on the crosscorrelation of sequences over gf p with short periods we investigate the crosscorrelation function cd t spl sigma sub i 1 p sup n 1 spl zeta a sup i t a sup di where spl zeta is a complex primitive pth root of unity a sub i i spl isin n sub 0 is a maximal linear shift register sequence of length p sup n 1 and p is an odd prime for p 3 n odd and d p sup n 1 4 p sup n 1 2 we show that 2 spl middot spl radic p sup n is an upper bound for the absolute value of 1 c sub d t for any odd prime p and p sup k 1 where n gcd n k is not divisible by 4 we determine the maximum absolute value of c sub d t and the number of values of c sub d t

discontinuous galerkin finite element methods for time dependent hamilton jacobi bellman equations with cordes coefficients we propose and analyse a fully discrete discontinuous galerkin time stepping method for parabolic hamilton jacobi bellman equations with cordes coefficients the method is consistent and unconditionally stable on rather general unstructured meshes and time partitions error bounds for both rough and regular solutions in terms of temporal regularity show that the method is arbitrarily high order with optimal convergence rates with respect to the mesh size time interval length and temporal polynomial degree and possibly suboptimal by an order and a half in the spatial polynomial degree numerical experiments on problems with strongly anisotropic diffusion coefficients and early time singularities demonstrate the accuracy and computational efficiency of the method with exponential convergence rates achieved under combined hp and tau q q refinement

convergent difference schemes for degenerate elliptic and parabolic equations hamilton jacobi equations and free boundary problems convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm explicit schemes endowed with an explicit cfl condition are built for time dependent equations and are used to solve stationary equations iteratively explicit and implicit formulations of monotonicity for first and second order equations are unified bounds on orders of accuracy are established an example of a scheme which is stable but nonmonotone and nonconvergent is presented schemes for hamilton jacobi equations obstacle problems one phase free boundary problems and stochastic games are built and computational results are presented

power penalty method for solving hjb equations arising from finance nonlinear hamilton jacobi bellman hjb equation commonly occurs in financial modeling implicit numerical scheme is usually applied to the discretization of the continuous hjb so as to find its numerical solution since it is generally difficult to obtain its analytic viscosity solution this type of discretization results in a nonlinear discrete hjb equation we propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term under some mild conditions we give the unique solvability of the penalized equation and show its convergence to the original discrete hjb equation moreover we establish a sharp convergence rate of the power penalty method which is of an exponential order with respect to the power of the penalty term we further develop a damped newton algorithm to iteratively solve the lower order penalized equation finally we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method we also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method

new constructions of quadratic bent functions in polynomial form new quadratic bent functions in polynomial form are constructed in this paper the constructions give new boolean bent generalized boolean bent and p ary bent functions based on z 4 valued quadratic forms a simple method provides several new constructions of generalized boolean bent functions from these generalized boolean bent functions a method is presented to transform them into boolean bent and semi bent functions moreover many new p ary bent functions can also be obtained by applying similar methods

duality based a posteriori error estimates for some approximation schemes for optimal investment problems abstract we consider a markov chain approximation scheme for utility maximization problems in continuous time which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments the error estimates previously derived in picarelli and reisinger 2018 are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only in the present paper we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds the theoretical results are confirmed by our numerical tests

narain gupta s three normal subgroup problem and group homology abstract this paper is about application of various homological methods to classical problems in the theory of group rings it is shown that the third homology of groups plays a key role in narain gupta s three normal subgroup problem for a free group f and its normal subgroups r s t and the corresponding ideals in the integral group ring z f r r 1 z f s s 1 z f t t 1 z f a complete description of the normal subgroup f 1 rst is given provided r t and the third fourth and fifth homology groups of r r s are torsion groups

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

a penalty method for the numerical solution of hamilton jacobi bellman hjb equations in finance we present a simple and easy to implement method for the numerical solution of a rather general class of hamilton jacobi bellman hjb equations in many cases classical finite difference discretizations can be shown to converge to the unique viscosity solutions of the considered problems however especially when using fully implicit time stepping schemes with their desirable stability properties one is still faced with the considerable task of solving the resulting nonlinear discrete system in this paper we introduce a penalty method which approximates the nonlinear discrete system to an order of o 1 rho where rho 0 is the penalty parameter and we show that an iterative scheme can be used to solve the penalized discrete problem in finitely many steps we include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results

stock trading an optimal selling rule trading in stock markets consists of three major steps select a stock purchase a number of shares and eventually sell them to make a profit the timing to buy and sell is extremely crucial a selling rule can be specified by two preselected levels a target price and a stop loss limit this paper is concerned with an optimal selling rule based on the model characterized by a number of geometric brownian motions coupled by a finite state markov chain such a policy can be obtained by solving a set of two point boundary value differential equations moreover the corresponding expected target period and probability of making money and that of losing money are derived analytic solutions are obtained in one and two dimensional cases finally a numerical example is considered to demonstrate the effectiveness of our method

almost perfect nonlinear power functions on gf 2 n the niho case almost perfect nonlinear apn mappings are of interest for applications in cryptography we prove for odd n and the exponent d 22r 2r 1 where 4r 1 0modn that the power functions xd on gf 2n is apn the given proof is based on a new class of permutation polynomials which might be of independent interest our result supports a conjecture of niho stating that the power function xd is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequences of degree n and a decimation of that sequence by d takes on precisely the three values 1 1 2 n 1 2

optimal consumption and portfolio with both fixed and proportional transaction costs we consider a market model with one risk free and one risky asset in which the dynamics of the risky asset are governed by a geometric brownian motion in this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets we suppose that these transfers involve a fixed transaction cost k 0 independent of the size of the transaction plus a cost proportional to the size of the transaction the objective is to maximize the cumulative expected utility of consumption over a planning horizon we formulate this problem as a combined stochastic control impulse control problem which in turn leads to a nonlinear quasi variational hamilton jacobi bellman inequality qvhjbi we prove that the value function is the unique viscosity solution of this qvhjbi finally numerical results are presented

free centre by abelian by exponent 2 groups abstract we study free centre by abelian by exponent 2 groups our main result is a complete description of the centre it is isomorphic to a direct sum of a free abelian group and a torsion subgroup the latter is a direct sum of cyclic groups of order two and cyclic groups of order four we exhibit a generating set consisting of elements of infinite order order 2 and order 4 such that the centre is the direct sum of cyclic subgroups generated by those generators our approach makes essential use of homological methods

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

new families of almost perfect nonlinear power mappings a power mapping f x x sup d over gf p sup n is said to be differentially k uniform if k is the maximum number of solutions x spl isin gf p sup n of f x a f x b where a b spl isin gf p sup n and a spl ne 0 a 2 uniform mapping is called almost perfect nonlinear apn we construct several new infinite families of nonbinary apn power mappings

nonlinear black scholes equations in finance associated control problems and properties of solutions we study properties of solutions to fully nonlinear versions of the standard black scholes partial differential equation these equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility we show that typical nonlinear black scholes equations can be viewed as dynamic programming equation of an associated control problem we establish existence and comparison results and show that the equation induces a convex risk measure on the set of all continuous terminal value claims moreover we study the asymptotic behavior of solutions as market frictions get large finally the pricing of individual contracts relative to a book of derivatives is discussed

pricing european options with proportional transaction costs and stochastic volatility using a penalty approach and a finite volume scheme abstract in this paper we propose a combination of a penalty method and a finite volume scheme for a four dimensional time dependent hamilton jacobi bellman hjb equation arising from pricing european options with proportional transaction costs and stochastic volatility the hjb equation is first approximated by a nonlinear partial differential equation containing penalty terms a finite volume method along with an upwind technique is then developed for the spatial discretization of the nonlinear penalty equation we show that the coefficient matrix of the discretized system is an m matrix an iterative method is proposed for solving the nonlinear algebraic system and a convergence theory is established for the iterative method numerical experiments are performed using a non trivial model pricing problem and the numerical results demonstrate the usefulness of the proposed method

a new family of ternary almost perfect nonlinear mappings a mapping f x from gf pn to gf pn is differentially k uniform if k is the maximum number of solutions x isin gf pn of f x a f x b where a b isin gf pn and a ne 0 a 2 uniform mapping is called almost perfect nonlinear apn this correspondence describes new families of ternary apn mappings over gf 3n n 3 odd of the form f x uxd xd 2 where d1 3n 1 2 1 and d2 3n 2

further results on generalized bent functions and their complete characterization this paper contributes to increase our knowledge on generalized bent functions including generalized bent boolean functions and generalized p ary bent functions with odd prime p by bringing new results on their characterization and construction in arbitrary characteristic more specifically we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions this enables us to completely characterize generalized bent functions and mathbb z p k bent functions by some affine space associated with the generalized bent functions we also present the relationship between generalized bent boolean functions with an odd number of variables and generalized bent boolean functions with an even number of variables based on the well known maiorana mcfarland class of boolean functions we present some infinite classes of generalized bent boolean functions in addition we introduce a class of generalized hyperbent functions that can be seen as generalized dillon s ps functions finally we solve an open problem related to the description of the dual function of a weakly regular generalized bent boolean function with an odd number of variables via the walsh hadamard transform of their component functions and we generalize these results to the case of odd prime

almost perfect nonlinear power functions in odd characteristic in this paper two new families of functions in odd characteristic are constructed and they are proved to be almost perfect nonlinear apn functions some of the open cases which were introduced by helleseth are answered by these new apn functions

cartesian authentication codes from functions with optimal nonlinearity in this paper we present several classes of authentication codes using functions with perfect nonlinearity and optimum nonlinearity some of the authentication codes are optimal on the other hand these authentication codes are easy to implement due to their simple algebraic structures

a fitted finite volume method for real option valuation of risks in climate change a large number of industries will experience climate change related damages with the climate change processes over the coming years for example the risks from sea level rising will be faced in addition there are a lot of uncertainties for the climate change in the future therefore making decisions when to invest in the long term the sea level rising risk related projects is important and complex the complexity of the decisions mainly lies in the evolving nature of the sea level rising risk particularly due to the global climate change but also the future socio economic development scenarios in this paper we first regard the sea level and the temperature as the underlying assets and then develop a real option model to evaluate potential sea level rising risk management opportunities in the case of american real options we reformulate the problem to a linear parabolic variational inequality vi in two spatial dimensions and develop a power penalty method to solve it it is shown that the nonlinear partial differential equation pde is uniquely solvable and the solution of the pde converges to that of the vi at the rate of order o k 2 a so called fitted finite volume method is proposed to solve the nonlinear pde in both cases of european and american options and the convergence of the fully discrete system of equations is obtained finally some numerical experiments are performed to illustrate the theoretical results of this method

new binomial bent functions over the finite fields of odd characteristic the p ary function f x mapping gf p4k to gf p and given by f x tr4k xp3k p2k pk 1 x2 is proven to be a weakly regular bent function and the exact value of its walsh transform coefficients is found this is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic the proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a limit approach to group homology abstract in this paper we consider for any free presentation g f r of a group g the coinvariance h 0 g r a b n of the nth tensor power of the relation module r a b and show that the homology group h 2 n g z may be identified with the limit of the groups h 0 g r a b n where the limit is taken over the category of these presentations of g we also consider the free lie ring generated by the relation module r a b in order to relate the limit of the groups n r n r f to the n torsion subgroup of h 2 n g z

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

injectivity theorems and algebraic closures of groups with coefficients abstractrecently cochran and harvey defined torsion free derived series of groups and proved an injectivity theorem on the associated torsion free quotients we show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem our result relates injectivity theorems to a certain homology localization of groups in order to give a concrete combinatorial description and existence proof of the necessary homology localization we introduce a new version of algebraic closures of groups with coefficients by considering certain types of equations

a semi lagrangian approach for natural gas storage valuation and optimal operation the valuation of a gas storage facility is characterized as a stochastic control problem resulting in a hamilton jacobi bellman hjb equation in this paper we present a semi lagrangian method for solving the hjb equation for a typical gas storage valuation problem the method is able to handle a wide class of spot price models that exhibit mean reverting seasonality dynamics and price jumps we develop fully implicit and crank nicolson timestepping schemes based on a semi lagrangian approach and prove the convergence of fully implicit timestepping to the viscosity solution of a modified hjb equation posed on a bounded domain provided that a strong comparison result holds the semi lagrangian approach avoids policy type iterations required by an implicit finite difference method without requiring additional cost numerical experiments are presented for several variants of the basic scheme

convergence of meshfree collocation methods for fully nonlinear parabolic equations we prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step a numerical experiment with a radial basis function demonstrates the convergence property

fast computational approach to the delta greek of non linear black scholes equations abstract in this paper we consider a class of non linear option pricing models the focus is on the numerical investigation of delta equation where the unknown solution is the first spatial derivative of the option value we construct and analyze monotone and sign preserving finite difference schemes for the problems newton s and picard s iterative procedures for solving the non linear systems of algebraic equations are proposed on this base in order to improve the computational efficiency we develop fast two grid algorithms numerical experiments using also richardson extrapolation in time are discussed in terms of accuracy convergence and efficiency

nonbinary kasami sequences over gf p the correlation values and the distribution of these correlation values for the small set of nonbinary kasami sequences over gf p p prime are presented the correlation results are an extension of the binary results and have p 2 correlation levels this nonbinary kasami set is asymptotically optimum with respect to its correlation properties these sequences are obtained as in the binary case from a large primitive polynomial of degree n 2 m and a small primitive polynomial of degree m that yields a sequence length of p sup n 1 and maximum nontrivial correlation value of 1 p sup m nonbinary kasami sequences are directly implemented using shift registers and are applicable for code division multiple access systems

almost perfect nonlinear power functions on gf 2 sup n the welch case we summarize the state of the classification of almost perfect nonlinear apn power functions x sup d on gf 2 sup n and contribute two new cases to prove these cases we derive new permutation polynomials the first case supports a well known conjecture of welch stating that for odd n 2m 1 the power function x sup 2m 3 is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequence of degree n and a decimation of that sequence by 2 sup m 3 takes on precisely the three values 1 1 spl plusmn 2 sup m 1

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

optimal investment problem under non extensive statistical mechanics abstract the optimal investment problem in which the asset price process is modeled by the non extensive statistical mechanics is studied in this paper by the methods of deterministic control and the dynamic programming we obtain the optimal strategy with logarithmic utility function power utility function and quadratic utility function respectively moreover the numerical results indicate that the optimal investment strategy is affected by the non extensive parameter the proportion invested in the risky asset decreases as the wealth increases under quadratic utility function but it remains unchanged under power utility and logarithmic utility function

generalized plateaued functions and admissible plateaued functions plateaued functions are very important crypto graphic functions due to their various desirable cryptographic characteristics we point out that plateaued functions are more general than bent functions that is functions with maximum nonlinearity some boolean plateaued functions have large nonlinearity which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers and contributes when they are the component functions of the substitution boxes in block ciphers to protection against linear cryptanalysis p ary plateaued functions have attracted recently some attention in the literature and many activities on generalized p ary functions have been carried out this paper increases our knowledge on plateaued functions in the general context of generalized p ary functions we first introduce two new versions of plateaued functions which we shall call generalized plateaued functions and admissible plateaued functions the generalized plateaued functions extend the standard notion of plateaued p ary functions to those whose outputs are in the ring mathbb z p k next we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components finally we provide for the first time two constructions of generalized plateaued functions in particular we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes

biological population management based on a hamilton jacobi bellman equation with boundary blow up establishment of a long term biological population management policy requires balancing its cost and utility disutility we approach this issue from the viewpoint of stochastic control as an effici

on the walsh transform of a class of functions from niho exponents in this paper a class of functions from niho exponents with four valued walsh transform is obtained for any prime by a uniform method and the distribution of the walsh transform values is also completely determined in particular this class of functions is proven to be bent for a special case although it is shown that the obtained bent functions are equivalent to the leander kholosha s class of bent functions a direct and much simpler proof for the bentness of this kind of niho functions is provided

ternary m sequences with three valued cross correlation function new decimations of welch and niho type we show that the cross correlation between two ternary m sequences of period 3 sup n 1 that differ by the decimation d 2 spl middot 3 sup m 1 where n 2m 1 takes on three different values we conjecture the same result for the decimation d 2 spl middot 3 sup r 1 where n is odd and r is defined by the condition 4r 1 spl equiv 0 mod n these two new cases form in a sense ternary counterparts of two previously confirmed binary cases the conjectures of welch and niho 1972

two constructions of approximately symmetric informationally complete positive operator valued measures symmetric informationally complete positive operator valued measures sic povms have many applications in quantum information however it is not easy to construct sic povms and there are only a few known classes of them and we do not even know whether there exists an infinite class of them thus constructing approximately symmetric informationally complete positive operator valued measures asic povms has its own meaning in this paper we use character sums over finite fields to present two constructions of asic povms we show that there are some classes of infinite families of asic povms by using some special functions over finite fields

irreducible trinomials over finite fields a necessary condition for irreducibility of a trinomial over a finite field based on classical results of stickelberger and swan is established it is applied in the special case f3 and some experimental discoveries are reported

prime phase sequences with periodic correlation properties better than binary sequences for the case where p is an odd prime n or 2 is an integer and omega is a complex primitive pth root of unity a construction is presented for a family of p sup n p phase sequences symbols of the form omega sup i where each sequence has length p sup n 1 and where the maximum nontrivial correlation value c sub max does not exceed 1 square root p sup n a complete distribution of correlation values is provided as a special case of this construction a previous construction due to sidelnikov 1971 is obtained the family of sequences is asymptotically optimum with respect to its correlation properties and in comparison with many previous nonbinary designs the present design has the additional advantage of not requiring an alphabet of size larger than three the new sequences are suitable for achieving code division multiple access and are easily implemented using shift registers they wee discovered through an application of deligne s bound 1974 on exponential sums of the weil type in several variables the sequences are also shown to have strong identification with certain bent functions

cubature on wiener space it is well known that there is a mathematical equivalence between solving parabolic partial differential equations pdes and the integration of certain functionals on wiener space monte carlo simulation of stochastic differential equations sdes is a naive approach based on this underlying principle in finite dimensions it is well known that cubature can be a very effective approach to integration we discuss the appropriate extension of this idea to wiener space in the process we develop high order numerical schemes valid for high dimensional sdes and semi elliptic pdes

a class of binomial bent functions over the finite fields of odd characteristic this paper studies a class of binomial functions over the finite fields of odd characteristic and characterizes their bentness in terms of the kloosterman sums numerical results show that the proposed class contains bent functions that are affinely inequivalent to all known monomial and binomial ones

vectorial bent functions and their duals abstract motivated by the observation that for two weakly regular bent functions f g for which also f g is bent the sum f g of their duals f and g is sometimes but not always bent we initiate the study of duality for vectorial bent functions we propose and investigate two concepts of self duality for vectorial bent functions self duality and weak self duality

on the rate of convergence of finite difference approximations for bellmans equations with variable coefficients the estimates presented here for parabolic bellman s equations with variable coefficients extend the ones earlier obtained for constant coefficients

maximal use of central differencing for hamilton jacobi bellman pdes in finance in order to ensure convergence to the viscosity solution the standard method for discretizing hamilton jacobi bellman partial differential equations uses forward backward differencing for the drift term in this paper we devise a monotone method which uses central weighting as much as possible in order to solve the discretized algebraic equations we have to maximize a possibly discontinuous objective function at each node nevertheless convergence of the overall iteration can be guaranteed numerical experiments on two examples from the finance literature show higher rates of convergence for this approach compared to the use of forward backward differencing only

semi lagrangian schemes for linear and fully non linear diffusion equations for linear and fully non linear diffusion equations of bellmanisaacs type we introduce a class of monotone approximation schemes relying on monotone interpolation as opposed to classical numerical methods these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices such schemes have to have a wide stencil in general besides providing a unifying framework for several known first order accurate schemes our class of schemes also includes more efficient versions and a new second order scheme that converges only for essentially monotone solutions the methods are easy to implement and analyze and they are more efficient than some other known schemes we prove stability and convergence of the schemes in the general case and provide error estimates in the convex case which are robust in the sense that they apply to degenerate equations and non smooth solutions the methods are extensively tested

a new class of balanced near perfect nonlinear mappings and its application to sequence design a mapping from zn to zm can be directly applied for the design of a sequence of period n with alphabet size m where zn denotes the ring of integers modulo n the nonlinearity of such a mapping is closely related to the autocorrelation of the corresponding sequence when m is a divisor of n the sequence corresponding to a perfect nonlinear mapping has perfect autocorrelation but it is not balanced in this paper we study balanced near perfect nonlinear npn mappings applicable for the design of sequence sets with low correlation we first construct a new class of balanced npn mappings from z p2 p to zp for an odd prime p we then present a general method to construct a frequency hopping sequence fhs set from a nonlinear mapping by applying it to the new class we obtain a new optimal fhs set of period p2 p with respect to the peng fan bound whose fhss are balanced and optimal with respect to the lempel greenberger bound moreover we construct a low correlation sequence set with size p period p2 p and maximum correlation magnitude p from the new class of balanced npn mappings which is asymptotically optimal with respect to the welch bound

a numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit gmwb in this paper we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit gmwb assuming the policyholder is allowed to withdraw funds continuously we develop a numerical scheme for solving the hamilton jacobi bellman hjb variational inequality corresponding to the impulse control problem we prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem provided a strong comparison result holds the scheme can be easily generalized to price discrete withdrawal contracts numerical experiments are conducted which show a region where the optimal control appears to be non unique

optimal portfolios with regime switching and value at risk constraint we consider the optimal portfolio selection problem subject to a maximum value at risk mvar constraint when the price dynamics of the risky asset are governed by a markov modulated geometric brownian motion gbm here the market parameters including the market interest rate of a bank account the appreciation rate and the volatility of the risky asset switch over time according to a continuous time markov chain whose states are interpreted as the states of an economy the mvar is defined as the maximum value of the vars of the portfolio in a short time duration over different states of the chain we formulate the problem as a constrained utility maximization problem over a finite time horizon by utilizing the dynamic programming principle we shall first derive a regime switching hamilton jacobi bellman hjb equation and then a system of coupled hjb equations we shall employ an efficient numerical method to solve the system of coupled hjb equations for the optimal constrained portfolio we shall provide numerical results for the sensitivity analysis of the optimal portfolio the optimal consumption and the var level with respect to model parameters these results are also used to investigating the effect of the switching regimes

impulse control of multidimensional jump diffusions in finite time horizon this paper analyzes a class of impulse control problems for multidimensional jump diffusions in the finite time horizon following the basic mathematical setup from stroock and varadhan multidimensional diffusion processes springer verlag heidelberg 2006 this paper first establishes rigorously an appropriate form of the dynamic programming principle it then shows that the value function is a viscosity solution for the associated hamilton jacobi bellman equation involving integro differential operators finally under additional assumptions that the jumps are of infinite activity but are of finite variation and that the diffusion is uniformly elliptic it proves that the value function is the unique viscosity solution and has w loc 2 1 p regularity for 1 p infty

two new measures for permutations ambiguity and deficiency we introduce the concepts of weighted ambiguity and deficiency for a mapping between two finite abelian groups of the same size then we study the optimum lower bounds of these measures for permutations of an abelian group a construction of permutations by modifying some permutation functions over finite fields is given their ambiguity and deficiency is investigated most of these functions are apn permutations we show that when they are not optimal the mobius function in the multiplicative group of bbfq is closer to being optimal in ambiguity than the inverse function in the additive group of bbfq we note that the inverse function over bbf28 is used in aes finally we conclude that a twisted permutation polynomial of a finite field is again closer to being optimal in ambiguity than the apn function employed in the safer cryptosystem

a nonstandard finite difference scheme for convection diffusion equations having constant coefficients in this note we derive using the subequation method a new nonstandard finite difference scheme nsfd for a class of convection diffusion equations having constant coefficients despite the fact that this scheme has nonlinear denominator functions of the step sizes even for linear pdes it has a couple of favourable properties it is explicit and due to its construction it reproduces important properties of the solution of the parabolic pde this proposed method conserves by construction the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the black scholes equation for the valuation of standard options since it avoids negative values for the calculated prices finally we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature

on the dual of certain ternary weakly regular bent functions in 2006 helleseth and kholosha conjectured and partially proved the existence of a class of ternary weakly regular monomial bent functions and also the expression for the dual bent function up to the sign value the bentness was finally proved later in 2009 using a complicated technique that employs stickelberger s theorem extensively using the previously found results and approaches in this paper a surprisingly short proof for the conjectured expression of the dual is given but without resolving the sign ambiguity furthermore we resolve the sign by finding the trace representation of the dual function

linear codes with two or three weights from weakly regular bent functions linear codes with a few weights have applications in consumer electronics communication data storage system secret sharing authentication codes association schemes and strongly regular graphs this paper first generalizes the method of constructing two weight and three weight linear codes of ding et al and zhou et al to general weakly regular bent functions and determines the weight distributions of these linear codes it solves an open problem proposed by ding et al furthermore this paper constructs new linear codes with two or three weights and presents their weight distributions they contain some optimal codes meeting certain bound on linear codes

convergence properties of policy iteration this paper analyzes asymptotic convergence properties of policy iteration in a class of stationary infinite horizon markovian decision problems that arise in optimal growth theory these problems have continuous state and control variables and must therefore be discretized in order to compute an approximate solution the discretization may render inapplicable known convergence results for policy iteration such as those of puterman and brumelle math oper res 4 1979 pp 60 69 under certain regularity conditions we prove that for piecewise linear interpolation policy iteration converges quadratically also under more general conditions we establish that convergence is superlinear we show how the constants involved in these convergence orders depend on the grid size of the discretization these theoretical results are illustrated with numerical experiments that compare the performance of policy iteration and the method of successive approximations

a fast preconditioned policy iteration method for solving the tempered fractional hjb equation governing american options valuation a fast preconditioned policy iteration method is proposed for the hamiltonjacobibellman hjb equation involving tempered fractional order partial derivatives governing the valuation of american options whose underlying asset follows exponential lvy processes an unconditionally stable upwind finite difference scheme with shifted grnwald approximation is first developed to discretize the established hjb equation under the tempered fractional diffusion models next the policy iteration method as an outer iterative method is utilized to solve the discretized hjb equation and proven to be convergent in finite steps to its numerical solution given the toeplitz like structure of the coefficient matrix in each policy iteration the resulting linear system can be fast solved by the krylov subspace method as an inner iterative method via fast fourier transform fft furthermore a novel preconditioner is proposed to speed up the convergence rate of the inner krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in o nlogn operations under some mild conditions where n is the number of spatial node points numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method

optimal ternary cyclic codes from monomials cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters 3m 1 3m 1 2m 4 by carlet ding and yuan in 2005 in this paper almost perfect nonlinear monomials and a number of other monomials over gf 3m are used to construct optimal ternary cyclic codes with the same parameters nine open problems on such codes are also presented

polynomial approximation of high dimensional hamilton jacobi bellman equations and applications to feedback control of semilinear parabolic pdes a procedure for the numerical approximation of high dimensional hamilton jacobi bellman hjb equations associated to optimal feedback control problems for semilinear parabolic equations is proposed its main ingredients are a pseudospectral collocation approximation of the pde dynamics and an iterative method for the nonlinear hjb equation associated to the feedback synthesis the latter is known as the successive galerkin approximation it can also be interpreted as newton iteration for the hjb equation at every step the associated linear generalized hjb equation is approximated via a separable polynomial approximation ansatz stabilizing feedback controls are obtained from solutions to the hjb equations for systems of dimension up to fourteen

systematic authentication codes from highly nonlinear functions recently highly nonlinear functions have been successfully employed to construct authentication codes with and without secrecy in this paper we construct four classes of systematic authentication codes from perfect nonlinear functions and almost perfect nonlinear functions the systematic authentication codes presented in this paper are either better than existing codes or as good as the best codes known

a numerical algorithm based on a variational iterative approximation for the discrete hamilton jacobi bellman hjb equation this paper presents a numerical algorithm based on a variational iterative approximation for the hamilton jacobi bellman equation and a domain decomposition technique based on this algorithm is also studied the convergence theorems have been established numerical results indicate the efficiency and accuracy of the methods

monomial and quadratic bent functions over the finite fields of odd characteristic considered are p ary bent functions having the form f x tr sub n spl sigma sub i 0 sup s a sub i x sup di a new class of ternary monomial regular bent function with the dillon exponent is discovered the existence of dillon bent functions in the general case is an open problem of deciding whether a certain kloosterman sum can take on the value 1 also described is the general gold like form of a bent function that covers all the previously known monomial quadratic cases the weak regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given finally some criteria for an arbitrary quadratic function to be bent are proven

impulse control of multidimensional jump diffusions this paper studies regularity properties of the value function for an infinite horizon discounted cost impulse control problem where the underlying controlled process is a multidimensional jump diffusion with possibly infinite activity jumps surprisingly despite these jumps we obtain the same degree of regularity as for the diffusion case at least when the jump satisfies certain integrability conditions

error bounds for monotone approximation schemes for hamilton jacobi bellman equations we obtain error bounds for monotone approximation schemes of hamilton jacobi bellman equations these bounds improve previous results of krylov and the authors the key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate almost smooth supersolutions for the hamilton jacobi bellman equation

constructions of quadratic and cubic rotation symmetric bent functions in this paper we consider constructions of rotation symmetric bent functions which are of the forms fc x i 1m 1 ci j 0n 1 xjxi j cm j 0m 1 xjxm j and ft x i 0n 1 xixt ixm i xixt i i 0m 1 xixm i where n 2m ci 0 1 the subscript u of xu in the previous expressions is taken as u modulo n for each case a necessary and sufficient condition is obtained to the best of our knowledge this class of cubic rotation symmetric bent functions is the first example of an infinite class of nonquadratic rotation symmetric bent functions

hamilton jacobi bellman quasi variational inequality arising in an environmental problem and its numerical discretization abstract a hamilton jacobi bellman quasi variational inequality hjbqvi for a river environmental restoration problem with wise use of sediment is formulated and its mathematical properties are analyzed a finite difference scheme with a penalization technique is then established for solving the hjbqvi the scheme is free from any iterative solvers and is unconditionally stable and convergent in the viscosity sense under certain conditions a demonstrative application example of the hjbqvi is finally presented

some estimates for finite difference approximations some estimates for the approximation of optimal stochastic control problems by discrete time problems are obtained in particular an estimate for the solutions of the continuous time versus the discrete time hamilton jacobi bellman equations is given the technique used is more analytic than probabilistic

smooth fit principle for impulse control of multidimensional diffusion processes value functions of impulse control problems are known to satisfy quasi variational inequalities qvis a bensoussan and j l lions impulse control and quasivariational inequalities heyden son philadelphia 1984 translation of controle impulsionnel et inequations quasi variationnelles gauthier villars paris 1982 this paper proves the smooth fit c 1 property of the value function for multidimensional controlled diffusions using a viscosity solution approach we show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions

probabilistic error analysis for some approximation schemes to optimal control problems abstract we introduce a class of numerical schemes for optimal stochastic control problems based on a novel markov chain approximation which uses in turn a piecewise constant policy approximation euler maruyama time stepping and a gaus hermite approximation of the gausian increments we provide lower error bounds of order arbitrarily close to 1 2 in time and 1 3 in space for lipschitz viscosity solutions coupling probabilistic arguments with regularization techniques as introduced by krylov the corresponding order of the upper bounds is 1 4 in time and 1 5 in space for sufficiently regular solutions the order is 1 in both time and space for both bounds finally we propose techniques for further improving the accuracy of the individual components of the approximation

a three valued walsh transform from decimations of helleseth gong sequences the walsh transform of two level autocorrelation sequences has played an important role in the construction of the set of sequences in which any two sequences are orthogonal for p ary sequences there are only two basic classes of two level autocorrelation sequences with no subfield structures for an arbitrary odd prime p one is the class of p ary m sequences and the other is the class of p ary helleseth gong sequences in this paper the walsh transform of a subclass of p ary helleseth gong sequences and the walsh transform of their particular decimations are completely determined and are shown to be three valued

several new classes of bent functions from dillon exponents several new classes of binary and p ary regular bent functions are obtained in this paper the bentness of all these functions is determined by some exponential sums over finite fields most of which have close relations with the well known kloosterman sums

group localization and two problems of levine a k bousfield s h mathbb z localization of groups inverts homologically two connected homomorphisms of groups j p levine s algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two connected and for which the image normally generates we resolve an old problem concerning bousfield h mathbb z localization of groups and answer two questions of levine regarding algebraic closure of groups in particular we show that the kernel of the natural homomorphism from a group g to its bousfield h mathbb z localization is not always a g perfect subgroup in the case of algebraic closure of groups we prove the analogous result that this kernel is not always a union of invisible subgroups