Omega theorems related to the general Euler totient function

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.     

Arakawa-Kaneko L-functions and generalized poly-Bernoulli polynomials

We introduce and investigate generalized poly-Bernoulli numbers and polynomials. We state and prove several properties satisfied by these polynomials. The generalized poly-Bernoulli numbers are algebraic numbers. We introduce and study the Arakawa-Kaneko L-functions. The non-positive integer values of the complex variable s of these L-functions are expressed rationally in terms of generalized poly-Bernoulli numbers and polynomials. Furthermore, we prove difference and Raabe?s type formulae for these L-functions.     

Central values of L-functions over CM fields

We prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are the L-functions attached to Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary quadratic extension of F. This formula generalizes our former result on L-functions twisted by finite CM characters.     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

On the weakened Siegel’s conjecture

In the paper, we prove that two definitions of E-functions present in mathematics are equivalent if E-functions satisfy second-order linear differential equations. For this set of functions, a weakened variant of the well-known Siegel conjecture about representability of any E-function by a polynomial of hypergeometric functions is also proved. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 33–39, 2005.     

Toda equation and special polynomials associated with the Garnier system

We prove that a certain sequence of τ-functions of the Garnier system satisfies Toda equation. We construct algebraic solutions of the system by the use of Toda equation; then show that the associated τ-functions are expressed in terms of the universal character which is a generalization of Schur polynomial attached to a pair of partitions.     

Construction of supersaturated design with large number of factors by the complementary design method

Supersaturated designs (SSDs) have been widely used in factor screening experiments. The present paper aims to prove that the maximal balanced designs are a kind of special optimal SSDs under the E(f _NOD) criterion. We also propose a new method, called the complementary design method, for constructing E(f _NOD) optimal SSDs. The basic principle of this method is that for any existing E(f _NOD) optimal SSD whose E(fNOD) value reaches its lower bound, its complementary design in the corresponding maximal balanced design is also E(f _NOD) optimal. This method applies to both symmetrical and asymmetrical (mixed-level) cases. It provides a convenient and efficient way to construct many new designs with relatively large numbers of factors. Some newly constructed designs are given as examples.     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

On edge domination numbers of graphs

Let and be the signed edge domination number and signed star domination number of G, respectively. We prove that holds for all graphs G without isolated vertices, where n=|V(G)|?4 and m=|E(G)|, and pose some problems and conjectures.     

A formula for the Mahler measure of axy+bx+cy+d

We introduce a formula for the Mahler measure of axy+bx+cy+d with complex coefficients a,b,c, and d and give examples which demonstrate a connection with L-functions. We then prove a generalization of Maillot's formula when the coefficients are real. Next, we discuss operations on the coefficients which fix the Mahler measure. Finally, we prove an alternate formulation of the main result in order to calculate the Mahler measure of a two-parameter family of polynomials in three variables.     

Congruences for Hecke eigenvalues of Siegel modular forms

We prove some congruences for Hecke eigenvalues of Klingen-Eisenstein series and those of cusp forms for Siegel modular groups modulo special values of automorphic L-functions.     

Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰ_ub(E), of allC ^∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰ_Nbc(E _v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ^∞-functions onE coincides with the projective limit of the spaces ℰ_Nbc(E _B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem. Research done during the stay of this author at the University of Bordeaux (France) in the academic year 1980–1981.     

Alternating permutations and symmetric functions

We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,…,n}. These classes include the following: (1) both w and w⁻¹ are alternating, (2) w has certain special shapes, such as (m−1,m−2,…,1), under the RSK algorithm, (3) w has a specified cycle type, and (4) w has a specified number of fixed points. We also enumerate alternating permutations of a multiset. Most of our formulas are umbral expressions where after expanding the expression in powers of a variable E, Ek is interpreted as the Euler number Ek. As a small corollary, we obtain a combinatorial interpretation of the coefficients of an asymptotic expansion appearing in Ramanujan's “Lost” Notebook.     

Bernstein-Sato ideals and hyperplane arrangements

We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice.We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the multivariable $n/d$ conjecture of Budur for complete factorizations of arbitrary hyperplane arrangements, which in turn proves the strong monodromy conjecture for the associated multivariable topological zeta functions.     

On real Paley-Wiener theorems for certain integral transforms

We prove real Paley-Wiener theorems for the (inverse) Jacobi transform, characterising the space of L²-functions whose image under the Jacobi transform are (smooth) functions with compact support.     

Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

In this paper, we introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland's variational principle, we derive generalized Caristi's (common) fixed point theorems, a generalized Takahashi's nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland's variational principle.     

On effective determination of modular forms by twists of critical L-values

We address the problem of identifying a newform f from the central values of the twisted L-functions ${L(1/2,f\otimes \chi)}We address the problem of identifying a newform f from the central values of the twisted L-functions L(1/2,f?c){L(1/2,f\otimes \chi)} where χ runs through the set of real characters. We prove a quantitative result in this direction.     

A uniform asymptotic formula for the second moment of primitive <Emphasis Type="Italic">L</Emphasis>-functions on the critical line

We prove an asymptotic formula for the second moment of primitive L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift, and twist.     

Continuation of separately analytic functions defined on part of the domain boundary

Let D ? ? ⁿ be a domain with smooth boundary ?D, let E??D be a subset of positive Lebesgue measure mes(E) > 0, and let F ? G be a nonpluripolar compact set in a strongly pseudoconvex domain D ? ? ^m . We prove that, under an additional condition, each function separately analytic on the set X = (D × F) ∪ (E × G) has a holomorphic contination to the domain $\rlap{--} X = \{ (z,w) \in D \times G:\omega _{in}^ * (z,E,D) + \omega ^ * (w,F,D) < 1\} $ , where ω* is the P-measure and ω*_in is the interior P-measure.     

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold-Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E_∞-algebras. We prove that it in fact sends algebras over arbitrary differential graded E_∞-operads to E_∞-algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.