Abstract
Vanishing theorems for the kernels of Lichnerowicz and Hodge Laplacians on a complete Riemannian manifold are proved, and the eigenvalues of a Lichnerowicz Laplacian on a closed Riemannian manifold are estimated.
Vanishing theorems for the kernels of Lichnerowicz and Hodge Laplacians on a complete Riemannian manifold are proved, and the eigenvalues of a Lichnerowicz Laplacian on a closed Riemannian manifold are estimated.
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.
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.
We prove that any strictly convex and closed set in (mathbb{R}^n) is an affine subspace if it contains a hyperplane as a subset. In other words, no hyperplane fits into a strictly convex and closed set (C) unless (C) is flat. We also present certain applications of this result in economic theory reminiscent of the separating and supporting hyperplane theorems.
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)).
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.
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.
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.
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).
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.
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