Acta Mathematica Hungarica最新文献

The precise steps of a procedure of going from Heegaard diagrams to framed link diagrams are introduced in this note.

Let (V) be an (n)-dimensional vector space over the finite field (mathbb{F}_{q}) and let (left[Vatop kright]_q) denote the family of all (k)-dimensional subspaces of (V). A family (mathcal{F}subseteq left[Vatop kright]_q) is called intersecting if for all (F), (F'inmathcal{F}), we have ( dim (Fcap F')geq 1). Let (delta_{d}(mathcal{F})) denote the minimum degree in (mathcal{F}) of all (d)-dimensional subspaces. In this paper we show that (delta_{d}(mathcal{F})leq left[ n -d -1atop k -d -1right]) in any intersecting family (mathcal{F}subseteq left[Vatop kright]_q), where (k>dgeq 2) and (ngeq 2k+1).

Let (f) be a Hecke-Maass cusp form for (mathrm{SL}_2(mathbb{Z})) with normalized Fourier coefficients (lambda_f(n)) and Laplace eigenvalue (1/4+mu_f^2). Let (g) be a Hecke-Maass cusp form for (mathrm{SL}_2(mathbb{Z})) with normalized Fourier coefficients (lambda_g(n)). In this paper, we study the asymptotic of (sum_{n leq X}lambda_{1boxplus(ftimes g)}(n)) and get the explicit dependence of the error term on the spectral parameter (mu_f).

Let (mathcal{A}) be a unital (mathbf{C}^*)-algebra with unit e.We develop several inequalities for a positive linear functional f on (mathcal{A}) and obtain several bounds for the numerical radius v(a) of an element (ain mathcal{A}).Among other inequalities, we show that if (a_k, b_k, x_kin mathcal{A}), (rin mathbb{N}) and (f(e)=1), then

We find several equivalent conditions for (v(a)=frac{|a|}{2}) and (v^2(a)={frac{1}{4}|a^*a+aa^*|}).We prove that (v^2(a)={frac{1}{4}|a^*a+aa^*|}) (resp., (v(a)=frac{|a|}{2})) if and only if

(resp., (mathbb{S}_{frac12 | a|} subseteq V(a) subseteq mathbb{D}_{frac12 | a|})),where V(a) is the numerical range of a and (mathbb{D}_k) (resp., (mathbb{S}_k)) denotes the circular disk (resp., semi-circular disk) with center at the origin and radius k. We also study inequalities for the ((alpha,beta))-normal elements in (mathcal{A}).

For a subfamily (mathcal{F}subseteq 2^{[n]}) of the Boolean lattice, consider the graph (G_mathcal{F}) on (mathcal{F}) based on the pairwise inclusion relations among its members. Given a positive integer t, how large can (mathcal{F}) be before (G_mathcal{F}) must contain some component of order greater than t?For t = 1, this question was answered exactly almost a century ago by Sperner: the size of a middle layer of the Boolean lattice. For t = 2ⁿ, this question is trivial. We are interested in what happens between these two extremes.For t = 2^g with g = g(n) being any integer function that satisfies (g(n)=o(n/log n)) as (ntoinfty), we give an asymptotically sharp answer to the above question: not much larger than the size of a middle layer.This constitutes a nontrivial generalisation of Sperner's theorem.We do so by a reduction to a Turán-type problem for rainbow cycles in properly edge-coloured graphs.Among other results, we also give a sharp answer to the question, how large can (mathcal{F}) be before (G_mathcal{F})must be connected?

For any commutative ring R, we show that the categories ofR-coalgebras and cocommutative R-coalgebras are locally(aleph_1)-presentable, while the categories of R-flatR-coalgebras are (aleph_1)-accessible. Similarly, for any associative ring R, the category of R-coringsis locally (aleph_1)-presentable, while the category ofR-R-bimodule flat R-corings is (aleph_1)-accessible. The cardinality of the ring R can be arbitrarily large. We also discuss R-corings with surjective counit and flat kernel. The proofs are straightforward applications of an abstractcategory-theoretic principle going back to Ulmer. For right or two-sided R-module flat R-corings, our cardinalityestimate for the accessibility rank is not as good. A generalization to comonoid objects in accessible monoidal categoriesis also considered.

Assume that (alpha>1) is an irrational number, (beta) and (c>1) are real numbers. The corresponding Beatty sequence and Piatetski–Shapiro sequence are defined as

respectively. Here, the symbol (lfloor yrfloor) denotes the largest integer not exceeding y. Let p be a prime, (gamma=c^{-1}), and let (F_{alpha,beta,c}(p)) be the least quadratic non-residue in the intersection of (mathcal{B}_{alpha,beta}) and (mathcal{N}^c). For (1<c<8/7), we obtain (F_{alpha,beta,c}(p)ll_c p^{1/((6gamma-5)4sqrt{e})+varepsilon}). As (crightarrow1^{+}), our result tends to the Burgess bound (p^{1/(4sqrt{e})+varepsilon}).

Under the assumption of small violations of choice with seed (S) ((SVC(S))), the failure of many choice principles reflect to local properties of (S), which can be a helpful characterisation for preservation proofs. We demonstrate the reflections of (DC), (AC_lambda), (PP), and other important forms of choice. As a consequence, we show that if (S) is infinite then (S) can be partitioned into (omega) many non-empty subsets.

Let (mathbb{P}) denote the set of all prime numbers, I be a set and (sigma = lbrace sigma_i mid i in I rbrace) be a partition of (mathbb{P}). A subgroup H of a finite group G is said to be (sigma)-subnormal in G if there is a chain (H = H_0 le H_1 le dots le H_n = G) of subgroups of G such that, for each (1 le j le n), the subgroup (H_{j-1}) is normal in H_j or (H_j/(H_{j-1})_{H_j}) is a (sigma_i)-group for some (i in I). If (sigma) is the partition of (mathbb{P}) into subsets of size one, then the concept of (sigma)-subnormality reduces to the familiar concept of subnormality. In recent years, many results about subnormal subgroups have been extended to results about (sigma)-subnormal subgroups. This line of research is continued in the present note by proving a (sigma)-version of Wielandt's zipper lemma.

We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence (B=(beta_n)_{ninmathbb{Z}}) of real numbers greater than one. We introduce the set of B-integers and code the sequence of gaps between consecutive B-integers by a symbolic sequence in general over the alphabet (mathbb{N}). We show that this sequence is S-adic. We focus on alternate base systems, where the sequence B of bases is periodic, and characterize alternate bases B in which B-integers can be coded by using a symbolic sequence (bf{v}_{it B}) over a finite alphabet. With these so-called Parry alternate bases we associate some morphisms and show that (bf{v}_{it B}) is a fixed point of their composition. We then provide two classes of Parry alternate bases B generating sturmian fixed points. The paper generalizes results of Fabre and Burdík et al. obtained for the Rényi numerations systems, i.e., in the case when the Cantor base B is a constant sequence.