相关论文: The compensation approach for walks with small ste…
Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are…
Quantum walks on graphs can model physical processes and serve as efficient tools in quantum information theory. Once we admit random variations in the connectivity of the underlying graph, we arrive at the problem of percolation, where the…
The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve,…
In this paper, we propose a kernel method for exact tail asymptotics of a random walk to neighborhoods in the quarter plane. This is a two-dimensional method, which does not require a determination of the unknown generating function(s).…
We provide a mathematical framework for identifying the shortest path in a maze using a Grover walk, which becomes non-unitary by introducing absorbing holes. In this study, we define the maze as a network with vertices connected by…
The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena…
This paper gives a general introduction to two-dimensional functional walks with particular attention to notation and definition. We also give applications of functional walks and a visual overview of some walks generated by $f(n)=n^2$ and…
The involution walk is the random walk on $S_n$ generated by involutions with a binomially distributed with parameter $1-p$ number of $2$-cycles. This is a parallelization of the transposition walk. The involution walk is shown in this…
We consider the problem of identifying people on the basis of their walk (gait) pattern. Classical approaches to tackle this problem are based on, e.g., video recordings or piezoelectric sensors embedded in the floor. In this work, we rely…
In this paper, we introduce a new iterative method which we call one step back approach: the main idea is to anticipate the consequence of the iterative computation per coordinate and to optimize on the choice of the sequence of the…
Recently, we provided a simple but accurate formula which closely approximates the first crossing distribution associated with random walks having correlated steps. The approximation is accurate for the wide range of barrier shapes of…
In this paper we consider a nonlocal evolution problem and obtain by a scaling method the first term in the asymptotic behavior of the solutions. The method employed treats in different way the smooth and the rough part of the solution.
We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell-Weil lattices, that, for each…
We present a novel Adaptive Distribution Generator that leverages a quantum walks-based approach to generate high precision and efficiency of target probability distributions. Our method integrates variational quantum circuits with…
Movement disorders, such as Parkinson's disease, affect more than 10 million people worldwide. Gait analysis is a critical step in the diagnosis and rehabilitation of these disorders. Specifically, step length provides valuable insights…
The article provides an explicit algebraic expression for the generating function of walks on graphs. Its proof is based on the scattering theory for the differential Laplace operator on non-compact graphs.
Recently, in ["The coin-turning walk and its scaling limit", Electronic Journal of Probability, 25 (2020)], the ``coin-turning walk'' was introduced on ${\mathbb Z}$. It is a non-Markovian process where the steps form a (possibly)…
In this expository note, we discuss spatially inhomogeneous quantum walks in one dimension and describe a genre of mathematical methods that enables one to translate information about the time-independent eigenvalue equation for the unitary…
The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for…
We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which…