1,086 research outputs found
Modules and Morita theorem for operads
Associative rings A, B are called Morita equivalent when the categories of
left modules over them are equivalent. We call two classical linear operads P,
Q Morita equivalent if the categories of algebras over them are equivalent. We
transport a part of Morita theory to the operadic context by studying modules
over operads. As an application of this philosophy, we consider an operadic
version of the sheaf of linear differential operators ona a (super) manifold M
and give a comparison theorem between algebras over this sheaf on M and
M_{red}. The paper is dedicated to A.N.Tyurin on the occasion of his 60th
birthday.Comment: Several revisions and corrections are made in this version. Some
topics got a more detailed presentation. 30 pp., no figure
Semantic spaces
Any natural language can be considered as a tool for producing large
databases (consisting of texts, written, or discursive). This tool for its
description in turn requires other large databases (dictionaries, grammars
etc.). Nowadays, the notion of database is associated with computer processing
and computer memory. However, a natural language resides also in human brains
and functions in human communication, from interpersonal to intergenerational
one. We discuss in this survey/research paper mathematical, in particular
geometric, constructions, which help to bridge these two worlds. In particular,
in this paper we consider the Vector Space Model of semantics based on
frequency matrices, as used in Natural Language Processing. We investigate
underlying geometries, formulated in terms of Grassmannians, projective spaces,
and flag varieties. We formulate the relation between vector space models and
semantic spaces based on semic axes in terms of projectability of subvarieties
in Grassmannians and projective spaces. We interpret Latent Semantics as a
geometric flow on Grassmannians. We also discuss how to formulate G\"ardenfors'
notion of "meeting of minds" in our geometric setting.Comment: 32 pages, TeX, 1 eps figur
Quantum Thetas on Noncommutative T^4 from Embeddings into Lattice
In this paper we investigate the theta vector and quantum theta function over
noncommutative T^4 from the embedding of R x Z^2. Manin has constructed the
quantum theta functions from the lattice embedding into vector space (x finite
group). We extend Manin's construction of the quantum theta function to the
embedding of vector space x lattice case. We find that the holomorphic theta
vector exists only over the vector space part of the embedding, and over the
lattice part we can only impose the condition for Schwartz function. The
quantum theta function built on this partial theta vector satisfies the
requirement of the quantum theta function. However, two subsequent quantum
translations from the embedding into the lattice part are non-additive,
contrary to the additivity of those from the vector space part.Comment: 20 pages, LaTeX, version to appear in J. Phys.
Gromov-Witten classes, quantum cohomology, and enumerative geometry
The paper is devoted to the mathematical aspects of topological quantum field
theory and its applications to enumerative problems of algebraic geometry. In
particular, it contains an axiomatic treatment of Gromov-Witten classes, and a
discussion of their properties for Fano varieties. Cohomological Field Theories
are defined, and it is proved that tree level theories are determined by their
correlation functions. Applications to counting rational curves on del Pezzo
surfaces and projective spaces are given.Comment: 44 p, amste
Quantum Cohomology of a Product
The operation of tensor product of Cohomological Field Theories (or algebras
over genus zero moduli operad) introduced in an earlier paper by the authors is
described in full detail, and the proof of a theorem on additive relations
between strata classes is given. This operation is a version of the Kuenneth
formula for quantum cohomology. In addition, rank one CohFT's are studied, and
a generalization of Zograf's formula for Weil-Petersson volumes is suggested.Comment: AMSTex file, 30 pages. Figures (hard copies) available from Yu.
Manin. Main paper by M. Kontsevich, Yu. Manin, appendix by R. Kaufman
Theta Vectors and Quantum Theta Functions
In this paper, we clarify the relation between Manin's quantum theta function
and Schwarz's theta vector in comparison with the kq representation, which is
equivalent to the classical theta function, and the corresponding coordinate
space wavefunction. We first explain the equivalence relation between the
classical theta function and the kq representation in which the translation
operators of the phase space are commuting. When the translation operators of
the phase space are not commuting, then the kq representation is no more
meaningful. We explain why Manin's quantum theta function obtained via algebra
(quantum tori) valued inner product of the theta vector is a natural choice for
quantum version of the classical theta function (kq representation). We then
show that this approach holds for a more general theta vector with constant
obtained from a holomorphic connection of constant curvature than the simple
Gaussian one used in the Manin's construction. We further discuss the
properties of the theta vector and of the quantum theta function, both of which
have similar symmetry properties under translation.Comment: LaTeX 21 pages, give more explicit explanations for notions given in
the tex
Limiting modular symbols and their fractal geometry
In this paper we use fractal geometry to investigate boundary aspects of the
first homology group for finite coverings of the modular surface. We obtain a
complete description of algebraically invisible parts of this homology group.
More precisely, we first show that for any modular subgroup the geodesic
forward dynamic on the associated surface admits a canonical symbolic
representation by a finitely irreducible shift space. We then use this
representation to derive an `almost complete' multifractal description of the
higher--dimensional level sets arising from Manin--Marcolli's limiting modular
symbols.Comment: 20 pages, 1 figur
Asymptotic bounds for spherical codes
The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square with coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximizing simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in the inventing explicit constructions of "good codes" and comparing new classes of codes with earlier ones. Less classical approach studies the geometry of the whole set of code points (R,delta) (with q fixed), at first independently of its computability properties, and only afterwords turning to the problems of computability, analogies with statistical physics etc. The main purpose of this article consists in extending this latter strategy to domain of spherical codes
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