I give a simple proof of Cramér’s large deviation for sums of independent, identically distributed real random variables.
This paper investigates the existence and qualitative properties of solutions to a nonlocal elliptic equation involving a fractional Kirchhoff-type operator, a critical Sobolev exponent, and a Hardy-type singular perturbation: (alpha + b integral(R3) vertical bar(- Delta)(s/2) u vertical bar (2) dx) (-Delta)(u)(s)+V(x)u=f(x,u)+lambda vertical bar u vertical bar p-2u/vertical bar x vertical bar(alpha) where 0 < s < 1, a, b > 0 are constants, 0 <= alpha < 2s characterizes the singular potential, and lambda > 0 is a parameter. The nonlinearity f (x, u) satisfies appropriate growth conditions, and the potential V(x) meets coercivity assumptions. We tackle several analytical difficulties arising from the combined effects of the nonlocal Kirchhoff operator, the critical Sobolev exponent, and the singular Hardy term. Employing variational methods, we prove the existence of at least two distinct positive solutions for sufficiently small lambda. Our approach relies on a detailed analysis of the associated Nehari manifold, which decomposes into two disjoint subsets corresponding respectively to ground state and mountain pass type solutions. We further establish a strict norm separation between these solutions, alongside uniform regularity and decay estimates. These findings extend the theory of fractional nonlocal equations by illuminating the interplay between Kirchhoff-type operators and Hardy potentials, with potential applications to models in continuum mechanics and anomalous diffusion. (c) 2026 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The pair consisting of a quantum group and its corresponding coideal subalgebra, known as a quantum symmetric pair, was developed independently by M. Noumi and G. Letzter through different approaches. The purpose of this paper is threefold. First, for symmetric pairs (sp2n, grn), we construct a coideal subalgebra Uqtw(grn) of the quantized enveloping algebra of type CI using the R-matrix presentation, based on the work of Noumi. Second, we derive a Poincar & eacute;- Birkhoff-Witt(PBW) basis for Uqtw(grn) by the A-form approach. As a consequence of the isomorphism between Utw q (grn) and the & imath; quantum group U & imath;, our method also yields the PBW basis for the & imath; quantum group of type CI. Finally, as an application of the R-matrix presentation, we construct a Poisson algebra Pn associated with Uqtw(grn), and explicitly describe the action of the braid group Bn on the elements of Pn. (c) 2026 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies. MSC 2010: primary 17B37
Let R be a semiprime ring with maximal right ring of quotients Q and extended centroid C. Suppose that f : R -> Q is an additive map satisfying [...[[ f (x), xn1 ], xn2 ], ..., xnk] = 0 for all x E R, where k, n1, n2, ..., nk are fixed positive integers. Then it can be proved that there exists an idempotent e E C such that e f (x) = lambda x + & micro;(x) for all x E R, where lambda E C and & micro; : R -> C is an additive map and (1-e)Q similar to= M2(E), the 2 & times; 2 matrix ring over a complete Boolean ring E. This result gives affirmative answers to several unsolved questions in the literature. (c) 2026 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
For suitable function kernels kappa on a locally compact space, we develop a theory of inner pseudo-balayage omega(A) of signed (Radon) measures omega of finite energy onto a quasiclosed set A, omega(A) being defined as the solution to the problem of minimizing the Gauss functional integral kappa(x, y) d(& micro; circle times & micro;)(x, y)-2 integral kappa(x, y) d(& micro; circle times omega)(x, y), where & micro; ranges over all positive measures of finite energy concentrated on the set A. If A is Borel, the concept of inner pseudo-balayage omega(A) is shown to coincide with that of outer pseudo-balayage, introduced in Fuglede's work (Fuglede, 2016), which was however only concerned with omega >= 0, whereas the investigation of signed omega requires essentially different methods and approaches. The theory of pseudo-balayage thereby established enables us to improve substantially our recent results on the well-known inner Gauss variational problem (Zorii, 2024), by strengthening their formulations and/or by extending the area of their validity. This study covers many interesting kernels in classical and modern potential theory, which looks promising for further applications.
We deduce a simple, compact formula for the number of representations of a positive integer as a sum of three squares in an elementary manner from a result of Gauss using only mathematics familiar to Gauss and Dirichlet. We prove that our formula is equivalent to the previously known formulas due to Bateman, Lomadze, Estermann, Siegel, Crandall, and Krammer. In contrast to some of these earlier formulas, our result does not involve any infinite sums or infinite products. We additionally obtain a formula for the number of representations of a nonnegative integer as a sum of three triangular numbers. Using this, we prove some conjectures due to Hirschhorn and Sellers on representations of a nonnegative integer as a sum of three triangular numbers. (c) 2026 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we prove that the equations 7x4 + 25y4 = 7z4 and x4 + 219y4 = z4 have no integer solutions with xyz =/ 0. These equations are open problems in the survey of Grechuk and Ratcliffe (2025) as well as in the book of Grechuk (2024). (c) 2026 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In "Extremal length and functional completion", Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings: (1) Extremal length and uniformization of metric surfaces. (2) Extremal length of families of surfaces and quasiconformal maps between n-dimensional spaces. (3) Schramm's transboundary extremal length and conformal maps between multiply connected plane domains. (c) 2024 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
In this paper, we give an expository presentation of a paper of Mathieu (1997). The paper of Mathieu proves that a certain Lie group-theoretic conjecture implies the Jacobian Conjecture. To give Mathieu's proof, we first discuss the required background on representation theory in an expository way. We continue to prove some results on the irreducible subrepresentations of the tensor algebra of the standard representation of SU(N). The last part of the paper is dedicated to Mathieu's proof. (c) 2026 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
We prove two unconditional upper bounds on the gaps between ordinates of consecutive non-trivial zeros of a general L-function L(s). This extends previous work of Hall and Hayman (2000) on the Riemann zeta-function and work of Siegel (1945) on Dirichlet L-functions. Interestingly, we observe that while Hall and Hayman's method gives a sharper estimate when the degree of L(s) is sufficiently small compared to the analytic conductor, Siegel's method does better in the other regime.
There is considerable current interest in applications of generalised Lie algebras graded by an abelian group P with a commutative factor w. This calls for a systematic development of the theory of such algebraic structures. We treat the representation theory and invariant theory of the P-graded general linear Lie w-algebra gl(V(P, w)), where V(P, w) is any finite dimensional Pgraded vector space. Generalised Howe dualities over symmetric (P, w)-algebras are established, from which we derive the first and second fundamental theorems of invariant theory, and a generalised Schur-Weyl duality. The unitarisable gl( V (P, w))-modules for two "compact" & lowast;- structures are classified, and it is shown that the tensor powers of V (P, w) and their duals are unitarisable for the two compact & lowast;-structures respectively. A Hopf (P, w)-algebra is constructed, which gives rise to a group functor corresponding to the general linear group in the P-graded setting. Using this Hopf (P, w)-algebra, we realise simple tensor modules and their dual modules by mimicking the classic Borel-Weil theorem. We also analyse in some detail the case with P = ZdimV(Gamma,w) and w depending on a complex parameter q =/ 0, where gl(V(P,w)) shares common features with the quantum general linear (super)group, but is better behaved especially when q is a root of unity. (c) 2026 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of countable groups which we call framenable. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups Fn, Aut(F2) and Aut(F3), all (countable) lattices of SL(2,R), the Baumslag–Solitar groups BSp,q, the braid groups Bn, and Thompson’s group F.
In Dragicevic et al. (2003, 2006) the Ahlfors-Beurling operator T and its powers Tn were represented as averages of two-dimensional martingale transforms. Our intention is to optimize the parameters of the averaging for the estimate of HT Hp from above. The second goal of ours is to give an estimate of HT nHp from below. (c) 2025 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We define the fine Laplacian 4 fu of a finely superharmonic function u on a regular (or more generally a Borel) finely open set U subset of Rn, n >= 2, as a Borel measure on U. If U is a Euclidean open set, then 4 fu is the classical Laplacian 4u of u in the distributional sense. Moreover, if u is finely of class C2 on U, then 4 fu =& sum;2 partial derivative 2u , where partial derivative 2u i=1 partial derivative x2 partial derivative x2 (i = 1, ... , n) are the fine partial i i derivatives of order 2 of u. (c) 2025 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies. 2010 Mathematics Subject Classification: 31B05; 31C40; 31C99
The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert C*-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert C(X)-modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators. (c) 2025 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
We look at explicit ways to bring one or two antiunitary symmetries into a standard form via unitary conjugation. We carefully reproduce Wigner's proof in two special cases, where the antiunitary operators square to +I, or to -I. Wigner's method is constructive and we show how it leads to two algorithms to compute the needed unitaries in small examples. We then adapt these algorithms to deal with two such antiunitary matrices that commute up to a sign. This leads to a proof a finite-dimensional version of the well-known ten-fold way of topological physics. This will allow physicists to perform a change of basis on any finitedimensional model in one of the Altland-Zirnbauer symmetry classes to a standardized version of that symmetry class in which time-reversal and particle-hole symmetry are standard operations that can be implemented efficiently in standard numerical software. This will also make it easier to find formulas for some key isomorphisms between graded real C & lowast;-algebras. (c) 2025 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we introduce the concept of complementary edge ideals of graphs and study their algebraic properties and invariants.
In the paper we study the Bott-Chern and Aeppli cohomologies of the twistor space of a compact self-dual 4-manifold and we characterize the validity of the 88-lemma. We also compute explicitly the Dolbeault cohomology of the twistor space Z of the flat 4-dimensional torus, which is known to not satisfy the 88 lemma. (c) 2026 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Let g be a classical simple Lie algebra over an algebraically closed field F of characteristic zero or large enough, and let n be a maximal nilpotent subalgebra of g. The main tool in representation theory of n is the orbit method, which classifies primitive ideals in the universal enveloping algebra U(n) and unitary representations of the unipotent group N=exp(n) in terms of coadjoint orbits on the dual space n∗. In the paper, we describe explicitly coadjoint orbits of low dimension for n as above. The answer is given in terms of subsets of positive roots. As a corollary, we provide a way to calculate the number of irreducible complex representations of dimensions q, q2 and q3 for a maximal unipotent subgroup N(q) in a classical Chevalley group G(q) over a finite field Fq with q elements. It turned out that this number is a polynomial in q−1 with nonnegative integer coefficients, which agrees with Isaac’s conjecture.
