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In this thesis we study convolutions that arise from noncommutative probability theory. We prove several regularity results for free convolutions, and for measures in partially defined one-parameter free convolution semigroups. We discuss…
Suppose V{\nu} is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+({\nu}) generated by a non degenerate probability measure {\nu} with support bounded from above. We determine the formula for pseudo-variance…
In this paper, a new criterion for the similarity of commuting tuples of operators on Hilbert spaces is introduced. As an application, we obtain a geometric similarity invariant of tuples in the Cowen-Douglas class which gives a partial…
We investigate the Mott transitions in two-band Hubbard models with different bandwidths. Applying dynamical mean field theory, we discuss the stability of itinerant quasi-particle states in each band. We demonstrate that separate Mott…
Interaction graphs provide an important qualitative modeling approach for System Biology. This paper presents a novel approach for construction of interaction graph with the help of Boolean function decomposition. Each decomposition part…
Functions satisfying the functional equation \begin{align*} \sum_{r=0}^{n-1} (-1)^r f(x+ry, ny) = f(x,y), \quad \text{for any positive odd integer $n$}, \end{align*} are named the alternating invariant functions. Examples of such functions…
The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic…
Boolean functions are mathematical objects used in diverse applications. Different applications also have different requirements, making the research on Boolean functions very active. In the last 30 years, evolutionary algorithms have been…
The properties of the square bias transformation are studied, in particular, the precise moment-type estimate for the $L_1$-metric between the transformed and the original distributions is proved, a relation between their characteristic…
The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class…
A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent…
Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions of exactly marginal operators. We show that the curvature of such conformal manifolds can be computed…
We investigate the problem of the superuniversality of the phase transition between different quantum Hall plateaus. We construct a set of models which give a qualitative description of this transition in a pure system of interacting…
We study uniform consistency in nonparametric mixture models as well as closely related mixture of regression (also known as mixed regression) models, where the regression functions are allowed to be nonparametric and the error…
Through exploring the embedded transnormal systems of codimension 1, we show the existence of a transnormal function on a connected complete Riemannian manifold requires the underlying manifold to have a vector bundle structure or a linear…
We explain the relationship between various characteristic classes for smooth manifold bundles known as ``higher torsion'' classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and…
Correlation-immune (CI) multi-output Boolean functions have the property of keeping the same output distribution when some input variables are fixed. Recently, a new application of CI functions has appeared in the system of resisting…
This paper is the survey of some of our results related to $q$-deformations of the Fock spaces and related to $q$-convolutions for probability measures on the real line $\mathbb{R}$. The main idea is done by the combinatorics of moments of…
If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz…
Recently, Keller and Pilpel conjectured that the influence of a monotone Boolean function does not decrease if we apply to it an invertible linear transformation. Our aim in this short note is to prove this conjecture.