the motivic nearby fiber and degeneration of stable rationality we prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities our proof is based on motivic specialization techniques and the criterion of larsen and lunts for stable rationality in the grothendieck ring of varieties

uniform asymptotics for spherical and cylindrical nonlinear acoustic waves generated by a sinusoidal source this paper is concerned with the nonlinear propagation of sound waves from spherical or cylindrical sinusoidal sources this problem is governed by the two dimensionless parameters e and r 0 and we seek asymptotic expansions valid uniformly in e r 0 as well as in space time away from the irreducible domain where both e and r 0 are of order unity various matching asymptotic domains of the parameter plane are defined and in each domain the space time asymptotic structure is given in many cases the leading term can actually be solved for analytically finally the application of the results for spherical waves to the important problem of nonlinear acoustic beam spreading are discussed

backlund transformations for nonlinear parabolic equations the general results backlund transformations bts are considered for nonlinear parabolic equations of the form u t u xx h u x u x t 0 the most general form of both the bt and the class is adopted and we show that the only nonlinear equations possessing bts are the slight generalizations of burgers equation obtained by adding a forcing term u t u xx 2 uu x a x t 0

iterated vanishing cycles convolution and a motivic analogue of a conjecture of steenbrink we prove a motivic analogue of steenbrink s conjecture 25 conjecture 2 2 on the hodge spectrum proved by m saito in 21 to achieve this we construct and compute motivic iterated vanishing cycles associated with two functions we are also led to introduce a more general version of the convolution operator appearing in the motivic thom sebastiani formula throughout the article we use the framework of relative equivariant grothendieck rings of varieties endowed with an algebraic torus action

exact n wave solutions for the non planar burgers equation an exact representation of n wave solutions for the non planar burgers equation u t uu x ju t u xx j m n m where m and n are positive integers with no common factors is given this solution is asymptotic to the inviscid solution for x q 0 t where q 0 is a function of the initial lobe area as lobe reynolds number tends to infinity and is also asymptotic to the old age linear solution as t tends to infinity the formulae for the lobe reynolds numbers are shown to have the correct behaviour in these limits the general results apply to all j m n m and are rather involved explicit results are written out for j 0 1 1 3 and 1 4 the case of spherical symmetry j 2 is found to be singular and the general approach set forth here does not work an alternative approach for this case gives the large time behaviour in two different time regimes the results of this study are com pared with those of crighton scott 1979

the generalized berg theorem and bdf theorem let a be a separable simple af algebra with finitely many extreme traces we give a necessary and sufficient condition for an essentially normal element x e m a i e ir x is normal r m a m a a is the quotient map having the form y a for some normal element y c m a and a e a we also show that a normal element x e m a can be quasidiagonalized if and only if the fredholm index ind a x 0 for all a 0 sp 7r x in the case that a is a simple c algebra of real rank zero with stable rank one and with continuous scale k1 a 0 and ko a has countable rank we show that a normal element x e m a with zero fredholm index can be written as

approximate solution methods for nonlinear acoustic propagation over long ranges the two cases of weak diffusivity and of high initial wave amplitude these solutions use results obtained earlier by nimmo crighton simpler approximate techniques to obtain similar solutions are then discussed the two approximate methods considered proposed by shooter et al and rudnick are based on physical considerations rather than asymptotic theory the validity of such methods is demonstrated for a broad though restricted range of physical situations in this paper we are concerned with the propagation of weakly nonlinear longitudinal waves subject both to the diffusive effects associated with viscosity and thermal conductivity and to the geometric effects associated with changing ray tube area we consider the signalling problem in which particle disturbance velocity is specified as a function of time at a source position the subsequent evolution of the disturbance under such conditions is modelled by the generalized burgers equations gbe of the form

the corona algebra of the stabilized jiang su algebra abstract let z be the jiang su algebra and k the c algebra of compact operators on an infinite dimensional separable hilbert space we prove that the corona algebra m z k z k has real rank zero we actually prove a more general result

a class of backlund transformations for equations of the type uxy f u ux the backlund transformation u f u u x uy is considered for equations of the form uxy f u ux the nonlinear forms of f u ux that admit such transformations are completely classified new backlund transformations for modified sine gordon modified sinh gordon and modified liouville equations are obtained

diagonality inc algebras soit a une c algebre et a sa les elements hermitiens soit i un ideal ferme separable dans une c algebre a et on suppose que les elements de a sa de spectres finis sont denses dans a sa si a a sa et e 0 alors il existe b a sa diagonal relativement a i et il existe x i sa tel que x e et a b x

on motivic zeta functions and the motivic nearby fiber we collect some properties of the motivic zeta functions and the motivic nearby fiber defined by denef and loeser in particular we calculate the relative dual of the motivic nearby fiber we give a candidate for a nearby cycle morphism on the level of grothendieck groups of varieties using the motivic nearby fiber

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

the ideal property the projection property continuous fields and crossed products abstract let a be the c algebra associated to an arbitrary continuous field of c algebras we give a necessary and sufficient condition for a to have the ideal property and if moreover a is separable we give a necessary and sufficient condition for a to have the projection property some applications of these results are given we also prove that many crossed products of commutative c algebras by discrete amenable groups have the projection property generalizing some of our previous results

equivariant motivic integration and proof of the integral identity conjecture for regular functions we develop denef loeser s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions in comparison with hartmann s work the equivariant grothendieck ring defined in this article is more elementary and it yields the application to the conjecture

abelian strict approximation in aw algebras and weyl von neumann type theorems in this paper for a c algebra a with m m a an aw algebra or equivalently for an essential norm closed two sided ideal a of an aw algebra m we investigate the strict approximability of the elements of m from commutative c subalgebras of a in the relevant case of the norm closed linear span a of all finite projections in a semi finite aw algebra m we shall give a complete description of the strict closure in m of any maximal abelian self adjoint subalgebra masa of a we shall see that the situation is completely different for discrete respectively continuous m in the discrete case for any masa c of a the strict closure of c is equal to the relative commutant c m while in the continuous case under certain conditions concerning the center valued quasitrace of the finite reduced algebras of m satisfied by all von neumann algebras c is already strictly closed thus in the continu ous case no elements of m which are not already belonging to a can be strictly approximated from commutative c subalgebras of a in spite of this pathology of the strict topology in the case of the norm closed linear span of all finite projections of a continuous semi finite aw algebra we shall prove that in general situations including also this case any normal y 2 m is equal modulo a to some x 2 m which belongs to an order theoretical closure of an appropriate commutative c subalgebra of a in other words if we replace the strict topology with order theoretical approximation weyl von neumann berg sikonia type theo rems will hold in substantially greater generality

compactifications of subvarieties of tori we study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non archimedean amoebas these compactifications have many nice properties for example any k boundary divisors intersect in codimension k we consider some examples including m 0 n subset overline m 0 n and more generally log canonical models of complements of hyperplane arrangements and compact quotients of grassmannians by a maximal torus

proofs of the integral identity conjecture over algebraically closed fields recently it has become well known that the conjectural integral identity is of crucial importance in the motivic donaldson thomas invariants theory for noncommutative calabi yau threefolds the purpose of this article is to consider different versions of the identity for regular functions and formal functions and to give them the positive answer for the algebraically closed ground fields technically the result on the motivic milnor fiber by hrushovski loeser using hrushovski kazhdan s motivic integration and nicaise s computations on motivic integrals on special formal schemes are main tools

geometric criteria for tame ramification we prove an a campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field k as a first application we relate the orders of the tame monodromy eigenvalues on the l adic cohomology of a k curve to the geometry of a relatively minimal sncd model and we show that the semi stable reduction theorem and saito s criterion for cohomological tameness are immediate consequences of this result as a second application we compute the error term in the trace formula for smooth and proper k varieties we see that the validity of the trace formula would imply a partial generalization of saito s criterion to arbitrary dimension

solutions of a system of forced burgers equation in this article we obtain explicit solutions of a system of forced burgers equation subject to some classes of bounded and compactly supported initial data and also subject to certain unbounded initial data in a series of papers rao and yadav 2010 1 3 obtained explicit solutions of a nonhomogeneous burgers equation in one dimension subject to certain classes of bounded and unbounded initial data earlier kloosterziel 1990 4 represented the solution of an initial value problem for the heat equation with initial data in l 2 r n e x 2 2 as a series of self similar solutions of the heat equation in r n here we express the solutions of certain classes of cauchy problems for a system of forced burgers equation in terms of self similar solutions of some linear partial differential equations

groups quasidiagonality and interpolation by multiplier projections we relate the following conditions on a a unital c algebra a with the fs property a k1 a 0 b every projection in m a a lifts c the general weyl von neumann theorem holds in m a any selfadjoint element h in m a can be written as h a lpi a for some selfadjoint element a in a some bounded real sequence ai and some mutually orthogonal projections pi in a with ez1 pi 1 d m a has fs and e interpolation by multiplier projections holds for any closed projections p and q in a with pq o there is a projection r in m a such that p b 4 a and that a x b if in addition a is stable combining the above results we obtain counterexamples to the conjecture of g k pedersen a has fs m a has fs for example the stabilized buncededdens algebras hence the generalized weyl von neumann theorem does not generally hold in l ha for a unital c algebras with fs

reciprocal auto backlund transformations invariant backlund transformations of reciprocal type are derived for a class of n 1 th order conservation laws a new auto backlund transformation for the harry dym equation is presented as a special case of the analysis

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

similarity solutions to three dimensional nonlinear diffusion equations the similarity transformation method has been used to solve three dimensional nonlinear diffusion equations the general lie group is calculated exact solutions to the cylindrical and spherical symmetry cases are found and their relations to some real physical processes are discussed