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This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder R n × R + {{\mathbf {R}}^n} \times {{\mathbf {R}}_ + } with initial data in L r {L^r} , n ≥ 3 n \geq 3 , 1 > r > ∞ 1 > r > \infty . An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].

The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows

In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together

Let A8 be the domain in the s-plane (s =o+it) defined by -gl(a) -Ah. The proofs of Theorems 1, I1, and III are the same, with the new function S(oI5.

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems., 50 pages. Originally part of arXiv:1703.09945 . We decided to split that paper following the recommendations of a referee. Updated subsequent to acceptance by Transactions of the American Mathematical Society

Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)

A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke., to appear in Trans. Amer. Math. Soc

In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata., The paper has been augmented considerably and the second version is now 81 pages long. This version has been accepted for publication in Transactions of the American Mathematical Society

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper., Minor change basically on the proof of Theorem 1.2 in Section 5

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages

The classical Hartman-Grobman Theorem states that a smooth diffeomorphism F ( x ) F(x) near its hyperbolic fixed point x ¯ \bar x is topological conjugate to its linear part D F ( x ¯ ) DF(\bar x) by a local homeomorphism Φ ( x ) \Phi (x) . In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth F ( x ) F(x) is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a C ∞ C^\infty diffeomorphism F ( x ) F(x) , the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a C 1 C^1 diffeomorphism F ( x ) F(x) with D F ( x ) DF(x) being α \alpha -Hölder continuous at the fixed point that the local homeomorphism Φ ( x ) \Phi (x) is differentiable at the fixed point. Here, α > 0 \alpha >0 depends on the bands of the spectrum of F ′ ( x ¯ ) F’(\bar x) for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on F ( x ) F(x) cannot be lowered to C 1 C^1 .

In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self-evident. The first is a genuine, 5-line proof, for the isoperimetric inequality for k k -cycles in R n \mathbb {R}^n , with a constant differing from the optimal constant by a factor of only k \sqrt {k} , as opposed to a factor of k k k^k produced by all of the other soft methods. The second is a 3-line proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of Giga and Yama-uchi (1993). We then turn to use the above-mentioned relation to prove a bound on the parabolic Hausdorff measure of the space-time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon isoperimetric inequality. To prove it, we are led to study the geometric measure theory of Euclidean rectifiable sets in parabolic space and prove a co-area formula in that setting. This formula, the proof of which occupies most of this paper, may be of independent interest.

In this paper, we introduce a notion of ampleness of a group action $G$ on a right noetherian graded algebra $A$, and show that it is strongly related to the notion of $A^G$ to be a graded isolated singularity introduced by the second author of this paper. Moreover, if $S$ is a noetherian AS-regular algebra and $G$ is a finite ample group acting on $S$, then we will show that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\cal D}^b(\operatorname{mod} \nabla S*G)$ where $\nabla S$ is the Beilinson algebra of $S$. We will also explicitly calculate a quiver $Q_{S, G}$ such that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\mathcal D}^b(\operatorname{mod} kQ_{S, G})$ when $S$ is of dimension 2.

In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A A which depends only on the Grothendieck ring of H . H. When H H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are.

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification.

Let G G be a compact connected Lie group. The question of when a weighted Fourier algebra on G G is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on G G with the order of growth strictly bigger than half of the dimension of the group. The case of S U ( n ) SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of the Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

In this paper, we prove that the quasihyperbolic metrics are quasiinvariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of above theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping.

We correct mistakes in a 2004 paper by the second author and report on recent new developments which settle cases left open in that paper.

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theor em for the cubic ruled surface in P 3 . We apply these results to limit the possible curves that can be s et-theoretic complete intersection in P 3 in characteristic zero. On a nonsingular variety, the study of divisors and linear systems is classical. In fact the entire theory of curves and surfaces is dependent on this study of codimension one subvarieties and the linear and algebraic families in which they move. This theory has been generalized in two directions: the Weil divisors on a normal variety, taking codimension one subvarieties as prime divisors; and the Cartier divisors on an arbitrary scheme, based on locally principal codimension one subschemes. Most of the literature both in algebraic geometry and commutative algebra up to now has been limited to these kinds of divisors. More recently there have been good reasons to consider divisors on non-normal varieties. Jaffe (9) introduced the notion of an almost Cartier divisor, which is locally principal off a subset of codimension two. A theory of generalized divisors was proposed on curves in (14), and extended to any dimension in (15). The latter paper gave a complete description of the generalized divisors on the ruled cubic surface in P 3 . In this paper we extend that analysis to an arbitrary integra l surface X, explaining the group APicX of linear equivalence classes of almost Cartier divisors on X in terms of the Picard group of the normalization S of X and certain local data at the singular points of X. We apply these results to give limitations on the possible curves that can b e set-theoretic compete intersections in P 3 in characteristic zero In section 2 we explain our basic set-up, comparing divisors on a variety X to its normalization S. In Section 3 we prove a local isomorphism that computes the group of almost Cartier divisors at a singular point of X in terms of the Cartier divisors along the curve of singulari ties and its inverse image in the normalization. In Section 4 we derive some global exact sequences for the groups PicX, APicX, and PicS, which generalize the results of (15, §6) to arbitrary surfaces These results are particularly transparent for surfaces wi th ordinary singularities, meaning a double curve with a finite number of pinch points and triple points.

In a previous paper, we applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincaré inequality on these graphs to obtain a dimension estimate for polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on the graph, we translate the problem to a polygonal surface by filling polygons into the graph with edge lengths 1. This polygonal surface then is an Alexandrov space of nonnegative curvature. From a harmonic function on the graph, we construct a function on the polygonal surface that is not necessarily harmonic, but satisfies crucial estimates. Using the arguments on the polygonal surface, we obtain the optimal dimension estimate for polynomial growth harmonic functions on the graph which is linear in the growth rate.

A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given Fomin and Zelevinsky.) The universal objects are closely related to a fan F_B called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of F_B for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces, and use it to construct universal geometric coefficients for these surfaces., Comment: 39 pages, 24 figures. Version 2: Stated explicitly several results that are implicit in the earlier version. Also minor expository changes. Version3: Expository changes (including additional figures) and changes to correct an error from arXiv:1209.3987 (see comments to arXiv:1209.3987v3)

The primary theorem of this paper concerns the Poincaré (Hilbert) series for the cohomology ring of a finite group G G with coefficients in a prime field of characteristic p p . This theorem is proved using the ideas of equivariant cohomology whereby one considers more generally the cohomology ring of the Borel construction H ∗ ( E G × G X ) H^*(EG \times _G X) , where X X is a manifold on which G G acts. This work results in a formula that computes the “degree” of the Poincaré series in terms of corresponding degrees of certain subgroups of the group G G . In this paper, we discuss the theorem and the method of proof.

This paper concerns homotopically stratified spaces. These were defined by Frank Quinn in his paper Homotopically Stratified Sets. His definition of stratified space is very general and relates strata by “homotopy rather than geometric conditions”. This makes homotopically stratified spaces the ideal class of stratified spaces on which to define and study stratified homotopy theory. We will define stratified analogues of the usual definitions of maps, homotopies and homotopy equivalences. Then we will provide an elementary criterion for deciding when a strongly stratified map is a stratified homotopy equivalence. This criterion states that a strongly stratified map is a stratified homotopy equivalence if and only if the induced maps on strata and holink spaces are homotopy equivalences. Using this criterion we will prove that any homotopically stratified space is stratified homotopy equivalent to a homotopically stratified space where neighborhoods of strata are mapping cylinders. Finally, we will develop categorical descriptions of the class of homotopically stratified spaces up to stratified homotopy.

This paper is inspired by the work of Nagel and Stein in which the L p L^p ( 1 > p > ∞ ) (1>p>\infty ) theory has been developed in the setting of the product Carnot-Carathéodory spaces M ~ = M 1 × ⋯ × M n \widetilde {M}=M_1\times \cdots \times M_n formed by vector fields satisfying Hörmander’s finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product B M O BMO space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.

The topic of this paper is the Mackey Cocycle Theorem: every Borel almost cocycle is equivalent to a Borel strict cocycle. This is a theorem about locally compact groups which is not true for arbitrary Polish groups. We discuss the theorem, the open question of whether the theorem generalizes to some nonlocally compact Polish groups, the generalization to non-Borel cocycles, and other subjects associated with the theorem. Traditionally, the subject of cocycles and related matters has been considered in the context of standard Borel G G -spaces. It is now known that a standard Borel G G -space has a topological realization as a Polish G G -space. This makes it possible to consider the subject from a topological point of view. The main theorem of this paper is that the conclusion of the Mackey Cocycle Theorem is equivalent to continuity properties of the almost cocycle. Even in the locally compact case, this continuity is a new result.

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Zn-trees give one a powerful tool to study groups. All finitely generated groups acting freely on R-trees also act freely on some Zn-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for Zn-actions (which is not the case for R-trees) comes along with a robust algorithmic theory. In this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on Zn-trees and give necessary and sufficient conditions for such actions.

In the late 1970s Mumford established Chow stability of smooth unpointed genus g curves embedded by complete linear systems of degree d ≥ 2g + 1, and at about the same time Gieseker established asymptotic Hilbert stability (that is, stability of m th Hilbert points for some large values of m) under the same hypotheses. Both of them then use an indirect argument to show that nodal Deligne-Mumford stable curves are GIT stable. The case of marked points lay untouched until 2006, when Elizabeth Baldwin proved that pointed Deligne-Mumford stable curves are asymptotically Hilbert stable. (Actually, she proved this for stable maps, which includes stable curves as a special case.) Her argument is a delicate induction on g and the number of marked points n; elliptic tails are glued to the marked points one by one, ultimately relating stability of an n-pointed genus g curve to Gieseker’s result for genus g + n unpointed curves. There are three ways one might wish to improve upon Baldwin’s results. First, one might wish to construct moduli spaces of weighted pointed curves or maps; it appears that Baldwin’s proof can accommodate some, but not all, sets of weights. Second, one might wish to study Hilbert stability for small values of m; since Baldwin’s proof uses Gieseker’s proof as the base case, it is not easy to see how it could be modified to yield an approach for small m. Finally, the Minimal Model Program for moduli spaces of curves has generated interest in GIT for 2, 3, or 4-canonical linear systems; due to its use of elliptic tails, Baldwin’s proof cannot be used to study these, as elliptic tails are known to be GIT unstable in these cases. In this paper I give a direct proof that smooth curves with distinct weighted marked points are asymptotically Hilbert stable with respect to a wide range of parameter spaces and linearizations. Some of these yield the (coarse) moduli space of Deligne-Mumford stable pointed curves M g,n and Hassett’s moduli spaces of weighted pointed curves M g,A, while other linearizations may give other quotients which are birational to these and which may admit interpretations as moduli spaces. The full construction of the moduli spaces is not contained in this paper, only the proof that smooth curves with distinct weighted marked points are stable, which is the key new result needed for the construction. For this I follow Gieseker’s approach to reduce to the GIT problem to a combinatorial problem, though the solution is very different.

As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis' BP-approach to the immersion problem of P^m. A form of the Euler class viewpoint is applied to show TC^S(P^3) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber's work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber's lead, this concept is connected to the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P^3., updated, final version to appear in Transactions of the American Mathematical Society

We improve some of the results in our 1999 paper concerning weighted norm estimates for homogeneous singular integrals with rough kernels. Using a representation of such integrals in terms of averages over starlike sets, we prove a two-weight L p L^{p} inequality for 1 > p > 2 1 > p > 2 which we were previously able to obtain only for p ≥ 2 p \geq 2 . We also construct examples of weights that satisfy conditions which were shown in our earlier paper to be sufficient for one-weight inequalities when 1 > p > ∞ 1>p>\infty .

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schutzenberger) could be used to dene a new basis for the ring of quasisymmetric functions we call \Quasisymmetric Schur functions" (QS functions for short). In this paper we develop the combinatorics of these polynomials futher, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coecients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis. A composition (weak composition) with n parts is a sequence of n positive (nonnegative) integers, respectively. A partition is a composition whose parts are monotone nonincreasing. If is a weak composition, composition, or partition, we let '( ) denote the number of parts of . Throughout this article is a weak composition with '( ) = n while and denote compositions and partitions, respectively, with '( ) n, '( ) n. The polynomials in this paper (Schur functions, Demazure atoms and characters, QS functions) depend on a

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.

In previous joint work, a theory introduced earlier by Tagare was developed for establishing certain kinds of correspondences, termed bimorphisms, between simple closed regular plane curves of differentiability class at least C 2 C^2 . A class of objective functionals was introduced on the space of bimorphisms between two fixed curves C 1 C_1 and C 2 C_2 , and it was proposed that one define a “best non-rigid match” between C 1 C_1 and C 2 C_2 by minimizing such a functional. In this paper we prove several theorems concerning the nature of the shape-space of plane curves and of spaces of bimorphisms as infinite-dimensional manifolds. In particular, for 2 ≤ j > ∞ 2\leq j>\infty , the space of parametrized bimorphisms is a differentiable Banach manifold, but the space of unparametrized bimorphisms is not. Only for C ∞ C^\infty curves is the space of bimorphisms an infinite-dimensional manifold, and then only a Fréchet manifold, not a Banach manifold. This paper lays the groundwork for a companion paper in which we use the Nash Inverse Function Theorem and our results on C ∞ C^\infty curves and bimorphisms to show that if Γ \Gamma is strongly convex, if C 1 C_1 and C 2 C_2 are C ∞ C^\infty curves whose shapes are not too dissimilar ( C j C^j -close for a certain finite j j ) and if neither curve is a perfect circle, then the minimum of a regularized objective functional exists and is locally unique.

By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R d . In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in R d can be approximated by a pair of homogeneous polynomials. This conjecture is not resolved yet but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) d = 2, 2) convex surfaces in R d with C 1+ǫ boundary.

We use the notion of rank defined in an earlier paper (2007) to introduce and study two correspondences between small irreducible unitary representations of the split real simple Lie groups of types E n , where n ∈ {6, 7}, and two reductive classical groups. We show that these correspondences classify all of the unitary representations of rank two (in the sense of our earlier paper) of these exceptional groups. We study our correspondences for a specific family of degenerate principal series representations in detail.

The aim of this paper is to give an abstract formulation of the classical reciprocity laws for function fields that could be generalized to the case of arbitrary (non-commutative) reductive groups as a first step to finding explicit non-commutative reciprocity laws. The main tool in this paper is the theory of determinant bundles over adelic Sato Grassmannians and the existence of a Krichever map for rank n vector bundles.

We introduce the categories of algebraic σ \sigma -varieties and σ \sigma -groups over a difference field ( K , σ ) (K,\sigma ) . Under a “linearly σ \sigma -closed" assumption on ( K , σ ) (K,\sigma ) we prove an isotriviality theorem for σ \sigma -groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski’s work (2001) and are also closely related to papers of Pink and Roessler (2002 and 2004).

We classify all homomorphisms between Weyl modules for SL3(k) when k is an algebraically closed field of characteristic at least three, and show that the Hom-spaces are all at most one-dimensional. As a corollary we obtain all homomorphisms between Specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three. We conclude by showing how a result of Fayers and Lyle on Hom-spaces for Specht modules is related to earlier work of Donkin for algebraic groups. Let G be a reductive algebraic group over an algebraically closed field of characteristic p > 0. An important class of modules for such a group are the Weyl modules �(λ), labelled by dominant weights; these can be constructed (relatively) explicitly, and their heads provide a full set of simple modules for G. (Equivalently one can study the duals of these modules, denoted ∇(λ) which have the advantage of being induced from one-dimensional modules for a Borel subgroup). In determining the structure of such modules, or indeed their cohomology, the calculation of Hom- spaces between them is a useful tool. Relatively little is known in general about such Hom-spaces. In type A, when λ andare related by a (suitable) single reflection, explicit non-zero homomorphisms from �(λ) to �(� ) were constructed (with some restrictions) by Carter and Lusztig (3), and (more generally) by Carter and Payne (4). The corresponding cases in other types were considered by Franklin (14). While it is clear that there should be a hierarchy of families of homomorphisms corresponding to different powers of p, the only case where the above results provide a complete classification is when G is SL2, where it is relatively easy to determine all Hom-spaces exactly (6). The only other general results in this area, by Andersen (1) and Koppinen (18), concern homo- morphisms between modules labelled by weights which are 'close together'. Typically such results show that certain Hom-spaces are non-zero, or in some cases one-dimensional. For weights which are far apart and not related by a single reflection almost nothing is known. In this paper we will determine all homomorphisms between Weyl modules for SL3 when p ≥ 3, and provide a recursive procedure for determining the composition factors arising in the image (or kernel) of such maps in most cases. From these results we will also classify all homomorphisms between Specht modules for the symmetric groups corresponding to three part partitions, when p ≥ 3. After a section of preliminaries, we review the SL3 data concerning p-filtrations that we will need from (21). This describes certain filtrations of induced modules which will allow us to proceed by induction, together with the set of p-good homomorphisms which will be fundamental in our later constructions. We also recall a theorem of Carter and Payne (4) on the existence of certain homomorphisms. These will be the two key sets of data which we need to determine all possible homomorphisms. With the notation developed up to that point in place, in Section 4 we can give the strategy to be followed in the remainder of the paper, and in particular the translation functor arguments

This paper solves in the affirmative, up to dimension n = 5, a question raised in an earlier paper by the authors. The equivalence of the problem with a conjecture of Shou-Wu Zhang is proved in the Appendix.

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω 1 \omega _1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah’s iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman’s (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah’s club–guessing sequences) that shows similar results do not hold for closed pre–images of ω 2 \omega _2 .

In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series E 6 ( τ ) E_6(\tau ) . In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.

Fix a prime p p , and let A A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A A induces the Steenrod operation P ~ 0 \widetilde {\mathscr {P}}^{0} on cohomology, and in this paper, we investigate this operation. We point out that if p = 2 p=2 , then for any element in the cohomology of A A , if one applies P ~ 0 \widetilde {\mathscr {P}}^{0} enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that “enough times” should be “once.” The bulk of the paper is a study of some quotients of A A in which the Frobenius is an isomorphism of order n n . We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about P ~ 0 \widetilde {\mathscr {P}}^{0} . The dual complete Steenrod algebra makes an appearance.

In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^\infty functions on the unit disc to problems concerning lifting onto H 2 H^2 of an operator T T that is defined on K = H 2 ⊖ ϕ H 2 \mathcal {K} =H^2\ominus \phi H^2 ( ϕ \phi is an inner function) and commutes with the (compressed) shift S S . In particular, he showed that interpolants (i.e., f ∈ H ∞ f\in H^\infty such that f ( S ) = T f(S)=T ) having norm equal to ‖ T ‖ \|T\| exist, and that in certain cases such an f f is unique and can be expressed as a fraction f = b / a f=b/a with a , b ∈ K a,b\in \mathcal {K} . In this paper, we study interpolants that are such fractions of K \mathcal {K} functions and are bounded in norm by 1 1 (assuming that ‖ T ‖ > 1 \|T\|>1 , in which case they always exist). We parameterize the collection of all such pairs ( a , b ) ∈ K × K (a,b)\in \mathcal {K}\times \mathcal {K} and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where ϕ \phi is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.