risk sensitive and robust escape criteria the problem of controlling a noisy process so as to prevent it from leaving a prescribed set has a number of interesting applications in this paper new approaches to this problem are considered first a risk sensitive criterion for a stochastic diffusion process model is examined and it is shown that the value is a classical solution of a related pde the qualitative properties of this criterion are favorably contrasted with those of existing criteria in the risk averse limit it is proved that in the risk averse limit the value of the risk sensitive criterion converges to a viscosity solution of a first order pde it is then demonstrated that the value function of a deterministic differential game is also a viscosity solution to the pde this game gives a robust control formulation of the escape time problem and is analogous to h infty control in particular the opposing player attempts to push the process out of the prescribed set and suffers an l2 cost for his efforts lower bounds on the escape time as a function of this cost are obtained

binary forms hypergeometric functions and the schwarz christoffel mapping formula in a previous paper it was shown that if f is a binary form with complex coefficients having degree n 3 and discriminant df 0 and if af is the area of the region if x y l 1 in the real affine plane then idlil n n i af 3b l i where b 1 1 denotes the beta function with arguments of 1 3 this inequality was derived by demonstrating that the sequence mn defined by mn maxidfi n n af where the maximum is taken over all forms of degree n with df 0 is decreasing and then by showing that m3 3b q 1 the resulting estimate af 3b 1 for such forms with integer coefficients has had significant consequences for the enumeration of solutions of thue inequalities this paper examines the related problem of determining precise values for the sequence mn by appealing to the theory of hypergeometric functions it is shown that m4 27 6b 2 and that m4 is attained for the form xy x2 y2 it is also shown that there is a correspondence arising from the schwarz christowfel mapping formula between a particular collection of binary forms and the set of equiangular polygons with the property that af is the perimeter of the polygon corresponding to f based on this correspondence and a representation theorem for idfii n n1 af it is conjectured that mn d lln n 1 f n where fn x y x sin kn y cos k i and that the limiting value of the sequence m is 27r

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

risk sensitive asset management and cascading defaults we consider an optimal risk sensitive portfolio allocation problem accounting for the possibility of cascading defaults default events have an impact on the distress state of the surviving stocks in the portfolio we study the recursive system of non lipschitz quasilinear parabolic hjb pdes associated with the value function of the control problem in the different default states of the economy we show the existence of a classical solution to this system via super sub solution techniques and give an explicit characterization of the optimal feedback strategy in terms of the value function we prove a verification theorem establishing the uniqueness of the solution a numerical analysis indicates that the investor significantly accounts for contagion effects when making investment decisions and that his strategy depends nonmonotonically on the aggregate risk level

reproducing kernel element method part iv globally compatible cn n 1 triangular hierarchy in this part of the work a globally compatible c n x triangular element hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary two dimensional domains in principle the smoothness of the globally conforming element can be made arbitrarily high np 1 the triangle interpolation field can interpolate the derivatives of an unknown function up to arbitrary mth order i m and it can reproduce complete kth order polynomials with k p m this is the first interpolation hierarchical structure that has ever been constructed with both minimal degrees of freedom and higher order smoothness and continuity over discretizations of a multiple dimensional domain the performance of the newly constructed compatible element is evaluated in solving several kirchhoff plate problems

approximation of escape time for lagrangian systems with fast noise this note is concerned with the large deviations asymptotics estimate of the mean escape time for lagrangian systems subjected to fast gaussian noise the solution to the hamilton jacobi equation of the associated variational problem is derived as a sum of two terms dependent on kinetic and potential energy of the system respectively a closed form solution for classes of linear and nonlinear systems is obtained an application to a controlled pointing problem is discussed as an example

reproducing kernel element method part ii globally conforming im cn hierarchies in this part of the work a minimal degrees of freedom arbitrary smooth globally compatible i m c n interpolation hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary multiple dimensional domains this is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher order smoothness or continuity over multi dimensional domain the proposed hierarchical structure possesses the generalized kronecker property i e o a w b i ox a x j dijdab jaj jbj 6 m

on thue s equation and more precisely for the number of primitive solutions to 1 1 that is solutions in coprime integers x y the first important results on the number of solutions of thue s equation were obtained by c l siegel in the case r 3 and in the binomial case f x y a x r b y r in view of his results and comments we should attribute to siegel the problem to decide whether the number of primitive solutions to 1 1 for irreducible f of degree r 3 could be bounded by a function depending only on r and h but otherwise independent of f in 1983 siegel s question was answered in the affirmative by j h evertse 2 as a special case of his results he also treats equations in number fields evertse obtains the bound

development of a novel meshless local kriging lokriging method for structural dynamic analysis this paper presents a novel meshless approach to study the dynamic responses of structures it is called local kriging lokriging method where the kriging interpolation is employed to construct the field approximation function and meshless shape functions and the local weak form of partial differential governing equations is derived by the weighted residual technique since the shape functions constructed by this interpolation possess the delta function property even for a set of randomly distributed points the essential boundary condition can be easily imposed in implementation of local weak form the spline function with high continuity is used as the weight function no mesh is required either for integration of the local weak form or for construction of the shape functions thus the present lokriging method is a truly meshless method several numerical examinations for the free forced vibrational analysis of structures and the dynamic performance of microelectromechanical systems mems device are carried out to demonstrate the accuracy and efficiency of the lokriging method the numerical results show that the presently developed method is quite easy to implement very accurate and highly efficient for the problems considered

reproducing kernel element method part i theoretical formulation in this paper and its sequels we introduce and analyze a new class of methods collectively called the reproducing kernel element method rkem the central idea in the development of the new method is to combine the strengths of both finite element methods fem and meshfree methods two distinguished features of rkem are the arbitrarily high order smoothness and the interpolation property of the shape functions these properties are desirable especially in solving galerkin weak forms of higher order partial differential equations and in treating dirichlet boundary conditions so unlike the fem there is no need for special treatment with the rkem in solving high order equations compared to meshfree methods dirichlet boundary conditions do not present any difficulty in using the rkem a rigorous error analysis and convergence study of the method are presented the performance of the method is illustrated and assessed through some numerical examples

a moving kriging interpolation based meshless method for numerical simulation of kirchhoff plate problems this paper mainly proposes an alternative way for numerical implementation of thin plates bending based on a new improvement of meshless method which is combined between the standard element free galerkin method and one different shape functions building technique the moving kriging mk interpolation is applied instead of the traditional moving least square approximation in order to overcome kronecker s delta property where the standard method does not satisfy obviously the deflection of the thin plates is approximated via the mk interpolation to illustrate this approach numerical analysis is examined in both regular and irregular systems three examples with different geometric shapes of thin plates undergoing a simply supported boundary are performed in addition two important parameters of the present method are also analyzed a good agreement can be found among the proposed analytical and finite element methods copyright 2008 john wiley sons ltd

a meshless method for kirchhoff plate bending problems in this work a meshless method for the analysis of bending of thin homogeneous plates is presented this meshless method is based on the use of radial basis functions to build an approximation of the general solution of the partial differential equations governing the kirchhoff plate bending problem in order to obtain a symmetric and non singular linear equation system the hermite collocation method is used to assess the formulation a series of plates with different boundary conditions are analysed comparisons are made with other results available in the literature copyright 2001 john wiley sons ltd

minimizing escape probabilities a large deviations approach this paper considers the problem of controlling a possibly degenerate diffusion process so as to minimize the probability of escape over a given time interval it is assumed that the control acts on the process through the drift coefficient and that the noise coefficient is small developing a large deviations type of theory for the controlled diffusion produces several results the limit of the normalized log of the minimum exit probability is identified as the value i of an associated deterministic differential game furthermore we identify a deterministic and varepsilon independent mapping g from the sample values varepsilon w s 0 leqq s leqq t into the control space such that if we define the control used at time t by u t g varepsilon w s 0 leqq s leqq t then the resulting control process is progressively measurable and delta optimal in the sense that the limit of the normalized log of the exit probability is within delta of i

moving kriging interpolation and element free galerkin method a new formulation of the element free galerkin efg method is presented in this paper efg has been extensively popularized in the literature in recent years due to its flexibility and high convergence rate in solving boundary value problems however accurate imposition of essential boundary conditions in the efg method often presents difficulties because the kronecker delta property which is satisfied by finite element shape functions does not necessarily hold for the efg shape function the proposed new formulation of efg eliminates this shortcoming through the moving kriging mk interpolation two major properties of the mk interpolation the kronecker delta property i s j i j and the consistency property n i i x i and n i i x x l i x i are proved some preliminary numerical results arc given

differential cubature method a solution technique for kirchhoff plates of arbitrary shape a treatment for bending analysis of kirchhoff plates using the differential cubature dc method is presented to the authors knowledge this represents the first known successful application of the dc method to this problem the plates treated are subjected to various combinations of boundary conditions and of arbitrary plate geometry a discussion on the implementation issues is presented and several numerical test examples are included to demonstrate the applicability efficiency and simplicity of the approximate procedures the accuracy of the procedures is verified by direct comparing the present numerical results of different plate shapes with the existing exact analytical solutions 1 and other sources of numerical values

a large deviations analysis of range tracking loops large deviations theory is applied to the analysis of a discrete time range tracking loop it is shown that the resulting asymptotics differ from those of the continuous time diffusion limit

differential cubature method for static solutions of arbitrarily shaped thick plates this paper presents the first endeavor to exploit the differential cubature method as an accurate and efficient global technique for fundamental solutions of arbitrarily shaped thick plates the method is examined here for its suitability for solving the boundary value problem of thick plates governing by the first order shear deformation theory using the method the governing differential equations and boundary conditions are transformed into sets of linear algebraic equations boundary conditions are implemented through discrete grid points by constraining displacements bending moments and rotations detailed discussion on the formulation and implementation of the method are presented the applicability efficiency and simplicity of the method are demonstrated through solving example plate problems of different shapes the accuracy of the method is verified by direct comparison with the known values

risk sensitive control for the parallel server model a markovian queueing model is considered in which servers of various types work in parallel to process jobs from a number of classes at rates mu ij that depend on the class i and the type j the problem of dynamic resource allocation so as to minimize a risk sensitive criterion is studied in a law of large numbers scaling letting x i t denote the number of class i jobs in the system at time t the cost is given by e exp n int 0 th bar x t dt g bar x t where t 0 h and g are given functions satisfying regularity and growth conditions and bar x bar x n n 1 x n cdot it is well known in an analogous context of controlled diffusion and has been shown for some classes of stochastic networks that the limit behavior as n to infty is governed by a differential game dg in which the state dynamics is given by a fluid equation for the formal limit varphi of bar x while the cost consists of int 0 th varphi t dt g varphi t and an additional term th

an escape time criterion for queueing networks asymptotic risk sensitive control via differential games we consider the problem of risk sensitive control of a stochastic network in controlling such a network an escape time criterion can be useful if one wishes to regulate the occurrence of large buffers and buffer overflow in this paper a risk sensitive escape time criterion is formulated which in comparison to the ordinary escape time criteria penalizes exits that occur on short time intervals more heavily the properties of the risk sensitive problem are studied in the large buffer limit and related to the value of a deterministic differential game with constrained dynamics we prove that the game has value and that the value is the viscosity solution of a pde for a simple network the value is computed demonstrating the applicability of the approach

a meshfree hermite type radial point interpolation method for kirchhoff plate problems a meshfree computational method is proposed in this paper to solve kirchhoff plate problems of various geometries the deflection of the thin plate is approximated by using a hermite type radial basis function approximation technique the standard galerkin method is adopted to discretize the governing partial differential equations which were derived from using the kirchhoff s plate theory the degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency accuracy and robustness of the method copyright 2005 john wiley sons ltd

the practical computation of areas associated with binary quartic forms we derive formulas for practically computing the area of the region f z y 1 defined by a binary quartic form f x y r x y these formulas which involve a particular hypergeometric function are useful when estimating the number of lattice points in certain regions of the type f x y h and will likely find application in many contexts we also show that for forms f of arbitrary degree the maximal size of the area of the region f x y 1 normalized with respect to the discriminant of f and taken with respect to the number of conjugate pairs of f x 1 increases as the number of conjugate pairs decreases and we give explicit numerical values for these normalized maxima when f is a quartic form

a risk sensitive maximum principle the case of imperfect state observation the risk sensitive maximum principle for optimal stochastic control derived by the author in an earlier work system control letters vol 15 1990 is restated this is an immediate generalization of the classic pontryagin principle to which it reduces in the deterministic case and is expressed immediately in terms of observables it is derived on the assumption that the criterion function is the exponential of an additive cost function and is exact under linear quadratic gaussian assumptions but is otherwise valid as a large deviation approximation the principle is extended to the case of imperfect state observation after preliminary establishment of a certainty equivalence principle the derivation yields as byproduct a large deviation version of the updating equation for nonlinear filtering the development is heuristic it is believed that the mathematical arguments given are the essential ones and provide a self contained treatment at this level

moving least squares differential quadrature method and its application to the analysis of shear deformable plates a moving least squares differential quadrature mlsdq method is developed and employed for the analysis of moderately thick plates based on the first order shear deformation theory fsdt to carry out the analysis the governing equations in terms of the generalized displacements transverse deflection and two rotations of the plate are formulated by employing the moving least squares approximation the weighting coefficients used in the mlsdq approximation are computed through a fast computation of shape functions and their derivatives numerical examples illustrating the accuracy stability and convergence of the mlsdq method are presented effects of support size order of completeness and node irregularity on the numerical accuracy are investigated copyright 2003 john wiley sons ltd