We study the spectra of random weighted bipartite graphs. We establish that, under specific assumptions on the edge probabilities, the symmetrized empirical spectral distribution function of the graph’s adjacency matrix converges to the symmetrized Marchenko-Pastur distribution function.
We study the bipolar type of the composition for pairs of endomorphisms of a groupoid and introduce the notion of an alternating pair of endomorphisms. For such a pair, the bipolar type of the composition is represented in terms of the bipolar types of the initial endomorphisms. We suggest an explicit formula for this representation. We also introduce alternating and special alternating semigroups of endomorphisms of a groupoid so that every pair of endomorphisms from an alternating semigroup is alternating. For every groupoid, we prove that the base set of endomorphisms of the first type is a special alternating semigroup with identity (i.e., a monoid). For isomorphic groupoids (G) and (G^{prime } ), we prove that every special alternating semigroup of endomorphisms of (G) is isomorphic to a suitable special alternating semigroup of endomorphisms of (G^{prime } ).
We study the problem on controlling solutions of nonlinear differential equations with unstable equilibrium states. We assume that the operator of the linearized problem is bounded and its spectrum is located in the right half-plane. We prove that there exists a control such that the solution remains in a prescribed neighborhood of an equilibrium state as long as required.
We consider the inverse problem of finding the kernel of the integral term in an integro-differential equation. The problem of finding the memory kernel in the wave process is reduced to a nonlinear Volterra integral equation of the first kind of convolution type, which is in turn reduced under some assumptions to a Volterra integral equation of the second kind. Using the method of contraction mappings, we prove the unique solvability of the problem in the space of continuous functions with weighted norms and obtain an estimate of the conditional stability of the solution.
We describe constructions that are used in the proof of the main result of the first part of the article. They are based on automorphisms and properties of the Cantor space.
The (n )-simplex equation was introduced by Zamolodchikov as a generalization of the Yang–Baxter equation which becomes the (2 )-simplex equation in this terms. In the present article, we suggest general approaches to construction of solutions of the (n )-simplex equation, describe certain types of solutions, and introduce an operation that allows us to construct, under certain conditions, a solution of the ((n + m + k))-simplex equation from solutions of the ((n + k) )-simplex equation and ((m + k) )-simplex equation. We consider the tropicalization of rational solutions and discuss its generalizations. We prove that a solution of the (n )-simplex equation on (G ) can be constructed from solutions of this equation on (H ) and (K ) if (G ) is an extension of a group (H ) by a group (K ). We also find solutions of the parametric Yang–Baxter equation on (H) with parameters in (K ). We introduce ternary algebras for studying the 3-simplex equation and present examples of such algebras that provide us with solutions of the 3-simplex equation. We find all elementary verbal solutions of the 3-simplex equation on a free group. (|| )
We prove a sub-Lorentzian analog of the area formula for intrinsically Lipschitz mappings of open subsets of Carnot groups of arbitrary depth with a sub-Lorentzian structure introduced on the image space.
We consider the problem on optimal quadrature formulas for curvilinear integrals of the first kind that are exact for constant functions. This problem is reduced to the minimization problem for a quadratic form in many variables whose matrix is symmetric and positive definite. We prove that the objective quadratic function attains its minimum at a single point of the corresponding multi-dimensional space. Hence, for a prescribed set of nodes, there exists a unique optimal quadrature formula over a closed smooth contour, i.e., a formula with the least possible norm of the error functional in the conjugate space. We show that the tuple of weights of the optimal quadrature formula is a solution of a special nondegenerate system of linear algebraic equations.
We consider the problem of constructing explicit consistent estimators of finite-dimensional parameters of nonlinear regression models using various nonparametric kernel estimators.
We consider the generating function (Phi ) for the number (f_{Gamma }(n) ) of rooted spanning forests in the circulant graph (Gamma ), where (Phi (x)= sum _{n=1}^{infty } f_{Gamma }(n) x^n) and either (Gamma =C_n(s_1,s_2,dots ,s_k) ) or (Gamma =C_{2n}(s_1,s_2,dots ,s_k,n) ). We show that (Phi ) is a rational function with integer coefficients that satisfies the condition (Phi (x)=-Phi (1/x) ). We illustrate this result by a series of examples.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：