Related papers: A probabilistic proof of Perron's theorem
In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook…
We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations…
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
Squaring and adding $\pm 1$ mod p generates a curiously intractable random walk. A similar process over the finite field $\mathbf{F}_q$ (with $q=2^d$) leads to novel connections between elementary Galois theory and probability.
Following the Perron-Frobenius theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix $A$, there is a positive rank one matrix $X$ such that $B = A \circ X$, where $\circ$ denotes…
It will be shown that Pascal's Theorem is equivalent to the associativity of a natural binary operation on conic sections. A novel proof for Pascal's Theorem will then be given by showing that this binary operation is associative…
Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only…
The Ruelle-Perron-Frobenius (RPF) theorem is a powerful tool in the study of equilibrium measures and their statistical properties. We prove a nonstationary version of this theorem under general conditions involving an invariant sequence of…
We present and prove a general form of Vandermonde's identity and use it as an alternative solution to a classic probability problem.
A new probalistic approach to general relativistic kinetic theory is proposed. The general relativistic Boltzmann equation is linked to a new Markov process in a completely intrinsic way. This treatment is then used to prove the causal…
We investigate the probability of shadowing of a random finite pseudotrajectory by an exact trajectory for linear skew products. We describe general conditions under which a random pseudotrajectory can be shadowed with polynomial (with…
We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem.
A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local "random-to-front" reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the…
The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit…
In this paper, we present a possible theoretical explanation for benford's law. We develop a recursive relation between the probabilities, using simple intuitive ideas. We first use numerical solutions of this recursion and verify that the…
We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an…
We prove several extensions of the Erdos-Fuchs theorem.
The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed…
We present a dynamical approach to the classical Perron-Frobenius theory by using some elementary knowledge on linear ODEs. It is completely self-contained and significantly different from those in the literature. As a result, we develop a…
We prove an ergodic theorem for Markov chains indexed by the Ulam-Harris-Neveu tree over large subsets with arbitrary shape under two assumptions: with high probability, two vertices in the large subset are far from each other and have…