The well-known Sensitivity Conjecture regarding combinatorial complexity measures on Boolean functions states that for any Boolean function f : {0, 1} → {0, 1}, block sensitivity of f is polynomially related to sensitivity of f (denoted by s(f)). From the complexity theory side, the Xor Log-Rank Conjecture states that for any Boolean function, f : {0, 1} → {0, 1} the communication complexity of...

Leptoquarks are hypothetical bosons which couple to a lepton and a quark via a Yukawa coupling (denoted λ). In the Standard model, both quarks and leptons occur in left-handed SU(2) doublets and right-handed SU(2) singlets. The symmetry between quarks and leptons leads to the cancellation of triangle anomalies which make the SM renormalizable. Leptoquarks appear in theories[1] in which this sym...

T he method of moments approach to parameter estimation dates back more than 100 years (Stigler, 1986). The notion of a moment is fundamental for describing features of a population. For example, the population mean (or population average), usually denoted m, is the moment that measures central tendency. If y is a random variable describing the population of interest, we also write the populati...

For points z = (z1, · · · , zn) and w = (w1, · · · , wn) in C we write 〈z, w〉 = z1w1 + · · ·+ znwn, |z| = √ |z1| + · · ·+ |zn|. Let B = {z ∈ C : |z| < 1} denote the open unit ball and let S = {ζ ∈ C : |ζ| = 1} denote the unit sphere in C. The normalized Lebesgue measures on B and S will be denoted by dv and dσ, respectively. Let H(B) denote the space of all holomorphic functions in B. Given 0 <...

Bidecompositions, i.e., solutions to r ◦ p = s ◦ q, play a central rôle in the study of uniqueness properties of complete decompositions with respect to functional composition. In [Rit22] all bidecompositions using polynomials over the complex number field have been characterized. Later the result was generalized to more general fields. All proofs tend to be rather long and involved. The object...

Quadtrees are a data structure used to represent two dimensional shapes[Sam84]. Each node in the tree represents a physical area, usually a square or triangle, which can be subdivided into four similar shapes. Nodes are marked with values representing whether the node is in the area, out of the area, or partially contain the area. These values are traditionally denoted, BLACK, WHITE, and GRAY, ...

1 Problem mathematical formulation Let us consider a time-varying system described by k, θ, Sk, Zk, where k is the time index, θ is a vector of parameters, Sk is a random vector corresponding to the time-varying state at time k (with the corresponding realization denoted by sk), and Zk is a random vector corresponding to the observation process at time k (with the corresponding realization deno...

We consider the problem of optimal stopping for a one dimensional diffusion process. Two classes of admissible stopping times are considered. The Þrst class consists of all non-anticipating stopping times that take values in [0,∞], while the second class further restricts the set of allowed values to the discrete grid {nh : n = 0, 1, 2, · · · ,∞} for some parameter h > 0. The value functions fo...