On the Davenport–Heilbronn theorems and second order terms

We give simple proofs of the Davenport–Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.     

On a Lyapunov-type inequality and the zeros of a certain Mittag–Leffler function

In this work, a Lyapunov-type inequality is obtained for the case when one is dealing with a fractional differential boundary value problem. We then use that result to obtain an interval where a certain Mittag–Leffler function has no real zeros.     

On the interlacing of the zeros of Poincaré series

The Ramanujan Journal - Rankin proved that the Poincaré series for $$mathbf{SL}(2,{{mathbb {Z}}})$$ that are not cusp forms have all their zeros on the unit circle in the standard...     

The zeros of the Weierstrass $$\wp$$–function and hypergeometric series

We express the zeros of the Weierstrass -function in terms of generalized hypergeometric functions. As an application of our main result we prove the transcendence of two specific hypergeometric functions at algebraic arguments in the unit disc. We also give a Saalschützian ₄ F ₃–evaluation. Research of W. Duke was supported in part by NSF Grant DMS-0355564. He wishes to acknowledge and thank the Forschungsinstitut für Mathematik of ETH Zürich for its hospitality and support.     

On Fourier–Haar coefficients of the function from the Marcinkiewicz space

The main objective of the paper is to describe asymptotic behaviour of Fourier–Haar coefficients of functions from Marcinkiewicz spaces. We also discuss the Cesàro summability and the almost convergence of sequences related to Fourier–Haar coefficients and generalise a result from [17]. Some analogues of Mercer's theorem for Fourier coefficients are proved.     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

On Analogues of Heilbronn’s Theorem

Mathematical Notes - Continued fractions with rational partial quotients arise in a natural way in the course of applying any $$k$$ -ary gcd algorithm to the ratio of natural numbers $$a$$ , $$b$$...     

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

In Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non-atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k-uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.     

On fluctuations of Riemann’s zeta zeros

It is shown that the normalized fluctuations of Riemann’s zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, $\gamma _{k}$ and $\gamma _{k+x},$ with $x\sim \left( \log k\right) ^{\beta }, \beta >0,$ for large $k$ follow the two-variate Gaussian distribution with correlation $\left( 1-\beta \right) _{+}\! .$      

On the M?bius function of a lower Eulerian Cohen–Macaulay poset

A certain inequality is shown to hold for the values of the M?bius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen–Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley’s nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen–Macaulay meet-semilattice and Adin’s nonnegativity conjecture for the cubical h-vector of a Cohen–Macaulay cubical complex.     

On the reducibility of the Kampé deFériet function

The existing list of reducible cases of double hypergeometric functions is supplemented by the use of summation theorems for nearly poised hypergeometric functions. A few new relations for single hypergeometric functions emerge as special cases. Relations of this type are useful for computing purposes.     

On the inverse problem relative to dynamics of the ω function

Let P be the set of prime numbers and P (n) denote the largest prime factor of integer n > 1. Write C3 = {p1p2p3 : pi ∈ P (i = 1, 2, 3), pi = pj (i = j)}, B3 = {p1p2p3 : pi ∈ P (i = 1, 2, 3), p1 = p2 or p1 = p3 or p2 = p3, but not p1 = p2 = p3}. For n = p1p2p3 ∈ C3 ∪ B3, we define the w function by ω(n) = P (p1 + p2)P (p1 + p3)P (p2 + p3). If there is m ∈ S - C3 ∪ B3 such that ω(m) = n, then we call m S-parent of n. We shall prove that there are infinitely many elements of C3 which have enough C3-parents an...     

On the dimension of the space of ℝ-places of certain rational function fields

We prove that for every n ∈ ? the space M(K(x ₁, …, x _n ) of ?-places of the field K(x ₁, …, x _n ) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x ₁, …, x _n )) ≤ n. For n = 2 the space M(K(x ₁, x ₂)) has covering and integral dimensions dimM(K(x ₁, x ₂)) = dim_? M(K(x ₁, x ₂)) = 2 and the cohomological dimension dim _G M(K(x ₁, x ₂)) = 1 for any Abelian 2-divisible coefficient group G.     

On the use of a general quadratic Lyapunov function for studying the stability of Takagi–Sugeno systems

We study the stability of the zero solution to a nonlinear system of ordinary differential equations on the base of its Takagi–Sugeno (TS) representation. As is known, the most constructive stability and stabilization conditions for TS systems stated as linear matrix inequalities are established with the help of a general quadratic Lyapunov function (GQLF). However, such conditions are often too rigid. Using a modification of the Lyapunov direct method, we propose asymptotic stability conditions with weaker requirements to GQLF. They allow an application to a wider class of systems. We also give some illustrative examples.