Acta Mathematica Hungarica最新文献

Let ((a(n) : n in mathbb{N})) denote a sequence of nonnegative integers. Let (0.a(1)a(2) ldots ) denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of ((a(n) : n in mathbb{N})). Research on digit expansions of this form has mainly to do with the normality of (0.a(1)a(2) ldots ) for a given base. Famously, the Copeland-Erdős constant (0.2357111317 ldots {}), for the case whereby (a(n)) equals the (n^{text{th}}) prime number (p_{n}), is normal in base 10. However, it seems that the “inverse” construction given by concatenating the decimal digits of ((pi(n) : n in mathbb{N})), where (pi) denotes the prime-counting function, has not previously been considered. Exploring the distribution of sequences of digits in this new constant (0.0122 ldots 9101011 ldots ) would be comparatively difficult, since the number of times a fixed (m in mathbb{N} ) appears in ((pi(n) : n in mathbb{N})) is equal to the prime gap (g_{m} = p_{m+1} - p_{m}), with the behaviour of prime gaps notoriously elusive. Using a combinatorial method due to Szüsz and Volkmann, we prove that Cramér’s conjecture on prime gaps implies the normality of (0.a(1)a(2) ldots ) in a given base (g geq 2), for (a(n) = pi(n)).

Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of (Theta(n^3)) triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them.For families of (Theta(n^{3-alpha})) triangles, with (0le alpha le 1), there might not be a point in more than (Theta(n^{3-2alpha})) of those triangles.An empty triangle of P is a triangle spanned by Pnot containing any point of P in its interior. Bárány conjectured that there exists an edgespanned by P that is incident to a super-constant number of empty triangles of P. The number of empty trianglesof P might be as low as (Theta(n^2)); in such a case, on average, every edge spanned by P is incident to a constant numberof empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper boundmight not hold. In this paper we show that, somewhat surprisingly,the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number (0leq alpha leq 1) there exists a point set of size n with (Theta(n^{3-alpha})) empty triangles such that any point of the plane is only in (O(n^{3-2alpha})) empty triangles.

We consider the Lipschitz class functions on [0, 1]and special series of their Fourier coefficients with respect to generalorthonormal systems (ONS).The convergence of classical Fourier series (trigonometric, Haar, Walsh systems) of Lip 1 class functions is a trivial problem and is well known. But general Fourier series, as it is known, even for the function f (x) = 1 does not converge.On the other hand, we show that such series do not converge with respect to general ONSs. In the paper we find the special conditions on the functions (varphi_{n}) of the system ((varphi_{n})) such that the above-mentioned series are convergent for any Lipschitz class function. The obtained result is the best possible.

We provide a non-vanishing region for an infinite sum of weight zero Hecke–Maass L-functions for the full modular group inside the critical strip. For given positive parameters T and (1 leq M ll frac{T}{log T}), T large, we also count the number of Hecke–Maass cusp forms whose L-values are non-zero at any point s in this region and whose spectral parameters (t_j) lie in short intervals.

We extend the concept of a G-Drazin inverse from the set (M_n) of all (ntimes n) complex matrices to the set (mathcal{R}^{D}) of all Drazin invertible elements in a ring (mathcal{R}) with identity. We also generalize a partial order induced by G-Drazin inverses from (M_n) to the set of all regular elements in (mathcal{R}^{D}), study its properties, compare it to known partial orders, and generalize some known results.

Let D be a division ring. The first aim of this paper is to describe all unipotent matrices of index 2 in the general linear group (mathrm {GL}_n(D)) of degree n and in the Vershik–Kerov group (mathrm{GL} _{rm VK}(D)). As a corollary, the subgroups generated by such matrices are investigated. The next aim is to seek a positive integer d such that every matrix in these groups is a product of at most d unipotent matrices of index 2. For example, we show that if every element in the derived subgroup (D') of (D^*=Dbackslash {0}) is a product of at most c commutators in (D^*), then every matrix in (mathrm{GL}_n(D)) (resp., (mathrm{GL} _{rm VK}(D)), which is a product of some unipotent matrices of index 2, can be written as a product of at most 4+3c (resp.,5 + 3c) of unipotent matrices of index 2 in (mathrm{GL}_n(D)) (resp., (mathrm{GL}_{rm VK}(D))).

It is consistent that the continuum be arbitrary large and no absolute (kappa)-Borel set X of density (kappa), (aleph_1<kappa<mathfrak{c}),condenses onto a compactum.

It is consistent that the continuum be arbitrary large and any absolute (kappa)-Borel set X of density (kappa), (kappaleqmathfrak{c}), containing a closed subspace of the Baire space of weight (kappa), condenses onto a compactum.

In particular, applying Brian's results in model theory, we get the following unexpected result. Given any (Asubseteq mathbb{N}) with (1in A), there is a forcing extension in which every absolute (aleph_n)-Borel set, containing a closed subspace of the Baire space of weight (aleph_n), condenses onto a compactum if and only if (nin A).

Let (S) be a semigroup, (Z(S)) the center of (S) and (sigma colon S rightarrow S) is aninvolutive automorphism. Our main results is that we describe the solutions ofthe Kannappan-Wilson functional equation

(int_{S} f(xyt), dmu(t) + int_{S} f(sigma(y)xt), dmu(t)= 2f(x)g(y), x,yin S,)

and the Van Vleck-Wilson functional equation

(int_{S} f(xyt), dmu(t) - int_{S} f(sigma(y)xt), dmu(t)= 2f(x)g(y), x,yin S,)

where (mu) is a measure that is a linear combination of Dirac measures ((delta_{z_i})_{iin I}),such that (z_iin Z(S)) for all (iin I). Interesting consequences of these results arepresented.

We introduce a discrete version of weighted Morrey spaces,and discuss the inclusion relations of these spaces. In addition, we obtain theboundedness of discrete weighted Hardy-Littlewood maximal operators on discreteweighted Lebesgue spaces by establishing a discrete Calderón-Zygmund decompositionfor weighted (l^1)-sequences. Furthermore, the necessary and sufficientconditions for the boundedness of the discrete Hardy-Littlewood maximal operatorson discrete weighted Morrey spaces are discussed. Particularly, the necessaryand sufficient conditions are also discussed for the discrete power weights.

A topological space is called submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces.