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Solving two-variable linear Diophantine equations has applications in many cryptographic protocols such as RSA and Elliptic curve cryptography. The Extended Euclid's algorithm is a well known algorithm to solve these equations. We revisit…

Bayesian inference problems require sampling or approximating high-dimensional probability distributions. The focus of this paper is on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely…

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…

The distributed Hill estimator is a divide-and-conquer algorithm for estimating the extreme value index when data are stored in multiple machines. In applications, estimates based on the distributed Hill estimator can be sensitive to the…

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…

In this paper we combine the k-means and/or k-means type algorithms with a hill climbing algorithm in stages to solve the joint stratification and sample allocation problem. This is a combinatorial optimisation problem in which we search…

Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer solutions and check for…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. If Matiyasevich's conjecture on finite-fold Diophantine representations is true, then for every computable function f:N->N there is a positive integer m(f) such that for…

The paper proposes a new algorithm for solving global univariate optimization problems. The algorithm does not require convexity of the target function. For a broad variety of target functions after performing (if necessary) several…

The continuation method is a popular approach in non-convex optimization and computer vision. The main idea is to start from a simple function that can be minimized efficiently, and gradually transform it to the more complicated original…

Sequential decision tasks with incomplete information are characterized by the exploration problem; namely the trade-off between further exploration for learning more about the environment and immediate exploitation of the accrued…

In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A table of arithmetic operations with digital roots is given. This method is incapable of…

Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-\alpha^2)(x^2-(\alpha+1)^2).$ We express their integer…

The present work includes some of the author's original researches on integer solutions of Diophantine liner equations and systems. The notion of "general integer solution" of a Diophantine linear equation with two unknowns is extended to…

Scalable algorithms of posterior approximation allow Bayesian nonparametrics such as Dirichlet process mixture to scale up to larger dataset at fractional cost. Recent algorithms, notably the stochastic variational inference performs local…

Several mathematical problems can be modeled as a search in a database. An example is the problem of finding the minimum of a function. Quantum algorithms for solving this problem have been proposed and all of them use the quantum search…

The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on…

In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are…

The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…

In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine…