This paper gives best bounds for the ratio ∫ b−t a f(x) f(x + t) dx/ ∫ b a f(x) dx for any square-summable real function f(x) on the interval (a, b]. Similarly, bounds are established for the autocorrelation of any pulse or finite-length sequence at any known lag, and the family of pulses and sequences attaining these bounds is identified. The form of this family is related to a half-cycle of a...

Let a < b, = [a, b]Z and H be the (formal) Hamiltonian defined on by H(η)= 1 2 ∑ x,y∈Zd J (x − y)(η(x)− η(y)), (1) where J : Z → R is any summable non-negative symmetric function (J (x) ≥ 0 for all x ∈ Z , x J (x) <∞ and J (x)= J (−x)). We prove that there is a unique Gibbs measure on associated to H . The result is a consequence of the fact that the corresponding Gibbs sampler is attractive an...

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L-summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki res...

Stochastic gradient descent (SGD), the workhorse of stochastic optimization, is slow in theory (sub-linear convergence) and in practice (thousands of iterations), intuitively for two reasons: 1) Its learning rate schedule is fixed a priori and decays rapidly enough to 0 that is square-summable. This learning rate schedule limits the step size and hence the rate of convergence for a Lipschitz ob...

The jth Rademacher function rj on [0, 1), j = 0, 1, 2, . . . , is defined as follows: r0 = 1, r1 = 1 on [0, 1/2) and r1 = −1 on [1/2, 1), r2 = 1 on [0, 1/4) ∪ [1/2, 3/4) and r2 = −1 on [1/4, 1/2) ∪ [3/4, 1), etc. The following is a classical result that can be found in Zygmund [10] (page 213): For every subset E of [0, 1] and every λ > 1, there is a positive integer N such that for all complex-...

Forming Limit Diagrams are useful tools for evaluation of formability in the sheet metals. In this paper the effects of yield criteria on predictions of the right and left-hand sides of forming limit diagrams (FLDs) are investigated. In prediction of FLD, Hosford 1979, “Karafillis-Boyce” (K-B) and BBC2000 anisotropy yield functions have been applied. Whereas the prediction of FLD is based on th...

In this paper, we study a double scaling limit of two multi-matrix models: the $U(N)^2 \times O(D)$-invariant model with all quartic interactions and bipartite $U(N) tetrahedral interaction ($D$ being here number matrices $N$ size each matrix). Those models admit double, large $D$ expansion. While tracks genus Feynman graphs, another quantity called grade. both models, rewrite sum over graphs a...

We show that many invariant subspaces M for d-shifts (S1, . . . , Sd) of finite rank have the property that the orthogonal projection PM onto M satisfies PMSk − SkPM ∈ L, 1 ≤ k ≤ d for every p > 2d, L denoting the Schatten-von Neumann class of all compact operators having p-summable singular value lists. In such cases, the d tuple of operators T̄ = (T1, . . . , Td) obtained by compressing (S1, ....

Let D lR N , 0 < (D) < +1 and f : D ! lR is an arbitrary summable function. Then the function F() := R fx2D:f(x)g (f(x) ?) dd (2 lR) is continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative F 0 () = ?f ]. Further on, it holds ess supf = supf 2 lR : F() > 0g, where ess supf denotes the essential supremum of f. These properties can be used for computing esssup...

In the rectangle D = (0, a) × (0, b) with the boundary Γ the Dirichlet problem ∂4u ∂x2∂y2 = p(x, y)u + q(x, y), u(x, y) = 0 for (x, y) ∈ Γ is considered, where p and q : D → R are locally summable functions and may have nonintegrable singularities on Γ. The effective conditions guaranteeing the unique solvability of this problem and the stability of its solution with respect to small perturbati...