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In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the L-stability operator and discuss the L-stability of some special self-expanders. Besides, we prove that the surfaces Γ×R with the product metric are the only complete self-expander surfaces immersed in R3 with constant scalar curvature, where Γ is a complete self-expander curve (properly) immersed in R2 .

Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.

We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21 , 143-153 1993 ). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.

We introduce a general framework for measuring risk in the context of Markov control processes with risk maps on general Borel spaces that generalize known concepts of risk measures in mathematical finance, operations research, and behavioral economics. Within the framework, applying weighted norm spaces to incorporate unbounded costs also, we study two types of infinite-horizon risk-sensitive criteria, discounted total risk and average risk, and solve the associated optimization problems by dynamic programming. For the discounted case, we propose a new discount scheme, which is different from the conventional form but consistent with the existing literature, while for the average risk criterion, we state Lyapunov-like stability conditions that generalize known conditions for Markov chains to ensure the existence of solutions to the optimality equation. [PUBLICATION ABSTRACT]

Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere Sn+1\\mathbb {S}^{n+1}. In this paper we prove that the weak index of any other compact constant mean curvature hypersurface MnM^n in Sn+1\\mathbb {S}{n+1} which is not totally umbilical and has constant scalar curvature is greater than or equal to n+2n+2, with equality if and only if MM is a constant mean curvature Clifford torus Sk(r)×Sn−k(1−r2)\\mathbb {S}^{k}(r)\\times \\mathbb {S}^{n-k}(\\sqrt {1-r^2}) with radius k/(n+2)⩽r⩽(k+2)/(n+2)\\sqrt {k/(n+2)}\\leqslant r\\leqslant \\sqrt {(k+2)/(n+2)}.

On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied problems the individual components of the vector of unknowns are coupled together and then splitting schemes are applied in order to get a simple problem for evaluating components at a new time level. Typically, the additive operator-difference schemes for systems of evolutionary equations are constructed for operators coupled in space. In this paper we investigate more general problems where coupling of derivatives in time for components of the solution vector takes place. Splitting schemes are developed using an additive representation for both the primary operator of the problem and the operator at the time derivative. Splitting schemes are based on a triangular two-component representation of the operators.

AbstractSchemes of the Samarskii alternating triangular method are based on splitting the problem operator into two operators that are conjugate to each other. When the Cauchy problem for a first-order evolution equation is solved approximately, this makes it possible to construct unconditionally stable two-component factorized splitting schemes. Explicit schemes are constructed for parabolic problems based on the alternating triangular method. The approximation properties can be improved by using three-level schemes. The main possibilities are indicated for constructing alternating triangular schemes for second-order evolution equations. New schemes are constructed based on the regularization of the standard alternating triangular schemes. The features of constructing alternating triangular schemes are pointed out for problems with many operator terms and for second-order evolution equations involving operator terms for the first time derivative. The study is based on the general stability (well-posedness) theory for operator-difference schemes.

Let be an immersion of a complete n -dimensional oriented manifold. For any v ∈ℝ n +2 , let us denote by ℓ v : M →ℝ the function given by ℓ v ( x )=〈 φ ( x ), v 〉 and by f v : M →ℝ, the function given by f v ( x )=〈 ν ( x ), v 〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v ≠0 and some real number λ , we have that ℓ v = λ f v , then, φ ( M ) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M n in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2 n +4.

Let M be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere$\\mathbb{S}^{n+1}$. In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of M in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called H(r)-tori in$\\mathbb{S}^{n+1}$, with$r^{2} \\leq (n - 1)/n$. This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.