pinecone/core-2020-05-10-deduplication huggingface dataset:
Positive Sampling in Wavelet Subspaces

AbstractThis paper is devoted to the discussion of a “hybrid” sampling series, a series of translates of a nonnegative summability function used in place of an orthogonal scaling function. The coefficients in the series are taken to be sampled values of the function to be approximated. This enables one to avoid the integration which arises in the other series. The approximations based on this hybrid series have certain desirable convergence properties: they are locally uniformly convergent for locally continuous functions, they have quadratic uniform convergence rate for functions in certain Sobolev spaces, they are locally bounded when the function is locally bounded and therefore, in particular, Gibbs' phenomenon is avoided. Numerical experiments are given to illustrate the theoretical results and to compare these approximations with the scaling function approximations
Scale Continuous, Scale Discretized, and Scale Discrete Harmonic Wavelets for the Outer and the Inner Space of a Sphere and Their Application to an Inverse Problem in Geomathematics

AbstractIn this paper, we construct a multiscale solution method for the gravimetry problem, which is concerned with the determination of the earth's density distribution from gravitational measurements. For this purpose, isotropic scale continuous wavelets for harmonic functions on a ball and on a bounded outer space of a ball, respectively, are constructed. The scales are discretized and the results of numerical calculations based on regularization wavelets are presented. The obtained solutions yield topographical structures of the earth's surface at different levels of localization ranging from continental boundaries to local structures such as Ayer's Rock and the Amazonas area
Multiwavelets on the Interval

AbstractSmooth orthogonal and biorthogonal multiwavelets on the real line with their scaling function vectors being supported on [−1,1] are of interest in constructing wavelet bases on the interval [0,1] due to their simple structure. In this paper, we shall present a symmetric C2 orthogonal multiwavelet with multiplicity 4 such that its orthogonal scaling function vector is supported on [−1,1], has accuracy order 4, and belongs to the Sobolev space W2.56288. Biorthogonal multiwavelets with multiplicity 4 and vanishing moments of order 4 are also constructed such that the primal scaling function vector is supported on [−1,1], has the Hermite interpolation properties, and belongs to W3.63298, while the dual scaling function vector is supported on [−1,1] and belongs to W1.75833. A continuous dual scaling function vector of the cardinal Hermite interpolant with multiplicity 4 and support [−1,1] is also given. All the wavelet filters constructed in this paper have closed form expressions. Based on the above constructed orthogonal and biorthogonal multiwavelets on the real line, both orthogonal and biorthogonal multiwavelet bases on the interval [0,1] are presented. Such multiwavelet bases on the interval [0,1] have symmetry, small support, high vanishing moments, good smoothness, and simple structures. Furthermore, the sequence norms for the coefficients based on such orthogonal and biorthogonal multiwavelet expansions characterize Sobolev norms ‖·‖Ws([0,1]) for s∈(−2.56288,2.56288) and for s∈(−1.75833,3.63298), respectively
Highly Nonstationary Wavelet Packets

AbstractWe introduce a new class of basic wavelet packets, called highly nonstationary wavelet packets, that generalize the class of nonstationary wavelet packets. We define the periodic Shannon wavelet packets and show how to obtain perturbations of this system using periodic highly nonstationary wavelet packets. Such perturbations provide examples of periodic wavelet packets that do form a Schauder basis for Lp[0,1) for 1<p<∞. We also consider the representation of the differentiation operator in such periodic wavelet packets
Analyzing the Weyl–Heisenberg Frame Identity

AbstractIn 1990, Daubechies proved a fundamental identity for Weyl–Heisenberg systems which is now called the Weyl–Heisenberg (WH)-frame identity. WH-frame identity: If g∈W(L∞,L1), then for all continuous, compactly supported functions f we have ∑m,n∣〈f,EmbTnag〉∣2=1b∑k∫Rf(t)f(t−k/b)∑ng(t−na)g(t−na−k/b)dt. It has been folklore that the identity will not hold universally. We make a detailed study of the WH-frame identity and show (1) The identity does not require any assumptions on ab (such as the requirement that ab≤1 have a frame); (2) As stated above, the identity holds for all f∈L2(R); (3) The identity holds for all bounded, compactly supported functions if and only if g∈L2(R); (4) The identity holds for all compactly supported functions if and only if ∑n∣g(x−na)∣2≤B a.e. Moreover, in (2)–(4) above, the series on the right converges unconditionally. We will also see that in general, symmetric, norm, and unconditional convergences of the series in the WH-frame identity are all different
Perturbation Stability of Coherent Riesz Systems under Convolution Operators

AbstractWe study the orthogonal perturbation of various coherent function systems (Gabor systems, Wilson bases, and wavelets) under convolution operators. This problem is of key relevance in the design of modulation signal sets for digital communication over time-invariant channels. Upper and lower bounds on the orthogonal perturbation are formulated in terms of spectral spread and temporal support of the prototype, and by the approximate design of worst-case convolution kernels. Among the considered bases, the Weyl–Heisenberg structure which generates Gabor systems turns out to be optimal whenever the class of convolution operators satisfies typical practical constraints
Hyperbolic Secants Yield Gabor Frames

AbstractWe show that (g2,a,b) is a Gabor frame when a>0, b>0, ab<1, and g2(t)=(12πγ)1/2(coshπγt)−1 is a hyperbolic secant with scaling parameter γ>0. This is accomplished by expressing the Zak transform of g2 in terms of the Zak transform of the Gaussian g1(t)=(2γ)1/4 exp(−πγt2), together with an appropriate use of the Ron–Shen criterion for being a Gabor frame. As a side result it follows that the windows, generating tight Gabor frames, that are canonically associated to g2 and g1 are the same at critical density a=b=1. Also, we display the “singular” dual function corresponding to the hyperbolic secant at critical density
Wavelet Characterizations for Anisotropic Besov Spaces

AbstractThe goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to Lp-spaces with 0<p<∞ are derived
A Framework for Image Deblurring Using Wavelet Packet Bases

AbstractWe show in this paper that the average over translations of an operator diagonal in a wavelet packet basis is a convolution. We also show that an operator diagonal in a wavelet packet basis can be decomposed into several operators of the same kind, each of them being better conditioned. We investigate the possibility of using such a convolution to approximate a given convolution (in practice an image blur). Then we use these approximations to deblur images. First, we show that this framework permits us to redefine existing deblurring methods. Then, we show that it permits us to define a new variational method which combines the wavelet packet and the total variation approaches. We argue and show by experiments that this permits us to avoid the drawbacks of both approaches which are, respectively, the ringing and the staircasing
On Generalized Gaussian Quadratures for Exponentials and Their Applications

AbstractWe introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions.We use a generalization of a representation theorem due to Carathéodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. For a given accuracy ϵ, selecting an eigenvalue close to ϵ yields an approximate quadrature with that accuracy. To compute its weights and nodes, we present a new fast algorithm.These new quadratures can be used to approximate and integrate bandlimited functions, such as prolate spheroidal wave functions, and essentially bandlimited functions, such as Bessel functions. We also develop, for a given precision, an interpolating basis for bandlimited functions on an interval
Nuclear Ground-State Masses and Deformations

We tabulate the atomic mass excesses and nuclear ground-state deformations of
8979 nuclei ranging from $^{16}$O to $A=339$. The calculations are based on the
finite-range droplet macroscopic model and the folded-Yukawa single-particle
microscopic model. Relative to our 1981 mass table the current results are
obtained with an improved macroscopic model, an improved pairing model with a
new form for the effective-interaction pairing gap, and minimization of the
ground-state energy with respect to additional shape degrees of freedom. The
values of only 9 constants are determined directly from a least-squares
adjustment to the ground-state masses of 1654 nuclei ranging from $^{16}$O to
$^{263}$106 and to 28 fission-barrier heights. The error of the mass model is
0.669~MeV for the entire region of nuclei considered, but is only 0.448~MeV for
the region above $N=65$.Comment: 50 pages plus 20 PostScript figures and 160-page table obtainable by
  anonymous ftp from t2.lanl.gov in directory masses, LA-UR-93-308
Weak Interaction Rates Of sd-Shell Nuclei In Stellar Environment
  Calculated in the Proton-Neutron Quasiparticle Random Phase Approximation

Allowed weak interaction rates for sd-shell nuclei in stellar environment are
calculated using a generalized form of proton-neutron quasiparticle RPA model
with separable Gamow-Teller forces. Twelve different weak rates are calculated
for each nucleus as a function of temperature and density. This project
consists of calculation of weak rates for a total of 709 nuclei with masses
ranging from A = 18 to 100. This paper contains calculated weak rates for
sd-shell nuclei. The calculated capture and decay rates take into consideration
the latest experimental energy levels and ft value compilations. The results
are also compared with earlier works. Particle emission processes from excited
states, previously ignored, are taken into account, and are found to
significantly affect some beta decay rates.Comment: 64 pages, 17 figures, rate tables are presented in an abbreviated
  form to save space. Complete rate tables can be seen in the original pape
A Symmetric Function Generalization of the Chromatic Polynomial of a Graph

AbstractFor a finite graph G with d vertices we define a homogeneous symmetric function XG of degree d in the variables x1, x2, ... . If we set x1 = ... = xn= 1 and all other xi = 0, then we obtain χG(n), the chromatic polynomial of G evaluated at n. We consider the expansion of XG in terms of various symmetric function bases. The coefficients in these expansions are related to partitions of the vertices into stable subsets, the Möbius function of the lattice of contractions of G, and the structure of the acyclic orientations of G. The coefficients which arise when XG is expanded in terms of elementary symmetric functions are particularly interesting, and for certain graphs are related to the theory of Hecke algebras and Kazhdan-Lusztig polynomials
A Basis for Polynomial Solutions to Systems of Linear Constant Coefficient PDE's

AbstractLetKrepresent either the real or the complex numbers. LetPk,k=1, 2, …, rbe constant coefficient (with coefficients fromk) polynomials innvariables and letN⩽M={u(x)∈K[x]⩽M∣Pk(∂/∂x1, …, ∂/∂xn) u(x)=0;k=1, 2, …, rbe the set of all polynomial solutions (of degree ⩽M) to this system of partial differential equations. We solve the problem of finding an easily computed basis for the vector spaceN⩽M. To do this we use a certain associative, and commutative algebra (defined overK), namelyK[β]=K[β1,β2, …,βn] where {Pk(β)=0∣k=1, …, r} and {β1m1β2m2…βnmn=0∣m1+…+mn=M+1}. Let the vector spaceK[β]⩽Mequal the span overKof {β1m1β2m2…βnmn∣m1+…+mn⩽M}. We show how the expression ∑Mj=0(x1⊗β1+…+xn⊗βn)j/j! can be used to find an easily computed basis forN⩽M
Projective Invariants of Four Subspaces

AbstractWe compute a set of generators of the ring of invariants for a set of 4 subspaces in a projective space. More precisely, letK[Gr4] be the ring of regular functions on the product of four Grassmannians, we present a set of generators forK[Gr4]SL(V), whereVis the underlying vector space. We also study syzygies among the generators. In most cases generators are algebraically independent. Only in the case where the affine dimensions of the four subspaces are given by the quadruple (k, k, k′, k′) andk+k′=dimV, there is, one and only one syzygy, the precise form of which is given
The Incidence Algebra of Polyhedra over the Minkowski Algebra

AbstractThe Minkowski algebra of polyhedra over an ordered field is introduced, and the incidence algebra of polyhedra over this algebra is considered. The Gram– Sommerville and Gauss–Bonnet theorems and their analogs for relatively open convex polyhedra are given by various Möbius inversion formulas. Cheeger's Chern–Gauss–Bonnet density formula for piecewise flat spaces is generalized to non-compact and unbounded polyhedra, and its non-numerical forms are naturally expressed in terms of interior cone chains. The tangential densities and interior cone functions are defined as the dual versions of the Chern–Gauss–Bonnet densities and exterior cone functions. The tangential density formulas and their non-numerical forms are similarly represented in terms of exterior curvatures and exterior cone chains
On Posets and Hopf Algebras

AbstractWe generalize the notion of the rank-generating function of a graded poset. Namely, by enumerating different chains in a poset, we can assign a quasi-symmetric function to the poset. This map is a Hopf algebra homomorphism between the reduced incidence Hopf algebra of posets and the Hopf algebra of quasi-symmetric functions. This work implies that the zeta polynomial of a poset may be viewed in terms Hopf algebras. In the last sections of the paper we generalize the reduced incidence Hopf algebra of posets to the Hopf algebra of hierarchical simplicial complexes
The Decidability of the Existential Theory of the Poset of Recursively Enumerable Degrees with Jump Relations

AbstractWe show that the existential theory of the recursively enumerable degrees in the language L containing predicates for order andn-jump comparability for alln, and constant symbols for least and greatest elements, is decidable. The decidability follows from our main theorem, where we show that any finite L-structure which is consistent with the order relation, the order-preserving property of the jump operator, and the property of the jump operator that the jump of an element is strictly greater than the element, can be embedded into the r.e. degrees
An Algebraic-Geometric Parametrization of the Canonical Basis

AbstractOne of the observations of [L4] has been that the canonical basis [L1] of the + part of a quantized enveloping algebra attached to a Cartan matrix of finite type is naturally in bijection with a collection of subsets of the set of “totally positive” elements inG(R((ε))) whereGis a semisimple simply connected algebraic group attached to the dual Cartan matrix andεis an indeterminate. (See also [L5].)This collection of subsets is not defined in algebraic-geometric terms since its definition involves positive real numbers. This paper arose from the desire to replace this collection of subsets by some algebraic-geometric objects which make sense over any field and still are in bijection with the canonical basis. For simplicity we will restrict ourselves to the simply laced case. (The general case can be easily reduced to it.) We replaceG(R((ε))) byGs=G(C[[ε]]/εsC[[ε]]) wheresis some large integer; this is regarded as an algebraic group overCof dimensionsdimG. We replace the collections of subsets above by a finite collection of closed, irreducible algebraic subvarieties ofGs. This collection of subvarieties ofGsis naturally in bijection with a large subset of the canonical basis which becomes the entire canonical basis ass→∞.The bijection is established at the combinatorial level. I do not know how to relate directly the canonical basis with the subvarieties above
Boundaries of Coxeter Matroids

AbstractThe purpose of this paper is to introduce, for a finite Coxeter groupW, the mod2 boundary operator on the space of all Coxeter matroids (also known asWP-matroids) forWandP, wherePvaries through all the proper standard parabolic subgroups ofW(Theorem 3 of the paper). A remarkably simple interpretation of Coxeter matroids as certain sets of faces of the generalized permutahedron associated with the Coxeter groupW(Theorem 1) yields a natural definition of the boundary of a Coxeter matroid. The latter happens to be a union of Coxeter matroids for maximal standard parabolic subgroupsQiofP(Theorem 2). These results have very natural interpretations in the case of ordinary matroids and flag-matroids (Section 3)
Relating a Locally Compact Abelian Group to Its Bohr Compactification

AbstractWe find several conditions on a locally compact Abelian groupGnecessary and sufficient thatG+, the groupGin the topology inherited from its Bohr compactification, is realcompact. We show further for suchGthat every continuous real-valued function from a closed subgroup ofG+extends continuously overG+; when in additionGis discrete, every continuous function from a subgroup ofG+to a complete metric space extends continuously overG+. These results respond to several of the questions posed by E. K. van Douwen [Topology and Its Applications34(1990), 69–91]
Self-Affine Tiles in Rn

AbstractAself-affine tilein Rnis a setTof positive measure withA(T)=∪d∈D(T+d), whereAis an expandingn×nreal matrix with |det(A)|=man integer, and D={d, d2, ..., dm}⊆Rnis a set ofmdigits. It is known that self-affine tiles always give tilings of Rnby translation. This paper extends known characterizations of digit sets D yielding self-affine tiles. It proves several results about the structure of tilings of Rnpossible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings
Order Analogues and Betti Polynomials

AbstractWe exhibit an order-preserving surjection from the lattice of subgroups of a finite abelianp-group of typeλonto the product of chains of lengths the parts of the partitionλ. Thereby, we establish the subgroup lattice as an order-theoretic, not just enumerative,p-analogue of the chain product. This insight underlies our study of the simplicial complexesΔS(p), whose simplices are chains of subgroups of orderspk, somek∈S. Each of these subgroup complexes is homotopy equivalent to a wedge of spheres of dimension |S|−1. The number of spheres in the wedge,βS(p), is known to have nonnegative coefficients as a polynomial inp. Our main result provides a topological explanation of this enumerative result. We use our order-preserving surjection to findβS(pmaximal simplices inΔS(pwhose deletion leaves a contractible subcomplex. This work suggests a definition of order analogue; our main result holds for any semimodular lattices that are order analogues of a semimodular lattice
Zeta Functions of Finite Graphs and Coverings

AbstractThree different zeta functions are attached to a finite connected, possibly irregular graphX. They originate with a zeta function of Ihara which is an analogue of Riemann's as well as Selberg's zeta function. The three zeta functions are associated to one vertex variable, two variables for each edge, and 2r(2r−1) path variables, respectively. Hereris the number of generators of the fundamental group ofX. We show how to specialize the variables of the last two zeta functions to obtain the first and we give elementary proofs of generalizations of Ihara's formula which says that the zeta function for a regular graph is the reciprocal of a polynomial. Many examples of covering graphs are also considered
Higher Dimensional Algebra I. Braided Monoidal 2-Categories

AbstractWe begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by previously Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the “center” of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2-categories
On Degenerations and Extensions of Finite Dimensional Modules

AbstractWe derive a cancellation theorem for degenerations of modules that says in particular, that projective or injective common direct summands can always be neglected. Combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease. For representation-directed algebras, whence in particular for path algebras of Dynkin quivers, we provide an alternative proof which shows in addition that any minimal degenerationNofMcomes from an exact sequence with middle termMwhose end terms add up toN. By a careful examination, the same is true for degenerations of matrix pencils. Having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences. In particular, we show that any non-simple indecomposable over a tame quiver is an extension of an indecomposable and a simple
Arguesian Identities in Invariant Theory

AbstractHaving been motivated by an example of Doubilet, Rota, and Stein [Stud. Appl. Math.56(1976), 185–216], we present a technique for constructing geometric identities in a Grassmann–Cayley algebra. Each identity represents a projective invariant closely related to the Theorem of Desargues in the plane and its generalizations to higher dimensional projective space. The construction employs certain combinatorial properties of matchings in bipartite graphs. We also prove a dimension independence result for Arguesian identities, thereby connecting the identities with lattice theory
The Margulis Lemma and the Thick and Thin Decomposition for Convex Real Projective Surfaces

AbstractConvex real projective surfaces are quotients of simply convex domains in the real projective planeRP2under the properly discontinuous and free action of a projective automorphism group. Such surfaces carry Hilbert metrics defined by logarithms of cross ratios. We prove an analogous proposition to the Margulis lemma in hyperbolic geometry holding for such a surfaceΣwith a Hilbert metric. This allows us to decomposeΣinto thick and thin components. We show a compactness result that given a certain collection of simple closed curvesα1, …, αnonΣ, the subset of the deformation space of convex structures P(Σ) corresponding to structures where the Hilbert lengths ofα1, …, αnare bounded above is compact