The paper presents the properties of generalized multiple multiplicative Fourier transforms. Also, upper and lower bounds are given for the integral modulus of continuity in terms of the mentioned Fourier transforms, and the bound in (L^2) is unimprovable. As a corollary, an analog of Titchmarsh’s equivalence theorem for the multiplicative Fourier transform is obtained.
The concepts of complete oscillation, rotation, and wandering as well as complete nonoscillation, nonrotation, and nonwandering of a system of differential equations (with respect to its zero solution) are introduced. A one-to-one relationship between these properties and the corresponding characteristics of the system is established. Signs of a guaranteed possibility of studying them using the first approximation system, as well as examples for which that is not possible, are given.
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)).
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).
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.
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.
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.
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.
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.
We obtain an asymptotic formula for the sum
$$H=sum_{0<gamma_kle T,,1le kle 4}h(gamma_1+gamma_2-gamma_3-gamma_4),$$where the (gamma_k) run over the imaginary parts of nontrivial zeros of the Riemann zeta function with multiplicities taken into account and the function (h) belongs to some special class of functions in (L^1(mathbb R)).
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