Mathematical Notes最新文献

Abstract

In this note, first, we describe the (minimal) hereditary saturated subsets of finite acyclic graphs and finite graphs whose cycles have no exits. Then we show that the Cartesian product (C_mtimes L_n) of an (m)-cycle (C_m) by an (n)-line (L_n) has nontrivial hereditary saturated subsets even though the graphs (C_m) and (L_n) themselves have only trivial hereditary saturated subsets. Tomforde (Theorem 5.7 in “Uniqueness theorems and ideal structure for Leavitt path algebras,” J. Algebra 318 (2007), 270–299) proved that there exists a one-to-one correspondence between the set of graded ideals of the Leavitt path algebra (L(E)) of a graph (E) and the set of hereditary saturated subsets of (E^0). This shows that the algebraic structure of the Leavitt path algebra (L(C_mtimes L_n)) of the Cartesian product is plentiful. We also prove that the invariant basis number property of (L(C_mtimes L_n)) can be derived from that of (L(C_m)). More generally, we also show that the invariant basis number property of (L(Etimes L_n)) can be derived from that of (L(E)) if (E) is a finite graph without sinks.

Abstract

We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem

where (lambdainmathbb{R}) is a parameter, (hin L^1(0,1)), and (f:[0,1]times mathbb{R}rightarrowmathbb{R}) is an (L^1)-Carathéodory function. Moreover, (f) is sublinear at (+infty) and nondecreasing with respect to the second variable. We obtain that if (lambda) is sufficiently close to (0) from the left or right, then the problem has at least one or two solutions, respectively. The proof of main results is based on bifurcation theory and the method of lower and upper solutions.

Abstract

We study the isometry group of the Grothendieck group (K_0(mathbb P_n)) equipped with a bilinear asymmetric Euler form. We prove several properties of this group; in particular, we show that it is isomorphic to the direct product of (mathbb Z/2mathbb Z) by the free Abelian group of rank ([(n+1)/2]). We also explicitly calculate its generators for (nle 6).

Abstract

The paper presents several results concerning the complexity of calculations in the model of vector addition chains. A refinement of N. Pippenger’s upper bound is obtained for the complexity of the class of integer (m times n) matrices with the constraint (q) on the size of the coefficients as (H=mnlog_2 q to infty) up to (min{m,n}log_2 q+(1+o(1))H/log_2 H+n). Next, we establish an asymptotically tight bound ((2+o(1))sqrt n) on the complexity of сomputation of the number (2^n-1) in the base of powers of (2). Based on generalized Sidon sequences, constructive examples of integer sets of cardinality (n) are constructed: sets, with polynomial size of elements, having the complexity (n+Omega(n^{1-varepsilon})) for any (varepsilon>0) and sets, with the size (n^{O(log n)}) of the elements, having the complexity (n+Omega(n)).

Abstract

We prove the periodicity of finite rank Somos sequences modulo (m). As an application, we prove the periodicity of the Somos-((6)(mathrm{mod} m)) sequence.

Abstract

It is well known that the lower quantization dimension (underline{D}(mu)) of a Borel probability measure (mu) given on a metric compact set ((X,rho)) does not exceed the lower box dimension (underline{dim}_BX) of (X). We prove the following intermediate value theorem for the lower quantization dimension of probability measures: for any nonnegative number (a) smaller that the dimension (zunderline{dim}_BX) of the compact set (X), there exists a probability measure (mu_a) on (X) with support (X) such that (underline{D}(mu_a)=a). The number (zunderline{dim}_BX) characterizes the asymptotic behavior of the lower box dimension of closed (varepsilon)-neighborhoods of zero-dimensional, in the sense of (dim_B), closed subsets of (X) as (varepsilonto 0). For a wide class of metric compact sets, the equality (zunderline{dim}_BX=underline{dim}_BX) holds.

Abstract

An open problem about integral representation of (zeta(2n)), proposed recently by Pain (2023), is resolved by integration by parts. More general integrals are examined by manipulating the beta integral and digamma function.

Abstract

This paper answers the question of V. M. Buchstaber on the growth function in case of certain (n)-valued group. This question is in close relation to specific discrete integrable systems. In the present paper, we find a specific formula for the growth function in the case of prime (n). We also prove a polynomial asymptotic estimate of the growth function in the general case. At the end, we pose new conjectures and questions regarding growth functions.

Abstract

The present paper is devoted to a lower bound for the number of critical points of the Lyapunov function for Morse–Smale 3-diffeomorphisms with fixed points with pairwise distinct indices. It is known that, in the presence of a single noncompact heteroclinic curve, the supporting manifold of the diffeomorphisms under consideration is a 3-sphere, and the class of topological conjugacy of such a diffeomorphism (f) is completely determined by the equivalence class (there exist infinitely many of them) of the Hopf knot (L_{f}), which is a knot in the generating class of the fundamental group of the manifold (mathbb S^2times mathbb S^1).

Moreover, any Hopf knot is realized by some diffeomorphism of the class under consideration. It is known that the diffeomorphisms defined by the standard Hopf knot (L_0={s}times mathbb S^1) have an energy function, which is a Lyapunov function whose set of critical points coincides with the chain recurrent set. However, the set of critical points of any Lyapunov function of a diffeomorphism (f) with a nonstandard Hopf knot is strictly greater than the chain recurrent set of the diffeomorphism.

In the present paper, for the diffeomorphisms defined by generalized Mazur knots, a quasi-energy function has been constructed, which is a Lyapunov function with a minimum number of critical points.

Abstract

New upper bounds are found in the problem of approximation to the (k)th derivatives of a function of (d) variables defined on a simplex by the derivatives of an algebraic polynomial of degree at most (n) ((0le kle n)) interpolating the values of the function at equidistant nodes of the simplex. The estimates are obtained in terms of the diameter of the simplex, the angular characteristic introduced in the paper, the dimension (d), the degree (n) of the polynomial, and the order (k) of the derivative to be estimated and do not contain unknown parameters. These estimates are compared with those most frequently used in the literature.