abstract
The classical Hartman-Grobman Theorem states that a smooth diffeomorphism F ( x ) F(x) near its hyperbolic fixed point x ¯ \bar x is topological conjugate to its linear part D F ( x ¯ ) DF(\bar x) by a local homeomorphism Φ ( x ) \Phi (x) . In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth F ( x ) F(x) is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a C ∞ C^\infty diffeomorphism F ( x ) F(x) , the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a C 1 C^1 diffeomorphism F ( x ) F(x) with D F ( x ) DF(x) being α \alpha -Hölder continuous at the fixed point that the local homeomorphism Φ ( x ) \Phi (x) is differentiable at the fixed point. Here, α > 0 \alpha >0 depends on the bands of the spectrum of F ′ ( x ¯ ) F’(\bar x) for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on F ( x ) F(x) cannot be lowered to C 1 C^1 .
- Published
- 2017
39. Isoperimetric properties of the mean curvature flow
- Author
-
Or Hershkovits
- Subjects
Pure mathematics ,Mean curvature flow ,Mean curvature ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Upper and lower bounds ,Geometric measure theory ,0103 physical sciences ,Hausdorff measure ,010307 mathematical physics ,0101 mathematics ,Isoperimetric inequality ,Constant (mathematics) ,Mathematics - Abstract
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self-evident. The first is a genuine, 5-line proof, for the isoperimetric inequality for k k -cycles in R n \mathbb {R}^n , with a constant differing from the optimal constant by a factor of only k \sqrt {k} , as opposed to a factor of k k k^k produced by all of the other soft methods. The second is a 3-line proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of Giga and Yama-uchi (1993). We then turn to use the above-mentioned relation to prove a bound on the parabolic Hausdorff measure of the space-time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon isoperimetric inequality. To prove it, we are led to study the geometric measure theory of Euclidean rectifiable sets in parabolic space and prove a co-area formula in that setting. This formula, the proof of which occupies most of this paper, may be of independent interest.
- Published
- 2017
40. On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility
- Author
-
Dilip Raghavan and Saharon Shelah
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Boolean algebra (structure) ,010102 general mathematics ,Ultrafilter ,Natural number ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,Embedding ,Continuum (set theory) ,0101 mathematics ,Partially ordered set ,Continuum hypothesis ,Axiom ,Mathematics - Abstract
The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.
- Published
- 2017
41. Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type
- Author
-
Wenxian Shen and Zhongwei Shen
- Subjects
Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Monotonic function ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Exponential growth ,Uniqueness ,0101 mathematics ,Exponential decay ,Mathematics - Abstract
The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile of the unique generalized traveling wave is of the same recurrence as the media. In particular, if the media is time almost periodic, then so is the wave profile of the unique generalized traveling wave.
- Published
- 2016
42. Derangements in subspace actions of finite classical groups
- Author
-
Robert M. Guralnick and Jason Fulman
- Subjects
Classical group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Codimension ,01 natural sciences ,Combinatorics ,Group of Lie type ,010201 computation theory & mathematics ,Simple group ,Bounded function ,Classification of finite simple groups ,CA-group ,0101 mathematics ,Subspace topology ,Mathematics - Abstract
This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.
- Published
- 2016
43. Classification of tile digit sets as product-forms
- Author
-
Chun-Kit Lai, Ka-Sing Lau, and Hui Rao
- Subjects
Polynomial (hyperelastic model) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Combinatorics ,Matrix (mathematics) ,Tree (descriptive set theory) ,Integer ,Product (mathematics) ,0101 mathematics ,Cyclotomic polynomial ,Mathematics - Abstract
Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.
- Published
- 2016
44. Ample group action on AS-regular algebras and noncommutative graded isolated singularities
- Author
-
Izuru Mori and Kenta Ueyama
- Subjects
Noetherian ,Pure mathematics ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Quiver ,Dimension (graph theory) ,Graded ring ,Isolated singularity ,Noncommutative geometry ,Nabla symbol ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper, we introduce a notion of ampleness of a group action $G$ on a right noetherian graded algebra $A$, and show that it is strongly related to the notion of $A^G$ to be a graded isolated singularity introduced by the second author of this paper. Moreover, if $S$ is a noetherian AS-regular algebra and $G$ is a finite ample group acting on $S$, then we will show that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\cal D}^b(\operatorname{mod} \nabla S*G)$ where $\nabla S$ is the Beilinson algebra of $S$. We will also explicitly calculate a quiver $Q_{S, G}$ such that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\mathcal D}^b(\operatorname{mod} kQ_{S, G})$ when $S$ is of dimension 2.
- Published
- 2015
45. Newton polyhedra and weighted oscillatory integrals with smooth phases
- Author
-
Toshihiro Nose and Joe Kamimoto
- Subjects
Weight function ,Explicit formulae ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Resolution of singularities ,Critical point (mathematics) ,Polyhedron ,symbols.namesake ,Newton fractal ,symbols ,Oscillatory integral ,Asymptotic expansion ,Mathematics - Abstract
In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.
- Published
- 2015
46. Probabilistically nilpotent Hopf algebras
- Author
-
Sara Westreich and Miriam Cohen
- Subjects
Discrete mathematics ,Pure mathematics ,Ring (mathematics) ,Quantum group ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,MathematicsofComputing_GENERAL ,Commutator (electric) ,Quasitriangular Hopf algebra ,Hopf algebra ,law.invention ,16T05 ,Nilpotent ,Invertible matrix ,law ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Nilpotent group ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A A which depends only on the Grothendieck ring of H . H. When H H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are.
- Published
- 2015
47. Bellman function for extremal problems in BMO
- Author
-
Paata Ivanisvili, Vasily Vasyunin, Dmitriy M. Stolyarov, Nikolay N. Osipov, and Pavel B. Zatitskiy
- Subjects
Large class ,42A05, 42B35, 49K20, 52A40 ,Applied Mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,Function (mathematics) ,Mathematical proof ,Space (mathematics) ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Homogeneous ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Applied mathematics ,Boundary value problem ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John--Nirenberg inequality and $L^p$ estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge--Amp\`ere equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way we have obtained an algorithm of constructing an exact Bellman function for a large class of integral functionals in the BMO space., Comment: 91 pages, 18 figures
- Published
- 2015
48. The cone spanned by maximal Cohen-Macaulay modules and an application
- Author
-
Kazuhiko Kurano and C. Y. Jean Chan
- Subjects
Combinatorics ,Noetherian ,Discrete mathematics ,Rational number ,Primary ideal ,Applied Mathematics ,General Mathematics ,Modulo ,Local ring ,Grothendieck group ,Finitely-generated abelian group ,Abelian group ,Mathematics - Abstract
The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain R R and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on R R , the Grothendieck group G 0 ( R ) ¯ \overline {G_0(R)} of finitely generated R R -modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R R is the cone in G 0 ( R ) ¯ R \overline {G_0(R)}_{\mathbb R} spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers ϵ i = 0 , ± 1 \epsilon _i = 0, \pm 1 ( d / 2 > i > d d/2 > i > d ), we shall construct a d d -dimensional Cohen-Macaulay local ring R R (of characteristic p p ) and a maximal primary ideal I I of R R such that the function ℓ R ( R / I [ p n ] ) \ell _R(R/I^{[p^n]}) is a polynomial in p n p^n of degree d d whose coefficient of ( p n ) i (p^n)^i is the product of ϵ i \epsilon _i and a positive rational number for d / 2 > i > d d/2 > i > d . The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist for it.
- Published
- 2015
49. Equidistribution in higher codimension for holomorphic endomorphisms of $\mathbb {P}^k$
- Author
-
Taeyong Ahn
- Subjects
Discrete mathematics ,Endomorphism ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,Holomorphic function ,Dynamical Systems (math.DS) ,Codimension ,FOS: Mathematics ,Computer Science::Symbolic Computation ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,Mathematics - Abstract
In this paper, we discuss the equidistribution phenomena for holomorphic endomorphisms over $\mathbb{P}^k$ in the case of bidegree $(p,p)$ with $1, Comment: Corrected an error in the statement of the main theorem and readability has been improved. This paper is based on the work in arXiv:math/0703702 and arXiv:0901.3000 by other authors; for precise estimates, we go over the proofs with modification
- Published
- 2015
50. Orthogonal symmetric affine Kac-Moody algebras
- Author
-
Walter Freyn
- Subjects
Symmetric algebra ,Pure mathematics ,Quantum affine algebra ,Jordan algebra ,Loop algebra ,Applied Mathematics ,General Mathematics ,Clifford algebra ,Kac–Moody algebra ,Affine Lie algebra ,Algebra ,High Energy Physics::Theory ,Mathematics::Quantum Algebra ,Mathematics::Representation Theory ,Generalized Kac–Moody algebra ,Mathematics - Abstract
Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification.
- Published
- 2015
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