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In this paper we discuss the local regularity of solutions of a nonlocal system of equations which describe the motion of a viscoelastic medium in several space dimensions. Our main tool is the microlocal analysis combined with MacCamy's trick and the argument of the classical energy method. 0. INTRODUCTION In this paper we discuss the local regularity of solutions of an integro-differential system which describes the motion of a linear viscoelastic medium in several space dimensions. The system is given by the following equations: ut4(t, X) C#f(t, X) a uj(t, X) +u(t, x) uJ(t, x) XaO aaaXl (0. 1) +1 ta.( )aa u'(s ,x) ds +; Ni9J(t, s, 'x)j90(s, x)ds, for i =1,..., n, in (0, oo) x Rn, (0.2) u(0, x) = uo(x), ut(0, x) = u(x), in R', where x = (xI, ... , x") E Rn and u = (ul, ..., u) denotes the displacement from equilibrium. We employ the convention of summation on repeated indices. The system (0.1) stands for the conservation of linear momentum. The integral terms are accountable for the effect of memory. Since these terms are nonlocal, the problem addressed in the present paper is a new kind in the subject of local regularity of solutions. The constituent law for a linear viscoelastic material with memory is given by t (0.3) oiA (t, tX) =jy KIJV (t-s, x)ejfl (s, X) ds, 00 Received by the editors August 21, 1992. 1991 Mathematics Subject Classification. Primary 35B65, 35L99, 73F15.

Here we study meromorphic foliations with singularities on a smooth compact complex surface. Aim: study the locations of the singularities, using vector bundle techniques and techniques introduced in the cohomological study of projective geometry. Roughly speaking, a meromorphic foliation with singularities on a complex connected manifold S is described by a subsheaf F of the tangent bundle TS with F closed under Lie brackets (see for instance [Gl, G2, G3, G4, G5, GK, GM] or, for the case dim(5') = 2, here in 1.8.1). The singularities of the foliations are the points of S at which F is not a subbundle of TS ; with this definition, the singularities may occur only in codimension at least 2 (but a meromorphic foliation without singularities is not a holomorphic foliation in the classical sense). Note that any rank one subsheaf of TS is involutive and it is contained in a line bundle contained in TS (in general not as subbundle). Thus a rank one meromorphic foliation G with singularities on S is given by a line bundle M on S and a nonzero map M -+ TS (which induces an inclusion as sheaves of M in TS) (up to a scalar factor, of course); one imposes that (s)o has no hypersurface as component (i.e. that M is a saturated subsheaf of TS) ; equivalent^, G is given by M and s e H°(S, TS®M*) with s ? 0 (up to a scalar factor); the singularities of G are the zero locus (s)o of s. Such a foliation is called also a foliation by curves. Thus, for fixed M e Pic(S), we are looking at a linear problem. Many papers are concerned with this case (see e.g. [G2, G3, G4, GK]) from different points of view and with several different kinds of problems in mind. When dim(5) = 2 (as always in this paper) every foliation is a foliation by curves. For the case dim(5) = 2, relevant references are (among others) [G2 and G3]. From now on we assume dim(5) = 2. It happens very seldom that S has a meromorphic foliation without singularities; by definition this happens if and only if TS is an extension of two line bundles. Here we are interested essentially on the possible location (for various M e Pic(iS)) of the singularities associated to some s e H°(S, TS ® M*); very often we will be interested in a general such s. The main problems considered here are: (a) given a zero dimensional scheme Z c S, find if there is seH°{S, TS®M*), 5^0

This paper proves the Alperin’s weight conjecture for the finite unitary groups when the characteristic r of modular representation is odd. Moreover, this paper proves the conjecture for finite odd dimensional special orthogonal groups and gives a combinatorial way to count the number of weights, block by block, for finite symplectic and even dimensional special orthogonal groups when r and the defining characteristic of the groups are odd.

We prove that any infinite group interpretable in a separably closed field F of finite Ersov-invariant is definably isomorphic to an F-algebraic group. Using this result we show that any infinite field K interpretable in a separably closed field F is itself separably closed; in particular, in the finite invariant case K is definably isomorphic to a finite extension of F . This paper answers a question raised by D. Marker, whose help and guidance made this work possible. I would like to thank A. Pillay for helpful discussions, and E. Hrushovski for pointing out mistakes in the first version of this paper.

The growth function of a group is a generating function whose coefficients a n {a_n} are the number of elements in the group whose minimum length as a word in the generators is n. In this paper we use finite state automata to investigate the growth function for the Baumslag-Solitar group of the form ⟨ a , b | a − 1 b a = a 2 ⟩ \langle a,b|{a^{ - 1}}ba = {a^2}\rangle based on an analysis of its combinatorial and geometric structure. In particular, we obtain a set of length-minimal normal forms for the group which, although it does not form the language of a finite state automata, is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational. The rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic. The results in this paper generalize to the groups ⟨ a , b | a − 1 b a = a m ⟩ \langle a,b|{a^{ - 1}}ba = {a^m}\rangle for all positive integers m.

A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields. Let m be a positive integer and let Cm = exp(27ri/m). Let Km = Q(4m) denote the mth cyclotomic field, and K+ its maximal real subfield. The ring of integers in Km (resp. K+) is Z[Cm] (resp. Z[4m + C-1]). In [2] Sinnott showed that the index of circular units in the full group of units of Q(Cm) equals the class number of Z[4m + Cm'] multiplied by a power of 2 which depends exclusively on the number of prime factors of m. Sinnott [3] subsequently generalized this result to an arbitrary abelian field. There is a parallel setup for function fields of characteristic p . Let Fq be the finite field with q elements, let RT = Fq[T] be the ring of polynomials over Fq (with T transcendental over Fq), and let Fq(T) be the field of rational functions over Fq . To each polynomial M E RT one can associate an extension KM, called the Mth cyclotomicfunctionfield, which enjoys properties analogous to those of the cyclotomic number field Km. In particular, Galovich and Rosen [1] proved the analogue of Sinnott's theorem in this setting. The purpose of this paper is to extend this unit index-class number formula to an arbitrary subfield of KM. Let k be any subfield of the Mth cyclotomic function field (M monic), G the Galois group of k over IFq(T), k+ the maximal subfield of k in which oo splits, Ok(Ok+) the integral closure of Fq[T] in k(k'), and Ok* the unit group of Ok. In ?3, we define a subgroup C of Ok*, which we call the circular units of k. Our main result is that C has finite index in Ok* and that this index may be written in the form [0k :C]=h(Ok+).c,k+ where h (Ok+) is the class number of Ok+, and ck+ is a rational number whose definition does not involve h(Ok+). We now briefly describe the contents of the rest of this paper. In ? 1, we present the relevant definitions and facts in the function field setting. We also state the analytic class number formula. In ?2, we review ordinary distributions on lFq(T)/RT, discuss an index notation, and obtain a preliminary result on the structure of a certain module. The circular units are introduced in ?3 and Received by the editors February 1, 1990 and, in revised form, November 27, 1991. 1991 Mathematics Subject Classification. Primary 1 lR58. (D 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

A skew lattice is a noncommutative associative analogue of a lattice and as such may be viewed both as an algebraic object and as a geometric ob- ject. Whereas recent papers on skew lattices primarily treated algebraic aspects of skew lattices, this article investigates their intrinsic geometry. This geometry is obtained by considering how the coset geometries of the maximal primitive subalgebras combine to form a global geometry on the skew lattice. While this geometry is derived from the algebraic operations, it can be given a description that is independent of these operations, but which in turn induces them. Var- ious aspects of this geometry are investigated including: its general properties; algebraic and numerical consequences of these properties; connectedness; the geometry of skew lattices in rings; connections between primitive skew lattices and completely simple semigroups; and finally, this geometry is used to help classify symmetric skew lattices on two generators. Recall that a band is a semigroup satisfying the idempotent law: xx = x. Upon examining bands which are multiplicative subsemigroups of rings, one uncovers classes of bands which also possess an idempotent countermultipli- cation. This leads one to define a skew lattice to be an algebra with a pair of associative idempotent binary operations, the join and the meet, which are con- nected by a set of absorption laws (see 1.1). While skew lattices of idempotents in rings remain important sources of motivation, the results of (10-12) make it clear that skew lattices can sustain mathematical life on their own. Perhaps the most natural way to think about a skew lattice is as a noncommutative ana- logue of a lattice. As such, a skew lattice is not only an algebraic object, but also a geometric object. Thus far most of the research given in (10-12) has emphasized the algebraic side of skew lattices. The purpose of this paper is to investigate their geometric aspects and in particular the role of the natural partial order in determining their algebraic structure, much as a lattice is deter- mined by its natural partial ordering. It is not our goal, however, to reinvent lattice theory. Hence the geometry of a skew lattice will be studied relative to the fixed structure of its underlying lattice. Saying this entails an implicit refer- ence to the fundamental Clifford-McLean Theorem which in effect provides a first sketch of a skew lattice: a congruence is defined on each skew lattice (called natural equivalence) which induces its maximal lattice image and whose equiv

Let A A be an n n -dimensional Abelian variety, n ≥ 2 n \geq 2 ; let CH 0 ( A ) {\text {CH}_0}(A) be the group of zero-cycles of A A , modulo rational equivalence; by regarding an effective, degree k k , zero-cycle, as a point on S k ( A ) {S^k}(A) (the k k -symmetric product of A A ), and by considering the associated rational equivalence class, we get a map γ : S k ( A ) → CH 0 ( A ) \gamma :{S^k}(A) \to {\text {CH}_0}(A) , whose fibres are called γ \gamma -orbits. For any n ≥ 2 n \geq 2 , in this paper we determine the maximal dimension of the γ \gamma -orbits when k = 2 k = 2 or 3 3 (it is, respectively, 1 1 and 2 2 ), and the maximal dimension of families of γ \gamma -orbits; moreover, for generic A A , we get some refinements and in particular we show that if dim ⁡ ( A ) ≥ 4 \dim (A) \geq 4 , S 3 ( A ) {S^3}(A) does not contain any γ \gamma -orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties: A t = E t × B {A_t} = {E_t} \times B ( E t {E_t} is an elliptic curve with varying moduli) and we have constructed suitable projections between S k ( A t ) {S^k}({A_t}) and S k ( B ) {S^k}(B) which preserve the dimensions of the families of γ \gamma -orbits; then we have done induction on n n . For n = 2 n = 2 the proof is based upon the papers of Mumford and Roitman on this topic.

This paper studies a minimax problem for functionals in Hilbert space in the form of G ( u ) = 1 2 ρ | | u | | 2 − g ( u ) G(u) = \frac {1} {2}\rho ||u|{|^2} - g(u) , where g ( u ) g(u) is Fréchet differentiable with weakly continuous derivative. If G G has a "mountain pass geometry" it does not necessarily have a critical point. Such a case is called, in this paper, a "mountain impasse". This paper states that in a case of mountain impasse, there exists a sequence u j ∈ H {u_j} \in H such that \[ g ′ ( u j ) = ρ j u j , ρ j → ρ , | | u j | | → ∞ , g\prime ({u_j}) = {\rho _j}{u_j},\quad {\rho _j} \to \rho ,||{u_j}|| \to \infty , \] and G ( u j ) G({u_j}) approximates the minimax value from above. If \[ γ ( t ) = sup | | u | | 2 = t g ( u ) \gamma (t) = \sup \limits _{||u|{|^2} = t} \;g(u) \] and \[ J 0 = ( 2 inf t 2 > t 1 > 0 γ ( t 2 ) − γ ( t 1 ) t 2 − t 1 , 2 sup t 2 > t 1 > 0 γ ( t 2 ) − γ ( t 1 ) t 2 − t 1 ) , {J_0} = \left ( {2\inf \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}},2\sup \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}}} \right ), \] then g ′ ( u ) = ρ u g\prime (u) = \rho u has a nonzero solution u u for a dense subset of ρ ∈ J 0 \rho \in {J_0} .

In his previous paper the author characterized free arrangements bythe vanishing of cohomology modules of a certain sheaf of graded modules overa polynomial ring. Thus the homogeneous components of these cohomologymodules can be viewed as obstructions to the freeness of an arrangement. Inthis paper the first two obstructions are studied in detail. In particular thecomponent of degree zero of the first nontrivial cohomology module has a closerelation to formal arrangements and to the operation of truncation. This enablesus to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement. IntroductionIn this paper we develop further the ideas from [Y] where certain sheaf co-homology was applied to the theory of arrangements.Let V be an /-dimensional (/ > 2) vector space over an arbitrary field Kand sf an arrangement of « (n > 2) hyperplanes in V. For each H £ sfwe fix a linear functional aH £ V* such that kera// = 77. Let S = ®d>0Sdbe the graded symmetric algebra of V* and Der S = Der^S, S) the gradedS-module of derivations of S. The following graded submodule of Deri' wasintroduced by H. Terao [TI, T3] and studied in many papers on arrangements:D = D(s/) = {6 £ Der(S)\0(aH) £ SaH , H £ sf}. If the module D is freethen the arrangement sf is called free.Let Lo be the lattice of all intersections of hyperplanes from sf except

The simplest model of the Cauchy problem considered in this paper is the following ( ∗ ) (\ast ) \[ u t = Δ u + u p , a m p ; x ∈ R n , t > 0 , u ≥ 0 , p > 1 , u | t = 0 = ϕ ∈ C B ( R n ) , a m p ; ϕ ≥ 0 , ϕ ≢ 0. \begin {array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} \hfill & {x \in {R^n},t > 0,u \geq 0,p > 1,} \hfill \\ {u{|_{t = 0}} = \phi \in {C_B}({R^n}),} \hfill & {\phi \geq 0,\phi \,\not \equiv \,0.} \hfill \\ \end {array} \; \] It is well known that when 1 > p ≤ ( n + 2 ) / n 1 > p \leq (n + 2)/n , the local solution of ( ∗ ) (\ast ) blows up in finite time as long as the initial value ϕ \phi is nontrivial; and when p > ( n + 2 ) / n p > (n + 2)/n , if ϕ \phi is "small", ( ∗ ) (\ast ) has a global classical solution decaying to zero as t → + ∞ t \to + \infty , while if ϕ \phi is "large", the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on ϕ \phi for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if n ≥ 3 n \geq 3 , p > n / ( n − 2 ) p > n/(n - 2) , \[ 0 ≤ ϕ ( x ) ≤ λ u s ( x ) = λ ( 2 ( n − 2 ) ( p − 1 ) 2 ( p − n n − 2 ) ) 1 / ( p − 1 ) | x | − 2 / ( p − 1 ) 0 \leq \phi (x) \leq \lambda {u_s}(x) = \lambda {\left ( {\frac {{2(n - 2)}} {{{{(p - 1)}^2}}}\left ( {p - \frac {n} {{n - 2}}} \right )} \right )^{1/(p - 1)}}|x{|^{ - 2/(p - 1)}} \] ( u s {u_s} is a singular equilibrium of ( ∗ ) (\ast ) ) where 0 > λ > 1 0 > \lambda > 1 , then ( ∗ ) (\ast ) has a (unique) global classical solution u u with 0 ≤ u ≤ λ u s 0 \leq u \leq \lambda {u_s} and \[ u ( x , t ) ≤ ( ( λ 1 − p − 1 ) ( p − 1 ) t ) − 1 / ( p − 1 ) . u(x,t) \leq {(({\lambda ^{1 - p}} - 1)(p - 1)t)^{ - 1/(p - 1)}}. \] (This result implies that u 0 ≡ 0 {u_0} \equiv 0 is stable w.r.t. to a weighted L ∞ {L^\infty } topology when n ≥ 3 n \geq 3 and p > n / ( n − 2 ) p > n/(n - 2) .) We also obtain some sufficient conditions on ϕ \phi for global nonexistence and those conditions, when combined with our global existence result, indicate that for ϕ \phi around u s {u_s} , we are in a delicate situation, and when p p is fixed, u 0 ≡ 0 {u_0} \equiv 0 is "increasingly stable" as the dimension n ↑ + ∞ n \uparrow + \infty . A slightly more general version of ( ∗ ) (\ast ) is also considered and similar results are obtained.

The purpose of this paper is to give an alternative proof of recent results of M. Arino and B. Muckenhoupt [1] and E. Sawyer [8], concerning Hardy’s inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general Lorentz spaces. Our approach will extend the results of [1,8] to the values of the parameters which are inaccessible by the methods of these papers.

Let A A be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of A A itself, but with respect to an arbitrary ideal of A A . The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that A A is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 0 are true.