It is shown by the author in 2017 that for the union of N orthants in the n-dimensional space there exists an efficient and systematic way to find the exact value, using a suitable partial order relation construction. In this paper our events are hyperrectangles (or n-orthotopes), the Cartesian product of intervals: another important sets (or events) in both theory and practice. We have discovered a new efficient algorithm for the union of such events. With other important Boolean functions we present optimization problem formulations for both hyperreectangles and orthants.
We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the tree size tends to infinity and derive a relation between the tree size complexity and the probability of a function. This is done by first expressing trees representing a particular function as expansions of minimal trees representing this fu…
In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.
Suppose Vν is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+(ν) generated by a non degenerate probability measure ν with support bounded from above. We determine the formula for pseudo-variance function (or variance function Vν in case of existence) under boolean additive convolution power. This formulas is used to identify the relation between variance functions under Boolean Bercovici-Bata Bijection between probability measures. We also gives the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measure…
The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of paraconsistent weak Kleene logic and whose elements are represented as Plonka sum of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Plonka sum representation and, the direct limit of these algebras. Moreover, we study completition of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric.
This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in this context. In particular we introduce a construction which defines a (finite) {\em Boolean algebra of conditionals} from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on condit…
Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz. We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension. Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probabi…
A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is obtained from $X^{n}$ by independently flipping each coordinate with probability $δ$. This paper is about self-predicting functions, which are those that coincide with their optimal predictor.
We prove that the Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ finite moment (in particular we do not assume that the distribution is bounded). More precisely, we prove that: -In the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance $n$: when the radius distribution has a finite exponential moment, the probability decays exponentially fast in $n$, and when the radius dist…
Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulan…
Semigroups related to additive and multiplicative, free and Boolean convolutions
Belinschi和Nica在概率测度集上引入了一个复合半群。利用这个半群，他们引入了一个自由整除指标，从这个指标可以知道一个概率测度是否是自由无限整除的。在本文中，我们进一步研究这个指标，介绍了一个乘法版本，它能够显示许多属性。具体来说，在上半年的文件，我们计算如何变化的指标相对于自由和布尔幂权力，我们证明了自由和布尔1/2稳定的法律有自由整除指标等于无穷大，我们推导出一个上界的指标的Jacobi参数。这个上界只能通过自由Meixner分布实现。我们还证明了Bozejko的猜想说，布尔幂的概率测度亩0 < t < 1是自由无限整除，如果亩是这样。在本文的另一半，我们引入了一个类似的复合半群乘法卷积和定义自由整除指标，这些卷积。证明了单位圆上的一个概率测度关于乘法自由卷积是自由无限可分的当且仅当指标不小于1.我们还证明了乘性整除指标在自由幂和布尔幂幂下的变化，以及Bozejko猜想的乘性模拟。我们包括一个附录，其中的柯西分布和点的措施被证明是唯一的不动点的布尔自由Bercovici-Pata双射。
We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $ε\ge 1/2 - δ$, for some absolute constant $δ> 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ε$, then, provided $ε\ge 1/2 - δ$, for any boolean function $f$ holds $I(f(X);Y) \le 1 - H(ε)$. This conjecture was previously known to hold only for balanced functions.
The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such credences. We then explain the relationship between credences, residual charges, and Borel probability measures. We show that a credence can be represented by a normal Borel measure, augmented with a `liminal structure', which specifies how two or more regular open sets share the pro…
Consider a monotone Boolean function $f:\{0,1\}^n\to\{0,1\}$ and the canonical monotone coupling $\{η_p:p\in[0,1]\}$ of an element in $\{0,1\}^n$ chosen according to product measure with intensity $p\in[0,1]$. The random point $p\in[0,1]$ where $f(η_p)$ flips from $0$ to $1$ is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large $n$, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switche…
We give sufficient conditions ensuring the strong ergodic property of unique mixing for $C^*$-dynamical systems arising from Yang-Baxter-Hecke quantisation. We discuss whether they can be applied to some important cases including monotone, Boson, Fermion and Boolean $C^*$-algebras in a unified version. The monotone and the Boolean cases are treated in full generality, the Bose/Fermi cases being already widely investigated. In fact, on one hand we show that the set of stationary stochastic processes are isomorphic to a segment in both the situations, on the other hand the Boolean processes enjo…
We present a correlated \textit{and} gate which may be used to propagate uncertainty and dependence through Boolean functions, since any Boolean function may be expressed as a combination of \textit{and} and \textit{not} operations. We argue that the \textit{and} gate is a bivariate copula family, which has the interpretation of constructing bivariate Bernoulli random variables following a given Pearson correlation coefficient and marginal probabilities. We show how this copula family may be used to propagate uncertainty in the form of probabilities of events, probability intervals, and probab…
In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties of a complex under this measure. We are able to determine or present bounds for a number of topological and combinatorial properties. We also study the random pure simplicial complex of dimension $d$, generated by letting any subset of size $d+1$ of a set of $n$ vertices be a facet with probability $p$ and considering the simplicial complex generated by thes…
Let $Z$ be a Boolean model based on a stationary Poisson process $η$ of compact, convex particles in Euclidean space ${\mathbb{R}}^d$. Let $W$ denote a compact, convex observation window. For a large class of functionals $ψ$, formulas for mean values of $ψ(Z\cap W)$ are available in the literature. The first aim of the present work is to study the asymptotic covariances of general geometric (additive, translation invariant and locally bounded) functionals of $Z\cap W$ for increasing observation window $W$, including convergence rates. Our approach is based on the Fock space representation asso…
In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theo…
How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena such as heterogeneous catalysis, permeation through semi-permeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behav…
Boundedness properties of operators associated with non-degenerate symmetric $α$-stable, $α\in (1,2)$, probability measures on $\mathbb{R}^d$ are investigated on appropriate, Euclidean or otherwise, $L^p$-spaces, $p \in (1,+\infty)$. Our approach is based on first obtaining Bismut-type formulae which lead to useful representations for various operators. In the Euclidean setting, the method of transference and one-dimensional multiplier theory combined with fine properties of stable distributions provide dimension-free estimates for the fractional Laplacian. In the non-Euclidean setting, we obt…
Fuzzy logic is a way to argue with boolean predicates for which we only have a confidence value between 0 and 1 rather than a well defined truth value. It is tempting to interpret such a confidence as a probability. We use Markov kernels, parametrised probability distributions, to do just that. As a consequence we get general fuzzy logic connectives from probabilistic computations on products of the booleans, stressing the importance of joint confidence functions. We discuss binary logic connectives in detail and recover the "classic" fuzzy connectives as bounds for the confidence for general …
