Binary networks are extremely efficient as they use only two symbols to define the network: {+1, −1}. One can make prior distribution of these a design choice. The recent IR-Net Qin et al. argues that imposing Bernoulli with equal priors (equal bit ratios) over binary weights leads maximum entropy and thus minimizes information loss. However, work cannot precisely control weight during training...

The Bernoulli–Barnes polynomials are defined as a natural multidimensional extension of 21 the classical Bernoulli polynomials. Many of the properties of the Bernoulli polynomials 22 admit extensions to this new family. A specific expression involving the Bernoulli–Barnes 23 polynomials has recently appeared in the context of self-dual sequences. The work pre24 sented here provides a proof of t...

The model of binary aggregation with constant kernel is subjected to stochastic resetting: aggregates any size explode into monomers at independent times. These resetting times are Poisson distributed, and the rate process called rate. master equation yields a Bernoulli-type in generating function concentration size, which can be solved exactly. This prescription leads non-equilibrium steady st...

In this paper we discuss loosely Bernoulli for Zd actions. In particular, we prove that extensions of zero entropy, ergodic, loosely Bernoulli Zd actions are also loosely Bernoulli.

Abstract Recently, B. A. Kupershmidt have constructed a reflection symmetries of q-Bernoulli polynomials (see [9]). In this paper we give another construction of a q-Bernoulli polynomials, which form Barnes’ multiple Bernoulli polynomials at q = 1, cf. [1, 13, 14]. By using q-Volkenborn integration, we can also investigate the properties of the reflection symmetries of these’ q-Bernoulli polyno...

The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...

I. INTRODUCTION Boltzmann machines are probability distributions on high dimensional binary vectors which are analogous to Gaussian Markov Random Fields in that they are fully determined by first and second order moments. A key difference however is that augmenting Boltzmann machines with hidden variables enlarges the class of distributions that can be modeled, so that in principle it is possib...