the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

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

linear preservers of weak majorization on lp i when p 1 abstract we present a necessary and sufficient conditions that a bounded linear map on l p i when p 1 may be considered as a linear preserver of weak majorization on l p i where i is an arbitrary infinite set we also prove that the class of all linear preservers of weak majorization is closed under the norm topology

linear preservers of circulant majorization on rn abstract let m n m be the set of all n m real matrices an n n real matrix a is called circulant doubly stochastic if it is a convex combination of circulant permutation matrices for x y r n x is said to be circulant majorized by y written as x c y if there exists a circulant doubly stochastic matrix d such that x d y in this paper the concept of circulant majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on r n

linear operators preserving directional majorization abstract a characterization of linear operators preserving directional majorization is given the result extends a theorem of t ando linear algebra appl 118 1989 163 and the proof also gives the characterization of linear operators preserving multivariate majorization recently obtained by beasley and lee

on motivic cohomology with z l coefficients in this paper we prove the conjecture of bloch and kato which relates milnor s k theory of a field with its galois cohomology as well as the related comparisons results for motivic cohomology with finite coefficients in the nisnevich and etale topologies

groups psl 3 p and nonorientable regular maps a 2 cell embedding of a graph on a nonorientable closed surface is called regular if its automorphism group acts regularly on its flags this paper gives a classification of nonorientable regular maps with the automorphism groups psl 3 p for a prime p equivalently we determine the representatives of orbits of aut psl 3 p acting on the set of involutory generating triples t r l of psl 3 p such that t l l t

weak matrix majorization abstract given x y r n m we introduce the following notion of matrix majorization called weak matrix majorization x w y if there exists a row stochastic matrix a r n n such that ax y and consider the relations between this concept strong majorization s and directional majorization it is verified that s w but none of the reciprocal implications is true nevertheless we study the implications w s and s under additional hypotheses we give characterizations of strong directional and weak matrix majorization in terms of convexity we also introduce definitions for majorization between abelian families of selfadjoint matrices called joint majorizations they are induced by the previously mentioned matrix majorizations we obtain descriptions of these relations using convexity arguments

on the residue fields of henselian valued stable fields abstract let k v be a henselian valued field satisfying the following conditions for a given prime number p i central division k algebras of finite p primary dimensions have schur indices equal to their exponents ii the value group v k properly includes its subgroup p v k the paper shows that if k is the residue field of k v and r is an intermediate field of the maximal p extension k p k then the natural homomorphism br k br r of brauer groups maps surjectively the p component br k p on br r p it proves that br k p is divisible if p 2 or k is a nonreal field and that br k 2 is of order 2 when k is formally real we also obtain that r embeds as a k subalgebra in a central division k algebra if and only if the degree r k divides the index of

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

the pseudo real genus of a group abstract a compact riemann surface is called pseudo real if it admits anti conformal orientation reversing automorphisms but no anti conformal automorphism of order 2 or equivalently if the surface is reflexible but not definable over the reals it is known that there exist pseudo real surfaces of genus g for every integer g 2 and the number of automorphisms of any such surface is bounded above by 12 g 1 in this paper we extend the concepts of symmetric genus strong symmetric genus and symmetric cross cap genus of a group by defining and investigating two new parameters as follows 1 the pseudo real genus g of a finite group g is the smallest genus of those pseudo real surfaces on which g acts faithfully as a group of automorphisms some of which might reverse orientation and 2 the strong pseudo real genus g of g is the smallest genus of those pseudo real surfaces on which g acts faithfully as a group of automorphisms some of which do reverse orientation when there exists such a surface for g our main theorem is that for every integer g 2 there exists a finite group g for which g g g and hence that the range of each of the functions and is the set of all integers g 2 we also give an example of a group g for which g is defined but g g

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

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

b majorization and its linear preservers abstract in this paper the concept of b majorization is studied and some facts about this relation are found next the linear preservers of b majorization on r n and m n m will be characterized it is also shown that the nonzero linear preservers and strong linear preservers of b majorization on r n are the same

on linear preservers of right matrix majorization abstract an n m matrix a is said to be matrix majorized or more precisely matrix majorized from the right by an n m matrix b and write a b if there exists a row stochastic matrix r such that a br we characterize the linear operators that preserve the matrix majorization

surface symmetries and psl_2 p we classify up to conjugacy all orientation preserving actions of psl 2 p on closed connected orientable surfaces with spherical quotients this classification is valid in the topological pl smooth conformal geometric and algebraic categories and is related to the inverse galois problem

row and column majorization on mn m abstract let m n m be the set of all n m matrices with entries in r for a b m n m it is said that a is row majorized respectively column majorized by b if every row respectively column of a is majorized by the corresponding row respectively column of b i e for every i 1 i n there exists a doubly stochastic matrix d i such that a i b i d i where a i and b i are the i th rows of a and b respectively in this paper the relations row and column majorization on m n m are studied and also all linear operators t m n m m n m preserving or strongly preserving row or column majorization will be characterized

linear operators preserving multivariate majorization abstract in this paper we characterize linear operators on real matrices that preserve multivariate majorization we place no additional restrictions on the linear operator

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

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