cyclic length in the tame brauer group of the function field of a p adic curve let f be the function field of a smooth curve over the p adic number field qp we show that for each prime to p number n the n torsion subgroup h 2 f n nbr f is generated by z n cyclic classes in fact the z n length is equal to two it follows that the brauer dimension of f is two first proved in 20 and any f division algebra of period n and index n 2 is decomposable

azumaya algebras with involutions res 2wr zbr s cores br r 1 one could ask whether this sequence is exact indeed if r and s are fields and char r 2 the exactness of the above sequence is proved in cl corollary 4 6 p 4731 let a be an azumaya s algebra whose class a belongs to kernel cores br s br r for a to belong to res br r it is necessary and sufficient that there exists an azumaya s algebra b brauer equivalent to a admitting involutions of first and second kinds which commute results of saltman assert lo that there exist algebras bi i 1 2 brauer equivalent to a which support involutions of the first and second kind respectively we construct section 4 a homomorphism h ker cores br s br r pic rjcores pit s whose kernel consists precisely of classes of azumaya s algebras which support commuting involutions of both kinds thus h is a measure of the obstruction to the sequence 1 being exact in particular if pit r cores pit s 1 is exact

second order linear differential equations with piecewise constant arguments subject to nonlocal boundary conditions abstract we analyze the existence of solution for the class of second order linear differential equations with functional dependence x t ax t bx t cx t dx t t t j 0 t where a b c d r t 0 and is a piecewise continuous function subject to a certain type of nonlocal mixed boundary conditions which include the value of the derivative of x at the initial point x 0

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

symbol length and stability index abstract we show that a pythagorean field more generally a reduced abstract witt ring has finite stability index if and only if it has finite 2 symbol length we give explicit bounds for the two invariants in terms of one another to approach the question whether those bounds are optimal we consider examples of pythagorean fields

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

the u invariant of the function fields of p adic curves the u invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field it is an open question whether the u invariant of function fields of p aidc curves is 8 in this paper we answer this question in the affirmative for function fields of nondyadic p adic curves

green s function and comparison principles for first order periodic differential equations with piecewise constant arguments in this paper we obtain the expression of the green s function related with a first order periodic differential equation with piecewise constant argument we derive comparison results for the treated linear operator by studying the sign of the obtained green s function

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

a continuous method for nonlocal functional differential equations with delayed or advanced arguments abstract in the previous works the authors presented the reproducing kernel method rkm for solving various differential equations however to the best of our knowledge there exist no results for functional differential equations the aim of this paper is to extend the application of reproducing kernel theory to nonlocal functional differential equations with delayed or advanced arguments and give the error estimation for the present method some numerical examples are provided to show the validity of the present method

green s function for second order periodic boundary value problems with piecewise constant arguments we obtain under suitable conditions the green s function to express the unique solution for a second order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function this expression is given in terms of the solutions for certain associated problems the sign of the solution is determined taking into account the sign of that green s function

nonlocal boundary value problems for second order functional differential equations abstract we study the existence of a solution to a nonlocal boundary value problem for a class of second order functional differential equations with piecewise constant arguments the equation studied includes terms depending on the derivative as well as delay arguments in the derivative

an upper bound on the essential dimension of a central simple algebra abstract we prove a new upper bound on the essential p dimension of the projective linear group pgl n

green s function for second order differential equations with piecewise constant arguments abstract in this paper the second order nonautonomous differential equations with piecewise constant arguments x t t t x t t t t x t c t x t d t x t t t 0 t x 0 x t x 0 x t is considered we obtain the green s function to express the unique solution for the system

on the reduction principle for differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a new type of differential equations with piecewise constant argument epcag more general than epca k l cooke j wiener retarded differential equations with piecewise constant delays j math anal appl 99 1984 265 297 j wiener generalized solutions of functional differential equations world scientific singapore 1993 the reduction principle v a pliss the reduction principle in the theory of the stability of motion izv akad nauk sssr ser mat 27 1964 1297 1324 in russian v a pliss integral sets of periodic systems of differential equations nauka moskow 1977 in russian is proved for epcag the structure of the set of solutions is specified we establish also the existence of global integral manifolds of quasilinear epcag in the so called critical case and investigate the stability of the zero solution

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

second order neutral delay differential equations with piecewise constant time dependence abstract we find conditions for the existence and uniqueness of almost periodic solutions of second order neutral delay differential equations with almost periodic time dependence of the form x t px t 1 qx t f t here is the greatest integer function p and q are nonzero constants and f t is bohr almost periodic

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

on the behaviour of brauer p dimensions under finitely generated field extensions abstract the present paper shows that if q p or q 0 where p is the set of prime numbers then there exist characteristic q fields e q k k n of brauer dimension brd e q k k and infinite absolute brauer p dimensions abrd p e q k for all p p not dividing q 2 q this ensures that brd p f q k p q 2 q for every finitely generated transcendental extension f q k e q k we also prove that each sequence a p b p p p satisfying the conditions a 2 b 2 and 0 b p a p equals the sequence abrd p e brd p e p p for a field e of characteristic zero

noncyclic division algebras over fields of brauer dimension one abstract let k be a complete discretely valued field of rank one with residue field q p it is well known that period equals index in br k we prove that when p 2 there exist noncyclic k division algebras of every 2 power degree divisible by four otherwise every k division algebra is cyclic this gives the first published example of a field whose brauer dimension and cyclic length are not equal

colliot thelene s conjecture and finiteness of u invariants let k be a number field and f a function field in one variable over k we prove that the ramification of a p torsion element in br f on a regular proper model over the ring of integers in k can be split in an extension of degree p 3 using this result we show that colliot thelene s conjecture on 0 cycles of degree 1 implies finiteness for the u invariant of the function field of a curve over a totally imaginary number field and period index bounds for the brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers

on the second order differential equation with piecewise constant argument and almost periodic coefficients in this paper we study the existence of almost and quasi periodic solutions to two classes of second order differential equations as a corollary it is shown that periodic and unbounded solutions can coexist for the equation x t 2x t bx t f t which is different from the case b 0 this phenomena is due to the piecewise constant argument and illustrates a crucial difference between ordinary differential equations and differential equations with piecewise constant argument the results are extended to nonlinear equations

local global principles for torsors over arithmetic curves we consider local global principles for torsors under linear algebraic groups over function fields of curves over complete discretely valued fields the obstruction to such a principle is a version of the tate shafarevich group and for groups with rational components we compute it explicitly and show that it is finite this yields necessary and sufficient conditions for local global principles to hold our results rely on first obtaining a mayer vietoris sequence for galois cohomology and then showing that torsors can be patched we also give new applications to quadratic forms and central simple algebras

degree four cohomological invariants for certain central simple algebras abstract let f be a field char f 2 in the first section of the paper we prove that if a a b c d is a biquaternion algebra divisible by 2 in the brauer group br f and 1 1 f 0 then the symbol a b c d h 4 f z 2 z is an invariant i e it does not depend on the decomposition of a into a sum of two quaternions in the second section we construct an invariant p in h 4 f z 2 z for elements c 4 br f where c is cyclic of degree at most 4 and 2 br f in the case 1 f we extend the invariant p to elements d 4 br f where ind d 4 and 2 br f

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

differential equations with state dependent piecewise constant argument abstract a new class of differential equations with state dependent piecewise constant argument is introduced it is an extension of systems with piecewise constant argument fundamental theoretical results for the equations the existence and uniqueness of solutions the existence of periodic solutions and the stability of the zero solution are obtained appropriate examples are constructed

comparison of some field invariants in all this paper p is a prime number and f is a field of characteristic 6 p the invariants we are interested in are the p cohomological dimension cdp f 16 the diophantine dimension dd f inf i f is ci 5 for p 2 the quadratic diophantine dimension ddq f inf i f is c i where c q i is the condition introduced by pfister 13 for p 2 the u invariant u f 8 ch 11 the p invariant 7 for an element c pbr f h f p p c inf n c is a sum of n classes of algebras of degree p p f sup p c c pbr f the p invariant 7 for c as above p c logp ind c where ind c is the schur index of any central simple algebra representing c p f sup p c c pbr f to muddy water a little more we shall also consider the stable p and p invariants p f sup p e e f finite separable e f p 1 p f sup p e e f finite separable e f p 1