Foundations, Volume 4, Issue 3 (September 2024) – 10 articles

In recent years, a variety of multiple zeta values (MZVs) variants have been defined and studied. One way to produce these variants is to restrict the indices in the definition of MZVs to some fixed parity pattern, which include Hoffman’s multiple t-values, Kaneko and Tsumura’s multiple T-values, and Xu and this paper’s author’s multiple S-values. Xu and this paper’s author have also considered the so-called multiple mixed values by allowing all possible parity patterns and have studied a few important relations among these values. In this paper, we turn to the finite analogs and the symmetric forms of the multiple mixed values, motivated by a deep conjecture of Kaneko and Zagier, which relates the finite MZVs and symmetric MZVs, and a generalized version of this conjecture by the author to the Euler sum (i.e., level two) setting. We present a few important relations among these values such as the stuffle, reversal, and linear shuffle relations. We also compute explicitly the (conjecturally smallest) generating set in weight one and two cases. In the appendix, we tabulate some dimension computations for various subspaces of the finite multiple mixed values and propose a conjecture. Full article

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solutions, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively. Full article

We follow the assumption that relativistic causality is a key element in the structure of quantum mechanics and integrate the speed of light, c, into quantum mechanics through the postulate that the (reduced) Planck constant is a function of c with a leading order of the form $\hslash \left(c\right)\sim \mathsf{\Lambda }/c^p$ for a constant $\mathsf{\Lambda }>0,$ and $p>1.$ We show how the limit $c\to \infty$ implies classicality in quantum mechanics and explain why p has to be larger than $1$. As the limit $c\to \infty$ breaks down both relativity theory and quantum mechanics, as followed by the proposed model, it can then be understood through similar conceptual physical laws. We further show how the position-dependent speed of light gives rise to an effective curved space in quantum systems and show that a stronger gravitational field implies higher quantum uncertainties, followed by the varied $c.$ We then discuss possible ways to find experimental evidence of the proposed model using set-ups to test the varying speed of light models and examine analogies of the model based on electrons in semiconductor heterostructures. Full article

Methods for controlling electromagnetic fields in materials are presented that mitigate effects such as electrostatic discharge and electromagnetic/radio frequency interference. The first method determines the effective response of composite materials using a d-dimensional effective medium theory. The material consists of inhomogeneous two-layer inclusions with hyperspherical geometry. Non-integer dimensions represent fractal limits. The material medium is composed of a low hypervolume fraction of inclusions that are randomly distributed inside it. The effective response of the dielectric function is obtained using a virial expansion of the Maxwell–Garnett theory. The other method uses the transformation medium theory and involves the transformation of the material’s permittivity and permeability tensors so that the material exhibits a predefined effective response. By selecting appropriate transformations, a homogeneous material medium is transformed into an inhomogeneous version, forcing the electromagnetic fields to propagate along geodesic paths. These geodesics determine the behaviour of the fields inside the material. As a result, the material can be made to exhibit similar physical characteristics as those of a material composed of hyperspherical inclusions. The theoretical analysis presented is further studied and validated via the use of full-wave numerical simulations of Maxwell’s equations. Full article

Zinc is integral to diverse biological functions, acting catalytically, structurally, and supportively in essential enzyme cycles, despite its limited amounts in the body. Targeting zinc enzymes with potent drugs, such as Vorinostat, demonstrates the therapeutic efficacy of zinc-binding ligands, notably in cutaneous T-cell lymphoma treatments. Our study merges experimental and theoretical approaches to analyze the coordination of 8-hydroxylquinoline (8HQ) inhibitors with biomimetic zinc complexes and human histone deacetylase 8 (HDAC8), a monozinc hydrolase enzyme. Assessing 10 8HQ derivatives for structural and electronic characteristics against these models, we observe minimal inhibition efficacy, corroborated through protein–ligand docking analyses, highlighting the complexities of inhibitor–zinc enzyme interactions and suggesting intricate noncovalent interactions that are important for ligand binding to enzymes not accounted for in model zinc hydrolase mimics. Full article

Computation of the solution of the nonlinear Caputo fractional differential equation is essential for using $q,$ which is the order of the derivative, as a parameter. The value of q can be determined to enhance the mathematical model in question using the data. The numerical methods available in the literature provide only the local existence of the solution. However, the interval of existence is known and guaranteed by the natural upper and lower solutions of the nonlinear differential equations. In this work, we develop monotone iterates, together with lower and upper solutions that converge uniformly, monotonically, and quadratically to the unique solution of the Caputo nonlinear fractional differential equation over its entire interval of existence. The nonlinear function is assumed to be the sum of convex and concave functions. The method is referred to as the generalized quasilinearization method. We provide a Caputo fractional logistic equation as an example whose interval of existence is $[0,\infty ).$ Full article

We explicitly calculate the value of the cosmological constant, $\mathsf{\Lambda }$, based on the recently developed theory connecting entropic gravity with quantum events induced by transactions, called transactional gravity. We suggest a novel interpretation of the cosmological constant and rigorously show its inverse proportionality to the squared radius of the causal universe $\mathsf{\Lambda }~R_U^{-2}$. Full article

The intracellular environment displays complex dynamics influenced by factors such as molecular crowding and the low Reynolds number of the cytoplasm. Enzymes exhibiting active matter properties further heighten this complexity which can lead to memory effects. Molecular simulations often neglect these factors, treating the environment as a “thermal bath” using the Langevin equation (LE) with white noise. One way to consider these factors is by using colored noise instead within the generalized Langevin equation (GLE) framework, which allows for the incorporation of memory effects that have been observed in experimental data. We investigated the structural and dynamic differences in Shikimate kinase (SK) using LE and GLE simulations. Our results suggest that GLE simulations, which reveal significant changes, could be utilized for assessing conformational motions’ impact on catalytic reactions. Full article

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras. Full article