pinecone/core-2020-05-10-deduplication huggingface dataset:
Maximal Averages over Surfaces

AbstractLetMf(x)=supt>0|f*δt(ψdσ)(x)|denote the maximal operator associated with surface measuredσon a smooth surfaceS. We prove that ifSis convex and has finite order contact with its tangent lines, then M is bounded onLp(Rn),p>2, if and only ifd(x,H)−1∈L1/loc(S) for all tangent planesHnot passing through the origin. LetM′f(x)=supt>0|f*δ′t(ψdσ)(x)|be the maximal operator associated with a nonisotropic dilationδ′tof surface measuredσ. We prove that M′ often behaves far better than M due to a rotational curvature in the time parametert
Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields

AbstractThe space of linear differential operators on a smooth manifoldMhas a natural one-parameter family of Diff(M)- (and Vect(M)-) module structures, defined by their action on the space of tensor densities. It is shown that, in the case of second-order differential operators, the Vect(M)-module structures are equivalent, for any degree of tensor densities except for three critical values; {0,12,1}. A second-order analogue of the Lie derivative appears as an intertwining operator between the spaces of second-order differential operators on tensor densities
Cohen–Macaulay Properties of Ring Homomorphisms

AbstractNumerical invariants which measure the Cohen–Macaulay character of homomorphismsϕ:R→Sof noetherian rings are introduced and studied. Comprehensive results are obtained for homomorphisms which are locally of finite flat dimension. They provide a point of view from which a variety of phenomena receive a unified treatment. The conceptual clarification and technical versatility of this approach leads, among other things, to a determination of those homomorphisms which preserve the Cohen–Macaulay character of the rings, to the discovery of new classes of homomorphisms with remarkable stability properties, and to solutions of some problems on flat homomorphisms, raised by Grothendieck
Whitney's Problem on Extendability of Functions and an Intrinsic Metric

AbstractWe consider the space of functions with bounded (k+1)th derivatives in a general domain in Rn. Is every such function extendible to a function of the same class defined on the whole Rn? H. Whitney showed that the equivalence of the intrinsic (=geodesic) metric in this domain to the Euclidean one is sufficient for such extendability. There was an old conjecture (going back to H. Whitney) that this equivalence is also necessary for extendability. We disprove this conjecture and construct examples of domains in R2such that the above extendability holds but the analogous property for smallerkfails. Our study is based on a duality approach
Equivariant Index and the Moment Map for Completely Integrable Torus Actions

AbstractConsider a completely integrable torus action on a compact Spincmanifold. The equivariant index of the Dirac operator is a virtual representation of the torus and is determined by the multiplicities of the weights which occur it in. We prove that these multiplicities are equal to values of the density function for the Duistermaat–Heckman measure, once this is defined appropriately. (The two-form that we take is half of the curvature of the line bundle which is associated to the Spincstructure. It is closed and invariant but not necessarily symplectic.) We deduce that these multiplicities are equal to the topological degree of the “descended moment map”Φ:M/T→t*, which in nice cases can be described as sums of certain winding numbers
Coding in the Partial Order of Enumerable Sets

AbstractWe develop methods for coding with first-order formulas into the partial order E of enumerable sets under inclusion. First we use them to reprove and generalize the (unpublished) result of the first author that the elementary theory of E has the same computational complexity as the theory of the natural numbers. Relativized versions of the coding methods show that the p.o. ofΣ0pandΣ0qsets are not elementarily equivalent for natural numbersp≠q. As a further application, definability of the class of quasimaximal sets in E is obtained. On the other side, we prove theorems limiting coding and definability in E, thereby establishing a sharp contrast between E and other structures occurring in computability theory
On Valuations, the Characteristic Polynomial, and Complex Subspace Arrangements

AbstractWe present a new combinatorial method to determine the characteristic polynomial of any subspace arrangement that is defined over an infinite field, generalizing the work of Blass and Sagan. Our methods stem from the theory of valuations and Groemer's integral theorem. As a corollary of our main theorem, we obtain a result of Zaslavsky about the number of chambers of a real hyperplane arrangement. The examples we consider include a family of complex subspace arrangements, which we call the divisor Dowling arrangement, whose intersection lattice generalizes that of the Dowling lattice. We also determine the characteristic polynomial of interpolations between subarrangements of the divisor Dowling arrangement, generalizing the work of Józefiak and Sagan
Higher-Dimensional Algebra III.n-Categories and the Algebra of Opetopes

AbstractWe give a definition of weakn-categories based on the theory of operads. We work with operads having an arbitrary setSof types, or “S-operads,” and given such an operadO, we denote its set of operations by elt(O). Then for anyS-operadOthere is an elt(O)-operadO+whose algebras areS-operads overO. LettingIbe the initial operad with a one-element set of types, and definingI0+=I,I(i+1)+=(Ii+)+, we call the operations ofI(n−1)+the “n-dimensional opetopes.” Opetopes form a category, and presheaves on this category are called “opetopic sets.” A weakn-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weakω-categories. Similarly, starting from an arbitrary operadOinstead ofI, we define “n-coherentO-algebras,” which arentimes categorified analogs of algebras ofO. Examples include “monoidaln-categories,” “stablen-categories,” “virtualn-functors” and “representablen-prestacks.” We also describe hown-coherentO-algebra objects may be defined in any (n+1)-coherentO-algebra
Symmetric Functions, Formal Group Laws, and Lazard's Theorem

AbstractLazard's theorem is a central result in formal group theory; it states that the ring over which the universal formal group law is defined (known as the Lazard ring) is a polynomial algebra over the integers with infinitely many generators. This ring also shows up in algebraic topology as the complex cobordism ring. The main aim of this paper is to show that the polynomial structure of the Lazard ring follows from the polynomial structure of a certain subalgebra of symmetric functions with integer coefficients. The connection between symmetric functions and the Lazard ring is provided by a certain Hopf algebra map from symmetric functions to the covariant bialgebra of a formal group law. We study this map by deriving formulas for the images of certain symmetric functions; in passing, we use this map to prove some symmetric function and Catalan number identities. Based on the above results, we prove Lazard's theorem, and present an application to the construction of certainp-typical formal group laws over the integers. Combinatorial methods play a major role throughout this paper
Borel–Weil Theorem for Configuration Varieties and Schur Modules

AbstractWe present a generalization of the classical Schur modules ofGL(n) exhibiting the same interplay among algebra, geometry, and combinatorics. A generalized Young diagramDis an arbitrary finite subset ofN×N. For eachD, we define the Schur moduleSDofGL(n). We introduce a projective variety FDand a line bundle LD, and describe the Schur module in terms of sections of LD. For diagrams with the “northeast” property(i1, j1), (i2, j2)∈D⇒(min(i1, i2),max(j1, j2))∈D,which includes the skew diagrams, we resolve the singularities of FDand show analogues of Bott's and Kempf's vanishing theorems. Finally, we apply the Atiyah–Bott Fixed Point Theorem to establish a Weyl-type character formula of the form:charSD(x)=∑txwt(t)∑i, j(1−xix−1j)dij(t),wheretruns over certain standard tableaux ofD. Our results are valid over fields of arbitrary characteristic
Symplectic Surgery and the Spinc–Dirac Operator

AbstractLetGbe a compact connected Lie group, and (M, ω) a compact HamiltonianG-space, with moment mapJ:M→g'. Under the assumption that these data are pre-quantizable, one can construct an associated Spinc–Dirac operator[formula], whose equivariant index yields a virtual representation ofG. We prove a conjecture of Guillemin and Sternberg that if 0 is a regular value ofJ, the multiplicityN(0) of the trivial representation in the index space[formula], is equal to the index of the Spinc–Dirac operator for the symplectic quotientM0=J−1(0)/G. This generalizes previous results for the case thatG=Tis abelian, i.e., a torus
Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

AbstractIn the phase space over a Riemann manifold we consider a submanifold 4 invariant with respect to a Hamilton flow, isotropic (i.e., the formpdq|Λis closed), and stable with respect to the first variation equation. For semiclassical wave- functions of the quantum Hamiltonian we propose a very simple global ansatz with an oscillation front onΛ. This ansatz has a form of an integral alongΛfrom Gaussian packets framed by an “amplitude.” The amplitude is a parallel section of a bundle of polynomials. The connection on this bundle is generated by a symplectic connection with zero curvature on the normal symplectic bundle overΛ. The coefficients of such a connection are calculated explicitly in terms of infinitesimal symmetries, and also in adiabatic approximation. We investigate topological and geometrical objects arising as corrections to the Poincaré–Cartan invariant in quantization rule onΛand calculate spectral series for the quantum Hamiltonian
A Bott Periodicity Theorem for Infinite Dimensional Euclidean Space

AbstractWe formulate and prove an equivariant Bott periodicity theorem for infinite dimensional Euclidean vector spaces. The main features of our argument are (i) the construction of a non-commutativeC*-algebra to play the role of the algebra of functions on infinite dimensional Euclidean space; and (ii) the construction of a certain index one elliptic partial differential operator which provides the basis for an inverse to the Bott periodicity map. These tools have applications to index theory and the Novikov conjecture, notably a proof of the Novikov conjecture for amenable groups (the applications will be considered elsewhere)
Limit Distributions of Directionally Reinforced Random Walks

AbstractAs a mathematical model for time and space correlations in ocean surface wave fields, Mauldin, Monticino, and von Weizsäcker (1996,Adv. in Math.117, 239–252), introduced directionally reinforced random walks and analyzed their recurrence properties. This paper develops limiting properties for these random walks inRd. In particular, it is shown that the limit distribution is a Brownian motion if the time between changes of directions has a finite second moment, and is a stable process if the time between changes of directions is in a domain of attraction of a stable law. The latter gives an affirmative answer to an open question posed by Mauldinet al.. The strong law of large numbers and strong approximations are also discussed
Limits of Compactified Jacobians and D-Modules on Smooth Projective Curves

AbstractLetSbe a smooth projective curve and DSits sheaf of differential operators. This paper classifies the rank one torsion-free DS-modules up to isomorphism. Such a module E has a degree which depends on the homological properties of E. Furthermore, the set of isomorphism classes with fixed degreed, say, bijects tolim→Jd+δ(X)(X),the set of points of the limit of compactified Jacobians of curves with injective normalizationπ:S→X. Here, if F is a rank one torsion-free OX-module (of degreed+δ(X)) representing the point of the limit corresponding to a torsion-free DS-module E then E≅DOS, Fthe sheaf of differential operators from OSto F. As an application, we classify the domains Morita equivalent to D(P1). We also study D(F), the ring of globally defined differential operators on a rank one torsion-free sheaf over a curve with injective normalization P1→X. It is shown that the finite-dimensional simple D(F)-modules are preciselyHi(X, F), when they are non-zero, and that the (non)vanishing ofHi(X, F) determines the Morita equivalence class of D(F)
Induction and Restriction of Kazhdan–Lusztig Cells

AbstractBarbasch and Vogan gave a beautiful rule for restricting and inducing Kazhdan–Lusztig representations of Weyl groups. In this paper we show that this rule implies and generalizes the Littlewood–Richardson rule for decomposing outer products of representations of the symmetric groups. A new recursive rule for computing characters of arbitrary Coxeter groups follows. Another application is a generalization of Garsia–Remmel's algorithm for decomposing certain tensor products of symmetric groups representations
Sequences of Symmetric Polynomials and Combinatorial Properties of Tableaux

AbstractIn 1977 G. P. Thomas showed that the sequence of Schur polynomials associated to a partitionλcan be comfortably generated from the sequence of variablesx=(x1, x2, x3, …) by the application of mixed Baxter/multiplication operators, which in turn can be easily computed from the set SYT(λ) of standard Young tableaux of shapeλ. We generalize this construction, thereby making possible the explicit and effective computation of the Hall–Littlewood, Jack, and Macdonald polynomials used in representation theory, combinatorics, multivariate statistics, and quantum algebra. These generalized formulas have a pleasing recursive structure with respect to the Young lattice and they can easily be specialized to yield “skew” forms in all cases and “super” forms in the Schur case. We introduce and investigate: (1) the “descent polynomial of a partitionλ,” which arises naturally in the enumeration of semistandard Young tableaux of shapeλ; (2) the Boolean latticeG(ζ) associated to anyζ∈SYT(λ), which is fundamental for the “weighted” generalization of Thomas' approach to Schur polynomials; and (3) an action of the symmetric groups on semistandard Young tableaux, which is connected with Knuth's combinatorial proof of the symmetry of Schur functions. Moreover, we argue that a generalization of Thomas' approach is a natural starting point in search of “universal weighted symmetric functions.
Intersection Bodies in R4

AbstractIntersection bodies were introduced in 1988 by Lutwak, who found a close connection between those bodies and the well-known 1956 Busemann–Petty problem, which asks whether origin symmetric convex bodies with larger central hyperplane sections also have greater volume. The author has recently shown that an origin symmetric star bodyKis an intersection body if and only if the function ‖x‖−1Kis a positive definite distribution, where ‖x‖K=min{a⩾0:x∈aK}. We use this result to prove that the unit balls of the spaces ℓ4q, 2<q⩽∞, are intersection bodies. Forn⩾5 the unit balls of the spaces ℓnq, 2<q⩽∞, are not intersection bodies. The technique of this paper allows to find precise expressions for the generating measures (signed measures) of these bodies. The result that the unit cube in R4is an intersection body shows that the negative solution to the Busemann–Petty problem forn=4 in [20] was incorrect
Second Derivative Test for Intersection Bodies

AbstractIn 1956, Busemann and Petty asked whether symmetric convex bodies in Rnwith larger central hyperplane sections also have greater volume. This question was answered in the negative forn⩾5 in a series of papers giving individual counterexamples. In 1988, Lutwak introduced the concept of an intersection body and proved that every smooth nonintersection body in Rnprovides a counterexample to the Busemann–Petty problem. In this article, we use the connection between intersection bodies and positive definite distributions, established by the author in an earlier paper, to give a necessary condition for intersection bodies in terms of the second derivative of the norm. This result allows us to produce a variety of counterexamples to the Busemann–Petty problem in Rn,n⩾5. For example, the unit ball of theq-sum of any finite dimensional normed spacesXandYwithq>2, dim(X)⩾1, dim(Y)⩾4 is not an intersection body, as well as the unit balls of the Orlicz spaces ℓnM,n⩾5, withM′(0)=M″(0)=0
Generalized Centers of Braided and Sylleptic Monoidal 2-Categories

AbstractRecent developments in higher-dimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic, and symmetric monoidal 2-categories, and a center construction for monoidal 2-categories which gives a braided monoidal 2-category. I give generalized center constructions for braided and sylleptic monoidal 2-categories which give sylleptic and symmetric monoidal 2-categories, respectively, and I correct some errors in the original center construction for monoidal 2-categories
A Geometric Condition for a Hyperplane Arrangement to be Free

AbstractLetG(r, 1, l) be the complex arrangement {xi, xj−ξhxk}, whereξis a primitiverth root of unity. The matroids of these arrangements are the Dowling matroidsQl(Zr), whereZris the group ofrth roots of unity. We show that ifEis a subset ofG(r, 1, l) which does not contain any matroid line, then deletingEgives an arrangementG(r, 1, l)\Ewhich is free. We also give necessary and sufficient conditions on a set E containing at least one matroid line but no matroid planes so that the deletionG(r, 1, l)\Ehas a characteristic polynomial which factors completely over the integers. Two types of arrangements can be obtained in this way. We show that one type is always non-free. This yields examples of non-free complex arrangements whose characteristic polynomials factor completely over the integers. The same ideas also yield examples of non-free arrangements over any sufficiently large field (and hence, over the reals) with characteristic polynomials which factor completely over the integers. The matroids of these arrangements are non-supersolvable matroids whose characteristic polynomials factor completely over the integers
Combinatorial Models for Coalgebraic Structures

AbstractWe introduce a convenient category of combinatorial objects, known as cell-sets, on which we study the properties of the appropriate free abelian group functor. We obtain a versatile generalization of the notion of incidence coalgebra, giving rise to an abundance of coalgebras, Hopf algebras, and comodules, all of whose structure constants are positive integers with respect to certain preferred bases. Our category unifies and extends existing constructions in algebraic combinatorics, providing proper functorial descriptions; it is inspired in part by the notion of CW-complex, and is also geared to future applications in algebraic topology and the theory of formal group laws
On the Expansion of Schur and Schubert Polynomials into Standard Elementary Monomials

AbstractMotivated by the recent discovery of a simple quantization procedure for Schubert polynomials we study the expansion of Schur and Schubert polynomials into standard elementary monomials (SEM). The SEM expansion of Schur polynomials can be described algebraically by a simple variant of the Jacobi–Trudi formula and combinatorially by a rule based on posets of staircase box diagrams. These posets are seen to be rank symmetric and order isomorphic to certain principal order ideals in the Bruhat order of symmetric groups ranging between the full symmetric group and the respective maximal Boolean sublattice. We prove and conjecture extensions of these results for general Schubert polynomials. The featured conjectures are: (1) an interpretation of SEM expansions as “alternating approximations” and (2) surprising properties of different numbers naturally associated to SEM expansions. This hints at as yet undiscovered deeper symmetry properties of the SEM expansion of Schubert polynomials
The Combinatorics of Steenrod Operations on the Cohomology of Grassmannians

AbstractThe study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber–Novikov algebra in complex cobordism theory; it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur functions. We also discuss applications of our formulas to the geometry of Grassmannians
Renormalization Group Analysis of Spectral Problems in Quantum Field Theory

AbstractIn this paper we present a self-contained and detailed exposition of the new renormalization group technique proposed in [1, 2]. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, or correlation functions. We apply this renormalization transformation to a Hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field, and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing
Quantum Electrodynamics of Confined Nonrelativistic Particles

AbstractWe consider a system of finitely many nonrelativistic, quantum mechanical electrons bound to static nuclei. The electrons are minimally coupled to the quantized electromagnetic field; but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives, and the interactions between the radiation field and electrons localized very far from the nuclei are turned off. For a class of Hamiltonians we prove exponential localization of bound states, establish the existence of a ground state, and derive sufficient conditions for its uniqueness. Furthermore, we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field. To this end we develop a novel renormalization transformation which acts directly on the space of Hamiltonians