The Weighted Basis for PHT-Splines

PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement properties.The original PHT-spline basis functions constructed by the truncation mechanism have a decay phenomenon,resulting in numerical instability.The non-decay basis functions are constructed as the B-splines that are defined on the 2×2 tensor product meshes associated with basis vertices in Kang et al.,but at the cost of losing the partition of unity.In the field of finite element analysis and topology optimization,forming the partition of unity is the default ingredient for constructing basis functions of approximate spaces.In this paper,we will show that the non-decay PHT-spline basis functions proposed by Kang et al.can be appropriately modified to form a partition of unity.Each non-decay basis function is multiplied by a positive weight to form the weighted basis.The weights are solved such that the sum of weighted bases is equal to 1 on the domain.We provide two methods for calculatingweights,based on geometric information of basis functions and the subdivision of PHT-splines.Weights are given in the form of explicit formulas and can be efficiently calculated.We also prove that the weights on the admissible hierarchical T-meshes are positive.

A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures

A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analogy to the solid-shell concept of the finite element method, discretization is carried out by the nodes located on the upper and lower surfaces of the structures. The approximation of all unknown field variables is carried out by using the moving least squares （MLS） approximation scheme in the in-plane directions, while the linear interpolation is applied through the thickness direction. Thus, different boundary conditions are defined only using displacements and penalty method is used to enforce the essential boundary conditions. The constrained Galerkin weak form, which incorporates only dis- placement degrees of freedom （d.o.f.s）, is derived. A modified 3D constitutive relationship is adopted in order to avoid or eliminate some self-locking effects. The numeric efficiency of the proposed meshless formulation is illustrated by the numeric examples.

A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference method and finite element method,the enforcement of boundary conditions in deep neural networks is highly nontrivial.One general strategy is to use the penalty method.In the work,we conduct a comparison study for elliptic problems with four different boundary conditions,i.e.,Dirichlet,Neumann,Robin,and periodic boundary conditions,using two representative methods:deep Galerkin method and deep Ritz method.In the former,the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter.Therefore,it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions.However,by a number of examples,we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides,in some cases,when the boundary condition can be implemented in an exact manner,we find that such a strategy not only provides a better approximate solution but also facilitates the training process.

BOUNDARY ELEMENT APPROXIMATION OF THE SEMI-DISCRETE PARABOLIC VARIATIONAL INEQUALITIES OF THE SECOND KIND

The boundary element approximation of the parabolic variational inequalities of the second kind is discussed. First, the parabolic variational inequalities of the second kind can be reduced to an elliptic variational inequality by using semidiscretization and implicit method in time; then the existence and uniqueness for the solution of nonlinear non-differentiable mixed variational inequality is discussed. Its corresponding mixed boundary variational inequality and the existence and uniqueness of its solution are yielded. This provides the theoretical basis for using boundary element method to solve the mixed variational inequality.

Asymptotic behavior for sums of non-identically distributed random variables

For any given positive integer m, let X_i, 1 ≤ i ≤ m be m independent random variables with distributions F_i, 1 ≤ i ≤ m. When all the summands are nonnegative and at least one of them is heavy-tailed, we prove that the lower limit of the ratio ■equals 1 as x →∞. When the summands are real-valued, we also obtain some asymptotic results for the tail probability of the sums. Besides, a local version as well as a density version of the above results is also presented.

The Dynamical Mechanisms of the Cell Cycle Size Checkpoint

Cell division must be tightly coupled to cell growth in order to maintain cell size,whereas the mechanisms of how initialization of mitosis is regulated by cell size remain to be elucidated.We develop a mathematical model of the cell cycle,which incorporates cell growth to investigate the dynamical properties of the size checkpoint in embryos of Xenopus laevis.We show that the size checkpoint is naturally raised from a saddle-node bifurcation,and in a mutant case,the cell loses its size control ability due to the loss of this saddle-node point.

Gelfand-Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Let g be a classical complex simple Lie algebra and q be a parabolic subalgebra.Let M be a generalized Verma module induced from a one dimensional representation of q.Such M is called a scalar generalized Verma module.In this paper,we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.

Irreducible Representations of GL_(n)(C)of Minimal Gelfand-Kirillov Dimension

In this article,by studying the Bernstein degrees and Goldie rank polynomials,we es-tablish a comparison between the irreducible representations of G=GL_(n)(C)possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type(n-1,1).We give the transition matrix between the two bases for the corresponding coherent families.

VECTORIAL EKELAND'S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS

By using the properties of w-distances and Gerstewitz＇s functions, we first give a vectorial Takahashi＇s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland＇s variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristfs fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland＇s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346（2008）, 9-16] are weakened or even completely relieved.

μ-Separations in generalized topological spaces

This paper takes some investigations on generalized topological spaces with some μ- separations. Some characterizations of μTi-spaces for i = 0, 1, 2, 3, 4, μTn-spaces and μR0-spaces are obtained and some relations among these spaces are established.

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle(see Qiu, 2014), we prove a new version of Ekeland variational principle(briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results(no matter scalar-valued case,or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.

Some questions on partial metric spaces

We show that the completion of a partial metric space can fail be unique,which answers a question on completions of partial metric spaces.In addition,to this paper discusses metrizability around partial metric spaces.

Counterparty risk valuation on credit-linked notes under a Markov Chain framework

A credit-linked note(CLN)is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity.In this paper,we study the counterparty risk on CLNs under a Markov chain framework,and introduce a Markov copula model to describe joint defaults between the reference entity underlying the CLN and CLN issuer.Assuming that the respective default intensities are directly and inversely proportional to the interest rate,which follows a CIR process,we obtain the explicit formulae for CLN values through a PDE approach.Finally,credit valuation adjustment(CVA)formula is derived to price counterparty credit risk.

NON-CONFLICTING ORDERING CONES AND VECTOR OPTIMIZATION IN INDUCTIVE LIMITS

Let （E,ξ）=ind（En,ξn） be an inductive limit of a sequence （En,ξn）n∈N of locally convex spaces and let every step （En,ξn） be endowed with a partial order by a pointed convex （solid） cone Sn. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence （Sn）n∈N of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.

Identification of vortex boundaries in two-dimensional incompressible flows based on the Liutex-shear interaction

According to the Liutex-shear decomposition,vorticity can be decomposed into a rotational part,i.e.,the Liutex vector,and a residual shear part.With this decomposition,the vorticity transport equation can be used to formulate a governing equation for Liutex easily for two-dimensional incompressible flows with a source term depending on the residual shear.The dynamics of Liutex-identified structures is then studied in a Taylor-Green vortex flow and a flow past a cylinder at Reynolds number of 200.It is revealed that such boundaries exist outside which the shear has trivial impact on the evolution of Liutex and inside which enhancing and weakening effects of shear on Liutex can be observed.In addition,there is a strong dissipation effect upon Liutex on these boundaries.Based on the interaction mechanism between Liutex and shear,we argue that the vortex boundaries can be identified by these highly dissipative boundaries.In contrast,traditional methods use iso-surfaces of arbitrarily selected thresholds to represent vortex boundaries.The current method of identifying vortex boundaries based on the Liutex-shear interaction has a clearer theoretical base and avoids the arbitrary selection of thresholds.Extensions to three-dimensional incompressible flows can be made in future following the same procedure but with a slightly more complex vorticity transport equation which includes the velocity gradient induced stretching or tilting term.

A NOTE ON g-CONCAVE FUNCTION

An equivalent condition is derived for g-concave function defined by (static) g-expectation. Several extensions including quadratic generators and (g,h)-concavity are also considered.

Simulated Annealing for the 0/1 Multidimensional Knapsack Problem

In this paper a simulated annealing(SA)algorithm is presented for the 0/1 mul- tidimensional knapsack problem.Problem-specific knowledge is incorporated in the algorithm description and evaluation of parameters in order to look into the perfor- mance of finite-time implementations of SA.Computational results show that SA per- forms much better than a genetic algorithm in terms of solution time,whilst having a modest loss of solution quality.

On complete convergence in Marcinkiewicz-Zygmund type SLLN for random variables

We consider a generalization of Baum-Katz theorem for random variables satisfying some cover conditions.Consequently,we get the results for many dependent structures,such as END,ϱ^(*)mixing,ϱ^(-)mixing andφ-mixing,etc.

Local Discontinuous Galerkin Methods with Novel Basis for Fractional Diffusion Equations with Non-smooth Solutions

In this paper,we develop novel local discontinuous Galerkin(LDG)methods for fractional diffusion equations with non-smooth solutions.We consider such problems,for which the solutions are not smooth at boundary,and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy.The novel LDG methods utilize a solution information enriched basis,simulate the problem on a paired special mesh,and achieve optimal order of accuracy.We analyze the L2 stability and optimal error estimate in L2-norm.Finally,numerical examples are presented for validating the theoretical conclusions.

Sensitive Group Actions on Regular Curves of Almost≤n Order

Let X be a regular curve and n be a positive integer such that for every nonempty open set U⊂X,there is a nonempty connected open set V⊂U with the cardinality|∂_(X)(V)|≤n.We show that if X admits a sensitive action of a group G,then G contains a free subsemigroup and the action has positive geometric entropy.As a corollary,X admits no sensitive nilpotent group action.