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We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…
We present expressions for the Weierstrass zeta-function and related elliptic functions by rapidly converging series. These series arise as triple products in the A-infinity category of an elliptic curve.
In this paper we consider a class of conjugate equations, which generalizes de Rham's functional equations. We give sufficient conditions for existence and uniqueness of solutions under two different series of assumptions. We consider…
We study monotonicity and convexity properties of functions arising in the theory of elliptic integrals, and in particular in the case of a Schwarz-Christoffel conformal mapping from a half-plane to a trapezoid. We obtain sharp monotonicity…
General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the "classical" special functions. In particular, an elliptic analogue of the Gauss hypergeometric…
The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function,…
The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral…
We give all possible holomorphic Eisenstein series on $\Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give…
In this paper we introduce new generalizations of the zeta function, the Tricomi functions; their main properties are studied. This opens the way to a deeper, better application of these functions both in the theory of special functions,…
A concise review of the notions of elliptic functions, modular forms, and theta-functions is provided, devoting most of the paper to applications to Conformal Field Theory (CFT), introduced within the axiomatic framework of quantum field…
We prove that the generalized Hilbert-Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form…
By employing contour integration the derivation of a generalized double finite series involving the Hurwitz-Lerch zeta function is used to derive closed form formulae in terms of special functions. We use this procedure to find special…
A $q$-analogue of the multiple gamma functions is introduced, and is shown to satisfy the generalized Bohr-Morellup theorem. Furthermore we give some expressions of these function.
We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modeling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of…
We study the elliptic maximal functions defined by averages over ellipses and rotated ellipses which are multi-parametric variants of the circular maximal function. We prove that those maximal functions are bounded on $L^p$ for some $p\neq…
We generalize our previous new definition of Euler Gamma function to higher Gamma functions. With this unified approach, we characterize Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson…
We generalize symplectic convexity theorems for Hamiltonian actions with proper momentum maps to symplectic actions on orbifolds with mod-$\Gamma$ proper momentum maps.
We analyze a class of sublinear functionals which characterize the interior and the exterior of a convex cone in a normed linear space.
A new classification of real functions and other related real objects defined within a compact interval is proposed. The scope of the classification includes normal real functions and distributions in the sense of Schwartz, referred to…
The choice of a homogeneous ideal in a polynomial ring defines a closed subscheme $Z$ in a projective space as well as an infinite sequence of cones over $Z$ in progressively higher dimension projective spaces. Recent work of Aluffi…