Acta Mathematica Hungarica最新文献

Elsner, Luca and Tachiya proved in [4] that the values of the Jacobi-theta constants (theta_3(mtau)) and (theta_3(ntau)) are algebraically independent over (mathbb{Q}) for distinct integers (m), (n) under some conditions on (tau). On the other hand, in [3] Elsner and Tachiya also proved that three values (theta_3(mtau),theta_3(ntau)) and (theta_3(ell tau)) are algebraically dependent over (mathbb{Q}). In this article we prove the non-vanishing of linear forms in (theta_3(mtau)), (theta_3(ntau)) and (theta_3(ell tau)) under various conditions on (m), (n), (ell), and (tau). Among other things we prove that for odd and distinct positive integers (m,n>3) the three numbers (theta_3(tau)), (theta_3(mtau)) and (theta_3(n tau)) are linearly independent over (overline{mathbb{Q}}) when (tau) is an algebraic number of some degree greater or equal to 3. In some sense this fills the gap between the above-mentioned former results on theta constants. A theorem on the linear independence over (mathbb{C(tau)}) of the functions (theta_3(a_1 tau), dots, theta_3(a_m tau))for distinct positive rational numbers (a_{1}, {dots}, a_{m}) is also established.

Suppose that (R_{alpha}) is a rhombus with side length (1) and with acute angle (alpha). Let ({S_{n}}) be any collection of squares. In this note a tight upper bound of the sum of the areas of squares from ({S_{n}}) that can be parallel packed into (R_{alpha}) is given.

By using the structure and some properties of extraspecial and generalized/almost extraspecial (p)-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic subgroups of any (generalized/almost) extraspecial group. For a finite group (G), the ratio of the number of cyclic subgroups to the number of subgroups is called the cyclicity degree of (G) and is denoted by cdeg ((G)). We show that the set containing the cyclicity degrees of all finite groups is dense in ([0, 1]). This is equivalent to giving an affirmative answer to the following question posed by Tóth and Tărnăuceanu: “For every (ain [0, 1]), does there exist a sequence ((G_n)_{ngeq 1}) of finite groups such that ( lim_{ntoinfty} text{cdeg} (G_n)=a)?”. We show that such sequences are formed of finite direct products of extraspecial groups of a specific type.

We introduce a new type of mappings in metric spaces which arethree-point analogue of the well-known Chatterjea type mappings, and call themgeneralized Chatterjea type mappings. It is shown that such mappings can be discontinuous as is the case of Chatterjea type mappings and this new class includesthe class of Chatterjea type mappings. The fixed point theorem for generalizedChatterjea type mappings is proven.

Ellipsephic or Kempner-like harmonic series are series of inverses of integers whose expansion in base B, for some (B geq 2), contains no occurrence of some fixed digit or some fixed block of digits. A prototypical example was proposed by Kempner in 1914, namely the sum inverses of integers whose expansion in base 10 contains no occurrence of a nonzero given digit. Results about such series address their convergence as well as closed expressions for their sums (or approximations thereof). Another direction of research is the study of sums of inverses of integers that contain only a given finite number, say k, of some digit or some block of digits, and the limits of such sums when k goes to infinity. Generalizing partial results in the literature, we give a complete result for any digit or block of digits in any base.

Related to a stochastic investment problem which aims to deter-mine when is it better to first invest a larger amount of money and afterwards asmaller one, in this paper we introduce a new preference relation between randomvariables. We investigate the link between this new relation and some well-knownstochastic order relations and present some characterization properties illustratedwith numerical examples.

We answer some questions about two cardinal invariants associated with separating and almost disjoint families and a partition relation involving indecomposable countable linear orderings.

Recently, Mangerel extended the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions in typical short intervals. In this paper, we combine Mangerel's result with Halász-type result recently established by Granville, Harper and Soundararajan to consider the distribution of a class of multiplicative functions in short intervals. First, we prove cancellation in the sum of the coefficients of the standard L-function of an automorphic irreducible cuspidal representation of (mathrm{GL}_m) over (mathbb{Q}) with unitary central character in typical intervals of length (h(log X)^c) with (h = h(X) rightarrow infty) and some constant (c > 0) (under Vinogradov–Korobov zero-free region and GRC). Then we also establish a non-trivial bound for the product of divisor-bounded multiplicative functions with the Liouville function in arithmetic progressions over typical short intervals.

The present note is an addition to the author’s recent paper[44], concerning general multiplicative systems of random variables. Using somelemmas and the methodology of [13], we obtain a general extremal inequality,with corollaries involving Rademacher chaos sums and those analogues for multiplicativesystems. In particular we prove that a system of functions generated bybounded products of a multiplicative system is a convergence system.

For the Borromean link, we determine its irreducible ({rm SL}(2,mathbb{C}))-character variety, and find a formula for the twisted Alexander polynomial as a function on the character variety.