On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

For a trigonometric series

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Critical Nonlinear Nonlocal Equations on a Half-Line

We study nonlinear nonlocal equations on a half-line in the critical case

Relaxation of an area-like functional for the function $$\frac{x}{|x|}$$

We compute the relaxation

Inequalities for the Volume of the Unit Ball in \mathbb {R}^{n}, II

We present several sharp inequalities for the volume of the unit ball in ,

Geometric second derivative estimates in Carnot groups and convexity

We prove some new a priori estimates for H ₂-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H ₂-convex function in vanishing on ∂Ω one has

The Fourier Formula for Discontinuous Functions of Several Variables

An analog of the classical Fourier formula for the characteristic function of a convex compact set is considered:

Jackson＇s Theorem on Bounded Symmetric Domains

Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson＇s theorem to several complex variables：Eκ（f,X）≤ω（1/k,f,X）, where Eκ（f,X） is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω（1/k,f,X） is the corresponding modulus of continuity.     

Nonradial solutions for a conformally invariant fourth order equation in \mathbb {R}^4

We consider the following Liouville equation in

Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the Dirichlet energy functional

Ratios of Norms for Polynomials and Connected n-width Problems

Let be a bounded simply connected domain with boundary Γ and let be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants:

A short proof for the sum formula and its generalization

For positive integers with a _r  = 2, the multiple zeta value or r-fold Euler sum is defined as [2]

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

ON FRT– CLIFFORD ALGEBRAS

We study the q-Clifford algebras

Nikolskii type inequality for cardinal splines

The Nikolskii type inequality for cardinal splines

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

Existence theory of abstract approximate deconvolution models of turbulence

This report studies an abstract approach to modeling the motion of large eddies in a turbulent flow. If the Navier-Stokes equations (NSE) are averaged with a local, spatial convolution type filter, , the resulting system is not closed due to the filtered nonlinear term . An approximate deconvolution operator D is a bounded linear operator which is an approximate filter inverse

Boundedness of Marcinkiewicz integral related to multilinear commutators with variable kernels on Hardy spaces

Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).     

A new algebraic approach to the Szegő-Widom limit theorem

We give another proof of the Szeg\H{o}–Widom Limit Theorem. This proof relies on a new Banach algebra method that can be directly applied to the asymptotic computation of the Toeplitz determinants. As a by-product, we establish an interesting identity for operator determinants of Toeplitz operators, namely if are certain matrix valued functions defined on the unit circle, then