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This article analyzes the periodic orbits of Syracuse dynamical systems in a novel algebraic setting: the commutative ring of graded $n$-adic integers. Within this context, this article introduces a dual-radix modular division algorithm for…
Particle tracking codes are one of the fundamental tools used in the design and the study of complex magnetic lattices in accelerator physics. For most practical applications, non-linear lenses are included and the Courant-Snyder formalism…
Lifting theorems are theorems that bound the communication complexity of a composed function $f\circ g^{n}$ in terms of the query complexity of $f$ and the communication complexity of $g$. Such theorems constitute a powerful generalization…
Some formulae are presented for finding two-integral distribution functions (DFs) which depends only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar…
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…
A formula is proposed for continuing physical correlation functions to non-integer numbers of dimensions, expressing them as infinite weighted sums over the same correlation functions in arbitrary integer dimensions. The formula is…
Integration of Hamiltonian systems by reduction to action-angle variables has proven to be a successful approach. However, when the solution depends on elliptic functions the transformation to action-angle variables may need to remain in…
The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical…
Functional methods can be applied to the quantum effective action to efficiently determine counterterms and matching conditions for effective field theories. We extend the toolbox to two-loop order and beyond and show how to evaluate the…
Using the theory of functions of several variables and $q$-calculus, we prove an expansion theorem for the analytic function in several variables which satisfies a system of $q$-partial differential equations. Some curious applications of…
Physical systems and signals are often characterized by complex functions of frequency in the harmonic-domain. The extension of such functions to the complex frequency plane has been a topic of growing interest as it was shown that specific…
We survey aspects of prediction theory in infinitely many dimensions, with a view to the theory and applications of functional time series.
A general statistical thermodynamic theory that considers given sequences of x-integers to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability…
In Physics, we are generally interested in real solutions involving natural phenomena, where knowledge of real functions of real variables is sufficient to obtain physically relevant results. However, the complexity of phenomena associated…
Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this…
An exact, one-to-one transform is presented that not only allows digital circular convolutions, but is free from multiplications and quantisation errors for transform lengths of arbitrary powers of two. The transform is analogous to the…
If we start from certain functional relations as definition of a quantum integrable theory, then we can derive from them a linear integral equation. It can be extended, by introducing dynamical variables, to become an equation with the form…
We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the Fourier…
This paper deals with the error analysis of the trapezoidal rule for the computation of Fourier type integrals, based on two double exponential transformations. The theory allows to construct algorithms in which the steplength and the…
The goal of this contribution is to introduce the Hamiltonian formalism of theoretical mechanics for analysing motion in generic linear and non-linear dynamical systems, including particle accelerators. This framework allows the derivation…