73,308 research outputs found
Quadratic functions on torsion groups
We investigate classification results for general quadratic functions on
torsion abelian groups. Unlike the previously studied situations, general
quadratic functions are allowed to be inhomogeneous or degenerate. We study the
discriminant construction which assigns, to an integral lattice with a
distinguished characteristic form, a quadratic function on a torsion group.
When the associated symmetric bilinear pairing is fixed, we construct an affine
embedding of a quotient of the set of characteristic forms into the set of all
quadratic functions and determine explicitly its cokernel. We determine a
suitable class of torsion groups so that quadratic functions defined on them
are classified by the stable class of their lift. This refines results due to
A.H. Durfee, V. Nikulin, C.T.C. Wall and E. Looijenga -- J. Wahl. Finally, we
show that on this class of torsion groups, two quadratic functions are
isomorphic if and only if they have equal associated Gauss sums and there is an
isomorphism between the associated symmetric bilinear pairings which preserves
the "homogeneity defects". This generalizes a classical result due to V.
Nikulin. Our results are elementary in nature and motivated by low-dimensional
topology.Comment: 15 pages; a few minor modifications (improved writing, lengthened
abstract
Homotopy-theoretically enriched categories of noncommutative motives
Waldhausen's -theory of the sphere spectrum (closely related to the
algebraic -theory of the integers) is a naturally augmented -algebra,
and so has a Koszul dual. Classic work of Deligne and Goncharov implies an
identification of the rationalization of this (covariant) dual with the Hopf
algebra of functions on the motivic group for their category of mixed Tate
motives over . This paper argues that the rationalizations of categories of
non-commutative motives defined recently by Blumberg, Gepner, and Tabuada
consequently have natural enrichments, with morphism objects in the derived
category of mixed Tate motives over . We suggest that homotopic descent
theory lifts this structure to define a category of motives defined not over
but over the sphere ring-spectrum .Comment: An attempt at a more readable version. Some reshuffling, a few new
references, small notational changes. Thanks to many for comments about
foolish blunders and obscuritie
Ring completion of rig categories
We offer a solution to the long-standing problem of group completing within
the context of rig categories (also known as bimonoidal categories). Given a
rig category R we construct a natural additive group completion R' that retains
the multiplicative structure, hence has become a ring category. If we start
with a commutative rig category R (also known as a symmetric bimonoidal
category), the additive group completion R' will be a commutative ring
category. In an accompanying paper we show how this can be used to prove the
conjecture from [BDR] that the algebraic K-theory of the connective topological
K-theory spectrum ku is equivalent to the algebraic K-theory of the rig
category V of complex vector spaces.Comment: There was a mathematical error in arXiv:0706.0531v2: the map T in the
purported proof of Lemma 3.7(2) is not well defined. Version 4 has been
edited for notational consistenc
Loop Spaces and Connections
We examine the geometry of loop spaces in derived algebraic geometry and
extend in several directions the well known connection between rotation of
loops and the de Rham differential. Our main result, a categorification of the
geometric description of cyclic homology, relates S^1-equivariant quasicoherent
sheaves on the loop space of a smooth scheme or geometric stack X in
characteristic zero with sheaves on X with flat connection, or equivalently
D_X-modules. By deducing the Hodge filtration on de Rham modules from the
formality of cochains on the circle, we are able to recover D_X-modules
precisely rather than a periodic version. More generally, we consider the
rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega
S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary
connection, with curvature given by their Omega S^3-equivariance.Comment: Revised versio
Invariance properties of random vectors and stochastic processes based on the zonoid concept
Two integrable random vectors and in are said
to be zonoid equivalent if, for each , the scalar products
and have the same first absolute
moments. The paper analyses stochastic processes whose finite-dimensional
distributions are zonoid equivalent with respect to time shift (zonoid
stationarity) and permutation of its components (swap invariance). While the
first concept is weaker than the stationarity, the second one is a weakening of
the exchangeability property. It is shown that nonetheless the ergodic theorem
holds for swap-invariant sequences and the limits are characterised.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ519 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
The equivalence of two graph polynomials and a symmetric function
The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them which also captures Tutte's universal V-function as a specialization. We show that the equivalence remains true for the strong functions thus
answering a question raised by Dominic Welsh
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