Related papers: On a problem \`a la Kummer-Vandiver for function f…

In this short note, we use Rudnev's point-plane incidence bound to improve some results on conditional expanding bounds for two-variable functions over arbitrary fields due to Hegyv\'ari and Hennecart.

We introduce reflection functors on quiver schemes in the sense of Hausel--Wong--Wyss, generalizing those on quiver varieties. Also we construct some isomorphisms between quiver schemes whose underlying quivers are different.

We quantize non-commutative Maxwell theory canonically in the background field gauge for weak and slowly varying background fields. We determine the complete basis for expansion under such an approximation. As an application, we derive the…

We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

We report a new analytical method for solution of a wide class of second-order differential equations with eigenvalues replaced by arbitrary functions. Such classes of problems occur frequently in Quantum Mechanics and Optics. This approach…

It is shown, that for quantum systems the vectorfield associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schr\"odinger and Heisenberg pictures. We illustrate these ambiguities in terms of…

We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by…

For large enough (but fixed) prime powers $q$, and trace functions to squarefree moduli in $\mathbb{F}_q[u]$ with slopes at most $1$ at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to…

We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case.

We prove the arithmetic fundamental lemma conjecture over a general $p$-adic field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to the one used by the second author during his proof of the AFL over $\mathbb{Q}_p$…

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to…

We extend the theorem of Liouville on integration in finite terms to include dilogarithmic integrals. The results provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension…

First we recall the notion of conxity and log-convexity for real-valued. Then we generalize the trick used by Artin in his famous paper on the Gamma function to find log-convex solutions to the functional equations f(x+1)=g(x)f(x). This…

An adaptation of the WKB method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between…

We study the quantum invariants of projective varieties over the number fields. Namely, explicit formulas for a functor $\mathscr{Q}$ on such varieties are proved. The case of abelian varieties with complex multiplication is treated in…

A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed in q-alg/9609011 In this paper we give an outline of the construction of a noncommutative analogy of the algebra of partial…

We present a new solution to the classification problem for the category of representations of a quiver of type $\widetilde{A}_{3}$. Our approach uses linear algebra techniques which lead us to a reduction that allows to use induction. As…

We study product sets of finite arithmetic progressions of polynomials over a finite field. We prove a lower bound for the size of the product set, uniform in a wide range of parameters. We apply our results to resolve the function field…

In the inverse problem of the calculus of variations one is asked to find a Lagrangian and a multiplier so that a given differential equation, after multiplying with the multiplier, becomes the Euler--Lagrange equation for the Lagrangian.…

We present an analysis of the vacuum Einstein equations for a recently proposed extension of the Kerr-Schild ansatz that includes a spacelike vector field as well as the usual Kerr-Schild null vector. We show that many, although not all, of…