Brief Survey Of Recent Results Research Articles

Connections between numerical integration, discrepancy, dispersion, and universal discretization

The main goal of this paper is to provide a brief survey of recent results, which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to the discrepancy theory. We discuss this connection in detail and provide a general view of this connection. It was established recently that the new concept of fixed volume discrepancy is very useful in proving the upper bounds for the dispersion. Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.

Word maps on perfect algebraic groups

We extend Borel’s theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel’s theorem to some words with constants. We also consider the surjectivity problem for particular words and groups, give a brief survey of recent results, present some generalizations and variations and discuss various approaches, with emphasis on new ideas, constructions and connections.

DNA origami: The bridge from bottom to top

Abstract

Renormalized Mori-Zwanzig-reduced models for systems without scale separation.

Model reduction for complex systems is a rather active area of research. For many real-world systems, constructing an accurate reduced model is prohibitively expensive. The main difficulty stems from the tremendous range of spatial and temporal scales present in the solution of such systems. This leads to the need to develop reduced models where, inevitably, the resolved variables do not exhibit (spatial and/or temporal) scale separation from the unresolved ones. We present a brief survey of recent results on the construction of Mori-Zwanzig-reduced models for such systems. The construction is inspired by the concepts of scale dependence and renormalization which first appeared in the context of high-energy and statistical physics.

A note on quasi-weakly almost periodic point

Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597–606 (2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493–502 (2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.

Linear preservers and quantum information science

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn × mn Hermitian matrices such that φ(A ⊗ B) and A ⊗ B have the same spectrum for any m × m Hermitian A and n × n Hermitian B. Such a map has the form A ⊗ B ↦ U(ϕ1(A) ⊗ ϕ2(B))U* for mn × mn Hermitian matrices in tensor form A ⊗ B, where U is a unitary matrix, and for j ∈ {1, 2}, ϕ j is the identity map X ↦ X or the transposition map X ↦ X t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A ⊗ B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.

An equivalent condition for the self similar sets on the real line to have best coverings

In this paper, an equivalent condition for the self similar sets on the real line to have best coverings is given. As a result, it partly gives answer to the conjecture which was posed by Zhou and Feng [Zhou, Z. L., Feng, L.: Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17(2), 493–502 (2004)].

Word equations in simple groups and polynomial equations in simple algebras

We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving new ways for solving older ones.

Upper convex density and similarly contractive fixed points

In this work, for the self similar sets satisfying the open set condition, we obtain an interesting character that the upper convex density at every similarly contractive fixed point equalling 1 implies that the upper convex densities at all the points are equal to 1. By using this result, we sufficiently answer the open problem 6 posed by Z. Zhou and L. Feng in [Twelve open problems on the exact value of the Hausdor measure and on topological entropy: A brief survey of recent results, Nonlinearity, 17(2004) 493–502.].

La géométrie de Bakry-Émery et l’écart fondamental

This is a brief survey of recent results culminating in the proof of the fundamental gap conjecture by Andrews and Clutterbuck [1]. Recalling the Bakry-Émery geometry and Laplacian, we present our joint results with Z. Lu [14] which demonstrate an intimate connection between the first non-trivial eigenvalue of a certain Bakry-Émery Laplacian and the fundamental gap. This is a special case of our more general results relating Dirichlet and Neumann eigenvalues and Bakry-Émery eigenvalues. Ideas particularly germane to the recent proof of the fundamental gap conjecture are discussed. In conclusion, we present recent results for the fundamental gap on the moduli spaces of n-simplices in general and triangles in particular.

Linear Groups of Finite Morley Rank

AbstractThis paper is a brief survey of recent results and some open problems related to linear groups of finite Morley rank, an area of research where Bruno Poizat's impact is very prominent. As a sign of respect to his strongly expressed views that mathematics has to be done, written and pulished only in the native tongue of the immediate author–the scribe, in effect–of the text, I insist on writing my paper in Russian, even if the results presented belong to a small but multilingual community of researchers of American, British, French, German, Kazakh, Russian, Turkish origin. To emphasise even further the linguistic subtleties involved, I use British spelling in the English fragments of my text.

Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: a brief survey of recent results**Project partially supported by the National Natural Science Foundation of China, the Foundation of Guangdong Province and the Foundation of Zhongshan University Advanced Research Center.

Twelve open problems and one conjecture are posed in this exposition. Among them, eight open problems and the conjecture are on calculation of the exact value of the Hausdorff measure of self-similar sets, and four open problems are on topological entropy. We give a brief survey of the recent research results related to these open problems and conjecture.

Statistics and causal inference: A review

This paper aims at assisting empirical researchers benefit from recent advances in causal inference. The paper stresses the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underly all causal inferences, the languages used in formulating those assumptions, and the conditional nature of causal claims inferred from nonexperimental studies. These emphases are illustrated through a brief survey of recent results, including the control of confounding, the assessment of causal effects, the interpretation of counterfactuals, and a symbiosis between counterfactual and graphical methods of analysis.

Causal Inference in the Health Sciences: A Conceptual Introduction

This paper provides a conceptual introduction to causal inference, aimed to assist health services researchers benefit from recent advances in this area. The paper stresses the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underlie all causal inferences, the languages used in formulating those assumptions, and the conditional nature of causal claims inferred from nonexperimental studies. These emphases are illustrated through a brief survey of recent results, including the control of confounding, corrections for noncompliance, and a symbiosis between counterfactual and graphical methods of analysis.

Relativizations of the P =? NP and Other Problems: Developments in Structural Complexity Theory

The notion of NP-completeness has helped researchers in several fields to argue that some problems are intrinsically difficult to solve computationally, even though proof of that is beyond the current state of our knowledge. The study of properties of complete sets and reducibilities has led to the development of a branch of complexity theory called structural complexity theory. This paper is a brief survey of recent results in structural complexity theory and is aimed at researchers who know something about computational complexity theory but do not work in structural complexity theory. To describe the theory in a way that gives the reader some concrete applications, the results are described in terms of their relationship to P =?NP and other problems.

Estimation of the drift for diffusion process

A brief survey of recent results in the area of parametric and nonparametric estimation of the drift coefficient of a diffusion process are presented. Some open problems are stated

Distributional and entire solutions of ordinary differential and functional differential equations

A brief survey of recent results on distributional and entire solutions of ordinary differential equations (ODE) and functional differential equations (FDE) is given. Emphasis is made on linear equations with polynomial coefficients. Some work on generalized‐function solutions of integral equations is also mentioned.

Bounded index, entire solutions of ordinary differential equations and summability methods

A brief survey of recent results on functions of bounded index and bounded index summability methods is given. Theorems on entire solutions of ordinary differential equations with polynomial coefficients are included.

Thermodynamics of metal/hydrogen systems

AbstractThe thermodynamics of metal/hydrogen systems are reviewed. Emphasis is placed on the thermodynamic properties of hydrogen dissolved in single solid phase regions. Methods of investigation are discussed. A brief survey of recent results on binary metal alloy/hydrogen systems is presented.

Variation of non-reductive geometric invariant theory

The wall-and-chamber structure of the dependence of the reductive GIT quotient on the choice of linearisation is well known. In this article, we first give a brief survey of recent results in non-reductive GIT, which apply when the unipotent radical is 'graded'. We then examine the dependence of these non-reductive quotients on the linearisation and an additional parameter, the choice of one-parameter subgroup grading the unipotent radical, and arrive at a picture similar to the reductive one.