In the article, we establish several inequalities for the Ramanujan constant function R ( x ) = − 2 γ − ψ ( x ) − ψ ( 1 − x ) on the interval ( 0 , 1 / 2 ] , where ψ ( x ) is the classical psi function and γ = 0.577215 ⋯ is the Euler-Mascheroni constant. MSC: 33B15, 26D07.

In this paper, we consider a variant of projected Tikhonov regularization method for solving Fredholm integral equations of the first kind. We give a theoretical analysis of this method in the Hilbert space L 2 ( a , b ) setting and establish some convergence rates under certain regularity assumption on the exact solution and the kernel k ( ⋅ , ⋅ ) . Some numerical results are also ...

In this paper we develop Caccioppoli-type estimates for arbitrary convex vectors and the vectors having both convex and concave arguments. To do this, we first develop these estimates for smooth convex vectors and then, through mollification, extend the results for arbitrary convex vectors. These types of estimates are valuable in the problems of financial mathematics for the ...

In this paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence space ℓ p ( r , s , t ; B ( m ) ) which is related to ℓ p spaces. By applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact. Further, we ...

In this paper, we get a criteria of weak Poincaré inequality by some integrability of hitting times for jump processes. In fact, integrability of hitting times on a subset F of state space E implies that the taboo process restricted on E ∖ F is decay, from which we get a weak Poincaré inequality with absorbing (Dirichlet) boundary. Using it and a local Poincaré inequality, we ...

In this paper, we provide a new type of study approach for the two-dimensional (2D) Sobolev equations. We first establish a semi-discrete Crank-Nicolson (CN) formulation with second-order accuracy about time for the 2D Sobolev equations. Then we directly establish a fully discrete CN finite volume element (CNFVE) formulation from the semi-discrete CN formulation about time and ...

In this paper, we study the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. First of all, we give a general inequality for eigenvalues of the eigenvalue problem of elliptic operators in weighted divergence form on compact smooth metric measure space with boundary (possibly empty). Then applying this general inequality, we get ...

In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem that models a turbulent flow in a porous medium. The obtained inequalities are used to obtain a lower bound for the eigenvalues of corresponding equations. MSC: 34A08, 15A42, 26D15, 76F70.

Our purpose is to introduce a two-parametric ( p , q ) -analogue of the Stancu-Beta operators. We study approximating properties of these operators using the Korovkin approximation theorem and also study a direct theorem. We also obtain the Voronovskaya-type estimate for these operators. Furthermore, we study the weighted approximation results and pointwise estimates for these ...

In the article, we present certain p , q ∈ R such that the Wilker-type inequalities 2 q p + 2 q ( sin x x ) p + p p + 2 q ( tan x x ) q > ( < ) 1 and ( π 2 ) p ( sin x x ) p + [ 1 − ( π 2 ) p ] ( tan x x ) q > ( < ) 1 hold for all x ∈ ( 0 , π / 2 ) . MSC: 26D05, 33B10.

Translation, dilation, and modulation are fundamental operations in wavelet analysis. Affine frames based on translation-and-dilation operation and Gabor frames based on translation-and-modulation operation have been extensively studied and seen great achievements. But dilation-and-modulation frames have not. This paper addresses a class of dilation-and-modulation systems in L 2 ( ...

In this paper, we propose a method to smooth the general lower-order exact penalty function for inequality constrained optimization. We prove that an approximation global solution of the original problem can be obtained by searching a global solution of the smoothed penalty problem. We develop an algorithm based on the smoothed penalty function. It is shown that the algorithm is ...

In this paper we derive a rank formula for the self-commutators of tuples of Toeplitz operators with matrix-valued rational symbols. MSC: 47B20, 47B35, 47A13, 30H10, 47A57.

Schuster introduced radial Blaschke-Minkowski homomorphisms. Recently, they were generalized to L p radial Blaschke-Minkowski homomorphisms by Wang et al. In this paper, we first establish Brunn-Minkowski type inequalities for some L q radial sums of L p radial Blaschke-Minkowski homomorphisms. Further, we consider monotonic inequalities for L p radial Blaschke-Minkowski ...

A complex matrix X is called an { i , … , j } -inverse of the complex matrix A, denoted by A ( i , … , j ) , if it satisfies the ith, …, jth equations of the four matrix equations (i) A X A = A , (ii) X A X = X , (iii) ( A X ) ∗ = A X , (iv) ( X A ) ∗ = X A . The eight frequently used generalized inverses of A are A † , A ( 1 , 3 , 4 ) , A ( 1 , 2 , 4 ) , A ( 1 , 2 , 3 ) , A ( 1 , ...

In this article, we first define a kind of generalized singular integral operator and discuss its properties. Then we propose a kind of boundary value problem for an inhomogeneous partial differential system in R 4 . Finally, the integral representation of the solution to a boundary value problem for the inhomogeneous partial differential system is obtained using the above singular ...

In this paper, we study a nonmatching grid finite element approximation of a class of elliptic variational inequalities with nonlinear source terms in the context of the Schwarz alternating domain decomposition. We show that the approximation converges optimally in the maximum norm, on each subdomain, making use of a Lipschitz continuous dependence with respect to both the boundary ...

This paper is concerned with the study of invariant subspace problems for nonlinear operators on Banach spaces/algebras. Our study reveals that one faces unprecedented challenges such as lack of vector space structure and unbounded spectral sets when tackling invariant subspace problems for nonlinear operators via spectral information. To bypass some of these challenges, we ...

We present a new class of nonsingular tensors (p-norm strictly diagonally dominant tensors), which is a subclass of strong H -tensors. As applications of the results, we give a new eigenvalue inclusion set, which is tighter than those provided by Li et al. (Linear Multilinear Algebra 64:727-736, 2016 ) in some case. Based on this set, we give a checkable sufficient condition for ...

An interplay between the sum of certain series related to harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits us to find sharp inequalities bounding these trigonometric sums. In particular, this answers positively an open problem of Chen (Excursions in ...

In the article, we prove that the function r ↦ E ( r ) / S 9 / 2 − p , p ( 1 , r ′ ) is strictly increasing on ( 0 , 1 ) for p ≤ 7 / 4 and strictly decreasing on ( 0 , 1 ) for p ∈ [ 2 , 9 / 4 ] , where r ′ = 1 − r 2 , E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ( t ) d t is the complete elliptic integral of the second kind, and S p , q ( a , b ) = [ q ( a p − b p ) / ( p ( a q − b q ) ) ] 1 ...

The spectral gradient method is one of the most effective methods for solving large-scale systems of nonlinear equations. In this paper, we propose a new trust region spectral method without gradient. The trust region technique is a globalization strategy in our method. The global convergence of the proposed algorithm is proved. The numerical results show that our new method is ...

This paper is concerned with the generalization of the homogeneous approximation property (HAP) for a continuous shearlet transform to higher dimensions. First, we give a pointwise convergence result on the inverse shearlet transform in higher dimensions. Second, we show that every pair of admissible shearlets possess the HAP in the sense of L 2 ( R d ) . Third, we give a ...

We prove new inequalities for general 2 × 2 operator matrices. These inequalities, which are based on classical convexity inequalities, generalize earlier inequalities for sums of operators. Some other related results are also presented. Also, we prove a numerical radius equality for a 5 × 5 tridiagonal operator matrix. MSC: 47A12, 47A30, 47A63, 47B15, 47B36.

Let C N J ( B ) and J ( B ) be the generalized von Neumann-Jordan and James constants of a quasi-Banach space B , respectively. In this paper we shall show the relation between C N J ( B ) , J ( B ) , and the modulus of convexity. Also, we show that if B is not uniform non-square then J ( B ) = C N J ( B ) = 2 . Moreover, we give an equivalent formula for the generalized von ...