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I advocate an extreme "shut-up-and-calculate" approach to physics, where our external physical reality is assumed to be purely mathematical. This brief essay motivates this "it's all just equations" assumption and discusses its…
Integration by parts (IBP) has acquired a bad reputation. While it allows us to compute a wide variety of integrals when other methods fall short, its implementation is often seen as plodding and confusing. Readers familiar with tabular IBP…
We give an Euler-Maclaurin formula with remainder for the weighted sum of the values of a smooth function on the integral points in a simple integral polytope. Our work generalizes the formula obtained by Karshon, Sternberg and Weitsman in…
This is a short survey on the recent developments made in the integration theory with effective formulas of algebraic structures stronger or higher than Lie algebras.
In this editorial survey we introduce the special issue of the journal Communications in Mathematics on the topic in the title of the article. Our main goal is to briefly outline some of the main aspects of this important area at the…
I use Supergravity as a test case to study the role and uses of elegance/simplicity in formulating and evaluating physical models, whose sole criterion is of course truth: an observationally verified description of Nature within a certain…
Maclaurin Integration is a new series-based technique for solving infamously difficult integrals in terms of elementary functions. It has fairly liberal conditions for sound use, making it one of the most versatile integration techniques.…
A new approach to special relativity is presented which introduces coordinate systems with imaginary time axes, observation systems, and coordinate bases.
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic…
Here we provide an overview of what is known, and what is not known, about an interesting dynamical system known as the Kepler-Heisenberg problem. The main idea is to pose a version of the classical Kepler problem of planetary motion, but…
Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An…
This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether…
We solve a problem concerning Euler's paper "Variae considerationes circa series hypergeometricas" (\cite{E661}) suggested by G. Faber in the preface to Volume 16,2 of the first series of Euler's Opera Omnia by methods proven by Euler on…
The physical models of a successful unified theory about the Universe must operate in different phase of matter evolution and different fields of physics. The attempts to build such wide range theory as a bunch of theories developed for…
In this article, we present a new method to study relative Cuntz-Krieger algebras for higher-rank graphs. We only work with edges rather than paths of arbitrary degrees. We then use this method to simplify the existing results about…
In this article, the weakest possible theorem providing a foundation for the Hilbert space formalism of quantum theory is stated. The necessary postulates are formulated, and the mathematics is spelt out in detail. It is argued that, from…
The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission…
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
In this paper, I argue, contrary to the prevailing opinion in the linguistics and philosophy literature, that a sortal approach to aspectual composition can indeed be explanatory. In support of this view, I develop a synthesis of competing…
A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.