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A simple proof of the Wiener-Ikehara Tauberian Theorem Expos. Math. (IF 0.7) Pub Date : 2024-03-18 M. Ram Murty, Jagannath Sahoo, Akshaa Vatwani
The Wiener-Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series . We provide a simple and elegant proof of the Wiener-Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener-Ikehara theorem with an
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Abstract embeddability ranks Expos. Math. (IF 0.7) Pub Date : 2024-03-18 Florent P. Baudier, Christian Rosendal
We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability
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Geometric criteria for the existence of capillary surfaces in tubes Expos. Math. (IF 0.7) Pub Date : 2024-02-28 Giorgio Saracco
We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected . These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section .
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Three families of matrices Expos. Math. (IF 0.7) Pub Date : 2024-02-22 Alexander Pushnitski
This paper has an expository nature. We compare the spectral properties (such as boundedness and compactness) of three families of semi-infinite matrices and point out similarities between them. The common feature of these families is that they can be understood as matrices of some linear operations on appropriate Hardy spaces.
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Reverse engineered Diophantine equations Expos. Math. (IF 0.7) Pub Date : 2024-02-09 Stevan Gajović
We answer a question of Samir Siksek, asked at the open problems session of the conference “Rational Points 2022”, which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set of perfect integer powers, using Mihăilescu’s theorem, we construct a polynomial such that the set contains a perfect integer power if and only if it belongs to . We first discuss
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The three harmonic homologies theorem Expos. Math. (IF 0.7) Pub Date : 2024-02-02 Fahimeh Heidari, Bijan Honari
In this paper, we give a complete answer to the question: “Under what conditions the product of three harmonic homologies of the real projective space is a harmonic homology again?” Among other things, we prove the three harmonic homologies theorem in by which the product of three harmonic homologies with collinear centers is again a harmonic homology if and only if the hyperplanes are polars of the
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On the Hopf problem and a conjecture of Liu–Maxim–Wang Expos. Math. (IF 0.7) Pub Date : 2024-01-17 Luca F. Di Cerbo, Rita Pardini
We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu et al. (2021). In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.
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Obituary: Bent Fuglede (1925–2023) Expos. Math. (IF 0.7) Pub Date : 2024-01-09 Liming Ge
Abstract not available
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Generalised solutions to linear and non-linear Schrödinger-type equations with point defect: Colombeau and non-Colombeau regimes Expos. Math. (IF 0.7) Pub Date : 2023-12-30 Nevena Dugandžija, Alessandro Michelangeli, Ivana Vojnović
For a semi-linear Schrödinger equation of Hartree type in three spatial dimensions, various approximations of singular, point-like perturbations are considered, in the form of potentials of very small range and very large magnitude, obeying different scaling limits. The corresponding nets of approximate solutions represent actual generalised solutions for the singular-perturbed Schrödinger equation
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A new theory of atomic [formula omitted] spaces with applications to smoothness of functions Expos. Math. (IF 0.7) Pub Date : 2023-12-23 Steven G. Krantz
We provide a new definition of Hardy space atoms that avoids use of coordinates to formulate the moment condition. Thus the new theory can be used in abstract settings such as spaces of homogeneous type. We give applications of this theory to the definition of and study of smooth functions on spaces of homogeneous type.
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The best constant in a Hilbert-type inequality Expos. Math. (IF 0.7) Pub Date : 2023-12-02 Ole Fredrik Brevig
We establish that ∑m=1∞∑n=1∞aman¯mn(max(m,n))3≤43∑m=1∞|am|2holds for every square-summable sequence of complex numbers a=(a1,a2,…) and that the constant 4/3 cannot be replaced by any smaller number. Our proof is rooted in a seminal 1911 paper concerning bilinear forms due to Schur, and we include for expositional reasons an elaboration on his approach.
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The generators of the K-groups of the sphere Expos. Math. (IF 0.7) Pub Date : 2023-11-10 Hermann Schulz-Baldes, Tom Stoiber
This note presents an elementary iterative construction of the generators for the complex K-groups Ki(C(Sd)) of the d-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.
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Unmarked trace spectrum rigidity on strictly convex real projective surfaces Expos. Math. (IF 0.7) Pub Date : 2023-11-10 Inkang Kim
We prove that for a given unmarked trace spectrum with multiplicity, there are only a finite number of convex real projective surfaces with that spectrum up to remarking.
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Homotopy groups of cubical sets Expos. Math. (IF 0.7) Pub Date : 2023-10-07 Daniel Carranza, Krzysztof Kapulkin
We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric realization functor. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy
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On Bott–Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms Expos. Math. (IF 0.7) Pub Date : 2023-09-14 Lorenzo Sillari, Adriano Tomassini
In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott–Chern and Aeppli cohomologies defined using the operators d, dc. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to d, dc, showing their relation with Bott–Chern and Aeppli cohomologies
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On Ramanujan’s continued fractions of order twenty-four Expos. Math. (IF 0.7) Pub Date : 2023-08-21 Shraddha Rajkhowa, Nipen Saikia
Two continued fractions U(q) and V(q) of order twenty-four are obtained from a general continued fraction identity of Ramanujan. Some theta-function and modular identities for U(q) and V(q) are established to prove general theorems for the explicit evaluations of U(±q) and V(±q). From the theta-function identities of U(q) and V(q), three colour partition identities are derived as application to partition
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Composition of Bhargava’s cubes over number fields Expos. Math. (IF 0.7) Pub Date : 2023-08-19 Kristýna Zemková
In this paper, the composition of Bhargava’s cubes is generalized to the ring of integers of a number field of narrow class number one, excluding the case of totally imaginary number fields. The exclusion of the latter case arises from the nonexistence of a bijection between (classes of) binary quadratic forms and an ideal class group. This problem, together with a related mistake in another paper
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Editorial for special issue in honor of B. Edixhoven (1962-2022) Expos. Math. (IF 0.7) Pub Date : 2023-08-09 Jennifer Balakrishnan, Ziyang Gao, Pierre Parent, Andrei Yafaev
Abstract not available
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The Riemann–Roch theorem for the Adams operations Expos. Math. (IF 0.7) Pub Date : 2023-08-01 A. Navarro, J. Navarro
We prove the classical Riemann–Roch theorems for the Adams operations ψj on K-theory: a statement with coefficients on Z[j−1], that holds for arbitrary projective morphisms, as well as another statement with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation
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On Briançon–Skoda theorem for foliations Expos. Math. (IF 0.7) Pub Date : 2023-07-20 Arturo Fernández-Pérez, Evelia R. García Barroso, Nancy Saravia-Molina
We generalize Mattei’s result relative to the Briançon–Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve
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Problèmes de type André-Oort en pinceau arithmétique Expos. Math. (IF 0.7) Pub Date : 2023-06-26 Rodolphe Richard
Nous proposons une “Conjecture d’André-Oort en pinceau arithmétique”. C’est une extension de la conjecture d’André-Oort, disons “classique”, formulée à l’origine par Y. André et F. Oort. La conjecture fait intervenir les modèles entiers des variétés de Shimura.
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On an identity of Sylvester Expos. Math. (IF 0.7) Pub Date : 2023-06-25 Bogdan Nica
We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.
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A matrix theory introduction to seaweed algebras and their index Expos. Math. (IF 0.7) Pub Date : 2023-06-19 Alex Cameron, Vincent E. Coll, Nicholas Mayers, Nicholas Russoniello
The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based on the connected components of a “meander”: a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract
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On real-analytic Levi-flat hypersurfaces and holomorphic Webs Expos. Math. (IF 0.7) Pub Date : 2023-06-13 Ayane Adelina da Silva, Arturo Fernández-Pérez
We investigate holomorphic webs tangent to real-analytic Levi-flat hypersurfaces on compact complex surfaces. Under certain conditions, we prove that a holomorphic web tangent to a real-analytic Levi-flat hypersurface admits a multiple-valued meromorphic first integral. We also prove that the Levi foliation of a Levi-flat hypersurface induced by an irreducible real-analytic curve in the Grassmannian
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An overview on problems of Unlikely Intersections in families of abelian varieties Expos. Math. (IF 0.7) Pub Date : 2023-06-01 Laura Capuano
This short survey is part of a minicourse I gave during the CMI-HIMR Summer School “Unlikely Intersections in Diophantine Geometry” on the Zilber–Pink conjecture, formulated independently by Zilber (2002), Bombieri, Masser and Zannier (1999) in the case of tori and by Pink (2005) in the more general setting of mixed Shimura varieties. This conjecture, which includes in its general formulation many
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Degeneration locus of Qp-local systems: Conjectures Expos. Math. (IF 0.7) Pub Date : 2023-05-19 Anna Cadoret
We introduce a conjecture on the arithmetic sparcity of the degeneration locus of a p-adic local system on a smooth variety over a number field and, modulo the Bombieri–Lang conjecture, show that it follows from a conjecture on the geometry of the level varieties attached to the local system. We present a few applications of our conjecture to classical problems in arithmetic geometry. Eventually, we
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On the Geometric Zilber–Pink theorem and the Lawrence–Venkatesh method Expos. Math. (IF 0.7) Pub Date : 2023-05-18 Gregorio Baldi, Bruno Klingler, Emmanuel Ullmo
Using our recent results on the algebraicity of the Hodge locus for variations of Hodge structures of level at least 3, we improve the results of Lawrence–Venkatesh in direction of the refined Bombieri–Lang conjecture.
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Geometric quadratic Chabauty and p-adic heights Expos. Math. (IF 0.7) Pub Date : 2023-05-18 Juanita Duque-Rosero, Sachi Hashimoto, Pim Spelier
Let X be a curve of genus g>1 over Q whose Jacobian J has Mordell–Weil rank r and Néron–Severi rank ρ. When r
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A unified view on the functorial nerve theorem and its variations Expos. Math. (IF 0.7) Pub Date : 2023-05-16 Ulrich Bauer, Michael Kerber, Fabian Roll, Alexander Rolle
The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems
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Fourfolds of Weil type and the spinor map Expos. Math. (IF 0.7) Pub Date : 2023-05-06 Bert van Geemen
Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.
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Introducing memory to a family of multi-step multidimensional iterative methods with weight function Expos. Math. (IF 0.7) Pub Date : 2023-05-06 Alicia Cordero, Eva G. Villalba, Juan R. Torregrosa, Paula Triguero-Navarro
In this paper, we construct a derivative-free multi-step iterative scheme based on Steffensen’s method. To avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence, we freeze the divided differences used from the second step and use a weight function on already evaluated operators. Therefore, we define a family of multi-step methods
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Logarithmic moduli of roots of line bundles on curves Expos. Math. (IF 0.7) Pub Date : 2023-04-25 David Holmes, Giulio Orecchia
We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a ‘double ramification cycle’ measuring the locus where the given root is isomorphic
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The Rochberg garden Expos. Math. (IF 0.7) Pub Date : 2023-04-12 Jesús M.F. Castillo, Raúl Pino
In 1996, it was published the seminal work of Rochberg “Higher order estimates in complex interpolation theory” (Rochberg, 1996). Among many other things, the paper contains a new method to construct new Banach spaces having an intriguing behaviour: they are simultaneously interpolation spaces and twisted sums of increasing complexity. The fundamental idea of Rochberg is to consider for each z∈S the
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A K3 surface related to Leonardo Pisano’s work on congruent numbers Expos. Math. (IF 0.7) Pub Date : 2023-04-12 Martin Djukanović, Jaap Top
This note recalls an early 13th century result on congruent numbers by Leonardo Pisano (“Fibonacci”), and shows how it relates to a specific much studied K3 surface and to an elliptic fibration on this surface. As an aside, the discussion reveals how, via explicit maps of degree two, the surface is covered by the Fermat quartic surface and also covers one of the two famous ‘most algebraic K3 surfaces’
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Relative plus constructions Expos. Math. (IF 0.7) Pub Date : 2023-04-12 Guille Carrión Santiago, Jérôme Scherer
Let h be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair (X,H), consisting of a connected space X and an h-perfect normal subgroup H of the fundamental group π1(X), an h-acyclic map X→XH+h inducing the quotient by H on the fundamental group. We show that this
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On Hasse’s inequality Expos. Math. (IF 0.7) Pub Date : 2023-03-28 M. Ram Murty
We give an elementary exposition of the little known work of Harold Davenport related to Hasse’s inequality. We formulate a new conjecture suggested by this proof that has implications for the classical Riemann hypothesis.
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Rational points on Atkin–Lehner quotients of geometrically hyperelliptic Shimura curves Expos. Math. (IF 0.7) Pub Date : 2023-03-17 Oana Padurariu, Ciaran Schembri
Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves X0(D,N). In this paper we compute the Q-rational points on the Atkin–Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.
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Extensions and torsors for finite group schemes Expos. Math. (IF 0.7) Pub Date : 2023-03-16 Peter Bruin
We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite locally free commutative group schemes, torsors under such group schemes and groups of isomorphism classes of these objects.
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Rings of tautological forms on moduli spaces of curves Expos. Math. (IF 0.7) Pub Date : 2023-03-16 Robin de Jong, Stefan van der Lugt
We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi–Zhang invariant as essentially the only smooth
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Splitting fields of Xn−X−1 (particularly for n=5), prime decomposition and modular forms Expos. Math. (IF 0.7) Pub Date : 2023-03-11 Chandrashekhar B. Khare, Alfio Fabio La Rosa, Gabor Wiese
We study the splitting fields of the family of polynomials fn(X)=Xn−X−1. This family of polynomials has been much studied in the literature and has some remarkable properties. In Serre (2003), Serre related the function on primes Np(fn), for a fixed n≤4 and p a varying prime, which counts the number of roots of fn(X) in Fp to coefficients of modular forms. We study the case n=5, and relate Np(f5) to
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Rational points on X0+(125) Expos. Math. (IF 0.7) Pub Date : 2023-03-09 Vishal Arul, J. Steffen Müller
We compute the rational points on the Atkin–Lehner quotient X0+(125) using the quadratic Chabauty method. Our work completes the study of exceptional rational points on the curves X0+(N) of genus between 2 and 6. Together with the work of several authors, this completes the proof of a conjecture of Galbraith.
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Computing the trace of an algebraic point on an elliptic curve Expos. Math. (IF 0.7) Pub Date : 2023-03-01 Nicolas Mascot, Denis Simon
We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.
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On three general forms of multiple zeta(-star) values Expos. Math. (IF 0.7) Pub Date : 2023-02-17 Kwang-Wu Chen, Minking Eie
In this paper, we investigate three general forms of multiple zeta(-star) values. We use these values to give three new sum formulas for multiple zeta(-star) values with height ≤2 and the evaluation of ζ⋆({1}m,{2}n+1). We also give a new proof of the sum formula of multiple zeta values.
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Cartan’s method and its applications in sheaf cohomology Expos. Math. (IF 0.7) Pub Date : 2023-02-13 Yuan Liu
This paper aims to use Cartan’s original method in proving Theorems A and B on closed cubes to provide a different proof of the vanishing of sheaf cohomology over a closed cube if either (i) the degree exceeds its real dimension or (ii) the sheaf is (locally) constant and the degree is positive. In the first case, we can further use Godement’s argument to show the topological dimension of a paracompact
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Extension of an inequality of Ramanujan Expos. Math. (IF 0.7) Pub Date : 2023-02-13 Horst Alzer
We prove that ∑k=1∞n+k−1k−1kk−2(x+k)n+k<1xn+1holds for all integers n≥0 and real numbers x>0. This extends a result of Ramanujan, who submitted the inequality with n=0 as a problem to the “Journal of the Indian Mathematical Society”.
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A theory of composites perspective on matrix valued Stieltjes functions Expos. Math. (IF 0.7) Pub Date : 2023-01-14 Graeme W. Milton, Mihai Putinar
A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.
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Approximations of the Riley slice Expos. Math. (IF 0.7) Pub Date : 2023-01-10 Alex Elzenaar, Gaven Martin, Jeroen Schillewaert
Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify ‘half-space’ neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating
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Krull-Remak-Schmidt decompositions in Hom-finite additive categories Expos. Math. (IF 0.7) Pub Date : 2023-01-10 Amit Shah
An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A Hom-finite category is an additive category A for which there is a commutative unital ring k, such that each Hom-set in A is a finite length k-module. The aim of this note is to provide
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Noncommutative Ck functions and Fréchet derivatives of operator functions Expos. Math. (IF 0.7) Pub Date : 2023-01-07 Evangelos A. Nikitopoulos
Fix a unital C∗-algebra A, and write Asa for the set of self-adjoint elements of A. Also, if f:R→ℂ is a continuous function, then write fA:Asa→A for the operator function a↦f(a) defined via functional calculus. In this paper, we introduce and study a space NCk(R) of Ck functions f:R→ℂ such that, no matter the choice of A, the operator function fA:Asa→A is k-times continuously Fréchet differentiable
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On Stiefel’s parallelizability of 3-manifolds Expos. Math. (IF 0.7) Pub Date : 2023-01-07 Valentina Bais, Daniele Zuddas
We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.
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A simple algorithm for expanding a power series as a continued fraction Expos. Math. (IF 0.7) Pub Date : 2022-12-29 Alan D. Sokal
I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet
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Thresholds for the monochromatic clique transversal game Expos. Math. (IF 0.7) Pub Date : 2022-11-24 Csilla Bujtás, Pakanun Dokyeesun, Sandi Klavžar
We study a recently introduced two-person combinatorial game, the (a,b)-monochromatic clique transversal game which is played by Alice and Bob on a graph G. As we observe, this game is equivalent to the (b,a)-biased Maker–Breaker game played on the clique-hypergraph of G. Our main results concern the threshold bias a1(G) that is the smallest integer a such that Alice can win in the (a,1)-monochromatic
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Groups of prime degree and the Bateman–Horn Conjecture Expos. Math. (IF 0.7) Pub Date : 2022-11-23 Gareth A. Jones, Alexander K. Zvonkin
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree (qn−1)/(q−1) of PSLn(q) is prime. We present heuristic arguments and computational evidence based on the Bateman–Horn Conjecture to support a conjecture that for each prime n≥3 there are infinitely many primes of this
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Criteria on the existence of limit cycles in planar polynomial differential systems Expos. Math. (IF 0.7) Pub Date : 2022-11-10 Jaume Giné, Maite Grau, Jaume Llibre
We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results. We give examples of systems which realize the maximum number of limit cycles provided by each criterion. In particular we consider the class of differential systems of the form ẋ=Pn(x,y)+Pm(x,y),ẏ=Qn(x,y)+Qm(x,y), where