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2. Addendum to the Paper Ëxistence of Weak Solutions for the Navier-Stokes Equations with Initial Data in L p '
- Author
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Calixto P. Calderón
- Subjects
Applied Mathematics ,General Mathematics ,Calculus ,Addendum ,Applied mathematics ,Navier–Stokes equations ,Mathematics - Published
- 1990
3. 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
4. Addendum to the paper on partially stable algebras
- Author
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A. Adrian Albert
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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
5. Remarks concerning the paper of W. L. Ayres on the regular points of a continuum
- Author
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Karl Menger
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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
6. Invariant means and fixed points: A sequel to Mitchell’s paper
- Author
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L. N. Argabright
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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
7. 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
8. Rectifications to the papers: Sets of uniqueness and sets of multiplicity, I and II
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R. Salem
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Algebra ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Multiplicity (mathematics) ,Uniqueness ,Mathematics - Published
- 1948
9. Remarks on a paper of Hermann
- Author
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Shlomo Sternberg and Victor Guillemin
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Applied Mathematics ,General Mathematics ,Classics ,Mathematics - Published
- 1968
10. 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
11. 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
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Uniformly convex space ,Mathematics - Published
- 1936
12. 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
13. An improvement on Furstenberg’s intersection problem
- Author
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Han Yu
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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
14. 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
15. 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
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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
16. 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
17. Extremal growth of Betti numbers and trivial vanishing of (co)homology
- Author
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Jonathan Montaño and Justin Lyle
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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
18. 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
19. Flow equivalence of G-SFTs
- Author
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Toke Meier Carlsen, Søren Eilers, and Mike Boyle
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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
20. 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
21. Positivity preservers forbidden to operate on diagonal blocks
- Author
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Prateek Kumar Vishwakarma
- Subjects
Power series ,Applied Mathematics ,General Mathematics ,Diagonal ,Monotonic function ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Combinatorics ,Mathematics - Classical Analysis and ODEs ,Converse ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,15B48, 26A21 (primary), 15A24, 15A39, 15A45, 30B10 (secondary) ,Schur product theorem ,Mathematics - Abstract
The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic., Minor edits in exposition, 19 pages. The paper now uses the style file of Trans. AMS (to appear)
- Published
- 2023
22. 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
23. 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
24. 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
25. 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
26. On the local time process of a skew Brownian motion
- Author
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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
27. The Groups of Steiner in Problems of Contact (Second Paper)
- Author
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Leonard Eugene Dickson
- Subjects
Combinatorics ,Group (mathematics) ,Simple (abstract algebra) ,Applied Mathematics ,General Mathematics ,Modulo ,Elementary proof ,Order (group theory) ,Abelian group ,Mathematics - Abstract
1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.
- Published
- 1902
28. Generalized Limits in General Analysis, First Paper
- Author
-
Charles N. Moore
- Subjects
Pure mathematics ,Series (mathematics) ,Basis (linear algebra) ,Simple (abstract algebra) ,Generalization ,Applied Mathematics ,General Mathematics ,Multiple integral ,Partial derivative ,Divergent series ,Equivalence (measure theory) ,Mathematics - Abstract
The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows
- Published
- 1922
29. Note Supplementary to the Paper 'On Certain Pairs of Transcendental Functions Whose Roots Separate Each Other'
- Author
-
Maxime Bôcher
- Subjects
Combinatorics ,General theorem ,Character (mathematics) ,Transcendental function ,Statement (logic) ,Applied Mathematics ,General Mathematics ,Order (group theory) ,Addendum ,Of the form ,Notation ,Mathematics - Abstract
In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions
- Published
- 1917
30. Concerning the Arc-Curves and Basic Sets of a Continuous Curve, Second Paper
- Author
-
W. Leake Ayres
- Subjects
Arc (geometry) ,Set (abstract data type) ,Pure mathematics ,Relation (database) ,Applied Mathematics ,General Mathematics ,Metric (mathematics) ,Point (geometry) ,Locally compact space ,Notation ,Separable space ,Mathematics - Abstract
In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together
- Published
- 1929
31. Correction to a Paper on the Moore-Kline Problem
- Author
-
Leo Zippin
- Subjects
TheoryofComputation_MISCELLANEOUS ,Discrete mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,Assertion ,Connected sum ,Mathematics - Abstract
It has been brought to my attention by Mr. N. E. Steenrod that the lemma of page 708 in the paper referred to is in error. The final assertion of the proof is false. It is therefore necessary to point out that the paper is not "disturbed" by this fault. For if one requires that the Pn, «= 1,2, • • •, of the lemma be arcs then the (altered) lemma does hold, since it is true that the connected sum of a perfect continuous curve and an arc is "perfect." One verifies that this restricted lemma is sufficient for the uses of the paper.
- Published
- 1933
32. A Note on the Preceding Paper
- Author
-
D. R. Curtiss
- Subjects
Combinatorics ,Corollary ,Applied Mathematics ,General Mathematics ,Root (chord) ,Laguerre polynomials ,Point (geometry) ,Proposition ,Slight change ,Notation ,Mathematics - Abstract
Walsh's Theorem II states that if the roots of f (z) are in the circular region C, and if z is exterior to C, then ai lies in C. This is precisely the result obtained by Laguerre as given on page 59 of volume I of his collected works, if expressed in non-homogeneous coordinates as on page 57. We shall use another form of Laguerre's Theorem (loc. cit., p. 57) to prove the underlying proposition on which Walsh bases his proof of his Theorem I, as follows: Every circle through any point z and its "derived point" a as defined by (1) either passes through all the roots off ( z ), or else has at least one root in the region interior to it, and at least one root in the exterior region. I have recently called attentiont to the fact that this is a corollary of a theorem of Bocher's on jacobians which has served as a starting point in Walsh's earlier papers. With a slight change in Walsh's notation, whereby we substitute z's for a's, the proposition from which Theorem I is deduced may be stated thus
- Published
- 1922
33. On Certain Families of Orbits with Arbitrary Masses in the Problem of Three Bodies (Second Paper)
- Author
-
F. H. Murray
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Computer Science::Computational Geometry ,Orbit (control theory) ,Equilateral triangle ,Constant (mathematics) ,Object (computer science) ,Stability (probability) ,Mathematics ,Characteristic exponent - Abstract
It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.
- Published
- 1926
34. Errata in My Paper 'On a Special Class of Polynomials'
- Author
-
Oystein Ore
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Line (text file) ,Term (logic) ,Special class ,Mathematics - Abstract
This paper contains a number of disturbing misprints: Equation (2) p. 560 should read Gpf(x) = aoxPfm+ a1xPf11 + *** + amixpf + amx. Line 17 p. 561 read Ap(x) XBp(x) instead of Ap(x)Bp(x). The term perfect (volkommen) in Theorem 1 is used in the sense of Steinitz, Algebraische Theorie der Korper, edited by Hasse and Baer, pp. 50-51. Line 21 p. 562 should read Fp(x) = Qp(x) X (xP ax) + Ax. Equation (9) p. 562 should read A = aoa(Pm-l)/(P-1) + aia(P'1)/(P-1) + + am,2a0P+1 + am-,a + am. In the expression line 9 p. 563 the last term should be A(")x. Equation (17) p. 564 should read F'"(x) = Fnl(x)P -Fn-1(Wn)PFn-1(X). Line 8 from below p. 574 should read
- Published
- 1934
35. On the Zeros of Dirichlet L-Functions.II (With Corrections to Ön the Zeros of Dirichlet L-Functions.I' and the Subsequent Papers)
- Author
-
Akio Fujii
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Dirichlet L-function ,Dirichlet's energy ,Dirichlet eta function ,Class number formula ,symbols.namesake ,Dirichlet kernel ,Dirichlet's principle ,symbols ,General Dirichlet series ,Dirichlet series ,Mathematics - Published
- 1981
36. On Extending a Continuous (1-1) Correspondence (Second Paper)
- Author
-
Harry Merrill Gehman
- Subjects
Applied Mathematics ,General Mathematics ,Calculus ,Mathematics - Published
- 1929
37. Correction to the Paper 'The Multinomial Solid and the CHI Test'
- Author
-
Burton H. Camp
- Subjects
Applied Mathematics ,General Mathematics ,Statistics ,Chi-square test ,Econometrics ,Multinomial distribution ,Mathematics - Published
- 1938
38. Algebraic Surfaces Invariant Under An Infinite Discontinuos Group of Birational Transformations: (Second Paper)
- Author
-
Virgil Snyder
- Subjects
Algebraic cycle ,Pure mathematics ,Function field of an algebraic variety ,Applied Mathematics ,General Mathematics ,Algebraic group ,Algebraic surface ,Dimension of an algebraic variety ,Geometric invariant theory ,Invariant (mathematics) ,Algebraic closure ,Mathematics - Published
- 1913
39. Errata in my paper: 'On a special class of polynomials' [Trans. Amer. Math. Soc. 35 (1933), no. 3, 559–584; 1501703]
- Author
-
Oystein Ore
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Calculus ,Special class ,Mathematics - Published
- 1934
40. Correction to the paper: 'A problem concerning orthogonal polynomials' [Trans. Amer. Math. Soc. 37 (1935), no. 1, 196–206; 1501782]
- Author
-
G. Szegö
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Orthogonal polynomials ,Mathematics - Published
- 1936
41. On the Order of Linear Homogeneous Groups: (Fourth Paper)
- Author
-
H. F. Blichfeldt
- Subjects
Order (business) ,Homogeneous ,Applied Mathematics ,General Mathematics ,Applied mathematics ,Mathematics - Published
- 1911
42. The Foundations of a Theory of the Calculus of Variations in the Large in m-Space (Second Paper)
- Author
-
Marston Morse
- Subjects
Applied Mathematics ,General Mathematics ,Calculus ,Space (mathematics) ,Mathematics - Published
- 1930
43. Correction to the Paper 'A Problem Concerning Orthogonal Polynomials'
- Author
-
G. Szegö
- Subjects
Gegenbauer polynomials ,Applied Mathematics ,General Mathematics ,Discrete orthogonal polynomials ,Mathematical analysis ,Classical orthogonal polynomials ,Algebra ,symbols.namesake ,Wilson polynomials ,Orthogonal polynomials ,Hahn polynomials ,symbols ,Jacobi polynomials ,Koornwinder polynomials ,Mathematics - Published
- 1936
44. Remarks on the Preceding Paper of James A. Clarkson
- Author
-
Nelson Dunford and Anthony P. Morse
- Subjects
Applied Mathematics ,General Mathematics ,Classics ,Mathematics - Published
- 1936
45. The Foundations of the Calculus of Variations in the Large in m-Space (First Paper)
- Author
-
Marston Morse
- Subjects
Applied Mathematics ,General Mathematics ,Calculus ,Space (mathematics) ,Mathematics - Published
- 1929
46. Correction to a paper: 'The Moore-Kline problem' [Trans. Amer. Math. Soc. 34 (1932), no. 3, 705–721; 1501658]
- Author
-
Leo Zippin
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Mathematics - Published
- 1933
47. Corrections to the Paper 'Integration in General Analysis'
- Author
-
Nelson Dunford
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Uniform continuity ,Class (set theory) ,Basis (linear algebra) ,Applied Mathematics ,General Mathematics ,Disjoint sets ,Function (mathematics) ,Finite set ,Mathematics ,Separable space - Abstract
This gives the desired result. It might be pointed out that Theorem 4 shows that Ilfn-gn I-->O In ?4 it is tacitly assumed that the measurable set E can be partitioned into measurable sets E,,. This is always the case in separable spaces. To proceed without this assumption it will not be necessary to assume that J is metric. The class So(E) is defined as the class of functions finitely valued on E. Such a function is one for which there is a decomposition of E into a finite number of disjoint measurable subsets on each of which it is constant. This basis necessitates only a slight rewording in a few places. In Lemma 1 the set E should be taken as a set in A. In Theorem 2 the words "functions uniformly continuous" should be replaced by "functions finitely valued." In the proof of Theorem 11 the sentence "Fix . . . continuous on e" should be worded "Fix e with 18(E-e)
- Published
- 1935
48. On the Order of Linear Homogeneous Groups (Second Paper)
- Author
-
H. F. Blichfeldt
- Subjects
Order (business) ,Homogeneous ,Applied Mathematics ,General Mathematics ,Applied mathematics ,Mathematics - Published
- 1904
49. Correction to a Paper on the Whitehead Huntington Postulates
- Author
-
A. H. Diamond
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics ,Epistemology - Published
- 1934
50. Cubic Curves and Desmic Surfaces; Second Paper
- Author
-
R. M. Mathews
- Subjects
Applied Mathematics ,General Mathematics ,Cubic form ,Geometry ,Mathematics - Published
- 1928
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