Finite Plane Research Articles

Sliding Viscoelastic Contacts: The Role of Adhesion, Boundary Conditions, and Finite Geometry

Sliding Viscoelastic Contacts: The Role of Adhesion, Boundary Conditions, and Finite Geometry

Conjunctive hierarchical secret sharing by finite geometry

AbstractSecret sharing is a general method for distributing sensitive data among the participants of a system such that only a collection of predefined qualified coalitions can recover the secret data. One of the most widely used special cases is threshold secret sharing, where every subset of participants of size above a given number is qualified. In this short note, we propose a general construction for a generalized threshold scheme, called conjunctive hierarchical secret sharing, where the participants are divided into disjoint levels of hierarchy, and there are different thresholds for all levels, all of which must be satisfied by qualified sets. The construction is the first method for arbitrary parameters based on finite geometry arguments and yields an improvement in the size of the underlying finite field in contrast with the existing results using polynomials.

Electrical damage mechanism analysis of traction motor bearings on high-speed trains considering overvoltage impulse

ABSTRACT As the critical rotating knuckle, traction motor bearings play a vital role in transmitting torque from motors to wheels for high-speed trains, while some electrochemical corrosion signs appear in traction motor bearings. To maintain the safety of the traction system, it is essential to explore the electrical damage mechanism of traction motor bearings considering the impact brought from current or overvoltage impulses. Herein, via analysing the electrochemical corrosion evolutionary process of bearings, main influencing factors are determined that both transient overvoltage impulse and long-term grounding current bring dramatic impact on bearings. To assess overvoltage distribution on different carriages, a ‘catenary-train-substation’ traction power supply model is launched based on practically collected impedances. A ‘carriage-bogie-motor’ circuit model is built to evaluate the overvoltage propagation path, meanwhile, the 3D finite element geometry of the bogie with bearings is launched to observe the overvoltage distribution in the bearings for evaluating breakdown risk.

Four new families of NMDS codes with dimension 4 and their applications

For an [n,k,d]q linear code C, the singleton defect of C is defined by S(C)=n−k+1−d. When S(C)=S(C⊥)=1, the code C is called a near maximum distance separable (NMDS) code, where C⊥ is the dual code of C. NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. In this paper, we present four new constructions of infinite families of NMDS codes with dimension 4 and completely determine their weight enumerators. As an application, we also determine the locality of the dual codes of these NMDS codes and obtain four families of distance-optimal and dimension-optimal locally recoverable codes.

Peculiarities of the Landau level collapse in graphene ribbons in crossed magnetic and in-plane electric fields

Employing the low-energy effective theory alongside a combination of analytical and numerical techniques, we explore the Landau level collapse phenomenon, uncovering previously undisclosed features. We consider both finite-width graphene ribbons and semi-infinite geometries subjected to a perpendicular magnetic field and an in-plane electric field, applied perpendicular to both zigzag and armchair edges. In the semi-infinite geometry the hole (electron)-like Landau levels collapse as the ratio of electric and magnetic fields reaches the critical value +(−)1. On the other hand, the energies of the electron (hole)-like levels remain distinct near the edge and deeply within the bulk, approaching each other asymptotically for the same critical value. In the finite geometry, we show that the electron (hole)-like levels become denser and merge, forming a band. Published by the American Physical Society 2024

The classifications of o-monomials and of 2-to-1 binomials are equivalent

AbstractWe observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials $$b=x^d+ux^e$$ b = x d + u x e for a large set of values of (d, e). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.

Exploiting finite geometries for better quantum advantages in Mermin-like games

Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no—communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum–classical success gap of 0.26― , larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of 1, beating the classical bound of 0.73― demonstrating the efficiency of the quantum strategy.

A finite geometry, inertia assisted coarsening-to-complexity transition in homogeneous frictional systems

The emergence of statistical complexity in frictional systems (where nonlinearity and dissipation are confined to an interface), manifested in broad distributions of various observables, is not yet understood. We study this problem in velocity-driven, homogeneous (no quenched disorder) unstable frictional systems of height H. The latter are described at the continuum scale within a realistic rate-and-state friction interfacial constitutive framework, where elasto-frictional instabilities emerge from rate-weakening friction. For large H, such frictional systems were recently shown to undergo continuous coarsening until settling into a spatially periodic traveling solution. We show that when the system’s height-to-length ratio becomes small — characteristic of various engineering and geophysical systems —, coarsening is less effective and the periodic solution is dynamically avoided. Instead, and consistently with previous reports, the system settles into a stochastic, statistically stationary state. The latter features slip bursts, whose slip rate is larger than the driving velocity, which are non-trivially distributed. The slip bursts are classified into two types: predominantly non-propagating, accompanied by small total slip and propagating, accompanied by large total slip. The statistical distributions emerge from dynamically self-generated heterogeneity, where both the non-equilibrium history of the interface and wave reflections from finite boundaries, mediated by material inertia, play central roles. Specifically, the dynamics and statistics of large bursts reveal a timescale ∼H/cs, where cs is the shear wave-speed. We discuss the robustness of our findings against variations of the frictional parameters, most notably affecting the magnitude of frictional rate-weakening, as well as against different interfacial state evolution laws. Finally, we demonstrate a reverse transition in which statistical complexity disappears in favor of the spatially periodic traveling solution. Overall, our results elucidate how relatively simple physical ingredients can give rise to the emergence of slip complexity.

Random walks and the “Euclidean” association scheme in finite vector spaces

Abstract In this paper, we provide an application to the random distance-t walk in finite planes and derive asymptotic formulas (as $q \to \infty $ ) for the probability of return to start point after $\ell $ steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

Revisiting logarithmic correction to five dimensional BPS black hole entropy

We compute logarithmic correction to the entropy of BPS black holes in asymptotically flat five dimensional space-time using finite temperature black hole geometry and find perfect agreement with the microscopic results and macroscopic computations based on zero temperature near horizon geometry. We also reproduce the Bekenstein-Hawking term for zero temperature black hole entropy from the corresponding term for finite temperature black hole.

Special directions on the finite affine plane

In this paper we study the number of special directions of sets of cardinality divisible by p on a finite plane of order p, where p is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly three special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that have exactly four special directions for small values of p.

Particle motion around traversable wormholes: Possibility of closed timelike geodesics

The present work investigates the general wormhole solution in Einstein gravity with an exponential shape function around an ultrastatic and a finite redshift geometry. The geodesic motion around the wormholes is studied in which the deflection angle of the orbiting photon sphere is found to be negative after a certain region, indicating the presence of repulsive effect of gravity in both the ultrastatic and finite redshift wormholes. Various unbounded and bounded timelike trajectories are presented on the wormhole embedding diagrams, in which some of the bound orbits involve intersection points that may lead to causality violating geodesics. Another class of closed timelike geodesics are obtained in the unstable circular trajectory that appeared at the wormhole throat. Finally, the trajectories are classified in terms of the family of CTG orbits.

Elliptic Rigid Inclusion in Soft Materials of Harmonic Type

Abstract We investigate finite plane deformations of an elliptic rigid inclusion embedded in a soft matrix that is made of a particular class of harmonic-type hyperelastic materials. The inclusion is assumed to be perfectly bonded to the matrix, which is subjected to a constant remote in-plane loading. Utilizing the Cauchy integral techniques associated with conformal mappings, we derive closed-form solutions for the full-field deformation, Piola stress, and Cauchy stress in the entire matrix. Numerical examples are presented to illustrate the current solutions in comparison with those established from linear elasticity theory. We find that in terms of the Cauchy stress around the inclusion, the maximum normal stress component always appears at the endpoints of the major axis of the inclusion, irrespective of the magnitude of the remote loading, while the maximum hoop stress component occurs not exactly at the above-mentioned endpoints when the remote loading exceeds a certain value. In particular, we identify an exact explicit formula for determining the relative rotation of the inclusion during deformation induced by a remote uniaxial loading of arbitrarily given magnitude and direction.

Genuinely nonabelian partial difference sets

AbstractStrongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

Random finite noncommutative geometries and topological recursion

In this paper, we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples {(\mathcal{A}, \mathcal{H}, D, \gamma, J)} , called random matrix geometries of type {(1,0)} , with a fixed fermion space {(\mathcal{A}, \mathcal{H}, \gamma, J)} and a distribution of the form {e^{- \mathcal{S} (D)}\, {\mathrm{d}} D} over the moduli space of Dirac operators. The action functional {\mathcal{S} (D)} is considered to be a sum of terms of the form {\prod_{i=1}^s \mathrm{Tr} ({D^{n_i}})} for arbitrary {s \geqslant 1} . The Schwinger–Dyson equations satisfied by the connected correlators {W_n} of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients {W_{g,n}} of the large N expansion of {W_n} ’s enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve {( {\Sigma, \omega_{0,1}, \omega_{0,2}} )} of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential {\omega_{0,2}} in terms of the formal parameters of the model.

Near MDS and near quantum MDS codes via orthogonal arrays

Near maximum distance separable (NMDS) codes are closely related to interesting objects in finite geometry and have nice applications in combinatorics and cryptography. But there are many unsolved problems about construction of NMDS codes. In this paper, by using symmetrical orthogonal arrays (OAs), we construct a lot of NMDS, m-MDS and almost extremal NMDS codes. Quantum error-correcting codes (QECCs) play a central role in quantum information processing and can protect quantum information from various quantum noises. We present a general method for constructing QECCs over mixed alphabets through asymmetrical OAs. Since quantum maximum distance separable (QMDS) codes over mixed alphabets with the dimension equal to one have not been found in all the literature so far, the definition of a near QMDS code over mixed alphabets is proposed. By using asymmetrical OAs, we obtain many such codes.

On affine blocking sets

We survey the known results on the minimal size of a blocking set in the finite affine geometries AG(n, q).

Adjoints of matroids

We first show that an adjoint of a loopless matroid is connected if and only if the original matroid is connected. By proving that the opposite lattice of a modular matroid is isomorphic to its extension lattice, we obtain that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice. This makes projective geometries become a key ingredient in characterizing the adjoint sequence ad0M,adM,ad2M,… of a connected matroid M. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to infinite projective geometries.

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR homogeneous.In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR transitive.We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix stepwise to refer to sets of MOLR with this property.Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

Positivity of Narayana polynomials and Eulerian polynomials

Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and number theory. Sagan and Tirrell (2020) [34] stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly γ-positive polynomials from γ-positive polynomials. We then prove the alternating γ-positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types A and B. In particular, by introducing the definition of colored 2×n Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type B. Finally, we present several identities involving the Eulerian polynomials of types A and B.