Results 11  20
of
33
Correlation Length, Isotropy, and Metastable States
, 1997
"... A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing paircorrelation function (and hence if it has a short correlation length). The "correlation length conjecture" allows to estimate the number of metastable states fro ..."
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Cited by 10 (6 self)
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A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing paircorrelation function (and hence if it has a short correlation length). The "correlation length conjecture" allows to estimate the number of metastable states from the correlation length, provided the landscape is "typical". Isotropy, originally introduced as a geometrical condition on the covariance matrix of a random field, can be reinterpreted as maximum entropy condition that lends a precise meaning to the notion of a "typical" landscape. The XYHamiltonian, which violates isotropy only to a relatively small extent, is an ideal model for investigating the influence of anisotropies. Numerical estimates for the number of local optima and predictions obtained from the correlation length conjecture indeed show deviations that increase with the extent of anisotropies in the model.
Complex Adaptations and the Structure of Recombination Spaces
 SCHOOL OF MATHEMATICS, UEA, NORWICH NR4 7TJ
, 1997
"... According to the Darwinian theory of evolution, adaptation results from spontaneously generated genetic variation and natural selection. Mathematical models of this process can be seen as describing a dynamics on an algebraic structure which in turn is defined by the processes which generate genetic ..."
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Cited by 9 (5 self)
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According to the Darwinian theory of evolution, adaptation results from spontaneously generated genetic variation and natural selection. Mathematical models of this process can be seen as describing a dynamics on an algebraic structure which in turn is defined by the processes which generate genetic variation (mutation and/or recombination). The theory of complex adaptive system has shown that the properties of the algebraic structure induced by mutation and recombination is more important for understanding the dynamics than the differential equations themselves. This has motivated new directions in the mathematical analysis of evolutionary models in which the algebraic properties induced by mutation and recombination are at the center of interest. In this paper we summarize some new results on the algebraic properties of recombination spaces. It is shown that the algebraic structure induced by recombination can be represented by a map from the pairs of types to the power set of the ty...
A lower bound for nodal count on discrete and metric graphs
, 2006
"... Abstract. We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we show that the number of nodal domains of the nth eigenfunction is bounded below by n − ℓ, ..."
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Cited by 7 (1 self)
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Abstract. We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we show that the number of nodal domains of the nth eigenfunction is bounded below by n − ℓ, where ℓ is the number of links that distinguish the graph from a tree. Our results apply to operators on both discrete (combinatorial) and metric (quantum) graphs. They complement already known analogues of a result by Courant who proved the upper bound n for the number of nodal domains. To illustrate that the genericity condition is essential we show that if it is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction. In the appendix we review the proof of the case ℓ = 0 on metric trees which has been obtained by other authors. 1.
Nodal Sets, Multiplicity and Superconductivity in Non Simply Connected Domains
, 2001
"... This is a survey on [HHOO] and further developments of the theory [He4]. We explain in detail the origin of the problem in superconductivity as first presented in [BeRu], recall the results of [HHOO] and explain the extension to the Dirichlet case. As illustration of the theory, we detail some semi ..."
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Cited by 5 (5 self)
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This is a survey on [HHOO] and further developments of the theory [He4]. We explain in detail the origin of the problem in superconductivity as first presented in [BeRu], recall the results of [HHOO] and explain the extension to the Dirichlet case. As illustration of the theory, we detail some semiclassical aspects and give examples where our estimates are sharp.
On the Relation Between Two MinorMonotone Graph Parameters
, 1997
"... We prove that for each graph (G) (G) + 2, where and are minormonotone graph invariants introduced by Colin de Verdi'ere [3] and van der Holst, Laurent and Schrijver [5]. It is also shown that a graph G exists with (G) ! (G). The graphs G with maximal planar complement and (G) = jV (G)j \Gamm ..."
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Cited by 4 (0 self)
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We prove that for each graph (G) (G) + 2, where and are minormonotone graph invariants introduced by Colin de Verdi'ere [3] and van der Holst, Laurent and Schrijver [5]. It is also shown that a graph G exists with (G) ! (G). The graphs G with maximal planar complement and (G) = jV (G)j \Gamma 4, characterised by Kotlov, Lov'asz and Vempala, are shown to be forbidden minors for fH j (H) ! jV (G)j \Gamma 4g. Introduction Given a graph G = (V; E) without loops or multiple edges, define OG as the collection of realvalued symmetric V \Theta V matrices M = (m ij ) satisfying 1. if ij 2 E, then m ij ! 0, and 2. if ij 62 E and i 6= j, then m ij = 0. There is no restriction on the diagonal entries. The elements of OG are sometimes called discrete Schrodinger operators. A matrix M 2 OG satisfies the Strong Arnol'd Hypothesis, SAH for short, if there is no nonzero symmetric matrix X = (x ij ) such that MX = 0, and such that x ij = 0 whenever i = j or ij 2 E. By i (M) we denote ...
A RELATION BETWEEN THE MULTIPLICITY OF THE SECOND EIGENVALUE OF A GRAPH LAPLACIAN, COURANT’S NODAL LINE THEOREM AND THE SUBSTANTIAL DIMENSION OF TIGHT POLYHEDRAL SURFACES
, 2007
"... A relation between the multiplicity m of the second eigenvalue λ2 of a Laplacian on agraphG, tight mappings of G and a discrete analogue of Courant’s nodal line theorem is discussed. For a certain class of graphs, it is shown that the mdimensional eigenspace of λ2 is tight and thus defines a tight ..."
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Cited by 4 (0 self)
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A relation between the multiplicity m of the second eigenvalue λ2 of a Laplacian on agraphG, tight mappings of G and a discrete analogue of Courant’s nodal line theorem is discussed. For a certain class of graphs, it is shown that the mdimensional eigenspace of λ2 is tight and thus defines a tight mapping of G into an mdimensional Euclidean space. The tightness of the mapping is shown to set Colin de Verdiére’s upper bound on the maximal λ2multiplicity, m ≤ chr(γ(G)) − 1, where chr(γ(G)) is the chromatic number and γ(G) is the genus of G.
Singular limits of Schrödinger operators and Markov processes
 J. OPERATOR THEORY
, 1999
"... After introducing the Γconvergence of a family of symmetric matrices, we study the limits in that sense, of Schrödinger operators on a finite graph. The main result is that any such limit can be interpreted as a Schrödinger operator on a new graph, the construction of which is described explicitly ..."
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Cited by 3 (2 self)
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After introducing the Γconvergence of a family of symmetric matrices, we study the limits in that sense, of Schrödinger operators on a finite graph. The main result is that any such limit can be interpreted as a Schrödinger operator on a new graph, the construction of which is described explicitly. The operators to which the construction is applied are reversible, almost reducible Markov generators. An explicit method for computing an equivalent of the spectrum is described. Among possible applications, quasidecomposable processes and lowtemperature simulated annealing are studied.
NODAL DOMAIN THEOREMS AND BIPARTITE SUBGRAPHS
, 2005
"... The Discrete Nodal Domain Theorem states that an eigenfunction of the kth largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sha ..."
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Cited by 3 (0 self)
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The Discrete Nodal Domain Theorem states that an eigenfunction of the kth largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.
Spectral Theory For The Dihedral Group.
"... Let H = \Gamma\Delta + V be a twodimensional Schrodinger operator defined on a bounded with Dirichlet boundary conditions on @ Suppose that H commutes with the actions of the dihedral group D 2n , the group of the regular ngone. We analyze completely the multiplicity of the groundstate eige ..."
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Cited by 3 (3 self)
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Let H = \Gamma\Delta + V be a twodimensional Schrodinger operator defined on a bounded with Dirichlet boundary conditions on @ Suppose that H commutes with the actions of the dihedral group D 2n , the group of the regular ngone. We analyze completely the multiplicity of the groundstate eigenvalues associated to the different symmetry subspaces related to the irreducible representations of D 2n . In particular we find that the multiplicities of these groundstate eigenvalues equal the degree of the corresponding irreducible representation. We also obtain an ordering of these eigenvalues. In addition we analyze the qualitative properties of the nodal sets of the corresponding ground state eigenfunctions.