Abstract
We consider special distance graphs and estimate the number of edges in their subgraphs. The estimates obtained improve some known results.
We consider special distance graphs and estimate the number of edges in their subgraphs. The estimates obtained improve some known results.
We consider estimates for the Fourier transform of measures concentrated on smooth surfaces (Ssubset mathbb{R}^3) given by the graph of a smooth function with simple Arnold singularities such that both principal curvatures of the surface vanish at some point. We prove that if the multiplicity of the critical point of the function does not exceed (7), then the Fourier transforms of the corresponding surface measures belong to (L^{p}(mathbb{ R}^3)) for any (p>3). Note that for any smooth surface the Fourier transform of a nontrivial surface measure with compact support does not belong to (L^3(mathbb{R}^3)); i.e., the (L^p(mathbb{R}^3))-estimate obtained is sharp. Moreover, there exists a function with an (E_8) singularity (the multiplicity of the critical point of the function is equal to (8)) such that the Fourier transform of the corresponding surface measure does not belong to (L^{22/7}(mathbb{R}^3)), which shows the sharpness of the results for the multiplicity of the critical point.
A prime (p) is a Sophie Germain prime if (2p+1) is prime as well. An integer (a) that is coprime to a positive integer (n>1) is a primitive root of (n) if the order of (a) modulo (n) is (phi(n).) Ramesh and Makeshwari proved that, if (p) is a prime primitive root of (2p+1), then (p) is a Sophie Germain prime. Since there exist primes (p) that are primitive roots of (2p+1), in this note we consider the following general problem: For what primes (p) and positive integers (k>1), is (p) a primitive root of (2^{k}p+1)? We prove that it is possible only if ((p,k)in {(2,2), (3,3), (3,4), (5,4)}.)
This article continues the study of the Hadamard–Bergman operators in the unit disk of the complex plane. These operators arose as a natural generalization of orthogonal projections and represent an integral realization of multiplier operators. However, the study of operators in integral form offers a number of advantages in the context of the application of the theory of integral operators as well as in the study of certain function spaces such as holomorphic Hölder functions to which the multiplier theory does not apply. As a main result, we prove boundedness theorems for the Hadamard–Bergman operators and variable Hadamard–Bergman operators using the technique of operators with homogeneous kernels earlier developed in real analysis.
The main objective of this work is to show the existence of solutions for quasilinear elliptic boundary value problem. In addition, we study compactness, directness of the solution set along with existence of smallest and biggest solutions in the set. The presence of dependence on the gradient and the Leray–Lions operator are the main novelties. We have used sub-supersolution technique in our work.
The survey is devoted to the complex WKB method which arose as an approach to describing the asymptotic behavior of solutions to one-dimensional ordinary differential equations with semiclassical parameter on the complex plane. Later this method was generalized to the case of difference equations. Related constructions arose when studying exponentially small effects in the problem concerning the adiabatic perturbation of the one-dimensional periodic Schrödinger operator. All these three branches of the method are discussed in the survey from a unified position. The main constructions of the method are described and the proofs are either provided or their ideas are described in detail. Some new finds are published for the first time.
In the paper, our main aim is to generalize the mixed radial Blaschke–Minkowski homomorphisms and Aleksandrov–Fenchel inequality for mixed radial Blaschke–Minkowski homomorphisms to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating Orlicz first order variation of dual quermassintegrals of the mixed radial Blaschke–Minkowski homomorphisms and call it Orlicz multiple radial Blaschke–Minkowski homomorphisms. The fundamental notions and conclusions of dual quermassintegrals of mixed radial Blaschke–Minkowski homomorphisms and Aleksandrov–Fenchel inequality for mixed radial Blaschke–Minkowski homomorphisms are extended to an Orlicz setting. The related concepts and inequalities of Orlicz mixed intersection bodies are also derived. The new Orlicz–Aleksandrov–Fenchel inequality for dual quermassintegrals of Orlicz multiple radial Blaschke–Minkowski homomorphisms in special case yield not only new (L_p) type Aleksandrov–Fenchel inequality and Orlicz–Minkowski inequality but also Orlicz–Aleksandrov–Fenchel inequality for Orlicz mixed intersection bodies.
In this note it is established that a finite family of positive linear operators acting from an Archimedean vector lattice into an Archimedean (f)-algebra with unit is disjointness preserving if and only if the polynomial presented in the form of the product of powers of these operators is orthogonally additive. A similar statement is established for the sum of polynomials represented as products of powers of positive operators.
By applying the “coefficient extraction method” to hypergeometric series, we establish several remarkable infinite series identities about harmonic numbers and five binomial coefficients, including three conjectured by Z.-W. Sun.
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