For an arbitrary initial compact and convex subset $K_{0}$ of $\mathbb{R}^{n},$ and for an arbitrary norm $\phi$ on $\mathbb{R}^{n}$, we construct a flat $\phi$ curvature flow $K\left(t\right)$ such that $K\left(t\right)$ is compact and convex throughout the evolution. Previously and using similar methods, R. McCann had shown that flat $\phi$ curvature flow in the plane preserves convex, balanced sets. More recently, G. Bellettini, V. Caselles, A. Chambolle, and M. Novaga showed that flat $\phi$ curvature flow in $\mathbb{R}^{n}$ preserves compact, convex sets. We also establish a new Hölder continuity estimate for the flow. Flat $\phi$ curvature flows, introduced by F. Almgren, J. Taylor, and L. Wang, model motion by $\phi$-weighted mean curvature. Under certain regularity assumptions, they coincide with smooth $\phi$-weighted mean curvature flows given by partial differential equations as long as the smooth flows exist.

Let $G/K$ be a Riemannian symmetric space of the complex type, meaning that $G$ is complex semisimple and $K$ is a compact real form. Now let $\Gamma$ be a discrete subgroup of $G$ that acts freely and cocompactly on $G/K$. We consider the Segal-Bargmann transform, defined in terms of the heat equation, on the compact quotient $\Gamma \backslash G/K$. We obtain isometry and inversion formulas precisely parallel to the results we obtained previously for globally symmetric spaces of the complex type. Our results are as parallel as possible to the results one has in the dual compact case. Since there is no known Gutzmer formula in this setting, our proofs make use of double coset integrals and a holomorphic change of variable.

To learn more of the phase distributions in unperturbed human tumour cells is a prerequisite prior to understanding of those in the perturbed cells. The work is important in understanding the efficiency of anti-cancer therapy. In this paper we investigate the existence, uniqueness and growth rate of the solution to a mathematical model of unperturbed human tumour cell line. At first, we construct the solution of this mathematical model by the method of continuation of solution, and then show the solution is unique. Finally, we find that the growth rate of the solution with respect to time is faster than exponential function. The basic mathematical techniques used here are variation of parameters and upper and lower solutions for differential equations. These results allowed one to estimate the cells population in each phase at specific time while one does not have cells mitosis DNA distribution data and it can also be used to compare with the perturbed cell lines.

We estimate the quantum query error of approximation to functions from the anisotropic Sobolev class ${\cal B}(W_p^{\bf r}([0,1]^d))$ and the Hölder-Nikolskii class ${\cal B}(H_p^{\bf r}([0,1]^d))$ in the $L_q([0,1]^d)$ norm for all $1 \leq p,q \leq \infty$. It turns out that for the class ${\cal B}(W_p^{\bf r}([0,1]^d))$ $({\bf r}\in {\mathbb{N}^d})$, when $p \lt q$, the quantum algorithms can essentially improve the rate of convergence of classical deterministic and randomized algorithms; while for the class ${\cal B}(H_p^{\bf r}([0,1]^d))$ and ${\cal B}(W_p^{\bf r}([0,1]^d))$ $({\bf r}\in {\mathbb{R}_+^d})$, when $p \geq q$, the optimal convergence rate is the same for all three settings.

In this paper, the Sadovskii fixed point theorem and the theory of strongly continuous cosine families of operators are used to investigate the controllability of damped second order neutral system with impulses. Examples are provided to show the application of the result.

We study a nonlocal parabolic problem airing in the modeling of linear friction welding. Using some a priori estimates, we derive the global in time existence of solution of this nonlocal problem.

The initial boundary value problem for a viscoelastic equation with linear damping and nonlinear source term in a bounded domain is considered. The decay rate of solution energy is discussed under some conditions on relaxation function $g$ and initial data by adopting the perturbed energy method of [4] and modifying the methods of [11, 17]. Decay estimates of the energy function are also given.

In this paper, we deal with a broad class of nonlinear mappings in a Hilbert space and a Banach space called generalized hybrid which contains the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings. Then, we prove fixed point theorems for these nonlinear mappings in a Hilbert space and a Banach space. Furthermore, we obtain duality theorems for nonlinear mappings in a Banach space.

In this paper, the authors give the boundedness of some multipliers satisfying Michlin condition on Herz-type Hardy spaces on the Heisenberg group.

In this paper, we consider strong form of a vector equilibrium problem and establish some existence results for solutions of such a problem in the setting of topological vector spaces. We provide several coercivity conditions under which strong vector equilibrium problem has a solution. Our results generalize and extend the results of Bianchi and Pini [10] to the topological vector space setting.

Let $A$ be an $n \times n$ Hermitian matrix and $A = U \Lambda U^H$ be its spectral decomposition, where $U$ is a unitary matrix of order $n$ and $\Lambda$ is a diagonal matrix. In this note we present the perturbation bound and condition number of the eigenvector matrix $U$ of $A$ with distinct eigenvalues. A perturbation bound of singular vector matrices is also given for a real $n \times n$ or $(n+1) \times n$ matrix. The results are illustrated by numerical examples.

The main purpose of this paper is to use the mean value theorems of Dirichlet $L$-function to study the distribution properties of the generalized Dedekind sums, and gives two interesting asymptotic formulas.

For an Orlicz function $\varphi$ and a Banach lattice $X$, let $\ell_\varphi$ denote the Orlicz sequence space associated to $\varphi$, ${\mathcal L}^r(\ell_\varphi, X)$ denote the space of regular operators from $\ell_\varphi$ to $X$, and ${\mathcal K}^r(\ell_\varphi, X)$ denote the linear span of positive compact operators from $\ell_\varphi$ to $X$. In this paper, we show that if $\varphi$ and its complementary function $\varphi^\ast$ satisfy the $\Delta_2$-condition, then (a) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $\ell_\infty$; and (b) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $c_0$ if and only if ${\mathcal L}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $c_0$ and each positive linear operator from $\ell_\varphi$ to $X$ is compact.

We present an accurate description of the limiting behavior for the partial sums and the weighted sums of independent and identically distributed random elements with heavy tail. A version of Chover-type law of the iterated logarithm is deduced.

In this paper, Levitin-Polyak well-posedness for generalized quasi-variational inclusion and disclusion problems are introduced and studied. Necessary and sufficient conditions for Levitin-Polyak well-posedness of these problems are proved. Moreover, Levitin-Polyak well-posedness for optimization problems with generalized quasi-variational inclusion problems, generalized quasi-variational disclusion problems and scalar generalized quasi-equilibrium problems as constraints are also given under some suitable conditions.

First we present the local $\Delta_2^E$ condition in generalized Calderón-Lozanovskiĭ spaces $E_\varphi$ and we discuss the relationships between the local and the global $\Delta_2^E$ condition in such spaces. We also give a full characterisation for a point of $B(E_\varphi)$ to have an order continuous norm. Then we apply the main result to particular spaces, i.e. Calderón-Lozanovskiĭ spaces and Orlicz-Lorentz spaces.

In [26, 27, 35], condition numbers and perturbation bounds were produced for the state feedback pole assignment problem (SFPAP), for the single- and multi-input cases with simple closed-loop eigenvalues. In this paper, we consider the same problem in a different approach with weaker assumptions, producing simpler condition numbers and perturbation results. For the SFPAP, we shall show that the absolute condition number $\kappa \leq c_0 \|B^{\dagger}\| $ $ \left[ \kappa_X + \left(1 + \|F\|^2 \right)^{1/2} \right]$, where the closed-loop system matrix $A+BF = X \Lambda X^{-1}$, the closed-loop spectrum in $\Lambda$ is pre-determined, $\kappa_X \equiv\|X\| \|X^{-1}\|$, the operators $P_c (\cdot) \equiv (A+BF)(\cdot) - (\cdot) \Lambda$ and $\mathcal{N}(\cdot) \equiv (I - BB^{\dagger}) P_c(\cdot)$, and $c_0 \equiv \|I(\cdot) -P_c\left[ \mathcal{N}^{\dagger}(I - BB^{\dagger}) (\cdot) \right]\|$. With $c_B \!\equiv\!\|B\| \|B^{\dagger}\|$ and $c_1 \!\equiv\! (\|B\| \|F\|)^{-1}$, the relative condition number $\kappa_r \!\leq c_0 c_B \left[ c_1 \kappa_X \|\Lambda\| + \right. \left. \left( c_1^2 \|A\|^2 + 1 \right)^{1/2} \right]$. With $B$ well-conditioned and $\Lambda$ well chosen, $\kappa$ and $\kappa_r$ can be small even when $\Lambda$ (not necessary in Jordan form) possesses defective eigenvalues, depending on $c_0$. Consequently, the SFPAP is not intrinsically ill-conditioned. Similar results were obtained in [23], although differentiability was not established for its local perturbation analysis. Simple as well as general multiple closed-loop eigenvalues are treated.

In this paper we give criteria of exposed points and strongly exposed points in Musielak-Orlicz sequence spaces endowed with Luxemburg norm.

We study in its homological class the periodic structure of the $\mathcal{C}^{1}$ self$-$maps on the manifolds $\mathbb{S}^{n}$ (the $n-$dimensional sphere), $\mathbb{S}^{n}\times \mathbb{S}^{m}$ (the product space of the $n-$dimensional with the $m-$dimensional spheres), $\mathbb{C}$P$^{n}$ (the $n-$dimensional complex projective space) and $\mathbb{H}$P$^{n}$ (the $n-$dimensional quaternion projective space), having all their periodic orbits hyperbolic.

In this paper, we introduce and study $p$-uniform subsmoothness of a collection of infinitely many closed sets in a Banach space. Using variational analysis and techniques, we mainly study linear regularity for a collection of infinitely many closed sets satisfying $p$-uniform subsmoothness. The necessary or/and sufficient conditions on the linear regularity are obtained in this case. In particular, we extend the characterizations of linear regularity for a collection of infinitely many closed convex sets to the nonconvex setting.

Sufficient conditions for a norm-to-weak$^*$ continuous mapping $f: X \rightarrow X^*$ being monotone or submonotone are established by its Fréchet and normal coderivatives, where $X$ is an Asplund space with its dual space $X^*$. Under some additional assumptions, they are also necessary conditions. Among other things, we obtain a criterion for the monotonicity of continuous mappings which extends the following classical result: a differentiable mapping $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is monotone if and only if for each $x \in \mathbb{R}^n$ the Jacobian matrix $\nabla F(x)$ is positive semi-definite; see [22, Proposition 12.3]. As a by-product, sufficient conditions for a function being convex or approximately convex are given.

We propose an Ulm-like Cayley transform method for solving inverse eigenvalue problems, which avoids solving approximate Jacobian equations comparing with other known methods. A convergence analysis of this method is provided and the R-quadratic convergence property is proved under the assumption of the distinction of the given eigenvalues. Numerical experiments are given in the last section and comparisons with the inexact Cayley transform method [1] are made.

The purpose of the present paper is to introduce and investigate the function classes $k$-${\cal SP}(\alpha,\beta)$ and $k$-${\cal UCV} (\alpha, \beta)$ of analytic functions associated with conic regions in the open unit disk $\mathbb{U}$, which generalize the function classes defined and studied in a series of earlier papers by Kanas et al. [11, 12, 13, 14]. In particular, we consider the extremal problems for each of the above-mentioned function classes. The Fekete-Szegö problem is also considered for functions in the class $k$-${\cal SP}(\alpha,\beta)$. Moreover, we investigate some mapping properties for each of the function classes $k$-${\cal SP}(\alpha, \beta)$ and $k$-${\cal UCV}(\alpha, \beta)$.