1,155 research outputs found
Residue currents associated with weakly holomorphic functions
We construct Coleff-Herrera products and Bochner-Martinelli type residue
currents associated with a tuple of weakly holomorphic functions, and show
that these currents satisfy basic properties from the (strongly) holomorphic
case, as the transformation law, the Poincar\'e-Lelong formula and the
equivalence of the Coleff-Herrera product and the Bochner-Martinelli type
residue current associated with when defines a complete intersection.Comment: 28 pages. Updated with some corrections from the revision process. In
particular, corrected and clarified some things in Section 5 and 6 regarding
products of weakly holomorphic functions and currents, and the definition of
the Bochner-Martinelli type current
Entire curves avoiding given sets in C^n
Let be a proper closed subset of and
at most countable (). We give conditions
of and , under which there exists a holomorphic immersion (or a proper
holomorphic embedding) with .Comment: 10 page
A Class of Topological Actions
We review definitions of generalized parallel transports in terms of
Cheeger-Simons differential characters. Integration formulae are given in terms
of Deligne-Beilinson cohomology classes. These representations of parallel
transport can be extended to situations involving distributions as is
appropriate in the context of quantized fields.Comment: 41 pages, no figure
Modular differential equations for characters of RCFT
We discuss methods, based on the theory of vector-valued modular forms, to
determine all modular differential equations satisfied by the conformal
characters of RCFT; these modular equations are related to the null vector
relations of the operator algebra. Besides describing effective algorithmic
procedures, we illustrate our methods on an explicit example.Comment: 13 page
Easing Legal News Monitoring with Learning to Rank and BERT
While ranking approaches have made rapid advances in the Web search, systems that cater to the complex information needs in professional search tasks are not widely developed, common issues and solutions typically rely on dedicated search strategies backed by ad-hoc retrieval models. In this paper we present a legal search problem where professionals monitor news articles with constant queries on a periodic basis. Firstly, we demonstrate the effectiveness of using traditional retrieval models against the Boolean search of documents in chronological order. In an attempt to capture the complex information needs of users, a learning to rank approach is adopted with user specified relevance criteria as features. This approach, however, only achieves mediocre results compared to the traditional models. However, we find that by fine-tuning a contextualised language model (e.g. BERT), significantly improved retrieval performance can be achieved, providing a flexible solution to satisfying complex information needs without explicit feature engineering
Wave Solutions of Evolution Equations and Hamiltonian Flows on Nonlinear Subvarieties of Generalized Jacobians
The algebraic-geometric approach is extended to study solutions of
N-component systems associated with the energy dependent Schrodinger operators
having potentials with poles in the spectral parameter, in connection with
Hamiltonian flows on nonlinear subvariaties of Jacobi varieties. The systems
under study include the shallow water equation and Dym type equation. The
classes of solutions are described in terms of theta-functions and their
singular limits by using new parameterizations. A qualitative description of
real valued solutions is provided
Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I
We define the partition and -point correlation functions for a vertex
operator superalgebra on a genus two Riemann surface formed by sewing two tori
together. For the free fermion vertex operator superalgebra we obtain a closed
formula for the genus two continuous orbifold partition function in terms of an
infinite dimensional determinant with entries arising from torus Szeg\"o
kernels. We prove that the partition function is holomorphic in the sewing
parameters on a given suitable domain and describe its modular properties.
Using the bosonized formalism, a new genus two Jacobi product identity is
described for the Riemann theta series. We compute and discuss the modular
properties of the generating function for all -point functions in terms of a
genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one
point function satisfies a genus two Ward identity.Comment: A number of typos have been corrected, 39 pages. To appear in Commun.
Math. Phy
The kernel of the edth operators on higher-genus spacelike two-surfaces
The dimension of the kernels of the edth and edth-prime operators on closed,
orientable spacelike 2-surfaces with arbitrary genus is calculated, and some of
its mathematical and physical consequences are discussed.Comment: 12 page
Enhanced Gauge Groups in N=4 Topological Amplitudes and Lorentzian Borcherds Algebras
We continue our study of algebraic properties of N=4 topological amplitudes
in heterotic string theory compactified on T^2, initiated in arXiv:1102.1821.
In this work we evaluate a particular one-loop amplitude for any enhanced gauge
group h \subset e_8 + e_8, i.e. for arbitrary choice of Wilson line moduli. We
show that a certain analytic part of the result has an infinite product
representation, where the product is taken over the positive roots of a
Lorentzian Kac-Moody algebra g^{++}. The latter is obtained through double
extension of the complement g= (e_8 + e_8)/h. The infinite product is
automorphic with respect to a finite index subgroup of the full T-duality group
SO(2,18;Z) and, through the philosophy of Borcherds-Gritsenko-Nikulin, this
defines the denominator formula of a generalized Kac-Moody algebra G(g^{++}),
which is an 'automorphic correction' of g^{++}. We explicitly give the root
multiplicities of G(g^{++}) for a number of examples.Comment: 33 pages, 3 figure
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