Related papers: A Randomized Version of the Collatz $3x + 1$ Probl…
This paper solves the Sylvester equation in the form of AX+XB=C in a distributed way, and proposes three distributed continuous-time algorithms for three cases. We start with the basic algorithm for solving a least squares solution of the…
The scope of the present work is to explain why it is true that all N have a distinct position in The Collatz Tree (The Collatz Graph)
This paper presents an algorithm for 3-SAT problems. First, logical formulas are transformed into elementary algebraic formulas. Second, complex trigonometric functions are assigned to the variables in the elementary algebraic formulas, and…
We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006 and Journal de Mathematiques Pures et Appliquees 2007]. The equation under consideration is a…
Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality whereby increases in the stochastic…
The technique of stochastic solutions, previously used for deterministic equations, is here proposed as a solution method for partial differential equations driven by distribution-valued noises.
Take an odd number x >0. Then 3x+1 is even and one can find an integer k> 0 so that y= 3x+1/2^k is again odd. We get in this way the mapping T, Tx=y. The paper contains two theorems describing statistical properties of T. The first…
The \textit{Collatz's conjecture} is an unsolved problem in mathematics. It is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse conjecture or problem. Take any positive integer $ n $. If $ n $ is even then divide it…
This paper proposes an algorithm for computing regularized solutions to linear rational expectations models. The algorithm allows for regularization cross-sectionally as well as across frequencies. A variety of numerical examples illustrate…
In this paper, we propose a quantized learning equation with a monotone increasing resolution of quantization and stochastic analysis for the proposed algorithm. According to the white noise hypothesis for the quantization error with dense…
In this paper, we study a stochastic strongly convex optimization problem and propose three classes of variable sample-size stochastic first-order methods including the standard stochastic gradient descent method, its accelerated variant,…
Solving linear systems of equations is a fundamental problem in mathematics. When the linear system is so large that it cannot be loaded into memory at once, iterative methods such as the randomized Kaczmarz method excel. Here, we extend…
The X-problem of number 3 for one dimension and related observations are discussed
We show an iterated function of which iterates oscillate wildly and grow at a dizzying pace. We conjecture that the orbit of arbitrary positive integer always returns to 1, as in the case of Collatz function. The conjecture is supported by…
One key challenge for solving a general stochastic optimization problem with expectations in the objective and constraint functions using ordinary stochastic iterative methods lies in the infeasibility issue caused by the randomness over…
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
The $3x+1$ Problem asks if whether for every natural number $n$, there exists a finite number of iterations of the piecewise function $$f(2n)=n, \quad f(2n-1)=6n-2, $$ with an iterate equal to the number $1$, or in other words, every…
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…
In this paper we define a small variation of the Taylor method and a formula for the global error of this new numerical method that allows us to keep track of the round-off error and does not require previous knowledge of the exact…
Stochastic linear combinations of some random vectors are studied where the distribution of the random vectors and the joint distribution of their coefficients are Dirichlet. A method is provided for calculating the distribution of these…