pinecone/core-2020-05-10-deduplication huggingface dataset:
Macdonald Polynomials and Algebraic Integrability

AbstractWe construct explicitly (nonpolynomial) eigenfunctions of the difference operators by Macdonald in the case t=qk, k∈Z. This leads to a new, more elementary proof of several Macdonald conjectures, proved first by Cherednik. We also establish the algebraic integrability of Macdonald operators at t=qk (k∈Z), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including the BCn case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of the An root system where the previously known methods do not work
The Lp-Busemann–Petty Centroid Inequality

AbstractThe ratio between the volume of the p-centroid body of a convex body K in Rn and the volume of K attains its minimum value if and only if K is an origin symmetric ellipsoid. This result, the Lp-Busemann–Petty centroid inequality, was recently proved by Lutwak, Yang, and Zhang. In this paper we show that all the intrinsic volumes of the p-centroid body of K are convex functions of a time-like parameter when K is moved by shifting all the chords parallel to a fixed direction. The Lp-Busemann–Petty centroid inequality is a consequence of this general fact
A New Algorithm for Solving the Word Problem in Braid Groups

AbstractOne of the most interesting questions about a group is whether its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists, and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometric one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition in order to give a new approach for solving its word problem. Our algorithm, although not better in complexity, is faster in comparison with known algorithms for short braid words, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms
Infinite Non-simple C*-Algebras: Absorbing the Cuntz Algebra O∞

AbstractThe first named author has given a classification of all separable, nuclear C*-algebras A that absorb the Cuntz algebra O∞. (We say that A absorbs O∞ if A is isomorphic to A⊗O∞.) Motivated by this classification we investigate here if one can give an intrinsic characterization of C*-algebras that absorb O∞. This investigation leads us to three different notions of pure infiniteness of a C*-algebra, all given in terms of local, algebraic conditions on the C*-algebra. The strongest of the three properties, strongly purely infinite, is shown to be equivalent to absorbing O∞ for separable, nuclear C*-algebras that either are stable or have an approximate unit consisting of projections. In a previous paper (2000, Amer. J. Math.122, 637–666), we studied an intermediate, and perhaps more natural, condition: purely infinite, that extends a well known property for simple C*-algebras. The weakest condition of the three, weakly purely infinite, is shown to be equivalent to the absence of quasitraces in an ultrapower of the C*-algebra. The three conditions may be equivalent for all C*-algebras, and we prove this to be the case for C*-algebras that are either simple, of real rank zero, or approximately divisible
Trace Functionals on Noncommutative Deformations of Moduli Spaces of Flat Connections

AbstractWe describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. The resulting algebra is a variant of the quantum moduli algebra introduced by Alekseev, Grosse, and Schomerus and Buffenoir and Roche. We construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of Fedosov and Nest and Tsygan via Verlinde's formula
Representation Dimension and Quasi-hereditary Algebras

AbstractWe study Auslander's representation dimension of Artin algebras, which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators. We show the following statements: (1) if an Artin algebra A is stably hereditary, then the representation dimension of A is at most 3. (2) If two Artin algebras are stably equivalent of Morita type, then they have the same representation dimension. Particularly, if two self-injective algebras are derived equivalent, then they have the same representation dimension. (3) Any incidence algebra of a finite partially ordered set over a field has finite representation dimension. Moreover, we use results on quasi-hereditary algebras to show that (4) the Auslander algebra of a Nakayama algebra has finite representation dimension
Fractal Measures for Parabolic IFS

AbstractLet h be the Hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d⩾2. In case the system of maps is finite, we provide necessary and sufficient conditions for the h-dimensional Hausdorff measure to be positive and finite and also, assuming the strong open set condition holds, characterize when the h-dimensional packing measure of the limit set is positive and finite. We also prove that the upper ball (box)-counting dimension and the Hausdorff dimension of this limit set coincide. As a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3
Non-commutative Poisson Boundaries and Compact Quantum Group Actions

AbstractWe discuss some relationships between two different fields, a non-commutative version of the Poisson boundary theory of random walks and the infinite tensor product (ITP) actions of compact quantum groups on von Neumann algebras. In contrast to the ordinary compact group case, the ITP action of a compact quantum group on a factor may allow non-trivial relative commutant of the fixed point subalgebra. We give a probabilistic description of the relative commutant in terms of a non-commutative Markov operator. In particular, we show that the following three objects can be naturally identified in the case of the quantum group SUq(2): (1) the relative commutant of the fixed point algebra under the action, (2) the space of harmonic elements for some non-commutative Markov operator on the dual quantum group of SUq(2), and (3) the weak closure L∞(T\SUq(2)) of one of the Podles quantum spheres. In view of the ordinary Poisson boundary theory of random walks on discrete groups, it shows that symbolically the quantum homogeneous space T\SUq(2) may be regarded as the “Poisson boundary” of a non-commutative random walk on the dual object of SUq(2). An analogy of the Poisson integral formula is also given
Ergodic Limits on the Conformal Repellers

AbstractLet J be the repeller of an expanding, C1+δ-conformal topological mixing map g. Let Φ:J→Rd be a continuous function and let α(x)=limn→∞1n∑n−1j=0Φ(gjx) (when the limit exists) be the ergodic limit. It is known that the possible α(x) are just the values ∫Φ dμ for all g-invariant measures μ. For any α in the range of the ergodic limits, we prove the following variational formula:dimx∈J:α(x)=α=maxμhg(μ)∫log∥Dxg∥dμ(x):∫Φ dμ=α, where μ is a g-invariant Borel probability measure on J, hg(μ) is the entropy of μ, ∥Dxg∥ is the operator norm of the differential Dxg, and dim is the Hausdorff dimension or the packing dimension. This result gives a substantial extension of the well-known case that Φ is Hölder continuous. We also prove that unless the same ergodic limit exists everywhere, the set of points whose ergodic limit does not exist has the same Hausdorff dimension as the whole space J
Absolute Continuity for Curvature Measures of Convex Sets, III

AbstractThis work is devoted to the investigation of the basic relationship between the geometric shape of a convex set and measure theoretic properties of the associated curvature and surface area measures. We study geometric consequences of and conditions for absolute continuity of curvature and surface area measures with respect to (d−1)-dimensional Hausdorff measure in Euclidean space Rd. Our main results are two “transfer principles” which allow one to translate properties connected with the absolute continuity of the rth curvature measure of a convex body to dual properties related to the absolute continuity of the (d−1−r)th surface area measure of the polar body, and conversely. Applications are also considered
A Cellular Nerve for Higher Categories

AbstractWe realise Joyal' cell category Θ as a dense subcategory of the category of ω-categories. The associated cellular nerve of an ω-category extends the well-known simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen's sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory ΘA of the category of A-algebras for each ω-operad A in Batanin's sense. Whenever A is contractible, the resulting homotopy category of A-algebras (i.e. weak ω-categories) is equivalent to the homotopy category of compactly generated spaces
Viro Method for the Construction of Real Complete Intersections

AbstractThe Viro method is a powerful construction method of real nonsingular algebraic hypersurfaces with prescribed topology. It is based on polyhedral subdivisions of Newton polytopes. A combinatorial version of the Viro method is called combinatorial patchworking and arises when the considered subdivisions are triangulations. B. Sturmfels has generalized the combinatorial patchworking to the case of real complete intersections. We extend his result by generalizing the Viro method to the case of real complete intersections
Integral Representations of Solutions of Differential Equations Free from Accessory Parameters

AbstractWe show that every accessory parameter free system of differential equations of Okubo normal form has integral representation of solutions. The proof is constructive; we study the change of solutions under the operations—the extension and the restriction, which have been introduced by Yokoyama [Construction of systems of differential equations of Okubo normal form with rigid monodromy, preprint] in order to construct every such system of differential equations. Several examples are given
A Unified Theory of Commutator Estimates for a Class of Interpolation Methods

AbstractA general family of interpolation methods is introduced which includes, as special cases, the real and complex methods and also the so-called ± or G1 and G2 methods defined by Peetre and Gustavsson–Peetre. Derivation operators Ω and translation operators R are introduced for all methods of this family. A theorem is proved about the boundedness of the commutators [T,Ω] and [T,R] for operators T which are bounded on the spaces of the pair to which the interpolation method is applied. This extends and unifies results previously known for derivation and translation operators in the contexts of the real and complex methods. Other results deal with higher order commutators and also include an “equivalence theorem,” i.e. it is shown that, as previously known only for real interpolation spaces, all these interpolation spaces have two different equivalent definitions in the style of the “J method” and “K method.” Auxiliary results which may also be of independent interest include the equivalence of Lions–Schechter complex interpolation spaces defined using an annulus with the same spaces defined in the usual way, using a strip
Special Transverse Slices and Their Enveloping Algebras

AbstractLet G be a simple, simply connected algebraic group over C, g=Lie G, N (g) the nilpotent cone of g, and (E,H,F) an sl2-triple in g. Let S=E+Ker ad F, the special transverse slice to the adjoint orbit Ω of E, and S0=S∩N (g). The coordinate ring C[S0] is naturally graded (See Slodowy, “Simple Singularities and Simple Algebraic Groups,” Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin/Heidelberg/New York, 1980). Let Z(g) be the centre of the enveloping algebra U(g) and η:Z(g)→C an algebra homomorphism. Identify g with g* via a Killing isomorphism and let χ denote the linear function on g corresponding to E. Following Kawanaka (Generalized Gelfand–Graev representations and Ennola duality, in “Algebraic Groups and Related Topics” Advanced Studies in Pure Mathematics, Vol. 6, pp. 175–206, North-Holland, Amsterdam/New York/Oxford, 1985), Moeglin (C.R. Acad. Sci. Paris, Ser. I 303 No. 17 (1986), 845–848), and Premet (Invent. Math.121 (1995), 79–117), we attach to χ a nilpotent subalgebra mχ⊂g of dimension (dim Ω)/2 and a 1-dimensional mχ-module Cχ. Let H̃χ denote the algebra opposite to Endg(U(g)⊗U(mχ)Cχ) and H̃χ,η=H̃χ⊗Z(g)Cη. It is proved in the paper that the algebra H̃χ,η has a natural filtration such that gr(H̃χ,η), the associated graded algebra, is isomorphic to C[S0]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of g
Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

AbstractUnder some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real–analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1–40; Stud. Math.97 (1991), 189–225) are obtained
Primitive Limit Algebras and C*-Envelopes

AbstractIn this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C*-envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C*-envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n-transitive for all n⩾1) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to *-representations of the C*-envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal
Triality, Exceptional Lie Algebras and Deligne Dimension Formulas

AbstractWe give a computer-free proof of the Deligne, Cohen and de Man formulas for the dimensions of the irreducible g-modules appearing in g⊗k, k⩽4, where g ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic chart. Our proofs use the triality model of the magic square, which we review and present a simplified proof of its validity. We conclude with some general remarks about obtaining “series” of Lie algebras in the spirit of Deligne and Vogel
On a q-Analogue of the McKay Correspondence and the ADE Classification of sl̂2 Conformal Field Theories

AbstractThe goal of this paper is to give a category theory based definition and classification of “finite subgroups in Uq(sl2)” where q=eπi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to sl̂2 at level k=l−2. We show that “finite subgroups in Uq(sl2)” are classified by Dynkin diagrams of types An,D2n,E6,E8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in (sl̂2)k conformal field theory.The results we get are parallel to those known in the theory of von Neumann subfactors, but our proofs are independent of this theory
Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

AbstractWe construct a unique G-equivariant graded star product on the algebra S(g)/I of polynomial functions on the minimal nilpotent coadjoint orbit Omin of G where G is a complex simple Lie group and g≠sl(2, C). This strengthens the result of Arnal, Benamor and Cahen.Our main result is to compute, for G classical, the star product of a momentum function μx with any function f. We find μx★f=μxf+12{μx, f}t+Λx(f)t2. For g different from sp(2n, C), Λx is not a differential operator. Instead Λx is the left quotient of an explicit order 4 algebraic differential operator Dx by an order 2 invertible diagonalizable operator. Precisely, Λx=−141E′(E′+1)Dx where E′ is a positive (half-form) shift of the Euler vector field. Thus μx★f is not local in f.Using ★ we construct a positive definite hermitian inner product on S(g)/I. The Hilbert space completion of S(g)/I is then a unitary representation of G. This quantizes Omin in the sense of geometric quantization and the orbit method
Complexity in Complex Analysis

AbstractWe show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable. Such domains turn out to have the property that their classical domain functions all extend to be meromorphic functions on a compact Riemann surface, and this condition will be shown to be equivalent to the condition that an Ahlfors map and its derivative are algebraically dependent. We also show how many of these results can be generalized to finite Riemann surfaces
On the Structure of Weak Hopf Algebras

AbstractWe study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo, a trivial automorphism. We find a sufficient condition in terms of Tr(S2) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple
Enumerating Representations in Finite Wreath Products II: Explicit Formulae

AbstractWe present several contributions to the enumerative theory of wreath product representations developed in a previous paper by the first named author (Adv. in Math.153 (2000), 118–154). Theorem 3.1 of the present paper establishes an explicit formula for one of the key ingredients in the description of the corresponding generating functions given in Müller (2000) (the exterior function ΦΓ). Building on Theorem 1 in Müller (2000) and the latter result, we derive explicit formulae for the exponential generating function of the series {∣Hom (Γ,Rn)∣} in the case where Γ is dihedral or a finite abelian group, and the representation sequence {Rn} is any of {H≀Sn} or {H≀An} with a fixed finite group H, or the sequence {∣Hom(G, Wn)}. Moreover, we verify a conjecture concerning the asymptotic behaviour of the sequence \{∣Hom(G,Wn)∣\} for finite groups G made in Müller (2000) in the case when G is dihedral or abelian
Species recognition by the sequence of discharge intervals in weakly electric fishes of the genus Campylomormyrus (Mormyridae, Teleostei)

In two Campylomormyrus species, tamandua and rhynchophorus from Central Africa, the electric organ discharge (EOD) activity was studied during the nocturnal activity phase in the laboratory. Both species have a pulse-type EOD of less than 200 μs duration and similar waveform; the sequence of inter-discharge intervals, however, differed characteristically: C. tamandua displayed a 'micro-pattern' encompassing nearly the whole range of intervals in a sequence of only three intervals (long-intermediate-short), while in C. rhynchophorus adjacent intervals tended to be more similar in duration. Four C. rhynchophorus and five C. tamandua were tested for their preference of conspecific pulse sequences rather than those of the other species, respectively, in playback experiments. These were performed in a T-maze paradigm such that two pulse patterns, one from each species, were simultaneously presented via two electric fish dipole models. Two pulse sequences recorded from different individuals were used to represent each species; the natural EOD waveform was replaced by a square-wave pulse of approximately natural duration. The pattern pairings and the sequence of stimulus patterns followed a randomized blocks design. Each of the four C. rhynchophorus had significantly higher preference scores for the conspecific pulse patterns than for those of C. tamandua, while in C. tamandua no such discrimination was observed. It is suggested that the juvenile C. tamandua used would probably have joined mixed-species schools as reported to exist in the wild, while this appears unlikely in the more mature C. rhynchophorus. One of the functions of the inter-discharge interval code of communication in mormyrids is species recognition although this may be seen only in sufficiently mature fish
Singing and mating success in water pipits: one specific song element makes all the difference.

Positive correlations between male vocalization and reproductive success have been documented for many animal species. They are usually based on differences between males in vocalization rate, duration or repertoire size. Here, we present probably the first field study linking differences in territorial overlap and mating status to differences in a single, clearly definable song element, the ‘Snarr’. Male water pipits,Anthus spinoletta, with high Snarr scores were mated more often than males with low scores, and their territories overlapped less with those of neighbours. Although correlating positively with male body condition, the frequency of the Snarr did not reflect male age, territory size, territory quality in terms of food and paternal performance. Therefore, it seems unlikely that the higher mating success of males with high Snarr scores results from active female choice of high-quality males; rather, high Snarr scores seem to signal dominance in males. Likely mechanisms that produce the link between vocalization and mating success, and potential costs that prevent some males from producing the Snarr at a higher rate, are discussed