A highly non-gaussian cosmological perturbation with a flat spectrum has unusual stochastic properties. We show that they depend on the size of the box within which the perturbation is defined, but that for a typical observer the parameters defining the perturbation ‘run’ to compensate for any change in the box size. Focusing on the primordial curvature perturbation, we show that an un-correlat...

We address the minimization of regularized convex cost functions which are customarily used for edge-preserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive half-quadratic reformulation of the original cost-function have been pioneered in Geman & Reynolds (1992) and Geman & Yang (1995). The alternate minimization o...

We present a generalization bound for feedforward neural networks with ReLU activations in terms of the product of the spectral norm of the layers and the Frobenius norm of the weights. The key ingredient is a bound on the changes in the output of a network with respect to perturbation of its weights, thereby bounding the sharpness of the network. We combine this perturbation bound with the PAC...

there are several kinds of hyperrings, for example, krasnerhyperrings, multiplicative hyperring, general hyperrings and$h_v$-rings. in a multiplicative hyperring, the multiplication isa hyperoperation, while the addition is a binary operation. in this paper, the notion of derivation on multiplicative hyperrings is introduced and some related properties are investigated. {bf keywords:} multiplic...

I recall the well-known sufficient conditions for the bound r < 16ǫ on the spectrum of the primordial tensor perturbation. Two recent papers claim a violation of this bound, without stating explicitely any violation of the sufficient conditions. The following conditions are sufficient for the bound r < 16ǫ on the primordial tensor perturbation. • Inflation is of the slow-roll variety. • The cur...

semilinear stochastic evolution equations with multiplicative l&apos;evy noise are considered‎. ‎the drift term is assumed to be monotone nonlinear and with linear growth‎. ‎unlike other similar works‎, ‎we do not impose coercivity conditions on coefficients‎. ‎we establish the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. ‎as corollarie...

The complexity lower bound (log N) for randomized computation trees is proved for recognizing an arrangement or a polyhedron with N faces. This provides in particular, the randomized lower bound (n log n) for the DISTINCTNESS problem and generalizes 11] where the randomized lower bound (n 2) was ascertained for the KNAPSACK problem. The core of the method is an extension of the lower bound from...

In the present note we prove an asymptotically tight relation between additive and multiplicative complexity of Boolean functions with respect to implementation by circuits over the basis {⊕,∧, 1}. To start, consider a problem of computation of polynomials over a semiring (K,+,×) by circuits over the arithmetic basis {+,×} ∪K. It’s a common knowledge that a polynomial of n variables with nonsca...