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These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states…

Automated Machine Learning (AutoML) is an area of research that focuses on developing methods to generate machine learning models automatically. The idea of being able to build machine learning models with very little human intervention…

This short note provides and proves an easy algorithm to find a basic feasible solution for the Simplex Algorithm. The method uses a rule similar to Bland's rule for the initial phase of the algorithm.

Quicksort algorithm with Hoare's partition scheme is traditionally implemented with nested loops. In this article, we present loop programming and refactoring techniques that lead to simplified implementation for Hoare's quicksort algorithm…

This work presents a new algorithm for matrix power series which is near-sparse, that is, there are a large number of near-zero elements in it. The proposed algorithm uses a filtering technique to improve the sparsity of the matrices…

Layer pruning has emerged as a widely used technique for compressing large language models (LLMs). However, existing layer pruning approaches often incur substantial performance degradation. We identify the majority of this degradation to a…

The SOM algorithm is very astonishing. On the one hand, it is very simple to write down and to simulate, its practical properties are clear and easy to observe. But, on the other hand, its theoretical properties still remain without proof…

Sorting is a common and ubiquitous activity for computers. It is not surprising that there exist a plethora of sorting algorithms. For all the sorting algorithms, it is an accepted performance limit that sorting algorithms are linearithmic…

In this short note, we present certain generalized versions of the commutator formulas of some natural operators on manifolds, and give some applications.

We present a new algorithm for iterating over all permutations of a sequence. The algorithm leverages elementary~$O(1)$ operations on recursive lists. As a result, no new nodes are allocated during the computation. Instead, all elements are…

In this paper, we give a brief overview of the rule-based programming system, called P$\rho$Log and illustrate its capabilities.

Consider the collection of all binary matrices having a specific sequence of row and column sums and consider sampling binary matrices uniformly from this collection. Practical algorithms for exact uniform sampling are not known, but there…

It is discussed and surveyed a numerical method proposed before, that alternative to the usual compression method, provides an approximation to the algorithmic (Kolmogorov) complexity, particularly useful for short strings for which…

We show how to "compile" human-readable programs into standard decoder-only transformer models. Our compiler, Tracr, generates models with known structure. This structure can be used to design experiments. For example, we use it to study…

A method of embedding partially ordered sets into linear spaces is presented. The problem of finding all orthocomplementations in a finite lattice is reduced to a linear programming problem.

In this note we introduce a representation of simple undirected graphs in terms of polynomials and obtain a unique code for a simple undirected graph.

Algorithms have been fundamental to recent global technological advances and, in particular, they have been the cornerstone of technical advances in one field rapidly being applied to another. We argue that algorithms possess fundamentally…

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least…

We describe a simple algorithm based on a Markov chain process to generate simply connected acyclic directed graphs over a fixed set of vertices. This algorithm is an extension of a previous one, designed to generate acyclic digraphs, non…

A simple algorithm to compute all the zeros of a generic polynomial is proposed.