a bound to kill the ramification over function fields let k be a field of characteristic zero let x be a geometrically integral k variety of dimension n and let k be its field of functions we prove that given an element hm k r m there exist n2 functions fi i 1 n2 such that becomes unramified in l k f11 r fn21 r

symbol length in the brauer group of a field we bound the symbol length of elements in the brauer group of a field k containing a cm field for example any field containing an algebraically closed field or a finite field and solve the local exponent index problem for a cm field f in particular for a cm field f we show that every f central simple algebra of exponent p is similar to the tensor product of at most len p f t pm 1 1 symbol algebras of degree p we then use this bound on the symbol length to show that the index of such algebras is bounded by p 1 which in turn gives a bound for any algebra of exponent n via the primary decomposition finally for a field k containing a cm field f we show that every f central simple algebra of exponent p and degree p is similar to the tensor product of at most len p p k len p l symbol algebras of degree p where l is a cm edl a ps t 1 field

the period index problem for fields of transcendence degree 2 using geometric methods we prove the standard period index conjecture for the brauer group of a field of transcendence degree 2 over fp

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

on nonlinear programming in complex spaces abstract the present approach is based essentially on theorems of the alternative which are exploited mainly in real spaces in this paper only finite dimensional spaces are considered making use of separation arguments and of the one to one correspondence between c n and r 2 n both the real and the complex case can be considered from the same point of view then optimality conditions are established both when the objective function is expressed in terms of real and imaginary parts and when it is expressed in terms of the real part only

the period index problem for the brauer group of an algebraic surface in this paper we show that the period equals the index for elements of brauer groups of function fields of surfaces a key idea of the proof is that any azumaya algebra over a surface can be transformed into an azumaya algebra that is unobstructed

nonstandard finite difference equations for odes and 1 d pdes based on piecewise linearization a method for the solution of initial and boundary value problems in nonlinear ordinary differential equations and for one dimensional partial differential equations which provides c 1 solutions is presented the method is based on the linearization of the differential equation in intervals which contain only two grid points and provides three point nonstandard finite difference equations for the nodal amplitudes the method is applied to steady reaction diffusion equations two point singularly perturbed boundary value problems and the steady burgers equation and compared with standard finite difference and finite element formulations for one dimensional partial differential equations the temporal derivatives are first discretized and the resulting ordinary differential equation accounts for both the temporal and spatial stiffnesses and is solved by means of piecewise linearization since the linearization includes a jacobian matrix it may be easily employed to refine the mesh where steep gradients occur

numerical solution of singular perturbation problems via deviating arguments we present a new approach to numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval the original problem is divided into outer and inner region problems a terminal boundary condition in the implicit form is derived then the outer region problem is solved as a two point boundary value problem tpbvp and an explicit terminal boundary condition is obtained in turn a new inner region problem is obtained and solved as a tpbvp using the explicit terminal boundary condition the proposed method is iterative on the terminal point of the inner region some numerical examples have been solved to demonstrate the applicability of the method

improvement of numerical solution by boundary value technique for singularly perturbed one dimensional reaction diffusion problem a class of singularly perturbed two point boundary value problems bvps for second order ordinary differential equations in self adjoint form arising in the study of chemical catalysis and michaelis menten process in biology is considered in order to solve them a numerical method as in appl math comput 55 1993 31 is proposed the essential idea in this method is to divide the domain of the differential equation into three non overlapping subdomains and solve the given equations over these regions separately as three two point bvps numerically the inner region problems are solved using a fitted operator method uniform numerical methods for problems with initial and boundary layers boole press dublin 1980 whereas the outer region problem is solved using a classical central difference method the boundary conditions at the transition points are obtained using the zero order asymptotic expansion approximation to the solution of the problem this method is well suited for parallel computing and an algorithm for the same is given error estimates are derived for the numerical solution some schemes for self adjoint equations in conservation form are given using newton s method of quasilinearization a class of semilinear problems are also solved numerical experiments are conducted it is found that the present method performs better than the fitted mesh method fitted numerical methods for singular perturbation problems world scientific singapore 1996 and higher order method numeriche mathematik 56 1990 675

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

unramified division algebras do not always contain azumaya maximal orders we show that in general over a regular integral noetherian affine scheme x of dimension at least 6 there exist brauer classes on x for which the associated division algebras over the generic point have no azumaya maximal orders over x despite the algebraic nature of the result our proof relies on the topology of classifying spaces of algebraic groups

symmetric duality for mathematical programming in complex spaces with f convexity abstract under f convexity f concavity f pseudoconvexity f pseudoconcavity appropriate duality results for a pair of wolfe and mond weir type symmetric dual nonlinear programming problems in complex spaces are established these results are then used to develop second order f convexity f concavity second order f pseudoconvexity f pseudoconcavity and appropriate second order symmetric dual nonlinear programming problems in complex spaces

a computational method for solving singular perturbation problems abstract a class of singularly perturbed two point boundary value problems for second order ordinary differential equations is considered in order to solve them a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods the proposed method is distinguished by the following facts first we divide the given interval the domain of definition of differential equation into two subintervals called inner and outer regions then we solve the differential equation over both the regions as two point boundary value problems the terminal boundary condition of the inner region is obtained using the zero order asymptotic expansion of the solution some numerical examples are given to illustrate the method

cyclic algebras over p adic curves abstract in this paper we study division algebras over the function fields of curves over q p the first and main tool is to view these fields as function fields over nonsingular s which are projective of relative dimension 1 over the p adic ring z p a previous paper showed such division algebras had index bounded by n 2 assuming the exponent was n and n was prime to p in this paper we consider algebras of prime degree and hence exponent q p and show these algebras are cyclic we also find a geometric criterion for a brauer class to have index q

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

optimality and duality for complex nondifferentiable fractional programming abstract both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of complex nondifferentiable fractional programming problems containing generalized convex functions subsequently these optimality criteria are utilized as a basis for constructing one parametric and two other parameter free dual models with appropriate duality theorems

complex minimax programming under generalized convexity we establish the kuhn tucker type sufficient optimality conditions for complex minimax programming under generalized invex functions subsequently we apply these optimality criteria to formulate two dual models we also establish weak strong and strict converse duality theorems

a computational method for solving quasilinear singular perturbation problems a class of quasilinear singularly perturbed two point boundary value problems for second order ordinary differential equations without interior turning points is considered to solve these problems newton s method of quasilinearization is adopted then the resultant linear problems are solved by the numerical method suggested in 1 the method presented in 1 is a combination of an exponentially fitted finite difference method and a classical numerical method further it is based on the boundary value technique 2 generally used to solve singularly perturbed boundary value problems error estimates for the numerical solution of linear problems are stated some examples are given to illustrate the method

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

on the method of modified equations vi asymptotic analysis of and asymptotic successive corrections techniques for two point boundary value problems in ode s the modified equation technique is extended to two point boundary value problems and a second order accurate implicit centered finite difference scheme for non homogeneous second order ordinary differential equations with linear boundary conditions is analyzed the first second and third modified equations or equivalent second equivalent and simply modified equations respectively for this scheme and its boundary conditions are presented it is shown that the three kinds of modified equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results for a linear problem multiple scales and summed up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original finite difference scheme asymptotic successive corrections techniques are also applied to the three kinds of modified equations to obtain higher order schemes higher order boundary conditions are easily treated in the asymptotic successive corrections technique although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries the methods introduced in this paper are applied to homogeneous and non homogeneous second order linear and non linear ordinary differential equations and yield very accurate results

period index bounds for arithmetic threefolds the standard period index conjecture for brauer groups of p adic surfaces s predicts that mathrm ind alpha mathrm per alpha 3 for every alpha in mathrm br mathbf q p s using gabber s theory of prime to ell alterations and the deformation theory of twisted sheaves we prove that mathrm ind alpha mathrm per alpha 4 for alpha of period prime to 6p giving the first uniform period index bounds over such fields

generalized α invexity and nondifferentiable minimax fractional programming in this paper we study a nondifferentiable minimax fractional programming problem under the assumptions of a invex function in this paper we utilize the concept of a invexity m a noor on generalized preinvex functions and monotonicities j inequalities pure appl math 5 2004 1 9 and pseudo a invexity s k mishra m a noor on vector variational like inequality problems j math anal appl 311 2005 69 75 we also introduce the concept of strict pseudo a invex and quasi a invex functions we derive karush kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different dual problems the results in this paper extend several known results in the literature

period index and u invariant questions for function fields over complete discretely valued fields let k be a complete discretely valued field with residue field and f the function field of a curve over k let p be the characteristic of and l a prime not equal to p if the brauer l dimensions of all finite extensions of are bounded by d and the brauer l dimensions of all extensions of of transcendence degree at most 1 are bounded by d 1 then it is known that the brauer l dimension of f is at most d 2 lieblich in j reine angew math 659 1 41 2011 saltman in j ramanujan math soc 12 25 47 1997 harbater et al in invent math 178 231 263 2009 in this paper we give a bound for the brauer p dimension of f in terms of the p rank of as an application we show that if is a perfect field of characteristic 2 then any quadratic form over f in at least 9 variables is isotropic this leads to the fact that every element in h 3 f mathbb z 2 mathbb z is a symbol if is a finite field of characteristic 2 u f 8 is a result of heath brown leep heath brown in compos math 146 271 287 2010 leep in j reine angew math 2013 to appear

generalized α univexity and duality for nondifferentiable minimax fractional programming abstract the aim of the present paper is to study a nondifferentiable minimax fractional programming problem under the assumptions of univex and related functions in this paper we introduce the concepts of univex pseudo univex strict pseudo univex and quasi univex functions respectively by unifying the notions of invex and univex functions we derive karush kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different dual problems the results in this paper extend several known results in the literature

complex minimax programming under generalized type i functions we establish the kuhn tucker type sufficient optimality conditions for complex minimax programming under generalized type i and related functions subsequently we apply these optimality criteria to formulate two dual models we also establish weak strong and strict converse duality theorems