Hyperbolic operator semigroups and Lyapunov’s equation

We obtain necessary and sufficient conditions for the hyperbolicity of a semigroup of operators. In so doing, we use Lyapunov’s equation in operator form constructed from its generator.     

Semicontinuity of Majorants and Minorants of Lyapunov’s Exponents as Functions of Complex Parameter

A parametric family of linear differential systems with continuous coefficients bounded on the semi-axis and analytically dependent on a complex parameter is considered. It is established that the majorant (minorant) of the Lyapunov exponent considered as a function of the parameter is upper (lower) semicontinuous.     

Lyapunov’s Convexity Theorem,zonoids, and bang-bang

This is a short overview of the connections of the Lyapunov Convexity Theorem with the modern sections of analysis, geometry, and optimal control.     

Green’s relations and regularity for semigroups of transformations that preserve reverse direction equivalence

Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let

$T_{\exists}(X)=\{\alpha\in T_X:\forall x,y\in X,(x\alpha,y\alpha)\in E\Rightarrow(x,y)\in E\}.$

Then T _?(X) is exactly the semigroup of mappings on the topological space X for which the collection of all E-classes is a basis. In this paper, we discuss regularity of elements and Green’s relations for T _?(X).

     

Itô’s calculus under sublinear expectations via regularity of PDEs and rough paths

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the $G$-expectation theory.     

Green’s relations and regularity for semigroups of transformations that preserve double direction equivalence

Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let $T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.$ Then $T_{E^{*}}(X)Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let

TE*(X)={a ? TX:"x,y ? X,(x,y) ? E?(xa,ya) ? E}.T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.      8. Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence      Lun-Zhi Deng  Ji-Wen Zeng  Tai-Jie You 《Semigroup Forum》2012,84(1):59-68 Let T X be the full transformation semigroup on a set X, TE*(X)={a ? TX:"x,y ? X, (x,y) ? E? (xa,ya) ? E}T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}      9. On regularity of sup-preserving maps: generalizing Zareckiĭ’s theorem      Ulrich Höhle  Tomasz Kubiak 《Semigroup Forum》2011,83(2):313-319 A sup-preserving map f between complete lattices L and M is regular if there exists a sup-preserving map g from M to L such that fgf=f. In the class of completely distributive lattices, this paper demonstrates a necessary and sufficient condition for f to be regular. When L=M is a power set, our theorem reduces to the well known Zareckiĭ’s theorem which characterizes regular elements in the semigroup of all binary relations on a set. Another application of our result is a generalization of Zareckiĭ’s theorem for quantale-valued relations.      10. On generalized Stokes’ and Brinkman’s equations with a pressure- and shear-dependent viscosity and drag coefficient      《Nonlinear Analysis: Real World Applications》2015 We study generalizations of the Darcy, Forchheimer, Brinkman and Stokes problem in which the viscosity and the drag coefficient depend on the shear rate and the pressure. We focus on existence of weak solutions to the problem, with the chief aim to capture as wide a group of viscosities and drag coefficients as mathematically feasible and to provide a theory that holds under minimal, not very restrictive conditions. Even in the case of generalized Stokes system, the established result answers a question on existence of weak solutions that has been open so far.      11. Estimation of intra-cluster correlation coefficient via the failure of Bartlett’s second identity      Tsung-Shan Tsou  Wan-Chen Chen 《Computational Statistics》2013,28(4):1681-1698 A new means of estimating the correlation coefficient for cluster binary data in the regression settings is introduced. The creation of this method is founded upon the violation of Bartlett’s second identity when adopting the binomial distributions to model binary data that are correlated. The new methodology applies to any sensible link functions that connect the success probability and covariates. One can easily implement the procedure by using any statistical software providing the naïve and the sandwich covariance matrices for regression parameter estimates. Simulations and real data analyses are used to demonstrate the efficacy of our new procedure.      12. On regularity for de Rham’s functional equations      Kazuki Okamura 《Aequationes Mathematicae》2016,90(6):1071-1085 We consider regularity for solutions of a class of de Rham’s functional equations. Under some smoothness conditions of functions making up the equation, we improve some results in Hata (Japan J Appl Math 2:381–414, 1985). Our results are applicable to some cases when the functions making up the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply the singularity of some well-known singular functions, in particular, Minkowski’s question-mark function, and, some small perturbed functions of the singular functions.      13. On solving the problems of stability by Lyapunov’s direct method      B. S. Kalitine 《Russian Mathematics (Iz VUZ)》2017,61(6):27-36 We consider problems of stability and instability of the trivial solution to nonautonomous systems of differential equations. We suggest new theorems of Lyapunov’s direct method with the use of semi-definite auxiliary functions. The idea is based on the use of the additional function that evaluates the rate of convergence of the solutions to the set, where Lyapunov’s function vanishes. We formulate theorems on the non-asymptotic stability and instability. The results are illustrated by examples, where we give a comparison with known results.      14. Study of Dieckmann’s equation with integral kernels having variable kurtosis coefficient      A. V. Kalistratova  A. A. Nikitin 《Doklady Mathematics》2016,94(2):574-577 An integral equation arising in a spatial model of stationary communities developed by the Austrian scientists Ulf Dieckmann and Richard Low is studied. A special case of this equation with integral kernels being Student distributions is considered. The existence of a solution in this case is proved, and the application of a method for reducing the dimension of a multidimensional integral equation previously proposed by the authors is described.      15. Multivariate versions of Blomqvist’s beta and Spearman’s footrule      Manuel Úbeda-Flores 《Annals of the Institute of Statistical Mathematics》2005,57(4):781-788 In this paper we define multivariate versions of the medial correlation coefficient and the rank correlation coefficient Spearman’s footrule in terms of copulas. We also present corresponding results for the sample statistic and provide a comparison of lower bounds among different measures of multivariate association.      16. Bykovskii’s theorem and a generalization of Larcher’s theorem      D. M. Ushanov 《Mathematical Notes》2012,91(5-6):746-750       17. Generalizations of Sherman’s Inequality Via Fink’s Identity and Green’s Function      Ivelić Bradanović  S.  Latif  N.  Pečarić  J. 《Ukrainian Mathematical Journal》2019,70(8):1192-1204 Ukrainian Mathematical Journal - New generalizations of Sherman’s inequality for n-convex functions are obtained with the help of Fink’s identity and Green’s function. By using...      18. Maluta’s coefficient and Opial’s properties in Musielak–Orlicz sequence spaces equipped with the Luxemburg norm      《Nonlinear Analysis: Theory, Methods & Applications》1999,35(4):475-485       19. On A.I. Koshelev’s work on regularity theory of generalized solutions of quasilinear systems of elliptic and parabolic type      M. A. Narbut 《Differential Equations》2011,47(13):1916-1928 This is a survey of A.I. Koshelev’s studies in the theory of regular solutions of boundary value problems, based on iterative processes converging in both the energy norm and the strong norm as well as on a priori estimates in weighted function spaces. In numerous cases, Koshelev’s estimates contain explicitly computable and sometimes sharp (unimprovable) constants. The results obtained for a broad class of problems were adapted by Koshelev to the study of boundary value problems of nonlinear elasticity and problems of hydrodynamics of viscous fluids.      20. Order-preserving transformations with restricted range: regularity,Green’s relations,and ideals      Worachead Sommanee  Jintana Sanwong 《Algebra Universalis》2015,74(3-4):277-291

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Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence

Let T _X be the full transformation semigroup on a set X,

On regularity of sup-preserving maps: generalizing Zareckiĭ’s theorem

A sup-preserving map f between complete lattices L and M is regular if there exists a sup-preserving map g from M to L such that fgf=f. In the class of completely distributive lattices, this paper demonstrates a necessary and sufficient condition for f to be regular. When L=M is a power set, our theorem reduces to the well known Zareckiĭ’s theorem which characterizes regular elements in the semigroup of all binary relations on a set. Another application of our result is a generalization of Zareckiĭ’s theorem for quantale-valued relations.     

On generalized Stokes’ and Brinkman’s equations with a pressure- and shear-dependent viscosity and drag coefficient

We study generalizations of the Darcy, Forchheimer, Brinkman and Stokes problem in which the viscosity and the drag coefficient depend on the shear rate and the pressure. We focus on existence of weak solutions to the problem, with the chief aim to capture as wide a group of viscosities and drag coefficients as mathematically feasible and to provide a theory that holds under minimal, not very restrictive conditions. Even in the case of generalized Stokes system, the established result answers a question on existence of weak solutions that has been open so far.     

Estimation of intra-cluster correlation coefficient via the failure of Bartlett’s second identity

A new means of estimating the correlation coefficient for cluster binary data in the regression settings is introduced. The creation of this method is founded upon the violation of Bartlett’s second identity when adopting the binomial distributions to model binary data that are correlated. The new methodology applies to any sensible link functions that connect the success probability and covariates. One can easily implement the procedure by using any statistical software providing the naïve and the sandwich covariance matrices for regression parameter estimates. Simulations and real data analyses are used to demonstrate the efficacy of our new procedure.     

On regularity for de Rham’s functional equations

We consider regularity for solutions of a class of de Rham’s functional equations. Under some smoothness conditions of functions making up the equation, we improve some results in Hata (Japan J Appl Math 2:381–414, 1985). Our results are applicable to some cases when the functions making up the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply the singularity of some well-known singular functions, in particular, Minkowski’s question-mark function, and, some small perturbed functions of the singular functions.     

On solving the problems of stability by Lyapunov’s direct method

We consider problems of stability and instability of the trivial solution to nonautonomous systems of differential equations. We suggest new theorems of Lyapunov’s direct method with the use of semi-definite auxiliary functions. The idea is based on the use of the additional function that evaluates the rate of convergence of the solutions to the set, where Lyapunov’s function vanishes. We formulate theorems on the non-asymptotic stability and instability. The results are illustrated by examples, where we give a comparison with known results.     

Study of Dieckmann’s equation with integral kernels having variable kurtosis coefficient

An integral equation arising in a spatial model of stationary communities developed by the Austrian scientists Ulf Dieckmann and Richard Low is studied. A special case of this equation with integral kernels being Student distributions is considered. The existence of a solution in this case is proved, and the application of a method for reducing the dimension of a multidimensional integral equation previously proposed by the authors is described.     

Multivariate versions of Blomqvist’s beta and Spearman’s footrule

In this paper we define multivariate versions of the medial correlation coefficient and the rank correlation coefficient Spearman’s footrule in terms of copulas. We also present corresponding results for the sample statistic and provide a comparison of lower bounds among different measures of multivariate association.     

Generalizations of Sherman’s Inequality Via Fink’s Identity and Green’s Function

Ukrainian Mathematical Journal - New generalizations of Sherman’s inequality for n-convex functions are obtained with the help of Fink’s identity and Green’s function. By using...     

On A.I. Koshelev’s work on regularity theory of generalized solutions of quasilinear systems of elliptic and parabolic type

This is a survey of A.I. Koshelev’s studies in the theory of regular solutions of boundary value problems, based on iterative processes converging in both the energy norm and the strong norm as well as on a priori estimates in weighted function spaces. In numerous cases, Koshelev’s estimates contain explicitly computable and sometimes sharp (unimprovable) constants. The results obtained for a broad class of problems were adapted by Koshelev to the study of boundary value problems of nonlinear elasticity and problems of hydrodynamics of viscous fluids.