Mathematical Notes最新文献

Abstract

In 2014, S. R. Nasyrov asked whether it is true that simple partial fractions (logarithmic derivatives of complex polynomials) with poles on the unit circle are dense in the complex space (L_2[-1,1]). In 2019, M. A. Komarov answered this question in the negative. The present paper contains a simple solution of Nasyrov’s problem different from Komarov’s one. Results related to the following generalizing questions are obtained: (a) of the density of simple partial fractions with poles on the unit circle in weighted Lebesgue spaces on ([-1,1]); (b) of the density in (L_2[-1,1]) of simple partial fractions with poles on the boundary of a given domain for which ([-1,1]) is an inner chord.

Abstract

Based on fixed point theory for condensing operators, an initial value problem for semilinear differential inclusions of fractional order (qin(1,2)) in Banach spaces is studied. It is assumed that the linear part of the inclusion generates a family of cosine operator functions and the nonlinear part is a multivalued map with nonconvex values. Local and global existence theorems for mild solutions of the initial value problem are proved.

Abstract

In the note, by a model example of a linear partial differential equation, it is demonstrated how the properties of continuation of germs of generalized solutions are changed depending on the type of differential system generated by the principal real-analytic symbol of the equation and on whether the infinitely differentiable coefficient at the lowest term of the equation belongs to the class of real-analytic functions.

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 complete the solution of the problem on the existence of generating triplets of involutions two of which commute for the special linear group (mathrm{SL}_n(mathbb{Z}+imathbb{Z})) and the projective special linear group (mathrm{PSL}_n(mathbb{Z}+imathbb{Z})) over the ring of Gaussian integers. The answer has only been unknown for (mathrm{SL}_5), (mathrm{PSL}_6), and (mathrm{SL}_{10}). We explicitly indicate the generating triples of involutions in these three cases, and we make a significant use of computer calculations in the proof. Taking into account the known results for the problem under consideration, as a consequence, we obtain the following two statements. The group (mathrm{SL}_n(mathbb{Z}+imathbb{Z})) (respectively, (mathrm{PSL}_n(mathbb{Z}+imathbb{Z}))) is generated by three involutions two of which commute if and only if (ngeq 5) and (nneq 6) (respectively, if (ngeq 5)).

Abstract

In this paper, the problem of (L^1)-convergence of Fourier series with quasi-monotone coefficients is handled by using the ((bar{N},p_n))-mean. Also, an example is given about the Fourier series of a signal (function) (f) and its ((bar{N},p_n)) mean.

Abstract

The stationary Kolmogorov equation with partially degenerate diffusion matrix and discontinuous drift coefficient is studied. Sufficient conditions for the existence of a probability solution are obtained. Examples demonstrating the sharpness of these conditions are given.

Abstract

We study the problem of the existence of a convex extension of any Boolean function (f(x_1,x_2,dots,x_n)) to the set ([0,1]^n). A convex extension (f_C(x_1,x_2,dots,x_n)) of an arbitrary Boolean function (f(x_1,x_2,dots,x_n)) to the set ([0,1]^n) is constructed. On the basis of the constructed convex extension (f_C(x_1,x_2,dots,x_n)), it is proved that any Boolean function (f(x_1,x_2,dots,x_n)) has infinitely many convex extensions to ([0,1]^n). Moreover, it is proved constructively that, for any Boolean function (f(x_1,x_2,dots,x_n)), there exists a unique function (f_{DM}(x_1,x_2,dots,x_n)) being its maximal convex extensions to ([0,1]^n).

Abstract

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.

Abstract

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.