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By adapting Salomaa's complete proof system for equality of regular expressions under the language semantics, Milner (1984) formulated a sound proof system for bisimilarity of regular expressions under the process interpretation he…
We study the trajectory of iterations and the convergence rates of the Expectation-Maximization (EM) algorithm for two-component Mixed Linear Regression (2MLR). The fundamental goal of MLR is to learn the regression models from unlabeled…
Balancing model complexity against the information contained in observed data is the central challenge to learning. In order for complexity-efficient models to exist and be discoverable in high dimensions, we require a computational…
This thesis is devoted to the study of a calculus that describes the application of conditional rewriting rules and the obtained results at the same level of representation. We introduce the rewriting calculus, also called the rho-calculus,…
In this short note, we give an elementary proof of a universal approximation theorem for neural networks with three hidden layers and increasing, continuous, bounded activation function. The result is weaker than the best known results, but…
Conventional extreme learning machines solve a Moore-Penrose generalized inverse of hidden layer activated matrix and analytically determine the output weights to achieve generalized performance, by assuming the same loss from different…
Recent technological advances allow us to view chemical mass-action systems as analog computers. In this context, the inputs to a computation are encoded as initial values of certain chemical species while the outputs are the limiting…
The level of abstraction at which application experts reason about linear algebra computations and the level of abstraction used by developers of high-performance numerical linear algebra libraries do not match. The former is conveniently…
The Extreme Learning Machine (ELM) is a growing statistical technique widely applied to regression problems. In essence, ELMs are single-layer neural networks where the hidden layer weights are randomly sampled from a specific distribution,…
The Magnus expansion provides an exponential representation of one-parameter operator families, expressed as a series expansion in its generators. This is useful for example in quantum mechanics for expressing a unitary evolution determined…
We obtain various characterizations of the fundamental operators of $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction. We also demonstrate some important relations between the fundamental operators of a…
A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear PDE to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order…
M. Lin defined a binary operation for two positive semi-definite matrices in studying certain determinantal inequalities that arise from diffusion tensor imaging. This operation enjoys some interesting properties similar to the operator…
Milner (1984) defined a process semantics for regular expressions. He formulated a sound proof system for bisimilarity of process interpretations of regular expressions, and asked whether this system is complete. We report conceptually on a…
The exponential generating functions of {n^(n+m)} for arbitrary integer m are expressed as rational functions of the e.g.f. of {n^(n-1)} [the tree function] and then of the e.g.f. of {n^n} [the endofunction function]. The coefficients in…
In this article, we prove that sequences generated by the functional calculus $(f(T)(e_n))_{n \in \mathbb{N}}$ can be equivalently written as function sequences $(f_n(T) g)_{n \in \mathbb{N}}$, when $T$ is normal and $g$ a cyclic vector for…
In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we…
A recast of the standard residue-class analysis of the 3x+1 (Collatz) map in terms of two elementary operators on arithmetic progressions. The resulting calculus (i) splits any progression into its even and odd subsequences in a single…
Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where $S$-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper…
Based on a transformer based sequence-to-sequence architecture combined with a dynamic batching algorithm, this work introduces a machine learning framework for automatically simplifying complex expressions involving multiple elliptic Gamma…