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We show that for a taut foliation F with one-sided branching of an atoroidal 3-manifold M, one can construct a pair of genuine laminations with solid torus complementary regions which bind every leaf of F in a geodesic lamination. These laminations come from a universal circle, a refinement of the universal circles proposed by Thurston, which maps monotonely and group-equivariantly to each of the circles at infinity of the leaves of the universal cover of F, and is minimal with respect to this property. This circle is intimately bound up with the extrinsic geometry of the leaves of F. In particular, for any pair of leaves m,l of the universal cover of F with m>l, the leaf l is asymptotic to m in a dense set of directions at infinity, where the leaf space is co-oriented so that the foliation branches in the negative direction. This is a macroscopic version of an infinitesimal result of Thurston's and gives much more drastic control over the topology and geometry of F. The pair of laminations can be used to produce a pseudo-Anosov flow transverse to F which is regulating in the non-branching direction. Rigidity results for these laminations in the R-covered case are extended to the case of one-sided branching. In particular, an R-covered foliation can only be deformed to a foliation with one-sided branching along one of the two laminations canonically associated to the R-covered foliation in math.GT/9903173, and these laminations become exactly the laminations constructed for the new branched foliation. Other corollaries include that the ambient manifold is d-hyperbolic in the sense of Gromov, and that a self-homeomorphism of this manifold homotopic to the identity is isotopic to the identity.
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arxiv:math/0101026
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We consider systems $(M,\omega,g)$ with $M$ a closed smooth manifold, $\omega$ a real valued closed one form and $g$ a Riemannian metric, so that $(\omega,g)$ is a Morse-Smale pair, Definition~2. We introduce a numerical invariant $\rho(\omega,g)\in[0,\infty]$ and improve Morse-Novikov theory by showing that the Novikov complex comes from a cochain complex of free modules over a subring $\Lambda'_{[\omega],\rho}$ of the Novikov ring $\Lambda_{[\omega]}$ which admits surjective ring homomorphisms $\ev_s:\Lambda'_{[\omega],\rho}\to\C$ for any complex number $s$ whose real part is larger than $\rho$. We extend Witten-Helffer-Sj\"ostrand results from a pair $(h,g)$ where $h$ is a Morse function to a pair $(\omega,g)$ where $\omega$ is a Morse one form. As a consequence we show that if $\rho<\infty$ the Novikov complex can be entirely recovered from the spectral geometry of $(M,\omega,g)$.
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arxiv:math/0101043
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Recall that a triangular Hopf algebra A is said to have the Chevalley property if the tensor product of any two simple A-modules is semisimple, or, equivalently, if the radical of A is a Hopf ideal. There are two reasons to study this class of triangular Hopf algebras: First, it contains all known examples of finite-dimensional triangular Hopf algebras; second, it can be, in a sense, "completely understood". Namely, it was shown in our previous work with Andruskiewitsch that any finite-dimensional triangular Hopf algebra with the Chevalley property is obtained by twisting of a finite-dimensional triangular Hopf algebra with R-matrix of rank at most 2 which, in turn, is obtained by "modifying" the group algebra of a finite supergroup. This provides a classification of such Hopf algebras. The goal of this paper is to make this classification more effective and explicit, i.e. to parameterize isomorphism classes of finite-dimensional triangular Hopf algebras with the Chevalley property by group-theoretical objects, similarly to how it was done in our previous work in the semisimple case. This is achieved in Theorem 2.2, where these classes are put in bijection with certain septuples of data. In the semisimple case, the septuples reduce to the quadruples, and we recover the result of our previous paper. In the minimal triangular pointed case, we recover a classification theorem from a previous paper of the second author.
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arxiv:math/0101049
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Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of `Idempotent Mathematics' with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.
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arxiv:math/0101080
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We give a surgery formula for the torsions and Seiberg-Witten invariants associated with $Spin^c$-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.
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arxiv:math/0101108
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The main result provides an algorithm for determining the minimal free resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8 general points with arbitrary multiplicities; the results hold over algebraically closed fields of any characteristic. The algorithm, which works by giving a formula in certain cases and a reduction to these cases otherwise, does not involve Gr\"obner bases, and so is very fast, even for very large multiplicities. Partial information is also obtained in certain cases with $n>8$.
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arxiv:math/0101110
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We give new estimates for the eigenvalues of the hypersurface Dirac operator in terms of the intrinsic energy-momentum tensor, the mean curvature and the scalar curvature. We also discuss their limiting cases as well as the limiting cases of the estimates obtained by X. Zhang and O. Hijazi in [13] and [10]. We compare these limiting cases with those corresponding to the Friedrich and Hijazi inequalities. We conclude by comparing these results to intrinsic estimates for the Dirac-Schr\"odinger operator D_f = D - f/2.
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arxiv:math/0101111
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A wide class of noncommutative spaces, including 4-spheres based on all the quantum 2-spheres and suspensions of matrix quantum groups is described. For each such space a noncommutative vector bundle is constructed. This generalises and clarifies various recent constructions of noncommutative 4-spheres.
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arxiv:math/0101129
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Let $K$ be a compact Lie group. We introduce the process of symplectic implosion, which associates to every Hamiltonian $K$-manifold a stratified space called the imploded cross-section. It bears a resemblance to symplectic reduction, but instead of quotienting by the entire group, it cuts the symmetries down to a maximal torus of $K$. We examine the nature of the singularities and describe in detail the imploded cross-section of the cotangent bundle of $K$, which turns out to be identical to an affine variety studied by Gelfand, Vinberg, Popov, and others. Finally we show that ``quantization commutes with implosion''.
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arxiv:math/0101159
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The sum in the title is a rational multiple of pi^n for all integers n=2,3,4,... for which the sum converges absolutely. This is equivalent to a celebrated theorem of Euler. Of the many proofs that have appeared since Euler, a simple one was discovered only recently by Calabi: the sum is written as a definite integral over the unit n-cube, then transformed into the volume of a polytope Pi_n in R^n whose vertices' coordinates are rational multiples of pi. We review Calabi's proof, and give two further interpretations. First we define a simple linear operator T on L^2(0,pi/2), and show that T is self-adjoint and compact, and that Vol(Pi_n) is the trace of T^n. We find that the spectrum of T is {1/(4k+1) : k in Z}, with each eigenvalue 1/(4k+1) occurring with multiplicity 1; thus Vol(Pi_n) is the sum of the n-th powers of these eigenvalues. We also interpret Vol(Pi_n) combinatorially in terms of the number of alternating permutations of n+1 letters, and if n is even also in terms of the number of cyclically alternating permutations of n letters. We thus relate these numbers with S(n) without the intervention of Bernoulli and Euler numbers or their generating functions.
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arxiv:math/0101168
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Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth differential forms. Out of this data one defines cohomology of the invariants and equivariant cohomology of $M$. We establish Koszul duality between each other.
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arxiv:math/0101180
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The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a $\Q$-Gorenstein singularity.
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arxiv:math/0101251
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Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient $PGL(2,\bold{Z})\setminus\bold{P}^1(\bold{R})$ should be considered as non-commutative modular curve, and show that the modular complex can be seen as a sequence of $K_0$-groups of the related crossed-product $C^*$-algebras. This paper is an expanded version of the previous "On the distribution of continued fractions and modular symbols". The main new features are Section 4 on non-commutative geometry and the modular complex and Section 1.2.2 on the Mixmaster Universe.
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arxiv:math/0102006
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We show that for stochastic measures with freely independent increments, the partition-dependent stochastic measures of math.OA/9903084 can be expressed purely in terms of the higher stochastic measures and the higher diagonal measures of the original.
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arxiv:math/0102062
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We describe all almost contact metric, almost hermitian and $G_2$-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the $\nabla$-parallel spinors. In particular, we obtain solutions of the type II string in dimension $n=5,6$ and 7.
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arxiv:math/0102142
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We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable 3-manifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in \pi_1(M).
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arxiv:math/0102154
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We prove that an arbitrary function, which is holomorphic on some neighbourhood of z=0 in $\mathbb{C}^N$ and vanishes at z=0, whose values are bounded linear operators mapping one separable Hilbert space into another one, can be represented as the transfer function of some multiparametric discrete time-invariant conservative scattering linear system with a Krein space of its inner states.
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arxiv:math/0102157
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We construct an explicit asymptotically good tower of curves over the field with eight elements. Its limit is 3/2.
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arxiv:math/0102158
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We give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety X_w for any element w in S_n. Our description of the irreducible components is computationally more efficient (O(n^6)) than the previously best known algorithms. This result proves a conjecture of Lakshmibai and Sandhya regarding this singular locus. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.
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arxiv:math/0102168
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We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R^2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows one to extend those results in a number of ways.
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arxiv:math/0102197
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We study the limiting behavior of eigenfunctions/eigenvalues of the Laplacian of a family of Riemannian metrics that degenerates on a hypersurface. Our results generalize earlier work concerning the degeneration of hyperbolic surfaces.
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arxiv:math/0102219
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A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the T-geometric one. T-geometry is the simplest geometric conception (essentially only finite point sets are investigated) and simultaneously it is the most general one. It is insensitive to the space continuity and has a new property -- nondegeneracy. Fitting the T-geometry metric with the metric tensor of Riemannian geometry, one can compare geometries, constructed on the basis of different conceptions. The comparison shows that along with similarity (the same system of geodesics, the same metric) there is a difference. There is an absolute parallelism in T-geometry, but it is absent in the Riemannian geometry. In T-geometry any space region is isometrically embeddable in the space, whereas in Riemannian geometry only convex region is isometrically embeddable. T-geometric conception appears to be more consistent logically, than the Riemannian one.
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arxiv:math/0103002
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We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T[(M_k,\theta_k)_{k=1}^{\ell}]$ with index $i(M_k)$ finite are either $c_0$ or $\ell_p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i(M_k)$ and the parameter $\theta_k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T[(A_k,\theta_k)_{k=1}^\infty]$ in terms of the asymptotic behaviour of the sequence $\Vert\sum_{i=1}^n e_i\Vert$ where $(e_i)$ is the canonical basis.
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arxiv:math/0103003
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This expository article proves some results of Ferguson, on the approximation of continuous functions on a compact subset of R by polynomials with integral coefficients.
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arxiv:math/0103004
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In this paper we derive a list of all the possible indecomposable normalized rank--two vector bundles without intermediate cohomology on the prime Fano threefolds and on the complete intersection Calabi Yau threefolds, say $V$, of Picard number $\rho=1$. For any such bundle $\E$, if it exists, we find the projective invariants of the curves $C \subset V$ which are the zero-locus of general global sections of $\E$. In turn, a curve $C \subset V$ with such invariants is a section of a bundle $\E$ from our lists. This way we reduce the problem for existence of such bundles on $V$ to the problem for existence of curves with prescribed properties contained in $V$. In part of the cases in our lists the existence of such curves on the general $V$ is known, and we state the question about the existence on the general $V$ of any type of curves from the lists.
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arxiv:math/0103010
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A broadly applicable geometric approach for constructing nef divisors on blow ups of algebraic surfaces at n general points is given; it works for all surfaces in all characteristics for any n. This construction is used to obtain substantial improvements for currently known lower bounds for n point Seshadri constants. Remarks are included about a range of applications to classical problems involving linear systems on P2.
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arxiv:math/0103029
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A suggestion is put forward regarding a partial proof of FLT(case1), which is elegant and simple enough to have caused Fermat's enthusiastic remark in the margin of his Bachet edition of Diophantus' "Arithmetica". It is based on an extension of Fermat's Small Theorem (FST) to mod p^k for any k>0, and the cubic roots of 1 mod p^k for primes p=1 mod 6. For this solution in residues the exponent p distributes over a sum, which blocks extension to equality for integers, providing a partial proof of FLT case1 for all p=1 mod 6. This simple solution begs the question why it was not found earlier. Some mathematical, historical and psychological reasons are presented. . . . . In a companion paper, on the triplet structure of Arithmetic mod p^k, this cubic root solution is extended to the general rootform of FLT (mod p^k) (case1), called "triplet". While the cubic root solution (a^3=1 mod p^k) involves one inverse pair: a+a^{-1} = -1 mod p^k, a triplet has three inverse pairs in a 3-loop: a+b^{-1} = b+c^{-1} = c+a^{-1} = -1 (mod p^k) where abc = 1 (mod p^k), which reduces to the cubic root form if a=b=c (\neq 1) mod p^k. The triplet structure is not restricted to p-th power residues (for some p \geq 59) but applies to all residues in the group G_k(.) of units in the semigroup of multiplication mod p^k.
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arxiv:math/0103051
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We explain the theory of refined cycle maps associated to arithmetic mixed sheaves. This includes the case of arithmetic mixed Hodge structures, and is closely related to work of Asakura, Beilinson, Bloch, Green, Griffiths, Mueller-Stach, Murre, Voisin and others.
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arxiv:math/0103116
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It is shown that any denumerable list L to which Cantor's diagonal method was applied is incomplete. However, this doesn't allow us to affirm that the cardinality of the real numbers of the interval [0, 1] is greater than the cardinality of the finite natural numbers. Paper withdrawn (its essential part is included in the version 3 of math.GM/0108119).
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arxiv:math/0103124
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Consider a closed analytic curve $\gamma$ in the complex plane and denote by > $D_+$ and $D_-$ the interior and exterior domains with respect to the curve. The point $z=0$ is assumed to be in $D_+$. Then according to Riemann theorem there exists a function $w(z)=\frac 1r z+\sum_{j=0}^\infty p_j z^{-j}$, mapping $D_-$ to the exterior of the unit disk $\{w\in C|| w | >1\}$. It is follow from [arXiv : hep-th /0005259] that this function is described by formula $\log w=\log z-\partial_{t_0} (\frac 12\partial_{t_0}+\sum\limits_{k\geqslant 1}\frac{z^{-k}}{k} \partial_{t_k})v$, where $v=v(t_0, t_1, \bar t_1, t_2, \bar t_2,...)$ is a function from the area $t_0$ of $D_+$ and the momemts $t_k$ of $D_-$. Moreover, this function satisfies the dispersionless Hirota equation for 2D Toda lattice hierarchy. Thus for an effectivisation of Riemann theorem it is sufficiently to find a representation of $v$ in the form of Taylor series $v=\sum N(i_0 | i_1,...,i_k| \bar i_1,...,\bar i_{\bar k})t_0 t_{i_1},...,t_k \bar t_{\bar i_1},...,\bar t_{\bar i_{\bar k}}$. The numbers $N(i_0 | i_1,...,i_k | \bar i_1, ..., \bar i_{\bar k})$ for $i_\alpha, \bar i_\beta\leqslant 2$ is found in [arXiv: hep-th/0005259]. In this paper we find some recurrence relations that give a possible to find all $N(i_0\bigl| i_1,...,i_k|\bar i_1,...,\bar i_{\bar k})$.
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arxiv:math/0103136
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Self-scaled barrier functions on self-scaled cones were introduced through a set of axioms in 1994 by Y.E. Nesterov and M.J. Todd as a tool for the construction of long-step interior point algorithms. This paper provides firm foundation for these objects by exhibiting their symmetry properties, their intimate ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and algebraic classification theory. In a first part we recall the characterisation of the family of self-scaled cones as the set of symmetric cones and develop a primal-dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then show that any self-scaled barrier function decomposes in an essentially unique way into a direct sum of self-scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self-scaled barrier functions using the correspondence between symmetric cones and Euclidean Jordan algebras.
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arxiv:math/0103196
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In this paper, we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications.
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arxiv:math/0103211
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Certain computable polynomials are described whose leading coefficients are equal to multiplicities in the tensor product decomposition for representations of a Lie algebra of ADE type.
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arxiv:math/0103212
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The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds'', Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map of an effective action of a p-adic group on compact connected n-manifolds with the aid of some new ideas. There is no attempt to use Smith Theory even though there may be similarities.
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arxiv:math/0103215
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Let \Omega X be the space of Moore loops on a finite, q-connected, n-dimensional CW complex X, and let R be a subring of Q containing 1/2. Let p(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M / Torsion M. We show that the inclusion of the sub-Lie algebra P of primitive elements of FH_*(\Omega X;R) induces an isomorphism of Hopf algebras UP = FH_*(\Omega X;R), provided p(R) > n/q - 1. Furthermore, the Hurewicz homomorphism induces an embedding of F(\pi_*(\Omega X)\otimes R) in P, with torsion cokernel. As a corollary, if X is elliptic, then FH_*(\Omega X;R) is a finitely-generated R-algebra.
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arxiv:math/0103223
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Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det^\prime\Lambda and det L as the metric on M varies, where \Delta denotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals denotes the conformal Laplacian. We see that the behavior of these functionals depends on the dimension. Indeed, every critical metric for (-1)^{(n-1)/2}det^\prime\Lambda or (-1)^{(n-1}/2}| det L| has finite index. Consequently there are no local maxima if n=4m+1 and no local minima if n=4m+3. We show that the standard 3-sphere is a local maximum for det^\prime\Lambda while the standard (4m-3)-sphere with m=1,2,...,4, is a saddle point. By contrast, for all odd n, the standard n-sphere is a local extremal for det L. An important tool in our work is the canonical trace on odd class operators in odd dimensions. This trace is related to the determinant by the formula det Q = TR log Q, and we prove some basic results on how to calculate the trace.
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arxiv:math/0103239
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We compute the space of global sections for the symmetric power of the tautological bundle on the punctual Hilbert scheme of a complex smooth projective surface.
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arxiv:math/0104005
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We study the completion of a group relative to a Zariski dense representation in a reductive algebraic group over a field $k$. The characteristic zero case was worked out previously by R. Hain; we extend his results to arbitrary characteristic. The primary application is to the study of the cohomology of groups such as $SL_n(k[[T]])$.
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arxiv:math/0104023
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This paper introduces a notion of integrality that is suitable for non-commutative varieties. It is compatible with the usual notion of integrality for schemes. The function field and generic point of a non-commutative integral space are also defined. These agree with the usual notion for noetherian schemes. It is shown that these notions behave as one would wish. For example, various non-commutative analogues of projective space are integral.
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arxiv:math/0104046
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Let $f_1, ..., f_n$ be homogeneous polynomials generating a generic ideal $I$ in the ring of polynomials in $n$ variables over an infinite field. Moreno-Soc\'ias conjectured that for the graded reverse lexicographic term ordering, the initial ideal ${\rm in}(I)$ is a weakly reverse lexicographic ideal. This paper contains a new proof of Moreno-Soc{\'\i}as' conjecture for the case $n=2$.
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arxiv:math/0104047
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We introduce a new construction of error-correcting codes from algebraic curves over finite fields. Modular curves of genus g -> infty over a field of size q0^2 yield nonlinear codes more efficient than the linear Goppa codes obtained from the same curves. These new codes now have the highest asymptotic transmission rates known for certain ranges of alphabet size and error rate. Both the theory and possible practical use of these new record codes require the development of new tools. On the theoretical side, establishing the transmission rate depends on an error estimate for a theorem of Schanuel applied to the function field of an asymptotically optimal curve. On the computational side, actual use of the codes will hinge on the solution of new problems in the computational algebraic geometry of curves.
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arxiv:math/0104115
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Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific categories. The corresponding Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. It is shown that commutation relations corresponding to a system with generalized statistics can be constructed from relations corresponding to Bolzman statistics.
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arxiv:math/0104141
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The study of different types of ideals in non self-adjoint operator algebras has been a topic of recent research. This study focuses on principal ideals in subalgebras of groupoid C*-algebras. An ideal is said to be principal if it is generated by a single element of the algebra. We look at subalgebras of r-discrete principal groupoid C*-algebras and prove that these algebras are principal ideal algebras. Regular canonical subalgebras of almost finite C*-algebras have digraph algebras as their building blocks. The spectrum of almost finite C*-algebras has the structure of an r-discrete principal groupoid and this helps in the coordinization of these algebras. Regular canonical subalgebras of almost finite C*-algebras have representations in terms of open subsets of the spectrum for the enveloping C*-algebra. We conclude that regular canonical subalgebras are principal ideal algebras.
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arxiv:math/0104163
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The aim of this paper is to prove the following result. For any commutative formal group ${\frak F}(x\otimes 1,1\otimes x),$ which is considered as a formal group over $H_\mathbb{Q},$ there exists a homomorphism to a formal group of the form ${\frak c}+x\otimes 1+1\otimes x,$ where $\frak c\in H_\mathbb{Q}{\mathop{\hat{\otimes}} \limits_{R_\mathbb{Q}}}H_\mathbb{Q}$ such that $(\id \otimes \epsilon){\frak c}=0= (\epsilon \otimes \id){\frak c}.$
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arxiv:math/0104167
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We construct universal Drinfel'd twists defining deformations of Hopf algebra structures based upon simple Lie algebras and contragredient simple Lie superalgebras. In particular, we obtain deformed and dynamical double Yangians. Some explicit realisations as evaluation representations are given for sl(N), sl(1|2) and osp(1|2).
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arxiv:math/0104181
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We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1, where the L-B operator is for the metric g = e^{2u}g_0, with $g_0$ the standard Euclidean metric on R^2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order, Delta u +K_g e^(2u)=0, Delta K_g + e^(2u)=0, with x in R^2, describing a conformally flat surface with a Gauss curvature function K_g that is generated self-consistently through the metric's conformal factor.
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arxiv:math/0104183
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It is shown that every linear surjective isometry between two right, full, Hilbert C*-modules is a sum of two maps : a (bi-) module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-) module actions. Every such isometry can be extended to an isometry of the C* linking algebras.
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arxiv:math/0104188
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We apply Tate's conjecture on algebraic cycles to study the N\'eron-Severi groups of varieties fibered over a curve. This is inspired by the work of Rosen and Silverman, who carry out such an analysis to derive a formula for the rank of the group of sections of an elliptic surface. For a semistable fibered surface, under Tate's conjecture we derive a formula for the rank of the group of sections of the associated Jacobian fibration. For fiber powers of a semistable elliptic fibration $E --> C$, under Tate's conjecture we give a recursive formula for the rank of the N\'eron-Severi groups of these fiber powers. For fiber squares, we construct unconditionally a set of independent elements in the N\'eron-Severi groups. When $E --> C$ is the universal elliptic curve over the modular curve $X_0(M)/\Q$, we apply the Selberg trace formula to verify our recursive formula in the case of fiber squares. This gives an analytic proof of Tate's conjecture for such fiber squares over $\Q$, and it shows that the independent elements we constructed in fact form a basis of the N\'eron-Severi groups. This is the fiber square analog of the Shioda-Tate Theorem.
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arxiv:math/0104200
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This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(Z/2.3) and the Hauptvermutung for manifolds.
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arxiv:math/0105047
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We shall give an axiomatic construction of Wess-Zumino-Witten actions valued in (G=SU(N)), (N\geq 3). It is realized as a functor ({WZ}) from the category of conformally flat four-dimensional manifolds to the category of line bundles with connection that satisfies, besides the axioms of a topological field theory, the axioms which abstract Wess-Zumino-Witten actions. To each conformally flat four-dimensional manifold (\Sigma) with boundary (\Gamma=\partial\Sigma), a line bundle (L=WZ(\Gamma)) with connection over the space (\Gamma G) of mappings from (\Gamma) to (G) is associated. The Wess-Zumino-Witten action is a non-vanishing horizontal section (WZ(\Sigma)) of the pull back bundle (r^{\ast}L) over (\Sigma G) by the boundary restriction (r). (WZ(\Sigma)) is required to satisfy a generalized Polyakov-Wiegmann formula with respect to the pointwise multiplication of the fields (\Sigma G). Associated to the WZW-action there is a geometric descrption of extensions of the Lie group (\Omega^3G) due to J. Mickelsson. In fact we shall construct two abelian extensions of (\Omega^3G) that are in duality.
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arxiv:math/0105090
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We construct in a Sonine Space of entire functions a subspace related to the Riemann zeta function and we show that the quotient contains vectors intrinsically attached to the non-trivial zeros and their multiplicities.
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arxiv:math/0105120
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In the early 1970's S.Tennenbaum proved that all countable models of PA^- + forall_1-Th(N) are embeddable into the reduced product N^omega/F, where F is the cofinite filter. In this paper we show that if M is a model of PA^- + forall_1-Th(N), and |M|= aleph_1, then M is embeddable into N^omega/D, where D is any regular filter on omega.
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arxiv:math/0105134
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Let T be a complete first order theory in a countable relational language L . We assume relation symbols have been added to make each formula equivalent to a predicate. Adjoin a new unary function symbol sigma to obtain the language L_sigma; T_sigma is obtained by adding axioms asserting that sigma is an L-automorphism. We provide necessary and sufficient conditions for T_sigma to have a model companion when T is stable. Namely, we introduce a new condition: T_sigma admits obstructions, and show that T_sigma has a model companion iff and only if T_sigma does not admit obstructions. This condition is weakening of the finite cover property: if a stable theory T has the finite cover property then T_sigma admits obstructions.
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arxiv:math/0105136
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We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic projective space or to the quaternionic hyperbolic space. We describe locally the hyper-Hermitian quaternionic Kaehler manifolds with closed Lee form and show that the only complete simply connected such manifold is the quaternionic hyperbolic space.
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arxiv:math/0105206
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We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, etc. I give a construction of \textit{doubles} for \textit{graded} $QS$- and \textit{graded $QP$-manifolds} (graded manifolds endowed with a homological vector field and a Schouten/Poisson bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded $QS$-manifolds can be considered, roughly, as ``generalized Lie bialgebroids''. The double for them is closely related with the analog of Drinfeld's double for Lie bialgebroids recently suggested by Roytenberg. Lie bialgebroids as a generalization of Lie bialgebras, over some base manifold, were defined by Mackenzie and P. Xu. Graded $QP$-manifolds give an {odd version} for all this, in particular, they contain ``odd analogs'' for Lie bialgebras, Manin triples, and Drinfeld's double.
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arxiv:math/0105237
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There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on $R^n$ for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, for an icis $V=f^{-1}(0)$, $f:(C^n, 0) \to (C^k, 0)$, $f$ is real, we define a complex analytic family of quadratic forms parameterized by the points $\epsilon$ of the image $(C^k, 0)$ of the map $f$, which become real for real $\epsilon$ and in this case their signatures defer from the "real" index by $\chi(V_\epsilon)-1$, where $\chi(V_\epsilon)$ is the Euler characteristic of the corresponding smoothing $V_\epsilon=f^{-1}(\epsilon)\cap B_\delta$ of the icis $V$.
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arxiv:math/0105242
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The number of comparisons X_n used by Quicksort to sort an array of n distinct numbers has mean mu_n of order n log n and standard deviation of order n. Using different methods, Regnier and Roesler each showed that the normalized variate Y_n := (X_n - mu_n) / n converges in distribution, say to Y; the distribution of Y can be characterized as the unique fixed point with zero mean of a certain distributional transformation. We provide the first rates of convergence for the distribution of Y_n to that of Y, using various metrics. In particular, we establish the bound 2 n^{- 1 / 2} in the d_2-metric, and the rate O(n^{epsilon - (1 / 2)}) for Kolmogorov-Smirnov distance, for any positive epsilon.
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arxiv:math/0105248
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The tensor product of vector and arbitrary representations of the nonstandard q-deformation U'_q(so(n)) of the universal enveloping algebra U(so(n)) of Lie algebra so(n) is defined. The Clebsch-Gordan coefficients of tensor product of vector and arbitrary classical or nonclassical type representations of q-algebra U'_q(so(n)) are found in an explicit form. The Wigner-Eckart theorem for vector operators is proved.
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arxiv:math/0105259
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Given a holomorphic selfmap f of the complex projective plane of algebraic degree at least 2, we give sufficient conditions on a positive closed (1,1) current S of unit mass under which the normalized pullbacks of S under iterates of f converge to the Green current T. In particular, we completely characterize the plane algebraic curves whose normalized pullbacks converge weakly to T.
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arxiv:math/0105260
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The purpose of this paper is to prove an essentially sharp L^2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature.
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arxiv:math/0105266
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Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $H^2 \times CP^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics.
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arxiv:math/0106077
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We define in a global manner the notion of a connective structure for a gerbe on a space X. When the gerbe is endowed with trivializing data with respect to an open cover of X, we describe this connective structure in two separate ways, which extend from abelian to general gerbes the corresponding descriptions due to J.- L. Brylinski and N. Hitchin. We give a global definition of the 3-curvature of this connective structure as a 3-form on X with values in the Lie stack of the gauge stack of the gerbe. We also study this notion locally in terms of more traditional Lie algebra-valued 3-forms. The Bianchi identity, which the curvature of a connection on a principal bundle satisfies, is replaced here by a more elaborate equation.
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arxiv:math/0106083
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Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics, or with connections with non-zero curvature). In this note we give some techniques to prove these estimates which rely more on physical space methods such as vector fields, tube localization, splitting into coarse and fine scales, and induction on scales (in the spirit of recent papers of Wolff).
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arxiv:math/0106091
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We prove uniform $L^p$ estimates for a family of paraproducts and corresponding maximal operators.
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arxiv:math/0106092
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We use the homological perturbation lemma to give an explicit proof of the cyclic Eilenberg-Zilber theorem for cylindrical modules.
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arxiv:math/0106167
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We study the theory of scattering for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling,in space dimension 3.We prove in particular the existence of modified wave operators for that system with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.The method consists in solving the wave equation, substituting the result into the Schr"odinger equation,which then becomes both nonlinear and nonlocal in time,and treating the latter by the method previously used for a family of generalized Hartree equations with long range interactions.
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arxiv:math/0107087
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Let X be a smooth projective complex curve, and let M be the moduli space of stable Higgs bundles on X (with genus g>1), with rank n and fixed determinant \xi, with n and deg(\xi) coprime. Let X' and \xi' be another such curve and line bundle, and M' the corresponding moduli space. We prove that if M and M' are isomorphic as algebraic varieties, then X and X' are isomorphic.
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arxiv:math/0107108
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We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal polyhedral maps on the Torus.
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arxiv:math/0107123
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We study the spectral properties of the linearized Euler operator obtained by linearizing the equations of incompressible two dimensional fluid at a steady state with the vorticity that contains only two nonzero complex conjugate Fourier modes. We prove that the essential spectrum coincides with the imaginary axis, and give an estimate from above for the number of isolated nonimaginary eigenvalues. In addition, we prove that the spectral mapping theorem holds for the group generated by the linearized 2D Euler operator.
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arxiv:math/0107125
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We use equivariant methods to establish basic properties of orbifold K-theory. We introduce the notion of twisted orbifold K-theory in the presence of discrete torsion, and show how it can be explicitly computed for global quotients.
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arxiv:math/0107168
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We give an example of a hypersurface in \C^2 through 0 whose stability group at 0 is determined by 3-jets, but not by jets of any lesser order. We also examine some of the properties which the stability group of this infinite type hypersurface shares with the 3-sphere in \C^2.
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arxiv:math/0107170
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Paper withdrawn due to errors (superseded by math.AG/0604303). Proposition 11.4 is false, Section 12 is false, and the main statement is true only for bundles $B$ with $c_1(B)$ SU(2)-invariant.
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arxiv:math/0107196
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Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure formalizing concepts of length and angle. The interplay of Riemannian metric and its metric connection with mechanical structures produces some features which are absent in the case of general (non-Riemannian) manifolds. The aim of present paper is to discuss these features and develop special language for describing Newtonian, Lagrangian, and Hamiltonian dynamical systems in Riemannian manifolds.
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arxiv:math/0107212
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q-Supernomial coefficients are generalizations of the q-binomial coefficients. They can be defined as the coefficients of the Hall-Littlewood symmetric function in a product of the complete symmetric functions or the elementary symmetric functions. Hatayama et al. give explicit expressions for these q-supernomial coefficients. A combinatorial expression as the generating function of ribbon tableaux with (co)spin statistic follows from the work of Lascoux, Leclerc and Thibon. In this paper we interpret the formulas by Hatayama et al. in terms of rigged configurations and provide an explicit statistic preserving bijection between rigged configurations and ribbon tableaux thereby establishing a new direct link between these combinatorial objects.
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arxiv:math/0107214
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We study the gap (= "projection norm" = "graph distance") topology of the space of (not necessarily bounded) self--adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show that the space is connected contrary to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.
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arxiv:math/0108014
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We define the notions of micro-support and regularity for ind-sheaves, and prove their invariance by contact transformations. We apply the results to the ind-sheaves of temperate holomorphic solutions of D-modules. We prove that the micro-support of such an ind-sheaf is the characteristic variety of the corresponding D-module and that the ind-sheaf is regular if the D-module is regular holonomic.
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arxiv:math/0108065
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Results are expoundd for the investigation of efficiency of the critical-component method for solving degenerate and ill-posed systems of linear algebraic equations
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arxiv:math/0108074
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Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g=2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+1}. Furthermore the geometric syzygies constructed by Green and Lazarsfeld [1984] from g^1_{k+1}'s form a non degenerate configuration of finitely many rational normal curves on this P^{k+1}. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in this case.
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arxiv:math/0108078
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If M is a monoid (e.g. the lattice Z^D), and G is a finite (nonabelian) group, then G^M is a compact group; a `multiplicative cellular automaton' (MCA) is a continuous transformation F:G^M-->G^M which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of G. We characterize when MCA are group endomorphisms of G^M, and show that MCA on G^M inherit a natural structure theory from the structure of G. We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.
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arxiv:math/0108084
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The random-to-top and the riffle shuffle are two well-studied methods for shuffling a deck of cards. These correspond to the symmetric group $S_n$, i.e., the Coxeter group of type $A_{n-1}$. In this paper, we give analogous shuffles for the Coxeter groups of type $B_n$ and $D_n$. These can be interpreted as shuffles on a ``signed'' deck of cards. With these examples as motivation, we abstract the notion of a shuffle algebra which captures the connection between the algebraic structure of the shuffles and the geometry of the Coxeter groups. We also briefly discuss the generalisation to buildings which leads to q-analogues.
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arxiv:math/0108094
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We give a short proof of Waldhausen's homeomorphism theorem for orientable Haken 3-manifolds.
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arxiv:math/0108116
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A proof that the set of real numbers is denumerable is given.
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arxiv:math/0108119
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We establish local $(L^p,L^q)$ mapping properties for averages on curves. The exponents are sharp except for endpoints.
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arxiv:math/0108137
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We describe algorithmic methods for the Gauss-Manin connection of an isolated hypersurface singularity based on the microlocal structure of the Brieskorn lattice. They lead to algorithms for computing invariants like the monodromy, the spectrum, and the spectral pairs. These algorithms use a normal form algorithm for the Brieskorn lattice, standard basis methods for localized polynomial rings, and univariate factorization. We give a detailed description of the algorithm to compute the monodromy.
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arxiv:math/0108145
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We prove several versions of Grothendieck's Theorem for completely bounded linear maps $T\colon E \to F^*$, when E and F are operator spaces. We prove that if E,F are $C^*$-algebras, of which at least one is exact, then every completely bounded $T\colon E \to F^*$ can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as $T=T_r+T_c$ where $T_r$ (resp. $T_c$) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on $C^*$-algebras. Moreover, our result holds more generally for any pair E,F of "exact" operator spaces. This yields a characterization of the completely bounded maps from a $C^*$-algebra (or from an exact operator space) to the operator Hilbert space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert operator spaces and their direct sums are the only operator spaces E such that both E and its dual $E^*$ are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class.
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arxiv:math/0108205
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We prove in this paper that the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of connected and simply connected Lie group, are isomorphic to the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of compact maximal subgroups, without localization. Some noncompact quantum groups and algebras were constructed and their irreducible representations were classified in recent works of Do Ngoc Diep and Nguyen Viet Hai, and Do Duc Hanh by using deformation quantization. In this paper we compute their K-groups, periodic cyclic homology groups and their Chern characters.
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arxiv:math/0109042
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In this article we defined and studied quasi-finite comodules, the cohom functors for coalgebras over rings. linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work characterizes equivalences between comodule categories generalizing the Morita-Takeuchi theory to coalgebras over rings. Morita-Takeuchi contexts in our setting is defined and investigated, a correspondence between strict Morita-Takeuchi contexts and equivalences of comodule categories over the involved coalgebras is obtained. Finally we proved that for coalgebras over QF-rings Takeuchi's representation of the cohom-functor is also valid.
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arxiv:math/0109061
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Nazarov and Tarasov recently generalized the notion of the rank of a partition to skew partitions. We give several characterizations of the rank of a skew partition and one possible characterization that remains open. One of the characterizations involves the decomposition of a skew shape into a minimal number of border strips, and we develop a theory of these MBSD's as well as of the closely related minimal border strip tableaux. An application is given to the value of a character of the symmetric group S_n indexed by a skew shape z at a permutation whose number of cycles is the rank of z.
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arxiv:math/0109092
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Given a cone pseudodifferential operator $P$ we give a full asymptotic expansion as $t\to 0^+$ of the trace $\Tr Pe^{-tA}$, where $A$ is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains $\log t$ and new $(\log t)^2$ terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals.
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arxiv:math/0109154
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We prove that for an induced CR structure on a compact, generic, regular 3-pseudoconcave CR submanifold ${\bold M}\subset{\bold G}$, of a complex manifold ${\bold G}$, satisfying condition $\dim H^1({\bold M}, T^{\prime}({\bold G})|_{\bold M})=0$ all the close CR structures are induced by close embeddings.
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arxiv:math/0109181
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Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.
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arxiv:math/0109188
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The 15 Gauss contiguous relations for ${}_2F_1$ hypergeometric series imply that any three ${}_2F_1$ series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also discuss contiguous relations of generalized and basic hypergeometric functions, and several applications of them.
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arxiv:math/0109222
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In this paper, we extend the Banach-Stone theorem to the non commutative case, i.e, we prove that the structure of the liminal $C^{*}$-algebras $\cal A$ determines the topology of its primitive ideal space.
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arxiv:math/0109226
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We address the problem of the determination of the images of the Galois representations attached to genus 2 Siegel cusp forms of level 1 having multiplicity one. These representations are symplectic. We prove that the images are as large as possible for almost every prime, if the Siegel cusp form is not a Maass spezialform and verifies two easy to check conditions.
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arxiv:math/0109228
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Let f:\Sigma_1 --> \Sigma_2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in \Sigma_1\times \Sigma_2. This article discusses a canonical way to deform f along area preserving diffeomorphisms. This deformation process is realized through the mean curvature flow of the graph of f in \Sigma_1\times \Sigma_2. It is proved that the flow exists for all time and the map converges to a canonical map. In particular, this gives a new proof of the classical topological results that O(3) is a deformation retract of the diffeomorphism group of S^2 and the mapping class group of a Riemman surface of positive genus is a deformation retract of the diffeomorphism group .
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arxiv:math/0110020
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We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to conformally compactifiable manifolds, and show that the mass is completion-independent. We also prove the result, closely related to the problem at hand, that conformal completions of conformally compactifiable manifolds are unique.
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arxiv:math/0110035
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In this note we give a p-adic proof of Hodge symmetry for smooth, projective threefolds over complex numbers.
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arxiv:math/0110066
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Here we present in a single essay a combination and completion of the several aspects of the problem of randomness of individual objects which of necessity occur scattered in our texbook "An Introduction to Kolmogorov Complexity and Its Applications" (M. Li and P. Vitanyi), 2nd Ed., Springer-Verlag, 1997.
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arxiv:math/0110086
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In case of Lebesgue measure zero of postcritical set the necessary and sufficient conditions (in terms of convergence of sequences of measures) of existence of invariant conformal structures on J(R) are obtained.
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arxiv:math/0110093
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We give a partial positive answer to a conjecture of Tyurin (\cite {Tyu}). Indeed we prove that on a general quintic hypersurface of $\Pj^4$ every arithmetically Cohen--Macaulay rank 2 vector bundle is infinitesimally rigid.
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arxiv:math/0110102
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