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93
Critical points of functions, sl2 representations and Fuchsian differential equations with only univalued solutions
, 2002
"... Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z1,..., zn with exponents (ρ1,1, ρ2,1),...,(ρ1,n, ρ2,n). Let the exponents at infinity be (ρ1,∞, ρ2,∞). Then for fixed generic z1,..., zn, the number of such Fuchsian equations is equal to ..."
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Cited by 49 (27 self)
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Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z1,..., zn with exponents (ρ1,1, ρ2,1),...,(ρ1,n, ρ2,n). Let the exponents at infinity be (ρ1,∞, ρ2,∞). Then for fixed generic z1,..., zn, the number of such Fuchsian equations is equal to the multiplicity of the irreducible sl2 representation of dimension ρ2, ∞ − ρ1,∞  in the tensor product of irreducible sl2 representations of dimensions ρ2,1 − ρ1,1,..., ρ2,n − ρ1,n. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the sl2 KZ equation and of the Bethe vectors in the sl2 Gaudin model. As a byproduct of this study we conclude that the set of Bethe vectors is a basis in the space of states for the sl2 inhomogeneous Gaudin model.
Rational functions with prescribed critical points
 Geom. Funct. Anal
, 2002
"... Abstract. A rational function is the ratio of two polynomials without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into another by the postcomposition with a linearfractional transformation. We ..."
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Cited by 24 (6 self)
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Abstract. A rational function is the ratio of two polynomials without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into another by the postcomposition with a linearfractional transformation. We prove that for a given degree d and given multiplicities m1,..., mn of given critical points, the number of classes equals the multiplicity of the trivial sl2 representation in the tensor product of the irreducible sl2 representations with highest weights m1,..., mn and 2d − 2 − m1 − · · · − mn, for generic positions of the critical points. The interpretation of the result in terms of the Wronski map is discussed. 1.
Statistical methods for polyploid radiation hybrid mapping. Genome Res
, 1995
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Braided Hopf Algebras
, 2005
"... The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics a ..."
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Cited by 19 (7 self)
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The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first stage is that integral and Maschke’s theorem are found. Maschke’s theorem gives a nice criterion of semisimplicity, which is due to M.E. Sweedler. The second stage is that the Lagrange’s theorem is proved, which is due to W.D. Nichols and M.B. Zoeller. The third stage is the research of the actions of Hopf algebras, which unifies actions of groups and Lie algebras. S. Montgomery’s “Hopf algebras and their actions on rings ” contains the main results in this area. S. Montgomery and R.J. Blattner show the duality theorem. S. Montegomery, M. Cohen, H.J. Schneider, W. Chin, J.R. Fisher and the author study the relation between algebra R and the crossed product R#σH of the semisimplicity, semiprimeness, Morita equivalence and radical. S. Montgomery and M. Cohen ask whether R#σH is semiprime when R is semiprime for a finitedimensional semiprime Hopf algebra H. This is a famous semiprime problem. The
Authorization and antichains
 University of London
, 2002
"... Access control has been an important issue in military systems for many years and is becoming increasingly important in commercial systems. There are three important access control paradigms: the BellLaPadula model, the protection matrix model and the rolebased access control model. Each of these ..."
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Cited by 18 (3 self)
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Access control has been an important issue in military systems for many years and is becoming increasingly important in commercial systems. There are three important access control paradigms: the BellLaPadula model, the protection matrix model and the rolebased access control model. Each of these models has its advantages and disadvantages. Partial orders play a significant part in the rolebased access control model and are also important in defining the security lattice in the BellLaPadula model. The main goal of this thesis is to improve the understanding and specification of access control models through a rigorous mathematical approach. We examine the mathematical foundations of the rolebased access control model and conclude that antichains are a fundamental concept in the model. The analytical approach we adopt enables us to identify where improvements in the administration of rolebased access control could be made. We then develop a new administrative model for rolebased access control based on a novel, mathematical interpretation of encapsulated ranges. We show that this model supports discretionary access control features which have hitherto been difficult to incorporate into rolebased access control frameworks.
On the Autoreducibility of Random Sequences
, 2001
"... A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addi ..."
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Cited by 16 (1 self)
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A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truthtableautoreducible.
Multifractal Structure of Convolution of the Cantor Measure
"... The multifractal structure of measures generated by iterated function systems (IFS) with overlaps is, to a large extend, unknown. In this paper we study the local dimension of the mtime convolution of the standard Cantor measure µ. By using some combinatoric techniques, we show that the set E of ..."
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Cited by 13 (5 self)
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The multifractal structure of measures generated by iterated function systems (IFS) with overlaps is, to a large extend, unknown. In this paper we study the local dimension of the mtime convolution of the standard Cantor measure µ. By using some combinatoric techniques, we show that the set E of attainable local dimensions of µ contains an isolated point. This is rather surprising because when the IFS satisfies the open set condition, the set E is an interval. The result implies that the multifractal formalism fails without the open set condition.
Network Restorability Design Using Preconfigured Trees, Cycles, and Mixtures of Pattern Types
 TRLABS
, 2000
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An Introduction to Bayesian Network Theory and Usage
, 2000
"... . I present an introduction to some of the concepts within Bayesian networks to help a beginner become familiar with this eld's theory. Bayesian networks are a combination of two dierent mathematical areas: graph theory and probability theory. So, I rst give the basic denition of Bayesian netwo ..."
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Cited by 12 (0 self)
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. I present an introduction to some of the concepts within Bayesian networks to help a beginner become familiar with this eld's theory. Bayesian networks are a combination of two dierent mathematical areas: graph theory and probability theory. So, I rst give the basic denition of Bayesian networks. This is followed by an elaboration of the underlying graph theory that involves the arrangements of nodes and edges in a graph. Since Bayesian networks encode one's beliefs for a system of variables, I then proceed to discuss, in general, how to update these beliefs when one or more of the variables' values are no longer unknown (i.e., you have observed their values). Learning algorithms involve a combination of learning the probability distributions along with learning the network topology. I then conclude Part I by showing how Bayesian networks can be used in various domains, such as in the timeseries problem of automatic speech recognition. In Part II I then give in more detail some ...