6,143 research outputs found
On the mean values of L-functions in orthogonal and symplectic families
Hybrid Euler-Hadamard products have previously been studied for the Riemann
zeta function on its critical line and for Dirichlet L-functions in the context
of the calculation of moments and connections with Random Matrix Theory.
According to the Katz-Sarnak classification, these are believed to represent
families of L-function with unitary symmetry. We here extend the formalism to
families with orthogonal & symplectic symmetry. Specifically, we establish
formulae for real quadratic Dirichlet L-functions and for the L-functions
associated with primitive Hecke eigenforms of weight 2 in terms of partial
Euler and Hadamard products. We then prove asymptotic formulae for some moments
of these partial products and make general conjectures based on results for the
moments of characteristic polynomials of random matrices
Random matrices and quantum spin chains
Random matrix ensembles are introduced that respect the local tensor
structure of Hamiltonians describing a chain of distinguishable spin-half
particles with nearest-neighbour interactions. We prove a central limit theorem
for the density of states when , giving explicit bounds on
the rate of approach to the limit. Universality within a class of probability
measures and the extension to more general interaction geometries are
established. The level spacing distributions of the Gaussian Orthogonal,
Unitary and Symplectic Ensembles are observed numerically for the energy levels
in these ensembles.Comment: Updated figures, as accepted in 'Markov Processes and Related Fields'
on 3 March 201
Spectra and eigenstates of spin chain Hamiltonians
We prove that translationally invariant Hamiltonians of a chain of qubits
with nearest-neighbour interactions have two seemingly contradictory features.
Firstly in the limit we show that any translationally
invariant Hamiltonian of a chain of qubits has an eigenbasis such that
almost all eigenstates have maximal entanglement between fixed-size sub-blocks
of qubits and the rest of the system; in this sense these eigenstates are like
those of completely general Hamiltonians (i.e. Hamiltonians with interactions
of all orders between arbitrary groups of qubits). Secondly in the limit
we show that any nearest-neighbour Hamiltonian of a chain
of qubits has a Gaussian density of states; thus as far as the eigenvalues
are concerned the system is like a non-interacting one. The comparison applies
to chains of qubits with translationally invariant nearest-neighbour
interactions, but we show that it is extendible to much more general systems
(both in terms of the local dimension and the geometry of interaction).
Numerical evidence is also presented which suggests that the translational
invariance condition may be dropped in the case of nearest-neighbour chains.Comment: Updated figures, as accepted in 'Communications in Mathematical
Physics' on 5 January 201
Passion & Purpose: Raising the Fiscal Fitness Bar for Massachusetts Nonprofits
Presents data on and financial analyses of the state's nonprofit sector by organization type, budget, focus area, and location. Recommends better financial stewardship, restructuring, repositioning, and reinvestment to enhance nonprofits' sustainability
On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class
We establish the equivalence of conjectures concerning the pair correlation
of zeros of -functions in the Selberg class and the variances of sums of a
related class of arithmetic functions over primes in short intervals. This
extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan
[11] for the Riemann zeta-function to other -functions in the Selberg class.
Our approach is based on the statistics of the zeros because the analogue of
the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic
functions we consider is not available in general. One of our main findings is
that the variances of sums of these arithmetic functions over primes in short
intervals have a different form when the degree of the associated -functions
is 2 or higher to that which holds when the degree is 1 (e.g. the Riemann
zeta-function). Specifically, when the degree is 2 or higher there are two
regimes in which the variances take qualitatively different forms, whilst in
the degree-1 case there is a single regime
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