Institute of Mathematics and Mechanics

In this paper we prove some Korovkin and Weierstrass type approximation theorems via statistical convergence. We are also concerned with the order of statistical convergence of a sequence of positive linear operators.

Abstract We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces 𝐿 𝑝(·), λ (·) (Ω) over a bounded open set Ω ⊂ ℝ 𝑛 and a Sobolev type 𝐿 𝑝(·), λ (·) → 𝐿 𝑞(·), λ (·) -theorem for potential operators 𝐼 α (·) , also of variable order. In the case of constant α , the limiting case is also studied when the potential operator 𝐼 α acts into BMO space.

Proceedings of the National Academy of Sciences (PNAS), a peer reviewed journal of the National Academy of Sciences (NAS) - an authoritative source of high-impact, original research that broadly spans the biological, physical, and social sciences.

The problem of boundedness of the fractional maximal operator M α, 0 ≤ α < n, in general local Morrey-type spaces is reduced to the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative non-decreasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

Early-life events influence brain development and evoke long-lasting behavioral consequences. Postweaning social isolation in rodents induces emotional and neurochemical alterations similar to those observed among some human psychopathologies. Central serotonergic neurotransmission is intimately involved in the observed adjustments, but the impact of social deprivation on serotonergic gene expression is unknown. We investigated the effects of prolonged early social isolation on emotion-related behaviors and 5-hydroxytryptamine (5-HT)-related gene transcription in mice. After weaning, male C57BL/6J mice were reared singly or in groups of four for 6 weeks. Gene expression of 5-HT(1A), 5-HT(1B), 5-HT(2A), 5-HT(2C), 5-HT(3A), 5-HT(6) and 5-HT(7) receptors and of 5-HT transporter and tryptophan hydroxylase-2 was determined by quantitative real-time polymerase chain reaction in distinct brain areas. Single-housed mice were hyperactive in a novel environment and showed signs of aggressive behavior. Housing condition did not alter weight gain or body temperature. Isolation markedly reduced transcription of all postsynaptic 5-HT receptors in the prefrontal cortex and reduced 5-HT(1B), 5-HT(2A) and 5-HT(2C) in both hypothalamus and midbrain. In contrast, the only alteration in the hippocampus was 5-HT(6) overexpression. Neither 5-HT transporter nor synthetic enzyme gene transcription differed between housing conditions. In conclusion, early social isolation in mice induces robust changes in postsynaptic 5-HT receptors gene transcription, motor hyperactivity and behavioral disinhibition. The overall pattern of decreased gene expression in the prefrontal cortex highlights its high vulnerability to environment. Furthermore, this is the first study to present a general representation of 5-HT-related gene expression in specific brain areas after social isolation and identifies novel candidates that may be critical for underlying molecular mechanisms.

Here, the very high thermal sensing capability of Er3+,Yb3+ doped LaF3 nanoparticles, where Er3+-to-Yb3+ energy transfer is used, is reported. Also Pr3+,Yb3+ doped LaF3 nanoparticles, with Pr3+-to-Yb3+ energy transfer, showed temperature sensing in the same temperature regime, but with lower performance. The investigated Er3+,Yb3+ doped LaF3 nanoparticles show a remarkably high relative sensitivity Sr of up to 6.6092% K-1 (at 15 K) in the near-infrared (NIR) region, in the cryogenic (15-105 K) temperature region opening a whole new thermometric system suitable for advanced applications in the very low temperature ranges. To date reports on NIR cryogenic sensors have been very scarce.

The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can approximate an arbitrary continuous function provided that an unlimited number of neurons in a hidden layer is permitted. In this note, we consider constructive approximation on any finite interval of [Formula: see text] by neural networks with only one neuron in the hidden layer. We construct algorithmically a smooth, sigmoidal, almost monotone activation function [Formula: see text] providing approximation to an arbitrary continuous function within any degree of accuracy. This algorithm is implemented in a computer program, which computes the value of [Formula: see text] at any reasonable point of the real axis.

Inverse problems of recovering the coefficients of Sturm–Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

Loaded partial differential equations are solved numerically. For illustrative purposes, a boundary value problem for a parabolic equation with various point loads is considered. By applying difference approximations, the problems are reduced to systems of algebraic equations of special structure, which are solved using a parametric representation involving solutions of auxiliary linear systems with tridiagonal matrices. Numerical results are presented and analyzed.

We study various direct and inverse spectral problems for the one-dimensional Schrödinger equation with distributional potential and boundary conditions containing the eigenvalue parameter.

The aim of this study is to investigate a new type boundary value problems which consist of the equation -y''(x) + (By)(x) = ?y(x) on two disjoint intervals (-1,0) and (0,1) together with transmission conditions at the point of interaction x = 0 and with eigenparameter dependent boundary conditions, where B is an abstract linear operator, unbounded in general, in the direct sum of Lebesgue spaces L2(-1,0)( L2(0,1). By suggesting an own approaches we introduce modified Hilbert space and linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. We establish such properties as isomorphism and coerciveness with respect to spectral parameter, maximal decreasing of the resolvent operator and discreteness of the spectrum. Further we examine asymptotic behaviour of the eigenvalues.

This paper considers the boundedness of Calderon–
\nZygmund singular integral operators in local and global
\nMorrey-type spaces. We formulate sufficient conditions
\nfor the boundedness of Calderon-Zygmund singular
\nintegral operators for all admissible parameter values.
\nIn the case of local Morrey-type spaces for a certain range of parameters, these sufficient conditions are also necessary for genuine Calderon–Zygmund singular
\nintegral operators.

Nematic elastomers dramatically change their shape in response to diverse stimuli including light and heat. In this paper, we provide a systematic framework for the design of complex three dimensional shapes through the actuation of heterogeneously patterned nematic elastomer sheets. These sheets are composed of nonisometric origami building blocks which, when appropriately linked together, can actuate into a diverse array of three dimensional faceted shapes. We demonstrate both theoretically and experimentally that the nonisometric origami building blocks actuate in the predicted manner, and that the integration of multiple building blocks leads to complex, yet predictable and robust, shapes. We then show that this experimentally realized functionality enables a rich design landscape for actuation using nematic elastomers. We highlight this landscape through examples, which utilize large arrays of these building blocks to realize a desired three dimensional origami shape. In combination, these results amount to an engineering design principle, which provides a template for the programming of arbitrarily complex three dimensional shapes on demand.

Contact-free manipulation of small objects (e.g., cells, tissues, and droplets) using acoustic waves eliminates physical contact with structures and undesired surface adsorption. Pioneering acoustic-based, contact-free manipulation techniques (e.g., acoustic levitation) enable programmable manipulation but are limited by evaporation, bulky transducers, and inefficient acoustic coupling in air. Herein, we report an acoustofluidic mechanism for the contactless manipulation of small objects on water. A hollow-square-shaped interdigital transducer (IDT) is fabricated on lithium niobate (LiNbO3), immersed in water and used as a sound source to generate acoustic waves and as a micropump to pump fluid in the ±x and ±y orthogonal directions. As a result, objects which float adjacent to the excited IDT can be pushed unidirectionally (horizontally) in ±x and ±y following the directed acoustic wave propagation. A fluidic processor was developed by patterning IDT units in a 6-by-6 array. We demonstrate contactless, programmable manipulation on water of oil droplets and zebrafish larvae. This acoustofluidic-based manipulation opens avenues for the contactless, programmable processing of materials and small biosamples.

Thin film strain sensors composed of GNWs grown by MPCVD, showing ultrahigh sensitivity which can be applied for acoustic signature recognition, as well as electronic skin devices to detect both subtle and large motions of the human body.

Nonlinear partial differential equations emerge in an extensive variants of physical problems inclusive of fluid dynamics, solid mechanics, plasma physics, quantum field theory as well as mathematics and engineering. It has also been noticed that systems of nonlinear partial differential equations arise in biological and chemical applications. This article presents the analytical investigation of a completely generalized (3 + 1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation which has applications in the fields of engineering and physics. The generalized version of the potential Yu-Toda-Sasa-Fukuyama equation is more comprehensively studied in this paper compared to other research work previously done on the equation, with various new solutions of interests achieved. The theory of Lie group is applied to the nonlinear partial differential equation to basically reduce the equation to an integrable form which consequently allows for direct integration of the result. The rigorous process involved in performing a comprehensive reduction of the model under consideration using its Lie algebra makes it possible to achieve various nontrivial solutions. Besides, more general solutions are found via a well-known standard technique. In consequence, we secured diverse solitons and solutions of great interest including topological kink solitons, singular solitons, algebraic functions, exponential function, rational function, Weierstrass function, Jacobi elliptic function as well as series solutions of the underlying equation. Moreover, the completeness of the result was ascertained by presenting the solutions graphically. In addition, discussions of the pictorial representations of the results are done. Conclusively, we constructed conserved quantities of the underlying equation via both the variational and non-variational approaches using the classical Noether’s theorem as well as the standard multiplier technique respectively. In addition, some pertinent observations made from the secured results via both techniques are analyzed.

In this study, we consider a new type boundary value problem consisting of a Sturm-Liouville equation on two disjoint intervals together with interaction conditions and with eigenvalue parameter in the boundary conditions. We suggest a special technique to reduce the considered problem into an integral equation by the use of which we define a new concept, the so-called weak eigenfunction for the considered problem. Then we construct some Hilbert spaces and define some self-adjoint compact operators in these spaces in such a way that the considered problem can be interpreted as a self-adjoint operator-pencil equation. Finally, it is shown that the spectrum is discrete and the set of weak eigenfunctions form a Riesz basis of the suitable Hilbert space.

In this paper, we discuss few existence result for solution of an infinite system of fractional differential equations of order ?(1 < ? < 2), with three point boundary value problem in the interval [0, T]. The problem is studied in the classical Banach sequence spaces c0 and lp (1 ? p < 1), using Hausdorff measure of noncompactness and Darbo type fixed point theorem. We also illustrate our results through some concrete examples.

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This work considers the Riemann boundary value problem with the piecewise continuous coefficient in Morrey-Hardy classes. Under some conditions on the coefficient, the Fredholmness of this problem is studied and the general solution of homogeneous and nonhomogeneous problems in Morrey-Hardy classes is constructed.