Simple Systems with Anomalous Dissipation and Energy Cascade

We analyze a class of linear shell models subject to stochastic forcing in finitely many degrees of freedom. The unforced systems considered formally conserve energy. Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients. The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes ($a_n$) with higher $n$; this is responsible for solutions with interesting energy spectra, namely \EE |a_n|^2 scales as $n^{-\alpha}$ as $n\to\infty$. Here the exponents $\alpha$ depend on the coupling coefficients $c_n$ and \EE denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.Comment: 32 Page

A lower bound for the Lindelöf function associated to generalized integers

AbstractIn this paper we study generalized prime systems for which the integer counting function NP(x) is asymptotically well-behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and β<12. For such systems, the associated zeta function ζP(s) has finite order for σ=Rs>β, and the Lindelöf function μP(σ) may be defined. We prove that for all such systems, μP(σ)⩾μ0(σ) for σ>β, whereμ0(σ)={12−σif σ<12,0if σ⩾12

Determinants of Multiplicative Toeplitz Matrices

In this paper we study matrices A = (aij) whose (i, j) th-entry is a function of i/j; that is, aij = f(i/j) for some f: Q + → C. We obtain a formula for the truncated determinants in the case where f is multiplicative, linking them to determinants of truncated Toeplitz matrices. We apply our formula to obtain several determinants of number-theoretic matrices

Well-behaved Beurling primes and integers

AbstractIn this paper, we study generalised prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately li(x) and ρx, respectively (where ρ is a positive constant), with error terms of order O(xθ1) and O(xθ2) for some θ1,θ2<1. We show that it is impossible to have both θ1 and θ2 less than 12