Related papers: Motzkin path decompositions of functionals in nonc…
We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the…
Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper we give the classification of the…
We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s \in…
Analytical work probability distributions for open classical systems are scarce; they can only be calculated in a few examples. In this work, I present a new method to derive such quantities for weakly driven processes in the overdamped…
In this paper, we derive explicit expressions for the moments and for the mixed moments of the compression of a free unitary Brownian motion by a free projection. While the moments of this non-normal operator are readily derived using…
We propose to identify process zones in heterogeneous materials by tailored statistical tools. The process zone is redefined as the part of the structure where the random process cannot be correctly approximated in a low-dimensional…
Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that…
We define a map from subspaces to Motzkin paths and show that the inverse image of every path is a disjoint union of symmetric Boolean subsets yielding an explicit symmetric Boolean decomposition of the subspace lattice.
We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that…
Stochastic treatments of magnetic resonance spectroscopy and optical spectroscopy require evaluations of functions like <exp(i int_0^t Q(s)ds)>, where t is time, Q(s) is the value of a stochastic process at time s, and the angular brackets…
In a recent work \cite{BG}, given a collection of continuous semimartingales, authors derive a semimartingale decomposition from the corresponding ranked processes in the case that the ranked processes can meet more than two original…
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…
A combinatorial theory for type $R_I$ orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are…
So called $S$-Motzkin paths are combined the concepts `catastrophes' and `air pockets. The enumeration is done by properly set up bivariate generating functions which can be extended using the kernel method.
The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from…
Microstructure reconstruction and compression techniques are designed to find a microstructure with desired properties. While the microstructure reconstruction searches for a microstructure with prescribed statistical properties, the…
Iterative linear solvers have gained recent popularity due to their computational efficiency and low memory footprint for large-scale linear systems. The relaxation method, or Motzkin's method, can be viewed as an iterative method that…
We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This…
Counting the number of permutations of a given total displacement is equivalent to counting weighted Motzkin paths of a given area (Guay-Paquet and Petersen, 2014). The former combinatorial problem is still open. In this work, we show that…
We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between…