23,248 research outputs found
Dispersions for linear differential equations of arbitrary order
summary:For linear differential equations of the second order in the Jacobi form O. Borvka introduced a notion of dispersion. Here we generalize this notion to certain classes of linear differential equations of arbitrary order. Connection with Abel’s functional equation is derived. Relations between asymptotic behaviour of solutions of these equations and distribution of zeros of their solutions are also investigated
Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
We investigate the asymptotic zero distribution of Heine-Stieltjes
polynomials - polynomial solutions of a second order differential equations
with complex polynomial coefficients. In the case when all zeros of the leading
coefficients are all real, zeros of the Heine-Stieltjes polynomials were
interpreted by Stieltjes as discrete distributions minimizing an energy
functional. In a general complex situation one deals instead with a critical
point of the energy. We introduce the notion of discrete and continuous
critical measures (saddle points of the weighted logarithmic energy on the
plane), and prove that a weak-* limit of a sequence of discrete critical
measures is a continuous critical measure. Thus, the limit zero distributions
of the Heine-Stieltjes polynomials are given by continuous critical measures.
We give a detailed description of such measures, showing their connections with
quadratic differentials. In doing that, we obtain some results on the global
structure of rational quadratic differentials on the Riemann sphere that have
an independent interest.Comment: 70 pages, 14 figures. Minor corrections, to appear in Comm. Math.
Physic
Oscillation Properties of Some Functional Fourth Order Ordinary Differential Equations
2010 Mathematics Subject Classification: 34A30, 34A40, 34C10.In this paper are considered oscillation properties of some classes of functional ordinary differential equations, namely equations of the type
ziv(t) + mz′′(t) + g(z(t), z′(t), z′′(t), z′′′(t)) +nXi=1_i(t)z(t − i) = f(t),
where m > 0 is constant, f(t) 2 C([T,1);R), T _ 0 is a large enough constant, g(z, _, _, _) 2 C(R4;R), _i(t) 2 C([0,1); [0,1)), 8 i = 1, n, n 2 N and {i}n i=1 are nonnegative constants. As a main result of this work we derive a sufficient condition for the distribution of the zeros of the above equations. Furthermore we discuss the complexity of the oscillation behavior of such equations and its relation to some properties of the corresponding solutions. Finally, we comment the oscillation behavior of a neutral fourth order ordinary differential equation, which appears in two papers of Ladas and Stavroulakis, as well as in a paper of Grammatikopoulos et al
Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros
Odlyzko has computed a data set listing more than successive Riemann
zeros, starting at a zero number beyond . The data set relates to
random matrix theory since, according to the Montgomery-Odlyzko law, the
statistical properties of the large Riemann zeros agree with the statistical
properties of the eigenvalues of large random Hermitian matrices. Moreover,
Keating and Snaith, and then Bogomolny and collaborators, have used random unitary matrices to analyse deviations from this law. We contribute
to this line of study in two ways. First, we point out that a natural process
to apply to the data set is to thin it by deleting each member independently
with some specified probability, and we proceed to compute empirical two-point
correlation functions and nearest neighbour spacings in this setting. Second,
we show how to characterise the order correction term to the spacing
distribution for random unitary matrices in terms of a second order
differential equation with coefficients that are Painlev\'e transcendents, and
where the thinning parameter appears only in the boundary condition. This
equation can be solved numerically using a power series method. Comparison with
the Riemann zero data shows accurate agreement.Comment: 22 pages, 10 figures, Version 2 added some new references in
bibliography, Version 3 corrected the scaling on the spacing distribution and
some typo
Complex Langevin Equations and Schwinger-Dyson Equations
Stationary distributions of complex Langevin equations are shown to be the
complexified path integral solutions of the Schwinger-Dyson equations of the
associated quantum field theory. Specific examples in zero dimensions and on a
lattice are given. Relevance to the study of quantum field theory phase space
is discussed.Comment: 23 pages, 4 figures, elsart, typos fixed, includes additional conten
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