The lack of measurability of the empirical process with respect to the sigma-field generated by the ‘natural’ l∞ metric, as illustrated in the previous notes, needs an extension of the standard weak convergence theory that can handle situations where the converging stochastic processes may no longer be measurable (though the limit will be a tight Borel measurable random element). Of course, an ...

We study the asymptotic behavior of the solution of semi-linear PDEs. Neither periodicity nor ergodicity assumptions are assumed. The coefficients admit only a limit in a C̀esaro sense. In such a case, the limit coefficients may have discontinuity. We use probabilistic approach based on weak convergence techniques for the associated backward stochastic differential equation in the S-topology. We...

In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type-2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel pro...

This report is a companion to another technical report [3] to present the formal proofs that could not fit into the conference proceedings of this article [4] due to space limitations. In this report we revisit the problem of Byzantine-robust topology discovery. We formally state the weak and strong versions of the problem. We focus on non-cryptographic solutions to these problems and explore t...

and Applied Analysis 3 The final solution to problems in 1.3 can yield optimal lower and upper bounds of eigenvalues. Actually, from 1.3 , we have In ∥q ∥ ∥ 1 ) ≤ λn ( q ) ≤ Mn ∥q ∥ ∥ 1 ) ∀q ∈ L1, 1.4 which are optimal. Several extremal problems on Dirichlet, Neumann, and periodic eigenvalues of the Sturm-Liouville operator and the p-Laplace operator have been studied recently, where the potent...

modeling and simulation of processing plants are widely used in industry. construction of a mathematical model for a plant is a time-consuming and error-prone task. in light of extensive advancements in computer science (both hardware and software), computers are becoming a necessary instrument in industrial activities. many software tools for modeling, simulation and optimization of processing...

In the context of ultra-low-power and ultra-low-voltage operational transconductance amplifiers (OTAs), there is the inconvenience of low unity-gain frequency and DC gain. In this paper, we presented a 0.5-V improved ultra-low-power OTA using the transistors operate in weak inversion. The proposed topology based on a bulk-driven input differential pair employed an under-weak-inversion gain-stag...

We study semicontinuity of multiple integrals ~ f~,u,Du)dx~ where the vector-valued function u is defined for xE ~C N with values in ~ . The function f(x,s,~) is assumed to be Carath@odory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology 1 N of the Sobolev space H 'P(~;~ ). The ...

In this paper we prove two results of nonsmooth analysis involving the Fréchet subdifferential. One of these results provides a necessary optimality condition for an optimization problem which arise naturally from a class of wide studied problems. In the second result we establish a sufficient condition for the metric regularity of a set-valued map without continuity assumptions. 1. Preliminari...

Using a perturbative expansion for weak synaptic weights and weak sources of randomness, we calculate the correlation structure of neural networks with generic connectivity matrices. In detail, the perturbative parameters are the mean and the standard deviation of the synaptic weights, together with the standard deviations of the background noise of the membrane potentials and of their initial ...