A Family of Generalized Legendre-Based Apostol-Type Polynomials

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

Human Folly and Border Fences: Looking to Non-Human Actors at the Indo–Bangladesh Border

The obsession with inter-state territorial borders and the associated paraphernalia of border management and security makes borders and their management a primarily human-centric discourse. This paper makes an attempt at introducing the agency of rivers as non-human actors—or rather as actants—in shaping and managing international borders. The paper looks specifically at the riverine sector of the Indo-Bangladesh border, where the international boundary has been re-negotiated each year by the transnational rivers, primarily the Brahmaputra (also the Gangadhar), through flooding, erosion, and deposition of sediment. By interrogating the role of rivers in shaping the border and border management strategies, the paper argues that humans, despite persisting as the primary agents in border management, are not the only actors. Drawing on Actor Network Theory (ANT), a case is made to appreciate the general symmetry between humans and non-humans as a-priori equal. Incorporating both in an actor-network may provide insights into border management in complex borderlands.

Rays, waves, SU(2) symmetry and geometry: toolkits for structured light

Structured light refers to the ability to tailor optical patterns in all its degrees of freedom, from conventional 2D transverse patterns to exotic forms of 3D,4D, and even higher-dimensional modes of light, which break fundamental paradigms and open new and exciting applications for both classical and quantum scenarios. The description of diverse degrees of freedom of light can be based on different interpretations, e.g. rays, waves, and quantum states, that are based on different assumptions and approximations. In particular, recent advances highlighted the exploiting of geometric transformation under general symmetry to reveal the "hidden" degrees of freedom of light, allowing access to higher dimensional control of light. In this tutorial, I outline the basics of symmetry and geometry to describe light, starting from the basic mathematics and physics of SU(2) symmetry group, and then to the generation of complex states of light, leading to a deeper understanding of structured light with connections between rays and waves, quantum and classical. The recent explosion of related applications are reviewed, including advances in multi-particle optical tweezing, novel forms of topological photonics, high-capacity classical and quantum communications, and many others, that, finally, outline what the future might hold for this rapidly evolving field.

Bandgaps of Atomically Precise Graphene Nanoribbons and Occam’s Razor

Rationalization of energy gaps of atomically precise AGNRs, “bulk” (ΔΕac) or “zigzag-end” (ΔΕzz), could be challenging and controversial concerning their magnitude, origin, substrate influence (ΔΕsb), and spin-polarization, among others. Hereby, a simple self-consistent and “economical” interpretation is presented, based on “appropriate” DFT (and TDDFT) calculations, general symmetry principles, and plausibility arguments, which is fully consistent with current experimental measurements for 5-, 7-, and 9-AGNRs within less than 1%, although at variance with some prevailing views or interpretations for ΔΕac, ΔΕzz, and ΔΕsb. Thus, an excellent agreement between experiment and theory emerges, provided some established stereotypes are reconsidered and/or abandoned. The primary source of discrepancies is the finite length of AGNRs together with inversion-symmetry conflict and topological end/edge states, which invariably mix with other “bulk” states making their unambiguous detection/distinction difficult. This can be further tested by eliminating end-states (and ΔΕzz), by eliminating empty (non-aromatic) end-rings

Fermionic duality: General symmetry of open systems with strong dissipation and memory

We consider the exact time-evolution of a broad class of fermionic open quantum systems with both strong interactions and strong coupling to wide-band reservoirs. We present a nontrivial fermionic duality relation between the evolution of states (Schrödinger) and of observables (Heisenberg). We show how this highly nonintuitive relation can be understood and exploited in analytical calculations within all canonical approaches to quantum dynamics, covering Kraus measurement operators, the Choi-Jamiołkowski state, time-convolution and convolutionless quantum master equations and generalized Lindblad jump operators. We discuss the insights this offers into the divisibility and causal structure of the dynamics and the application to nonperturbative Markov approximations and their initial-slip corrections. Our results underscore that predictions for fermionic models are already fixed by fundamental principles to a much greater extent than previously thought.

Exploring the landscape for soft theorems of nonlinear sigma models

We generalize soft theorems of the nonlinear sigma model beyond the $$ \mathcal{O} $$ O (p2) amplitudes and the coset of SU(N) × SU(N)/SU(N). We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known $$ \mathcal{O} $$ O (p2) single soft theorem for SU(N) × SU(N)/SU(N) in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of SO(N), where a special flavor ordering of the “pair basis” is available. We provide novel amplitude relations and a Cachazo-He-Yuan formula for such a basis, and derive the corresponding single soft theorem. Next, we extend the single soft theorem for a general group representation to $$ \mathcal{O} $$ O (p4), where for at least two specific choices of the $$ \mathcal{O} $$ O (p4) operators, the leading non-vanishing pieces can be interpreted as new extended theory amplitudes involving bi-adjoint scalars, and the corresponding soft factors are the same as at $$ \mathcal{O} $$ O (p2). Finally, we compute the general formula for the double soft theorem, valid to all derivative orders, where the leading part in the soft momenta is fixed by the $$ \mathcal{O} $$ O (p2) Lagrangian, while any possible corrections to the subleading part are determined by the $$ \mathcal{O} $$ O (p4) Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.

Fuzzy Order Acceptance and Scheduling on Identical Parallel Machines

In this study, we focus on the fuzzy order acceptance and scheduling problem in identical parallel machines (FOASIPM), which is a scheduling and optimization problem to decide whether the firm should accept or outsource the order. In general, symmetry is a fundamental property of optimization models used to represent binary relations such as the FOASIPM problem. Symmetry in optimization problems can be considered as an engineering tool to support decision-making. We develop a fuzzy mathematical model (FMM) and a Genetic Algorithm (GA) with two crossover operators. The FOASIPM is formulated as an FMM where the objective is to maximize the total net profit, which includes the revenue, the penalty of tardiness, and the outsourcing. The performance of the proposed methods is tested on the sets of data with orders that are defined by fuzzy durations. We use the signed distance method to handle the fuzzy parameters. While FMM reaches the optimal solution in a reasonable time for datasets with a small number of orders, it cannot find a solution for datasets with a large number of orders due to the NP-hard nature of the problem. Genetic algorithms provide fast solutions for datasets with a medium and large number of orders.

Existence of Solutions for a Singular Fractional q-Differential Equations under Riemann–Liouville Integral Boundary Condition

We investigate the existence of solutions for a system of m-singular sum fractional q-differential equations in this work under some integral boundary conditions in the sense of Caputo fractional q-derivatives. By means of a fixed point Arzelá–Ascoli theorem, the existence of positive solutions is obtained. By providing examples involving graphs, tables, and algorithms, our fundamental result about the endpoint is illustrated with some given computational results. In general, symmetry and q-difference equations have a common correlation between each other. In Lie algebra, q-deformations can be constructed with the help of the symmetry concept.

Symmetry properties of a Brownian motor with a sawtooth potential perturbed by harmonic fluctuations

We present a study of general symmetry properties of a Brownian ratchet model. The study is based both on constructing chains of symmetry transformations reflecting explicit and hidden symmetries of the average ratchet velocity as a functional of the spatially periodic potential energy of a nanoparticle and on taking into account the symmetry types of periodic functions that are components of the potential energy of an additive-multiplicative form. A ratchet with a sawtooth stationary potential profile, dichotomously perturbed by a spatially harmonic signal, is investigated. Conclusions are made on both the possibility of occurrence of the ratchet effect and its direction for given values of the asymmetry parameter of the sawtooth profile, phase shifts of the control component, and frequencies of temporal fluctuations. These conclusions have been obtained only on the basis of symmetry transformations; that demonstrates the predictive value of the approach presented. The results of the symmetry analysis are confirmed by numerical simulation of the functioning of a ratchet with dichotomous stochastic spatially periodic fluctuations of the nanoparticle potential energy.