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.
