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Spectra, Pseudospectra, and Localization for Random Bidiagonal Matrices
 Comm. Pure Appl. Math
"... There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding ..."
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Cited by 13 (4 self)
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There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the nonperiodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the in nitedimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").
Large Deviations for the Weighted Height of an Extended Class of Trees
 Algorithmica
, 2006
"... We use large deviations to prove a general theorem on the asymptotic edgeweighted height H ⋆ n of a large class of random trees for which H ⋆ n ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary ..."
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Cited by 13 (6 self)
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We use large deviations to prove a general theorem on the asymptotic edgeweighted height H ⋆ n of a large class of random trees for which H ⋆ n ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary search trees [11], [13], random recursive trees [12] and plane oriented trees [23] for instance. New applications include the heights of some random lopsided trees [19] and of the intersection of random trees.
Quantum Probability applied to the Damped Harmonic Oscillator
"... Contents 1. The Framework of Quantum Probability 1.1. Making probability noncommutative 1.2. Events and random variables 1.3. Interpretation of quantum probability 1.4. The quantum coin toss: `spin' 1.5. Positive denite kernels 2. Some Quantum Mechanics 2.1. Position and momentum 2.2. Energy and ..."
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Cited by 9 (0 self)
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Contents 1. The Framework of Quantum Probability 1.1. Making probability noncommutative 1.2. Events and random variables 1.3. Interpretation of quantum probability 1.4. The quantum coin toss: `spin' 1.5. Positive denite kernels 2. Some Quantum Mechanics 2.1. Position and momentum 2.2. Energy and time evolution 2.3. The harmonic oscillator 2.4. The problem of damping 3. Conditional Expectations and Operations 3.1. Conditional expectations in nite dimension 3.2. Operations in nite dimension 3.3. Operations on quantum probability spaces 3.4. Quantum stochastic processes 3.5. Conditional expectations and transition operators 3.6. Markov processes 4. Second Quantisation 4.1. The functor 4.2. Fields 4.3. Quanta 5. Unitary dilations of spiralling motion 6. The Damped Harmonic Oscillator 6.1. Stochastic behaviour of the oscillator 6.2. The driving eld 6.3. Excitations of the oscillator 6.4. Emitted quanta
Continuity for MultiType Branching Processes With Varying Environments: Example
, 1997
"... Gamma1 ; 2 n\Gamma1 p 3) + Vn ] and En+1 = En [ [(2 n ; 0) +En ] [ [(2 n\Gamma1 ; 2 n\Gamma1 p 3) +En ] taking the sums elementwise over the given sets. Let V = V1 [ [\GammaV 1 ] and E = E1 [ [\GammaE 1 ] and write G 0 for the graph (V; E) and Gn for 2 \Gamman G 0 . The Sierpinski gas ..."
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Cited by 6 (2 self)
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Gamma1 ; 2 n\Gamma1 p 3) + Vn ] and En+1 = En [ [(2 n ; 0) +En ] [ [(2 n\Gamma1 ; 2 n\Gamma1 p 3) +En ] taking the sums elementwise over the given sets. Let V = V1 [ [\GammaV 1 ] and E = E1 [ [\GammaE 1 ] and write G 0 for the graph (V; E) and Gn for 2 \Gamman G 0 . The Sierpinski gasket G is the closure of the set [ 1 n=0 2 \Gamman V
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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Cited by 4 (3 self)
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
THE GROWTH OF THE INFINITE LONGRANGE PERCOLATION CLUSTER
, 2009
"... We consider longrange percolation on Zd, where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 − exp[−λ(r)] ∈ (0, 1) and the presence or absence of different edges are independent. Here λ(r) is a strictly positive, nonincreasing regularly varying f ..."
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Cited by 2 (1 self)
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We consider longrange percolation on Zd, where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 − exp[−λ(r)] ∈ (0, 1) and the presence or absence of different edges are independent. Here λ(r) is a strictly positive, nonincreasing regularly varying function. We investigate the asymptotic growth of the size of the kball around the origin, Bk, i.e. the number of vertices that are within graphdistance k of the origin, for k → ∞ for different λ(r). We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying λ(r) exist for which respectively • Bk  1/k → ∞ almost surely, • there exist 1 < a1 < a2 < ∞ such that lim k→ ∞ P(a1 < Bk  1/k < a2) = 1, • Bk  1/k → 1 almost surely. This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, R0, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.
Large Deviations In The Random Siev
"... . The proportion ae k of gaps with length k between squarefree numbers is shown to satisfy log ae k = \Gamma \Gamma 1 + o(1) \Delta (6=ß 2 )k log k as k ! 1. Such asymptotics are consistent with Erdos's challenge to prove that the gap following the squarefree number t is smaller than c log t ..."
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Cited by 1 (0 self)
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. The proportion ae k of gaps with length k between squarefree numbers is shown to satisfy log ae k = \Gamma \Gamma 1 + o(1) \Delta (6=ß 2 )k log k as k ! 1. Such asymptotics are consistent with Erdos's challenge to prove that the gap following the squarefree number t is smaller than c log t= log log t, for all t and some constant c satisfying c ? ß 2 =12. The results of this paper are achieved by studying the probabilities of large deviations in a certain `random sieve', for which the proportions ae k have representations as probabilities. The asymptotic form of ae k may be obtained in situations of greater generality, when the squared primes are replaced by an arbitrary sequence (sr ) of relatively prime integers satisfying P r 1=sr ! 1, subject to two further conditions of regularity on this sequence. 1. Introduction A positive integer is called squarefree if it is divisible by no squared prime. The sequence of squarefree numbers has density 6=ß 2 , but the gaps be...
Selfaffine Asperity Model for earthquakes
, 1995
"... A model for fault dynamics consisting of two rough and rigid brownian profiles that slide one over the other is introduced. An earthquake occurs when there is an intersection between the two profiles. The energy release is proportional to the overlap interval. Our model exhibits some specific featur ..."
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Cited by 1 (0 self)
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A model for fault dynamics consisting of two rough and rigid brownian profiles that slide one over the other is introduced. An earthquake occurs when there is an intersection between the two profiles. The energy release is proportional to the overlap interval. Our model exhibits some specific features which follow from the fractal geometry of the fault: (1) nonuniversality of the exponent of the GutenbergRichter law for the magnitude distribution; (2) presence of local stress accumulation before a large seismic event; (3) non1 trivial spacetime clustering of the epicenters. These properties are in good agreement with various observations and lead to specific predictions that can be experimentally tested. PACS NUMBERS: 91.30.Px, 05.40.+j 2 Many forms of scaling invariance appear in seismic phenomena: the celebrated GutenbergRichter law for the magnitude distribution [1], the Omori law for the time correlations of aftershocks [2], spacetime clustering of the epicenters [3] are a common mark of the earthquake statistics. Unfortunately, the complexity of modelling the motion of a fault system, even in rather well controlled situation such as the San Andreas fault in California, is a
Linear Goal Programming and Experience Rating.
, 2001
"... This paper is devoted to the explanation of a new methodology in bonus malus system design, capable of taking into account very well known theoretical conditions like fairness and …nancial equilibrium of the portfolio, in addition to market conditions that could …t the resulting scale of premiums in ..."
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This paper is devoted to the explanation of a new methodology in bonus malus system design, capable of taking into account very well known theoretical conditions like fairness and …nancial equilibrium of the portfolio, in addition to market conditions that could …t the resulting scale of premiums into competitive commercial settings. This is done through the resolution of a classical Bayesian decision problem, by means of minimization of the absolute error instead of the classical quadratic error. It is at this stage that we apply Goal Programming methods, which are linear thanks to the equivalence between the minimization of the absolute error and the minimization of the sum of some deviation variables which have a natural interpretation as rating errors. We show in an example how does the new methodology work. All the linear programs have been solved using the simplex method.