pinecone/core-2020-05-10-deduplication huggingface dataset:
Singular Perturbation Series in Quantum Mechanics

AbstractThis is the first in a series of articles on singular perturbation series in quantum mechanics. In these papers we will compute transition amplitudes non-perturbatively for several simple quantum mechanical systems. This paper treats the behavior of a harmonic oscillator as a function of the spring constant. All the systems to be considered exhibit the same phenomenon: there is, in general, no power series expansion, even of an asymptotic nature, in powers of the perturbation parameter. For reasonable initial and final states there are, however, expansions of a more complicated type. In the case of the harmonic oscillator there is a log-power series expansion. This example is sufficiently simple and explicit that one can see how the logarithmic terms arise mathematically. How they arise physically is rather more mysterious. One point however is clear; the standard techniques of perturbation theory, whether expressed in terms of Dyson series, Feynman diagrams, successive approximation, or path integrals, are all structured so as to produce only power series and thus fail to detect logarithmic terms
CR Invariants of Weight Five in the Bergman Kernel

AbstractFefferman's program (1979, C. Fefferman,Adv. Math.31, 131–262) of getting a biholomorphically invariant asymptotic expansion of the Bergman kernel for smoothly bounded strictly pseudoconvex domains is realized in dimension 2 with the identification of universal constants. According to the program, the expansion is in terms of an approximately invariant smooth defining function of the domain, which we refer to as Fefferman's defining function, and the coefficients are functions in the domain constructed by using derivatives of Fefferman's defining function. Consequently, the invariant expansion is necessarily a finite sum with a remainder term and the ambiguity estimate is crucial in the problem. We get an expansion such that the boundary values of the coefficients are CR invariants of weight ⩽5. This refines earlier results of C. R. Graham (1987,in“Lecture Notes in Math.,” Vol. 1276, pp. 108–135, Springer–Verlag, New York/Berlin) and the authors (1993, K. Hirachi, G. Komatsu, and N. Nakazawa,in“Lecture Notes in Pure and Appl. Math.,” Vol. 143, pp. 77–96, Dekker, New York). The refinement becomes possible by appropriate extensions inside the domain of the CR invariants of weight 4. Due to the ambiguity estimate of these extensions, our expansion is optimal as far as Fefferman's defining function is used. A similar result for the Szegö kernel is also obtained
Noncommutative Schubert Calculus and Grothendieck Polynomials

AbstractIn this paper we extend the work of Fomin and Greene on noncommutative Schur functions by defining noncommutative analogs of Schubert polynomials. If the variables satisfy certain relations (essentially the same as those needed in the theory of noncommutative Schur functions), we prove a Pieri-type formula and a Cauchy identity for our noncommutative polynomials. Our results imply the conjecture of Fomin and Kirillov concerning the expansion of an arbitrary Grothendieck polynomial in the basis of Schubert polynomials; we also present a combinatorial interpretation for the coefficients of the expansion. We conclude with some open problems related to it
Thin-G Theory and Local Moves for Gems

AbstractUsing the thin presentation of ann-gem, forn=3, 4, we prove that a set of local moves for crystallizations are sufficient in the sense that any two crystallizations inducing the samen-manifold are related by a finite sequence of moves taken from this set. In dimension 3, we apply the moves to prove an identity among quantum 6j-symbols, and we propose a new model for a quantum topological invariant. In dimension 4, we observe that the thin presentation reduces the description of a 4-manifold to a 3-dimensional one
A Class of Invariant Unitary Operators

AbstractLet H=L2((0, ∞), dx), andKλf(x)=f(λx), forλ>0,f∈H. Aninvariant operatoron H is one commuting with all theKλ. Askew rootis a self-adjoint, unitary operator on H satisfyingT2=I, andTKλ=K*λT, for allλ>0. Ageneratorg is an element of H such that the smallest, closed subspace containing {Kλg}λ>0is equal to H. We show that for any skew rootTand any real-valued generatorgthere is a unique, invariant, unitary operatorWsatisfyingWg=Tg. It turns out thatW−1=TWT. This construction is related to an approximation problem in H arising from a theorem due to A. Beurling (1955,Proc. Nat. Acad. Sci. U.S.A.41, 312–314) and B. Nyman (1950, “On Some Groups and Semigroups of Translations,” Thesis, Uppsala) which shows the Riemann hypothesis is equivalent to a closure problem in Hilbert space
Residue Complex for Sklyanin Algebras of Dimension Three

AbstractWe describe the residue complex for three-dimensional Sklyanin algebras, which are the interesting special cases of quantum polynomial rings in three variables. In particular, we show that the multiplicities of the point modules in the residue complex are all one, just as in the classical case of commutative polynomial rings in three variables. We explain why the residue complex in the quantum case has the same multiplicities of point modules as that in the commutative case (even if there are fewer point modules in the quantum case) by pointing out two quantum anomalies
The State Reduction and Related Algorithms and Their Applications to the Study of Markov Chains, Graph Theory, and the Optimal Stopping Problem

AbstractWe discuss the State Reduction/GTH (Grassmann, Taksar, Heyman) algorithm for recursively finding invariant measure. We demonstrate the relationship between this algorithm and the Freidlin–Wentzell “tree decomposition” approach to study the characteristics of Markov chains. The structure of the State Reduction/GTH algorithm suggests the natural idea for finding the distribution of a Markov chain at the moment of first visit to a given set, and some similar characteristics. We study the possible range of such algorithms. We also present a new algorithm for solving the classical problem of optimal stopping of a Markov chain based on a similar idea of sequential elimination of some states. We give shorter and more transparent proofs of some previously known results, and improve the bounds of Freidlin–Wentzell in the perturbation theory of Markov chains. Some applications to graph theory are also discussed
Arguesian Identities in Linear Lattices

AbstractA class of identities in the Grassmann–Cayley algebra which yields a large number of geometric theorems on the incidence of subspaces of projective spaces was found by Hawrylycz (“Geometric Identities in Invariant Theory,” Ph.D. thesis, Massachusetts, Institute of Technology, 1994). In this paper we establish a link between such identities in the Grassmann–Cayley algebra and a class of inequalities in the class of linear lattices, i.e., the lattices of commuting equivalence relations. We prove that a subclass of identities found by Hawrylycz, namely, the Arguesian identities of order 2, can be systematically translated into inequalities holding in linear lattices. As a consequence, we obtain a family of geometric theorems on the incidence of subspaces that are characteristic-free and independent of dimensions
Domains with Non-Compact Automorphism Group: A Survey

AbstractWe survey results arising from the study of domains in Cn with non-compact automorphism group. Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even non-pseudoconvex) domains with particular emphasis on characterizations of (i) smoothly bounded domains with non-compact automorphism group and (ii) the Levi geometry of boundary orbit accumulation points. Particular attention will be paid to results derived in the past ten years
Structures and Diagrammatics of Four Dimensional Topological Lattice Field Theories

AbstractCrane and Frenkel proposed a state sum invariant for triangulated 4-manifolds. They sketched the definition of a Hopf category that was to be used in their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4-manifolds using Crane–Yetter cocycles as Boltzmann weights. Our invariant is analogous to the 3-dimensional invariants defined by Dijkgraaf and Witten and the invariants that are defined via Hopf algebras. We present diagrammatic methods for the study of such invariants that illustrate connections between Hopf categories and moves to triangulations
On the Classification of k-Involutions

AbstractLet G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ, and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk is called a symmetric k-variety. These varieties occur in many problems in representation theory, geometry, and singularity theory. Over the last few decades the representation theory of these varieties has been extensively studied for k=R and C. As most of the work in these two cases was completed, the study of the representation theory over other fields, like local fields and finite fields, began. The representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space, like the restricted root systems with Weyl groups, etc. Thus it is essential to study first this structure and the related geometry. In this paper we give a characterization of the isomorphy classes of these symmetric k-varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers, p-adic numbers, finite fields and number fields
Enumerating Representations in Finite Wreath Products

AbstractLet Γ be a group (finite or infinite), H a finite group, and let Rn denote the sequence H≀Sn of symmetric wreath products as well as certain variants of it (including in particular H≀An and Wn, the Weyl group of type Dn). We compute the exponential generating function for the number |Hom(Γ, Rn)| of Γ-representations in Rn and for some refinements of this sequence under very mild finiteness assumptions on Γ (always met for instance if Γ is finitely generated). This generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group Γ and the enumeration of Γ-actions on finite sets on the one hand, and the recent results of Chigira concerning solutions of the equation xm=1 in the groups H≀Sn, H≀An, and Wn on the other. We also study the asymptotics of the function |Hom(G, H≀Sn)| for arbitrary finite groups G and H
On Disjoint Borel Uniformizations

AbstractLarman showed (1973, Mathematika20, 233–246) that any closed subset of the plane with uncountable vertical cross-sections has ℵ1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable cross sections which do not have more than ℵ1 disjoint Borel uniformizations, even if the continuum is much larger than ℵ1. This negatively answers some questions of Mauldin (1990, “Open Problems in Topology” (J. van Mill and G. M. Reed, Eds.), 617–629). The proof is based on a result of Stern, stating that certain Borel sets cannot be written as a small union of low-level Borel sets (1978, C. R. Acad. Sci. Pan's Ser. A–B286, A855–A857). The proof of the latter result uses Steel's method of forcing with tagged trees (1978, Ann. Math. Logic15, 55–74); a full presentation of this method, written in terms of Baire category rather than forcing, is given here
Relative Convolutions I. Properties and Applications

AbstractTo study operator algebras with symmetries in a wide sense we introduce a notion of relatives convolution operators induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already studied (operators of multiplication, usual group convolutions, two-sided convolution, etc.), and their various combinations. Basic properties of relative convolutions are given and a connection with usual convolutions is established. The examples presented show that relative convolutions provide us with a base for systematic application of harmonic analysis to pseudodifferential operator theory, complex and hypercomplex analysis, coherent states, wavelet transform, and quantum theory
Morphisms of Extensions of C*-Algebras: Pushing Forward the Busby Invariant

AbstractWe study completions of diagrams of extensions of C*-algebras in which all three C*-algebras in one of the rows and either the ideal or the quotient in the other are given, along with the three morphisms between them. We find universal solutions to all four of these problems under restrictions of varying severity, on the given vertical maps and describe the solutions in terms of push-outs and pull-backs of certain diagrams. Our characterization of the universal solution to one of the diagrams yields a concrete description of various amalgamated free products. This leads to new results about the K-theory of amalgamated free products, verifying the Cuntz conjecture in certain cases. We also obtain new results about extensions of matricial fieldC*-algebras, verifying partially a conjecture of Blackadar and Kirchberg. Finally, we show that almost commuting unitary matrices can be uniformly approximated by commuting unitaries when an index obstruction vanishes
Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

AbstractA real-valued function f defined on a convex set K is an approximately convex function iff it satisfiesfx+y2⩽f(x)+f(y)2+1. A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function E vanishing on the vertices of a simplex. A set A in a normed space is an approximately convex set iff for all a, b∈A the distance of the midpoint (a+b)/2 to A is ⩽1. The bounds on approximately convex functions are used to show that in Rn with the Euclidean norm, for any approximately convex set A, any point z of the convex hull of A is at a distance of at most [log2(n−1)]+1+(n−1)/2[log2(n−1)] from A. Examples are given to show this is the sharp bound. Bounds for general norms on Rn are also given
Strong Exact Borel Subalgebras of Quasi-hereditary Algebras and Abstract Kazhdan–Lusztig Theory

AbstractStrong exact Borel subalgebras and strong Δ-subalgebras are shown to exist for quasi-hereditary algebras which possess exact Borel subalgebras and Δ-subalgebras. This implies that the algebras associated with blocks of category O have strong exact Borel subalgebras and strong Δ-subalgebras. The structure of these subalgebras is shown to be closely related to abstract Kazhdan–Lusztig theory. The main technical tool in this paper is a construction which has an exact Borel subalgebras (of a given quasi-hereditary algebra) as input and a strong exact Borel subalgebra as output. From this result, Morita invariance of the existence of exact Borel subalgebras is derived
Hyperbolic Structures on the Configuration Space of Six Points in the Projective Line

AbstractThe oriented configuration space X+6 of six points on the real projective line is a noncompact three-dimensional manifold which admits a unique complete hyperbolic structure of finite volume with ten cusps. On the other hand, it decomposes naturally into 120 cells each of which can be interpreted as the set of equiangular hexagons with unit area. Similar hyperbolic structures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X+6 is at the center of a five parameter family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhibit two real analytic maps from the set of prescribed angles of hexagons into R10 whose components are the traces of the monodromies at the ten cusps. We show that this map has maximal rank 5 at the center
Gibbs Measures for Fibred Systems

AbstractWe consider a topological dynamical system T:Y→Y on a metric space Y which forms a fibre bundle over another dynamical system. If T is fibrewise expanding and exact along fibres and if ϕ is a Hölder continuous function we prove the existence of a system of conditional measures (called a family of Gibbs measures) where the Jacobian is determined by ϕ. This theorem reduces to Ruelle's Perron–Frobenius theorem when the base of the fibred system consists of a single point. The method of proof does not use any form of symbolic representation. We also study continuity properties of a family of Gibbs measures (over the base) and give applications to the equilibrium theory of higher dimensional complex dynamics
On the Cohomology of Galois Groups Determined by Witt Rings

AbstractLet F denote a field of characteristic different from two. In this paper we describe the mod2 cohomology of a Galois group GF (called the W-group of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H*(GF, F2) contains the mod2 Galois cohomology of F and that its structure will reflect important properties of the field. We construct a space XF endowed with an action of an elementary abelian group E such that the computation of the cohomology of GF reduces to calculating the equivariant cohomology H*E(XF, F2). For the case of a field which is not formally real this amounts to computing the cohomology of an explicit Euclidean space form, an object which is interesting in its own right. We provide a number of examples and a substantial combinatorial computation for the cohomology of the universal W-groups
Vector Fields and Their Duals

AbstractA standard way of realizing a Lie algebra is as a family of vector fields closed under commutation. Using the action on the universal enveloping algebra, one finds a realization in a dual form—the double dual. This is an algebraic Fourier transform of a “vector fields realization” of the Lie algebra. On the other hand, in the subject of umbral calculus (canonical boson calculus) the duals-to-vector fields play a primary role. It is shown that the double dual realizations of Lie algebras provide a rich source of examples for the umbral calculus, which, complementarily, provides a canonical construction of polynomial systems associated to the Lie algebra. For any finite-dimensional Lie algebra, take an element in the local Lie group it generates. Then there is an abelian family of operators such that acting on a canonical vacuum state, the abelian group gives the same result as the group element constructed via the given Lie algebra. In other words, they yield the same coherent states
The Bruhat–Chevalley Order of Parabolic Group Actions in General Linear Groups and Degeneration for Δ-Filtered Modules

AbstractLet k be an algebraically closed field and V a finite dimensional k-space. Let GL(V) be the general linear group of V and P a parabolic subgroup of GL(V). Now P acts on its unipotent radical Pu and on pu=LiePu, the Lie algebra of Pu, via the adjoint action. More generally, we consider the action of P on the lth member of the descending central series of pu denoted by p(l)u. All instances when P acts on p(l)u for l⩾0 with a finite number of orbits are known. In this note, we give a complete combinatorial description of the closure relation on the set of P-orbits on p(l)u, i.e., the Bruhat–Chevalley order, for every finite case. There is a canonical bijection between the set of P-orbits on p(l)u and the set of isomorphism classes of Δ-filtered modules of a particular dimension vector e of a certain quasi-hereditary algebra A(t, l). These isomorphism classes in turn are given by the orbits of a reductive group G(e) on the variety R(Δ)(e) of all A(t, l)-modules with Δ-filtration and dimension vector e. The subcategory of A(t, l)-mod of all Δ-filtered A(t, l)-modules of dimension vector e is denoted by F(Δ)(e). In our chief result, Theorem 1.1, we show that provided there is only a finite number of isomorphism classes of indecomposable modules in F(Δ) the following three posets coincide: (1)the Bruhat–Chevalley order on the set of P-orbits on p(l)u; (2)the Bruhat–Chevalley order on the set of G(e)-orbits on R(Δ)(e); (3)the poset opposite to the so called hom-order on the set of isomorphism classes of F(Δ)(e). The advantage of this hom-order is that it is given purely by discrete invariants and that it can be computed explicitly for any given finite case. Theorem 1.1 then in turn allows us to explicitly determine the closure relations for the P-orbits on p(l)u with the aid of this hom-order. We present some examples of Hasse diagrams in an Appendix
Unrectifiable 1-Sets with Moderate Essential Flatness Satisfy Besicovitch's 12-Conjecture

AbstractIn this paper we show that for a wide class of totally unrectifiable 1-sets in the plane (or even a Hilbert space) satisfying a mild measure-theoretic flatness condition almost everywhere, at sufficiently small scales, the lower spherical density is bounded above by 12 at almost every point, thereby affirming Besicovitch's conjecture, which states that for all totally unrectifiable 1-sets in the plane (or possibly even in Rn, or a Hilbert space), the lower spherical density is bounded above by 12 at almost every point
Galois Theory for Braided Tensor Categories and the Modular Closure

AbstractGiven a braided tensor *-category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C⋊S. This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over VectC with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between C and C⋊S and closed subgroups of the Galois group Gal(C⋊S/C)=AutC(C⋊S) of C, the latter being isomorphic to the compact group associated with S by the duality theorem of Doplicher and Roberts. Denoting by D⊂C the full subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S⊂D. Under this condition, C⋊S has no non-trivial degenerate objects iff S=D. If the original category C is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C≡C⋊D is called the modular closure of C since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2, Z). If all simple objects of S have dimension one the structure of the category C⋊S can be clarified quite explicitly in terms of group cohomology