Algebraic Relations Research Articles

The capacity curve framework of EN 1993-1-6 (2025): Cylindrical shells under uniform meridional compression vs uniform bending

The capacity curve framework of the Eurocode EN 1993-1-6 on the strength and stability of metal shells offers a standardised algebraic relationship between the dimensionless buckling strength and the dimensionless slenderness of a shell. It is the basis of the buckling resistance assessment when used in conjunction with algebraic stress design, reference resistance design or the semi-computational LBA-MNA design. The capacity curve for a cylinder under uniform meridional compression has long been treated as the most conservative by default, recommended when no other curve can be justified. However, the most recent EN 1993-1-6, likely to be published in 2025, cautions that the most conservative choice for systems of intermediate slenderness may in fact be the curve for uniform bending. As this has come as a surprise to some in the community, this paper intends to justify the rationale behind this codified recommendation through a quantitative comparison of both systems’ computed capacity curves.This paper illustrates that the historical perception of uniform meridional compression as the ‘most conservative’ cylindrical shell system is based solely on its more severe imperfection-sensitive elastic buckling behaviour at high slendernesses. However, at intermediate slendernesses it is the uniform bending system that exhibits a more conservative dimensionless capacity curve, owing to the mechanics of local buckling under only partial plastification of the circumference. This, as well as other interesting properties of computationally generated capacity curves, is discussed in detail herein. This paper is intended to act as a background document to the recently evolved EN 1993-1-6 and will be of significant interest to practitioners and researchers aiming to apply this new state-of-the-art Eurocode to buckling assessments.

Dirac oscillator in a symmetric sextic anharmonic double-well potential

This paper investigates the relativistic Dirac analogue of a one-dimensional Schrödinger oscillator in a symmetric sextic anharmonic double-well potential, constructed via a non-minimal substitution technique. We show that the large component of the Dirac spinor obeys a Schrödinger-like equation, whose eigenvalues are used to determine the relativistic energy levels. These eigenvalues are computed using high-order Bender-Wu perturbation theory, resulting in an algebraic recursion relation for the expansion coefficients. A proper analytic formula describing the large-order behavior of these coefficients is derived, revealing their asymptotic factorial divergence. Additionally, a conjecture illustrating the dependence of these coefficients on the energy quantum numbers is proposed and confirmed. To overcome the challenge posed by the divergent nature of the perturbative eigenvalue expansion, Borel resummation with conformal mapping and Padé approximation are employed, leading to accurate results of the relativistic energy levels and their non-relativistic counterparts as functions of the coupling constants.

Non-equilibrium dynamics of symmetry-resolved entanglement and entanglement asymmetry: exact asymptotics in Rule 54*

Abstract Symmetry resolved entanglement and entanglement asymmetry are two measures of quantum correlations sensitive to symmetries of the system. Here we discuss their non-equilibrium dynamics in the Rule 54 cellular automaton, a simple, yet interacting, integrable model. Both quantities can be expressed in terms of the more analytically tractable ‘charged moments’, i.e. traces of powers of a suitably deformed density matrix, via a replica trick. We express them in terms of a tensor network, which we contract in space using a system of local algebraic relations. This gives the asymptotic form for the charged moments, valid in the regime of large but finite time that is shorter than all the relevant subsystem sizes. In this regime the charge moments decay exponentially with the rate given by the leading solution to a cubic equation.

On modeling flow between adjacent surfaces where the fluid is governed by implicit algebraic constitutive relations

We consider pressure-driven flow between adjacent surfaces, where the fluid is assumed to have constant density. The main novelty lies in using implicit algebraic constitutive relations to describe the fluid’s response to external stimuli, enabling the modeling of fluids whose responses cannot be accurately captured by conventional methods. When the implicit algebraic constitutive relations cannot be solved for the Cauchy stress in terms of the symmetric part of the velocity gradient, the traditional approach of inserting the expression for the Cauchy stress into the equation for the balance of linear momentum to derive the governing equation for the velocity becomes inapplicable. Instead, a non-standard system of first-order equations governs the flow. This system is highly complex, making it important to develop simplified models. Our primary contribution is the development of a framework for achieving this. Additionally, we apply our findings to a fluid that exhibits an S-shaped curve in the shear stress versus shear rate plot, as observed in some colloidal solutions.

The paramodular Hecke algebra

We give a presentation via generators and relations of the local graded paramodular Hecke algebra of prime level. In particular, we prove that the paramodular Hecke algebra is isomorphic to the quotient of the free Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Z}}}$$\\end{document}-algebra generated by four non-commuting variables by an ideal generated by seven relations. Using this description, we derive rationality results at the level of characters and give a characterization of the center of the Hecke algebra. Underlying our results are explicit formulas for the product of any generator with any double coset.

On algebraic dependencies between Poincaré functions

Abstract Let A be a rational function of one complex variable of degree at least two, and $z_0$ its repelling fixed point with the multiplier $\unicode{x3bb} .$ A Poincaré function associated with $z_0$ is a function meromorphic on ${\mathbb C}$ such that , and In this paper, we study the following problem: given Poincaré functions and , find out if there is an algebraic relation between them and, if such a relation exists, describe the corresponding algebraic curve $f(x,y)=0.$ We provide a solution, which can be viewed as a refinement of the classical theorem of Ritt about commuting rational functions. We also reprove and extend previous results concerning algebraic dependencies between Böttcher functions.

A 3-D Near-Field Source Localization Approach Based on the Combination of a Phase Interferometer, the Centroid Algorithm and the Perpendicular Foot Algorithm.

In this study, several 3-dimensional (3-D) parameter estimation and localization algorithms for wireless near-field (NF) sources are proposed employing the uniform circular array (UCA) structure. In the single-base-station case, the algebraic relation is demonstrated between the azimuth angle under the far-field (FF) assumption and the actual NF source firstly. Secondly, two groups of antenna pairs are selected with distances less than half the wavelength, which are called short baselines in the interferometer method. The foregoing short-baseline method is qualified to localize an NF source. In addition, a long-baseline method is also proposed with further research. Two groups of antenna pairs with distances greater than half the wavelength are selected as two long baselines. In the multiple-base-stations case, another two novel algorithms are also proposed. The first one is the centroid algorithm, which is based on the centroid calculation of three estimated source locations. And the second one is the perpendicular foot algorithm, which takes the perpendicular foot within three estimated source locations as the final positioning location. Simulation results illustrate that the proposed algorithms can achieve higher localization accuracy than the conventional 3-D Root MUSIC method. Moreover, the long-baseline method performs better than the short-baseline method. And it is also shown that the proposed perpendicular foot algorithm shows better performance than the proposed centroid algorithm.

Algebraic structure of the renormalization group in the renormalizable QFT theories

In this paper, we consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant, its generators [Formula: see text] with [Formula: see text] satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the [Formula: see text]-function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map.

Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic σ functions

Abstract It is known that the elliptic function solutions of the nonlinear Schrödinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, [ − i ∂ ∂ t + α ∂ ∂ u ] Ψ − 1 2 ∂ 2 ∂ u 2 Ψ + ( Ψ ∗ Ψ ) Ψ = 1 2 ( 2 β + α 2 − 3 ℘ ( v ) ) Ψ , where Ψ ( u ; v , t ) := e α u + i β t + c e − ζ ( v ) u σ ( u + v ) σ ( u ) σ ( v ) , its dual Ψ ∗ ( u ; v , t ) , and certain complex numbers α , β and c. In this paper, we generalize the algebraic differential relation to those of genera two and three in terms of the hyperelliptic sigma functions.

ADHM wilson line defect indices

The Coulomb and Higgs indices of the 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 U(N) ADHM theories can be decorated by line defect operators as the line defect correlators. We obtain exact closed-form expressions and various non-trivial algebraic relations for the correlators of the Wilson lines in the fundamental and (anti)symmetric representations by means of the Hall-Littlewood expansion, the Fermi-gas method and the residue calculation. From the large N limit of the correlators we obtain the single particle gravity indices which are expected to encode the spectra of fluctuation modes on the gravity duals of line operators in M2-brane SCFTs.

Anomaly inflow for dipole symmetry and higher form foliated field theories

In accordance with recent progress of fracton topological phases, unusual topological phases of matter hosting fractionalized quasiparticle excitations with mobility constraints, new type of symmetry is studied — multipole symmetry, associated with conservation of multipoles. Based on algebraic relation between dipole and global charges, we introduce a series of (d + 1)-dimensional BF theories with p-form gauge fields, which admit dipole of spatially extended excitations, and study their physical properties. We elucidate that gauge invariant loops have unusual form, containing linear function of the spatial coordinate, which leads to the position dependent braiding statistics and unusual ground state degeneracy dependence on the system size. We also show that the theories exhibit a mixed ’t Hooft anomaly between p-form and (d − p)-form dipole symmetries, which is canceled by an invertible theory defined in one dimensional higher via anomaly inflow mechanism.

New partial key exposure attacks on RSA with additive exponent blinding

Partial key exposure attacks present a significant threat to RSA-type cryptosystems. These attacks factorize the RSA modulus by utilizing partial knowledge of the decryption exponent, which is typically revealed by side-channel attacks, cold boot attacks, etc. In practice, the RSA implementations typically employ countermeasures to resist physical attacks, such as additive exponent blinding d′=d+rφ(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d' = d + r \\varphi (N)$$\\end{document} with unknown random blinding factor r. Although there are a couple of partial key exposure attacks on blinding RSA, these attacks require a considerable amount of leakage and fail to work when e is up to full size. In this paper, we propose new partial key exposure attacks on RSA with additive exponent blinding, focusing on leakage scenarios where the Most Significant Bits (MSBs) or Least Significant Bits (LSBs) of d′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d'$$\\end{document} are revealed. For the case where e is small, we first recover partial information of p by solving the quadratic congruence equation, and then find the small roots of the integer equation to recover entire private key. Our method relaxes the attack requirements, for instance, we reduce the amount of MSBs for a successful attack from 75 to 25% when e≈N0.25\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e \\approx N^{0.25}$$\\end{document} and r≈N0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\approx N^{0}$$\\end{document}. Furthermore, we propose new attacks using the unique algebraic relationship in blinding RSA, which extend the attack to the case where e is of full size.

Revisiting the Relationship of Arithmetical Thinking and Letter-Symbolic Algebra

Revisiting the Relationship of Arithmetical Thinking and Letter-Symbolic Algebra

Timestep-dependent element interpolation functions in the method of matched sections on the example of heat conduction problem

The paper is devoted to further elaboration of the method of matched sections as a new technique within the finite element method. Like FEM it supposes that: a) the complex domain is represented as a mesh of nonintersecting simple elements; b) algebraic relations between the main parameters of an element are established from the governing differential equations; c) all relationships from all elements are assembled into one global matrix. On the other hand, it has two distinct features. The first one is that relations between kinematic and inner force parameters (called Connection equations) are derived from the approximate analytical solution of the governing equations rather than by the application of minimization techniques. The second one consists in that the conjugation between elements is provided between the adjacent sides (sections) rather than in the nodes of the elements. In application to the transient 2D heat conduction, it is assumed that for each small rectangular element, the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, T, and heat flux Q. In practical realization for rectangular finite elements, the method is reduced to the determination of eight unknowns for each element – two unknowns on each side, which are related by the connection equations, and the requirement of the temperature continuity at the center of the element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step becomes the parameter of the element interpolation function within the element, i.e. it determines the behavior of the connection equations. This method was initially proposed by the first author for several 1D problems, and here for the first time, it is applied to 2D problems. The number of tests for rectangular plates exhibits the remarkable properties of the proposed time integration scheme concerning stability, accuracy, and absence of any restrictions as to increasing the time step.

Algebraic rough sets via algebraic relations

Algebraic rough sets via algebraic relations

The in-out formalism for in-in correlators

Cosmological correlators, the natural observables of the primordial universe, have been extensively studied in the past two decades using the in-in formalism pioneered by Schwinger and Keldysh for the study of dissipative open systems. Ironically, most applications in cosmology have focused on non-dissipative closed systems. We show that, for non-dissipative systems, correlators can be equivalently computed using the in-out formalism with the familiar Feynman rules. In particular, the myriad of in-in propagators is reduced to a single (Feynman) time-ordered propagator and no sum over the labelling of vertices is required. In de Sitter spacetime, this requires extending the expanding Poincaré patch with a contracting patch, which prepares the bra from the future. Our results are valid for fields of any mass and spin but assuming the absence of infrared divergences.We present three applications of the in-out formalism: a representation of correlators in terms of a sum over residues of Feynman propagators in the energy-momentum domain; an algebraic recursion relation that computes Minkowski correlators in terms of lower order ones; and the derivation of cutting rules from Veltman’s largest time equation, which we explicitly develop and exemplify for two-vertex diagrams to all loop orders.The in-out formalism leads to a natural definition of a de Sitter scattering matrix, which we discuss in simple examples. Remarkably, we show that our scattering matrix satisfies the standard optical theorem and the positivity that follows from it in the forward limit.

Optimized topological quantum compilation of three-qubit controlled gates in the Fibonacci anyon model: A controlled-injection approach

A method, termed controlled injection, is proposed for compiling three-qubit controlled gates within the non-Abelian Fibonacci anyon model. Building on single-qubit compilation techniques with three Fibonacci anyons, the approach showcases enhanced accuracy and reduced braid length compared to the conventional decomposition method for the controlled three-qubit gates. This method necessitates only four two-qubit gates for decomposition, a notable reduction from the conventional five. In conjunction, the study introduces a class of controlled three-qubit gates and conducts a numerical simulation of the topological iToffoli gate to validate the approach. In addition, we propose an optimization method for single-qubit gate approximation using algebraic relations and numerical methods, including distributed computing. Published by the American Physical Society 2024

Synthesis of Itˆo Equations for a Shaping Filter with a Given Spectrum

The analytical method for the synthesis of a generator of a random process with a given spectrum in the form of a linear system of Ito’s equations is proposed. The stationarity of a random process is assumed, the spectral and corresponding transfer functions of which are defined in the form of rational fractions. The coefficients of the system of Ito’s equations of the generator are found from recurrent algebraic relations. The method is focused on working with mathematical models of nature random processes, such as the Dryden’s wind model. The transformation of the spectra of the wind gust model in three directions is presented in detail and the corresponding stochastic equations are given.

Conformal Graphs as Twisted Partition Functions.

We show that a class of L-loop conformal ladder graphs are intimately related to twisted partition functions of free massive complex scalars in d=2L+1 dimensions. The graphs arise as four-point functions in certain two- and four-dimensional conformal fishnet models. The twisted thermal two-point function of the scalars becomes a generator of conformal ladder graphs for all loops. We argue that this correspondence is seeded by a system of two decoupled harmonic oscillators twisted by an imaginary chemical potential. We find a number of algebraic and differential relations among the conformal graphs that mirror the underlying free dynamics.

A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ-dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.