sums of squares in real analytic rings let a be an analytic ring we show 1 a has finite pythagoras number if and only if its real dimension is 2 and 2 if every positive semidefinite element of a is a sum of squares then a is real and has real dimension 2

the blow up rate for a system of heat equations with nonlinear boundary conditions this paper deals with the blow up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions the upper and lower bounds of blow up rate are obtained

continuous functions on the plane regular after one blowing up we study rational functions admitting a continuous extension to the real affine space first of all we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives afterwards since these functions correspond to rational functions which become regular after some blowings up we work on the plane where it suffices to blow up points and then we can count the number of stages of blowings up necessary in the latest parts of the paper we investigate the ring of rational continuous functions on the plane regular after one stage of blowings up in particular we prove a positivstellensatz without denominator in this ring

some remarks on the computation of pythagoras numbers of real irreducible algebroid curves through gram matrices we clarify a difficulty that appears in r quarez j algebra 238 2001 139 to bound the pythagoras number of a real irreducible algebroid curve by its multiplicity

some remarks on pythagorean real curve germs let k be a real closed field a real ap curve over k is a 1 dimensional excellent henselian local real domain with residue field k a 1 dimensional noetherian local ring is arf if emb dim b mult b for every local ring b infinitely near to a j lipman amer j math 93 1971 649 685 for n 1 the 2nth pythagoras number p2n of a commutative ring a is the least p 1 p such that any sum of 2nth powers in a is a sum of no more than p2nth powers in a a main purpose of this paper is to affirm the following conjectures proposed by ruiz j algebra 94 1985 no 1 126 144 let a be a real ap curve and let a be pythagorean i e p2 1 then i a is arf ii every local ring infinitely near to a is pythagorean actually the authors obtain a finer result for a real ap curve a the following assertions are equivalent 1 a is arf 2 a is pythagorean 3 p2n 1 for some n 4 p2n 1 for all n here 2 1 is exactly conjecture i and 1 2 reduces conjecture ii to the obvious fact that if a is arf every local ring infinitely near to a is arf too of course the result contains some additional insight into the study of pythagoras s numbers even of higher order of real curve germs

sums of two squares in analytic rings we study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions this is a basic useful property of the plane but difficult to check in other cases in particular what about z 2 xy z 2 yx 2 y 3 z 2 x 3 y 4 or z 2 x 3 xy 3 in fact the unique positive examples we can find are the brieskorn singularity the union of two planes in 3 space and the whitney umbrella conversely we prove that a complete intersection with that property other than the seven embedded surfaces already mentioned must be a very simple deformation of the two latter namely in particular except for the stems z 2 x 2 and z 2 yx 2 all singularities are real rational double points

impossibility of c variation or formal power series variation in solutions to hilbert s 17th problem abstract no matter how a positive semidefinite polynomial f r x 1 x n is represented according to e artin s 1926 solution to hilbert s 17th problem in the form f piri2 with 0 p i r and r i r x 1 x n the pi and the coefficients of the ri cannot be chosen to depend in a c i e infinitely differentiable manner upon the coefficients of f unless degf 2 formal powers series variation is also impossible this answers a question we had raised in 1990 contemp math vol 155 amer math soc 1994 pp 107 117 where we had already shown that real analytic variation was impossible and gonzalez vega and lombardi math z 225 3 1997 427 451 then showed that for every fixed finite r n cr variation is possible improving upon their and the author s result that continuous piecewise polynomial variation is possible

on the nullstellensatze for stein spaces and c analytic sets in this work we prove the real nullstellensatz for the ring o x of analytic functions on a c analytic set x rn in terms of the saturation of ojasiewicz s radical in o x the ideal i a of the zero set a of an ideal a of o x coincides with the saturation formula presented of ojasiewicz s radical formula presented if a has good properties concerning hilbert s 17th problem then i a formula presented where formula presented stands for the real radical of a the same holds if we replace formula presented with the real analytic radical formula presented of a which is a natural generalization of the real radical ideal in the c analytic setting we revisit the classical results concerning hilbert s nullstellensatz in the framework of complex stein spaces let a be a saturated ideal of o rn and yrn the germ of the support of the coherent sheaf that extends aorn to a suitable complex open neighborhood of rn we study the relationship between a normal primary decomposition of a and the decomposition of yrn as the union of its irreducible components if a p is prime then i p p if and only if the complex dimension of yrn coincides with the real dimension of p

sums of squares on real algebraic curves given an affine algebraic variety v over with real points v compact and a non negative polynomial function f v with finitely many real zeros we establish a local global criterion for f to be a sum of squares in v we then specialize to the case where v is a curve the notion of virtual compactness is introduced and it is shown that in the local global principle compactness of v can be relaxed to virtual compactness the irreducible curves on which every non negative polynomial is a sum of squares are classified all results are extended to the more general framework of preorders moreover applications to the k moment problem from analysis are given in particular schmudgen s solution of the k moment problem for compact k is extended for dim k 1 to the case when k is virtually compact

eigenvalues of symmetric matrices over integral domains abstract given an integral domain a we consider the set of all integral elements over a that can occur as an eigenvalue of a symmetric matrix over a we give a sufficient criterion for being such an element in the case where a is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element the latter result settles a questions raised by bass estes and guralnick

a going down theorem for real spectra they also proved a positive result theorem 1 1 with the extra assumption that is finitely presented cr 6 3ii unfortunately that assumption was too restrictive it excluded for instance completions of polynomial rings in any case counterexample 1 2 already suggests that going down need not be related to finiteness conditions on thus our goal in theorem 1 1 is to drop any restriction on ii except regularity a nice immediate consequence is

approximation of maps into spheres by piecewise regular maps of class c k the aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise regular maps of class mathcal c k where k is an arbitrary nonnegative integer

blow up analysis for a system of heat equations with nonlinear flux which obey different laws abstract we consider a system of heat equations u t u and v t v in 0 t completely coupled by nonlinear boundary conditions u e p v u v u q e v on 0 t we prove that the solutions always blow up in finite time for non zero and non negative initial values also the blow up only occurs on with c 1 t t p 2 p q max u x t c 2 t t p 2 p q log c 3 t t q 1 2 p q max v x t log c 4 t t q 1 2 p q for p q 0 0 1 and 0 p

fibers of flat morphisms and weierstrass preparation theorem abstract we characterize flat extensions of commutative rings satisfying the weierstrass preparation theorem using this characterization we prove a variant of the weierstrass preparation theorem for rings of functions on a normal curve over a complete local domain of dimension one this generalizes recent works of harbater hartmann and krashen with a different method of proof

a convex optimization model for finding non negative polynomials this paper presents a convex optimization model for the problem of finding some polynomials for which certain linear combinations are non negative polynomials this model is then applied to solve several filter design problems we first reformulate some low pass filter design problems with finite or infinite impulse response as optimization problems over non negative real or complex polynomials whose feasibility problems can be solved by applying our model the whole optimization problems are then solved by using a combination of a bisection search procedure on an appropriate parameter and our convex optimization model to solve the feasibility problems some numerical examples illustrate the method

the length and other invariants of a real field the length of a field is the smallest integer m such that any totally positive quadratic form of dimension m represents all sums of squares we investigate this field invariant and compare it to others such as the u invariant the pythagoras number the hasse number and the mordell function related to sums of squares of linear forms

pythagoras numbers of real irreducible algebroid curves and gram matrices abstract using elementary techniques of linear algebra in relation with gram matrices we bound the pythagoras number of a real irreducible algebroid curve by its multiplicity

weighted sums of squares in local rings and their completions i let a be an excellent regular local ring of dimension two let t be a finitely generated preordering in a and let widehat t be the preordering generated by t in the completion widehat a of a we study the question when the property of being saturated descends from widehat t to t and establish conditions of geometric nature which allow to decide this question as an application we classify all principal preorderings in a of degree 3 which are saturated in the case where a has a real closed residue field these results have direct implications for nonnegativity certificates for real polynomials on two dimensional semi algebraic sets

on the pythagoras numbers of real analytic rings abstract we show that the pythagoras number of a real analytic ring of dimension 2 is finite bounded by a function of the multiplicity and the codimension

multiple blow up rates in a coupled heat system with mixed type nonlinearities abstract this paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed type nonlinearities at first the critical exponent is obtained and simply described via a characteristic algebraic system introduced by us then as the main results of the paper three blow up rates are established under different dominations of nonlinearities for the one dimensional case and represented in another characteristic algebraic system in particular it is observed that unlike those in previous literature on parabolic models with absorptions two of the multiple blow up rates obtained here do depend on the absorption exponents in the known works the absorptions affect the blow up criteria the blow up time as well as the initial data required for the blow up of solutions all without changing the blow up rates to our knowledge this is the first example of absorption dependent blow up rates exploiting the significant interactions among diffusions inner absorptions and nonlinear boundary fluxes in the coupled system it is also proved that the blow up of solutions in the model occurs on the boundary only

roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition in this paper we investigates the blow up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition we obtain the blow up condition and its blow up rate estimate

non simultaneous blow up in a parabolic system with three components abstract this paper deals with heat equations coupled via nonlinear boundary flux u 1 u 1 p 11 u 2 p 12 u 2 u 2 p 22 u 3 p 23 u 3 u 3 p 33 u 1 p 31 a necessary and sufficient condition for the existence of only one component blowing up for nondecreasing in time and radially symmetric solutions three kinds of exponent regions are obtained as follows i only one component blows up for every initial data ii the existence of two components blowing up simultaneously while the third one remains bounded also with two kinds of blow up rates 1 2 p 22 1 1 2 p 33 1 and p 23 1 p 33 2 p 33 1 1 2 p 33 1 iii e g u 1 remains bounded and u 2 u 3 blow up simultaneously with blow up rate p 23 1 p 33 2 p 33 1 1 2 p 33 1 for every initial data moreover the eight kinds of simultaneous blow up rates and blow up sets are discussed

on the approximation property of excellent rings where d is a smooth finite type a algebra a theorem which would imply the existence of an embedding of the above type yields the positive solution of the bass quillen conjecture so far this theorem is only known in special cases there is the following result by artin 3 4

blow up phenomena for a system of semilinear heat equations with nonlinear boundary flux abstract this paper deals with the blow up for a system of semilinear r laplace heat equations with nonlinear boundary flux it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time and when blow up does occur upper and lower bounds for the blow up time are obtained

the hamiltonian structure of a complex version of the burgers hierarchy in this paper we construct a class of integrable hamiltonian nonlinear evolution equations generated by a purely differential recursion operator it turns out that this hierarchy is a complex version of the burgers hierarchy and can be linearized through a generalization of the cole hopf transformation

multiple absorption related blow up rates to a coupled heat system this paper deals with asymptotic behavior of solutions to a heat system with absorptions and coupling positive multi nonlinearities it is known that although absorption mechanisms may affect such as blow up criteria blow up time and initial data required for blow up solutions they cannot change blow up rates of solutions in general it has been reported in the current literature that blow up rates for scalar equations with absorptions are all absorption independent in a previous paper of the authors four absorption independent simultaneous blow up rates were obtained already for the same problem under weak absorptions the present paper will furthermore prove that if the absorptions are unbalanced in the model i e the absorption is stronger for one component and weaker for another then there are in addition eight possible absorption related blow up rates for the model besides the four absorption independent ones this exposes a significant difference between scalar and coupled nonlinear parabolic equations with absorptions

sums of squares in real rings let a be an excellent ring we show that if the real dimension of a is at least three then a has in finite pythagoras number and there exists a positive semidefinite element in a which is not a sum of squares in a

positive semidefinite germs in real analytic surfaces we find all real analytic surface germs in mathbb r 3 on which every positive semidefinite function germ is a sum of squares in fact of two squares of analytic function germs

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

existence of positive solutions for a class of degenerate or singular equations abstract in this paper we consider the class of equations div x 2 a u a x 2 a u u x b p x r n 0 where a b r n 2 a r 0 and 0 p 2 n n 2 1 a b in various range of the parameters involved we obtain both results of existence and nonexistence of positive solutions by combining the mountain pass lemma and the pohozaev type identity

the blow up rate for a strongly coupled system of semilinear heat equations with nonlinear boundary conditions abstract the paper deals with the blow up rate of positive solutions to the system u t u xx u l 11 v l 12 v t v xx u l 21 v l 22 with boundary conditions u x 1 t u p 11 v p 12 1 t and v x 1 t u p 21 v p 22 1 t under some assumptions on the matrices l l ij and p p ij and on the initial data u 0 v 0 the solution u v blows up at finite time t and we prove that max x 0 1 u x t resp max x 0 1 v x t goes to infinity as t t 1 2 resp t t 2 2 where i p id 1 2 t 1 1 t

positive solutions for a system of fractional differential equations with coupled integral boundary conditions we investigate the existence of positive solutions for a system of nonlinear riemann liouville fractional differential equations with coupled integral boundary conditions

blow up for a semilinear reaction diffusion system coupled in both equations and boundary conditions we study the blow up behavior for a semilinear reaction diffusion system coupled in both equations and boundary conditions the main purpose is to understand how the reaction terms and the absorption terms affect the blow up properties we obtain a necessary and sufficient condition for blow up derive the upper bound and lower bound for the blow up rate and find the blow up set under certain assumptions

blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions abstract this paper concerns with blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space we establish the rate estimates for blow up solutions and prove that the blow up set is r n under proper conditions on initial data furthermore for n 1 more complete conclusions about such two topics are given

semi algebraic geometry with rational continuous functions let x be a real algebraic subset of mathrm mathbb r n we investigate the theory of algebraically constructible functions on x and the description of the semi algebraic subsets of x when we replace the polynomial functions on x by some rational continuous functions on x

piecewise regular maps let v w be real algebraic varieties that is up to isomorphism real algebraic sets and x subseteq v some subset a map from x into w is said to be regular if it can be extended to a regular map defined on some zariski locally closed subvariety of v that contains x furthermore a continuous map f x rightarrow w is said to be piecewise regular if there exists a stratification mathscr s of v such that for every stratum s in mathscr s the restriction of f to each connected component of x cap s is a regular map by a stratification of v we mean a finite collection of pairwise disjoint zariski locally closed subvarieties whose union is equal to v assuming that the subset x of v is compact we prove that every continuous map from x into a grassmann variety or a unit sphere can be approximated by piecewise regular maps as an application we obtain a variant of the algebraization theorem for topological vector bundles if the variety v is compact and nonsingular we prove that each continuous map from v into a unit sphere is homotopic to a piecewise regular map of class mathscr c k where k is an arbitrary nonnegative integer

non simultaneous blow up in coupled heat equations with multi nonlinearities abstract this paper studies coupled heat equations with multi nonlinearities of six nonlinear parameters the critical blow up exponent is established via a complete classification for all the six nonlinear parameters where a precise analysis on the geometry of and the absorption coefficients is given for the balanced interaction situation among the multi nonlinearities the main attention is contributed to non simultaneous phenomena in the model to determine the necessary and sufficient conditions of non simultaneous blow up with suitable initial data as well as the conditions under which any blow up must be non simultaneous finally the model of the paper is characterized by a comparison with the known results for other models

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

on the pythagoras numbers of real analytic curves we show that the pythagoras number of a real analytic curve is the supremum of the pythagoras numbers of its singularities or that supremum plus 1 this includes cases when the pythagoras number is infinite

approximation in compact nash manifolds let rn be a compact nash manifold a b the rings of nash analytic global functions on the main result of this paper is the following theorem 1 let be a pair of nash submanifolds of some rn rq and let us suppose is compact let f1 fq r be nash functions then every analytic solution y f x of the system f1 x y fq x y 0 can be approximated in the whitney topology by the global nash solutions y g x the main tool used to prove the above results is this version of neron s desingularisation theorem any homomorphism of a algebras c b with c finitely generated over a factorizes through a finitely generated a algebra d such that a d is regular using theorem 1 the authors are able to solve several interesting problems that have been open for many years for example they prove i every analytic factorization of a global nash function defined over is equivalent to a nash factorization ii every semialgebraic subset of which is a global analytic subset is also a global nash subset iii every prime ideal of a generates a prime ideal in b iv every coherent ideal subsheaf of the sheaf n of nash functions on is generated by its global sections the case where is noncompact is only partially studied in this paper in the reviewer s opinion this article makes crucial progress in the theory of global nash functions

distinguished representations of non negative polynomials abstract let g 1 g r r x 1 x n such that the set k g 1 0 g r 0 in r n is compact we study the problem of representing polynomials f with f k 0 in the form f s 0 s 1 g 1 s r g r with sums of squares s i with particular emphasis on the case where f has zeros in k assuming that the quadratic module of all such sums is archimedean we establish a local global condition for f to have such a representation vis a vis the zero set of f in k this criterion is most useful when f has only finitely many zeros in k we present a number of concrete situations where this result can be applied as another application we solve an open problem from s kuhlmann et al positivity sums of squares and the multi dimensional moment problem ii adv geometry in press on one dimensional quadratic modules

on the positive extension property and hilbert s 17th problem for real analytic sets in this work we study the positive extension pe property and hilbert s 17th problem for real analytic germs and sets a real analytic germ x 0 of r 0 n has the pe property if every positive semidefinite analytic function germ on x 0 has a positive semidefinite analytic extension to r 0 n analogously one states the pe property for a global real analytic set x in an open set q of r 0 n these pe properties are natural variations of hilbert s 17th problem here we prove that 1 a real analytic germ x 0 subset of r 0 3 has the pe property if and only if every positive semidefinite analytic function germ on x 0 is a sum of squares of analytic function germs on x 0 and 2 a global real analytic set x of dimension 2 and local embedding dimension 3 has the pe property if and only if it is coherent and all its germs have the pe property if that is the case every positive semidefinite analytic function on x is a sum of squares of analytic functions on x moreover we classify the singularities with the pe property

an existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions the existence and uniqueness of positive solutions for a second order nonlinear system with coupled integral boundary value conditions are investigated the main tools used in the proofs are a priori estimate method and a maximal principle

sum of squares length of real forms for n d ge 1 let p n 2d denote the smallest number p such that every sum of squares of degree d forms in mathbb r x 1 ldots x n is a sum of p squares we establish lower bounds for p n 2d that are considerably stronger than the bounds known so far combined with known upper bounds they give p 3 2d in d 1 d 2 in the ternary case assuming a conjecture of iarrobino kanev on dimensions of tangent spaces to catalecticant varieties we show that p n 2d sim const cdot d n 1 2 for d rightarrow infty and all n ge 3 for ternary sextics and quaternary quartics we determine the exact value of the invariant showing p 3 6 4 and p 4 4 5

blow up estimates of the positive solution of a parabolic system this paper establishes the blow up estimates for the systems ut u 0 vt v 0 in br 0 t br rn with the nonlinear boundary conditions u um1vn1 and v um2vn2 on sr 0 t where 0 m1 1 m2 and 0 n2 1 n1 we prove that c t t 2 max u x t c t t 2 and c t t 2 max v x t c t t 2 under some monotonicity assumptions on the initial values where n1 n2 1 m2 m1 1 and n1m2 1 m1 1 n2

the pythagoras number and the u invariant of laurent series fields in several variables abstract we show that every sum of squares in the three variable laurent series field r x y z is a sum of 4 squares as was conjectured in a paper of choi dai lam and reznick in the 1980 s we obtain this result by proving that every sum of squares in a finite extension of r x y is a sum of 3 squares it was already shown in choi dai lam and reznick s paper that every sum of squares in r x y itself is a sum of two squares we give a generalization of this result where r is replaced by an arbitrary real field our methods yield similar results about the u invariant of fields of the same type

sums of squares in function fields over henselian local fields we give upper bounds for the level and the pythagoras number of function fields over fraction fields of integral henselian excellent local rings in particular we show that the pythagoras number of mathbb r x 1 dots x n is le 2 n 1 which answers positively a question of choi dai lam and reznick

blow up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition this paper is devoted to a nonlinear parabolic equation subject to nonlocal boundary condition and with nonlocal reaction source some criteria on this problem which determine whether the solutions blow up in a finite time or exist globally are given moreover the blow up rate estimates are also obtained

global blow up for a localized nonlinear parabolic equation with a nonlocal boundary condition abstract this paper deals with the blow up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition under certain conditions the blowup criteria is established furthermore when f u u p 0 p 1 the global blowup behavior is shown and the blowup rate estimates are also obtained

analytic surface germs with minimal pythagoras number we determine all complete intersection surface germs whose pythagoras number is 2 and find that they are all embedded in 3 and have the property that every positive semidefinite analytic function germ is a sum of squares of analytic function germs in addition we discuss completely these properties for mixed surface germs in 3 finally we find in higher embedding dimension three different families with these same properties

the weierstrass approximation theorem for maps between real algebraic varieties p n r for some n endowed with the zariski topology and the sheaf of rvalued regular functions morphisms between real algebraic varieties will be called regular maps every real algebraic variety carries also the euclidean topology that is the topology induced by the usual metric topology on r unless explicitly stated otherwise all topological notions related to real algebraic varieties will refer to the euclidean topology given two real algebraic varieties x and y we regard the set r x y of all regular maps from x into y as a subset of the space c x y of all continuous maps fromx into y endowed with the compact open topology if x and y are nonsingular then we also regard r x y as a subset of the space c x y of all c maps from x into y endowed with the c compact open topology the weak c topology in the terminology used in 6