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Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field \(\mathbb {F}\) that outputs the evaluations of an m-variate polynomial of degree less than d in each variable at N points in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\). A previous work of Kedlaya and Umans (FOCS 2008 and SICOMP 2011) achieved the same time complexity when the number of variables m is at most \(d^{o(1)}\) and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar, and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than \(d^{o(1)}\). In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski, and Williams (IPEC 2017 and Algorithmica 2019), and that of Bhargava, Ghosh, Kumar, and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and, in particular, avoids the use of the Bombieri–Vinogradov theorem. However, it requires a mild assumption that the field size is bounded by an exponential tower in d of bounded height. More specifically, our second algorithm solves the multivariate multipoint evaluation problem over a finite field \(\mathbb {F}\) in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\) , provided that the size of the finite field \(\mathbb {F}\) is at most \((\exp (\exp (\exp (\cdots (\exp (d)))))\) , where the height of this tower of exponentials is fixed. [ABSTRACT FROM AUTHOR]

In this paper, we report on the implementation of CC2 and CC3 in the context of molecules in finite magnetic fields. The methods are applied to the investigation of atoms and molecules through spectroscopic predictions and geometry optimizations for the study of the atmosphere of highly magnetized White Dwarf stars. We show that ground-state finite-field (ff) CC2 is a reasonable alternative to CCSD for energies and, in particular, for geometrical properties. For excited states, ff-CC2 is shown to perform well for states with predominant single-excitation character. Yet, for cases in which the excited state wavefunction has double-excitation character with respect to the reference, ff-CC2 can easily lead to completely unphysical results. Ff-CC3, however, is shown to reproduce the CCSDT behavior very well and enables the treatment of larger systems at a high accuracy. [ABSTRACT FROM AUTHOR]

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [7], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [25] and Kedlaya & Umans [16] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most do(1) where the degree of the input polynomial in every variable is less than d. They also stated the question of designing fast algorithms for the large variable case (i.e., n x do(1)) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field Fq of characteristic p, which evaluates an n-variate polynomial of degree less than d in each variable on N inputs in time provided that p is at most do(1), and q is at most (exp(exp(exp(· · · (exp(d))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n x do(1)), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: --We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [21] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program [26]. More precisely, for every fixed prime p, we show that for every constant e > 0, and large enough n, the rank of any n × n Vandermonde matrix V over the field Fpa can be reduced to (n/exp(O(poly(e)log0.53 n))) by changing at most nT(e) entries in every row of V, provided a = poly(logn). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [9]. [ABSTRACT FROM AUTHOR]

We present a numerical approach to magnetic optical rotation based on real-time time-dependent electronic-structure theory. Not relying on perturbation expansions in the magnetic field strength, the formulation allows us to test the range of validity of the linear relation between the rotation angle per unit path length and the magnetic field strength that was established empirically by Verdet 160 years ago. Results obtained from time-dependent coupled-cluster and time-dependent current density-functional theory are presented for the closed-shell molecules H2, HF, and CO in magnetic fields up to 55 kT at standard temperature and pressure conditions. We find that Verdet's linearity remains valid up to roughly 10–20 kT, above which significant deviations from linearity are observed. Among the three current density-functional approximations tested in this work, the current-dependent Tao–Perdew–Staroverov–Scuseria hybrid functional performs the best in comparison with time-dependent coupled-cluster singles and doubles results for the magnetic optical rotation. [ABSTRACT FROM AUTHOR]

With the increasing use of digital images, there is a growing need for secure and efficient image encryption algorithms to ensure their confidentiality during transmission. In this paper, we propose an image encryption scheme for grayscale and color images that is based on primitive polynomial and Kronecker product of two invertible Suslin matrices. In the proposed scheme, primitive polynomial to map the pixel values of plain image to the elements of finite field is used, which results in pixel permutation. Then, the shuffled pixels are further diffused using a matrix obtained from the Kronecker product of two invertible Suslin matrices. To enhance security, the proposed scheme incorporates the logistic chaotic map. Further, we add authentication with the help of digital signature. Finally, the proposed scheme is implemented using Python and analyzed using various security parameters. The proposed methodology outperforms existing techniques in several critical aspects, including resistance to noise and data loss attacks, key space size, time complexity, correlation coefficient of adjacent pixels and entropy. Notably, the proposed scheme showcases a significant improvement in performance with PSNR values approaching ideal levels for noisy images and a vast 256-bit key space, bolstering security. These advancements along with the disruption of pixel correlations and increased randomness collectively makes encryption scheme a promising and secure choice for safeguarding grayscale and color images during transmission. [ABSTRACT FROM AUTHOR]

In this paper, we study a new graph structure, called the directsumgraph on a finite-dimensional vector space. We investigate the connectivity, diameter and the completeness of ΓU⊕W(핍). We also determine the degree of each vertex in case the base field is finite. We establish some results about Eulerian and triangulation of the graph ΓU⊕W(핍). Finally, we find the domination number and independence number, size, girth, edge-connectivity, clique number and the chromatic number of ΓU⊕W(핍). [ABSTRACT FROM AUTHOR]

In this article, we study a certain Galois property of subextensions of k(Ators)$k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety A$A$ defined over a number field k$k$. Concretely, we show that each subfield of k(Ators)$k(A_{\mathrm{tors}})$ that is Galois over k$k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of k$k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, that is, does not contain any infinite set of algebraic numbers of bounded height. [ABSTRACT FROM AUTHOR]

Let S be a standard graded polynomial ring over a field K in a finite set of variables, and let 픪 be the graded maximal ideal of S. It is known that for a finitely generated graded S-module M and all integers k ≫ 0, the module 픪kM is componentwise linear. For large k we describe the pattern of the Betti table of 픪kM when depthM > 0. Moreover, we show that for any k ≫ 0, 픪kI has linear quotients if I is a monomial ideal. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group G over a finite field K are determined such that the set of derivations of the group algebra form an associative K -algebra. The derivations of K G form a nontrivial associative K -algebra if and only if K has characteristic 2 and G is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting K -algebra is determined. [ABSTRACT FROM AUTHOR]

Polynomials of the form ∑ a i j x p i + p j + L (x) are called quadratic polynomials over a finite field p n and the quadratic polynomials of the form ∑ a i j x p i + p j are called Dembowski–Ostrom polynomials. In this paper, we propose four classes of permutation trinomials of Dembowski–Ostrom type polynomials over finite fields 2 n and two classes of quadratic permutation polynomials of the form ∑ a i j x 2 i + 2 j + L (x) over 2 n . [ABSTRACT FROM AUTHOR]

In this paper, we introduce several new construction techniques of linear complimentary dual (LCD) codes. First, we show that if a LCD code is fixed by a transitive automorphism group, then the punctured code is also LCD. We show that several important families such as anti-primitive BCH codes and extended Gabidulin codes are LCD. Finally relationships among self-dual codes, λ-orthogonal matrices and LCD codes are studied. As an application we give LCD codes from Paley-type bipartite graph and from punctured MacDonalds codes. Further, we give a bound on some LCD codes. [ABSTRACT FROM AUTHOR]

The objective of this study is to assess the impact of spatial variability in the subgrade layer on the critical response of pavements and the effectiveness of geogrid reinforcement, employing the random field finite difference analysis (RFFDA). A comprehensive parametric study was conducted to examine the influence of two crucial factors: the coefficient of variation ( COV E ) and scale of fluctuation (SOF) of the subgrade modulus. Further investigation was conducted to uncover the statistical and mechanical mechanisms underlying the impact of subgrade spatial variability with emphasis on the critical strain distributions and their correlation with both the overall modulus and the local spatial variability of the key influence zone. Furthermore, this study explored the influence of subgrade spatial variability on the effectiveness of geogrid in reducing critical strains, considering various placement positions and geogrid moduli. The following main conclusions are drawn: (a) subgrade spatial variability has a substantial amplifying effect on critical pavement strains due to low modulus dominating effect, (b) there exists a worst value of SOF that results in the most unfavorable statistics of critical subgrade strain, (c) the effect of subgrade spatial variability on critical subgrade strain is more pronounced compared to its effect on critical asphalt strain, (d) the mean value of critical subgrade strain in RFFDA can be significantly underestimated when assuming fixed location for the strain, and (e) the effectiveness of geogrid in reducing critical strains is impacted by subgrade spatial variability, with the impact varying with the type of critical strain and geogrid location. Specifically, when placed at the base course–subgrade interface, the ability of geogrid to reduce critical subgrade strain is significantly compromised due to the subgrade spatial variability. [ABSTRACT FROM AUTHOR]

In this paper, the deformation response of the stratum in the vicinity of a tunnel is investigated based on a case study in Tianjin. A method to predict the surface settlement in the case of double-track curvature shield tunneling is also proposed. The research methodology consists of analytical calculations, finite element analysis as well as field monitoring. Straight shield tunneling and curvature shield tunneling are modeled, each with a single-track shield and a double-track shield, and their operational parameters are compared. The influence of overcutting and existing tunnel on ground movement is also discussed. The salient observations from the series of analyses are as follows: (a) The maximum settlement induced by curvature shield tunneling is much greater than that induced by straight shield tunneling, while the existing tunnel has little impact on stratum settlement in contrast, (b) the proposed method of predicting the surface settlement induced by double-track curvature shield tunneling fits well with the field-monitored data in the Tianjin case study, and (c) the location of the maximum ground settlement is deviated by the overcutting phenomenon, whereas the settlement of the soil above the excavated tunnel is reduced due to the presence of the existing tunnel. [ABSTRACT FROM AUTHOR]

Let V$V$ be a finite‐dimensional vector space equipped with a nondegenerate Hermitian form over a field K${\mathbb {K}}$. Let G(V)${\mathcal {G}}(V)$ be the graph with vertex set the one‐dimensional nondegenerate subspaces of V$V$ and adjacency relation given by orthogonality. We give a complete description of when G(V)${\mathcal {G}}(V)$ is connected in terms of the dimension of V$V$ and the size of the ground field K${\mathbb {K}}$. Furthermore, we prove that if dim(V)>4$\dim (V) > 4$, then the clique complex F(V)${\mathcal {F}}(V)$ of G(V)${\mathcal {G}}(V)$ is simply connected. For finite fields K${\mathbb {K}}$, we also compute the eigenvalues of the adjacency matrix of G(V)${\mathcal {G}}(V)$. Then, by Garland's method, we conclude that H∼m(F(V);k)=0$\tilde{H}_m({\mathcal {F}}(V);{\mathbb {k}}) = 0$ for all 0⩽m⩽dim(V)−3$0\leqslant m\leqslant \dim (V)-3$, where k${\mathbb {k}}$ is a field of characteristic 0, provided that dim(V)2⩽|K|$\dim (V)^2 \leqslant |{\mathbb {K}}|$. Under these assumptions, we deduce that the barycentric subdivision of F(V)${\mathcal {F}}(V)$ deformation retracts to the order complex of the certain rank selection of F(V)${\mathcal {F}}(V)$ that is Cohen–Macaulay over k${\mathbb {k}}$. Finally, we apply our results to the Quillen poset of elementary abelian p$p$‐subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of V$V$ and the poset of orthogonal decompositions of V$V$. [ABSTRACT FROM AUTHOR]

(p, q)-Pentadiagonal matrices have attracted considerable attention in the past few years, which are one of the generalizations of pentadiagonal matrices. In the current paper, we present an algorithm for tridiagonalization of (p, q)-pentadiagonal matrices using permutation matrices. Moreover, we give the explicit representations of the permutation matrices and describe the nonzero structure of the obtained tridiagonal matrix. As applications, the tridiagonalization provides us with some useful results such as a breakdown-free algorithm of the matrix determinants and permanents, efficient parallel matrix multiplications and integer powers, and a property of the (p, q)-pentadiagonal matrix over the finite field. Some other applications on numerical linear algebra are also discussed. [ABSTRACT FROM AUTHOR]

Let Γ (L , g) be a Goppa code over F q , where L ⊂ F q m is a support and g (x) ∈ F q m [ x ] is a polynomial with s distinct roots in F q m . In [Couvreur A, Otmani A, Tillich JP (2014) New identities relating wild Goppa codes. Finite Field Appl 29: 178–197.], Couvreur at al. gave the bound: dim F q Γ (L , g e) - dim F q Γ (L , g e + 1) ≤ s , where e = q m - 1 + q m - 2 + ⋯ + q . In this paper, we give the conditions such that dim F q Γ (L , g e) = dim F q Γ (L , g e + 1) . [ABSTRACT FROM AUTHOR]

By exploiting symmetries of finite fields, covering perfect hash families provide a succinct representation for covering arrays of index one. For certain parameters, this connection has led to both the best current asymptotic existence results and the best known efficient construction algorithms for covering arrays. The connection generalizes in a straightforward manner to arrays in which every t-way interaction is covered λ > 1 times, i.e., to covering arrays of index more than one. Using this framework, we focus on easily computed, explicit upper bounds on numbers of rows for various parameters with higher index. [ABSTRACT FROM AUTHOR]

In this paper, the controllability of multi-agent systems over finite fields is studied based on the high-order fully actuated (HOFA) system approaches. The concept of HOFA systems over finite fields is introduced. Necessary and sufficient conditions for the controllability of linear systems over finite fields are established. It is proved that a leader–follower multi-agent system over finite fields is controllable if and only if it can be transformed into an HOFA system. Then the stabilisation control protocol is proposed based on the full-actuation property, and the upper bound of settling time for the system can be determined. The uncontrollable leader–follower multi-agent system over finite fields can be transformed into a controllable subsystem and an autonomous subsystem via structural decomposition, the conditions for the system to be stabilisable are provided. Finally, an illustrative example is shown to verify the validity of the main results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of r-primitive elements, a generalisation of primitive elements, in arithmetic progression by using a new formulation of the characteristic function for r-primitive elements in F q . In fact, we find a condition on q for the existence of α ∈ F q × for a given n ⩾ 2 and β ∈ F q × such that each of α , α + β , α + 2 β , ⋯ , α + (n - 1) β ⊂ F q × is r-primitive in F q ×. This result is utilized with the help of an inequality due to Robin also to produce an explicit bound on q; this, in turn, shows that for any n , r ∈ N , for all but finitely many prime powers q, for any β ∈ F q × , there exists α ∈ F q such that α , α + β , ⋯ , α + (n - 1) β are all r-primitive whenever r ∣ q - 1 . The number of arithmetic progressions in F q consisting of r-primitive elements of length n, is asymptotic to q (q - 1) n φ (q - 1 r) n . [ABSTRACT FROM AUTHOR]

We generalize a theorem of Ding relating the generalized Loewy length g ℓ ℓ (R) and index of a one-dimensional Cohen-Macaulay local ring (R , m , k). Ding proved that if R is Gorenstein, the associated graded ring is Cohen-Macaulay, and k is infinite, then the generalized Loewy length and index of R are equal. However, if k is finite, equality may not hold. We prove that if the index of a one-dimensional Cohen-Macaulay local ring is finite and the associated graded ring has a homogeneous nonzerodivisor of degree t , then g ℓ ℓ (R) ≤ index (R) + t − 1. Next we prove that if R is a one-dimensional hypersurface ring with a witness to the generalized Loewy length that induces a regular initial form on the associated graded ring, then the generalized Loewy length achieves this upper bound. We then compute the generalized Loewy lengths of several families of examples of one-dimensional hypersurface rings over finite fields. Finally, we study a graded version of the generalized Loewy length and determine its value for numerical semigroup rings. [ABSTRACT FROM AUTHOR]

Asymmetric orthogonal arrays are very useful in theoretical studies and practical applications. However, it is difficult to construct such designs of high strength. In this article, we use the generator matrix method to obtain some asymmetric orthogonal arrays of strength three. Then we combine it with the general (further) replacement procedure to get more general asymmetric orthogonal arrays of high strength. [ABSTRACT FROM AUTHOR]

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result. [ABSTRACT FROM AUTHOR]

We investigate the rationality of Weil sums of binomials of the form W u K , s = ∑ x ∈ K ψ (x s − u x) , where K is a finite field whose canonical additive character is ψ , and where u is an element of K × and s is a positive integer relatively prime to | K × | , so that x ↦ x s is a permutation of K. The Weil spectrum for K and s , which is the family of values W u K , s as u runs through K × , is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if s is nondegenerate (i.e., if s is not a power of p modulo | K × | , where p is the characteristic of K). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where | K | = 5 and s ≡ 3 (mod 4). [ABSTRACT FROM AUTHOR]

We give an axiomatic characterization of multiple Dirichlet series over the function field F q (T) , generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature. [ABSTRACT FROM AUTHOR]

In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for \mathbb {Z}, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough [Ann. of Math. (2) 181 (2015), no. 1, pp. 361–382] in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed 10^{16}. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [Invent. Math. 228 (2022), pp. 377–414] developed a versatile method called the distortion method and significantly reduced Hough's bound to 616,000. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for \mathbb {F}_q[x] of multiplicity s is bounded by a constant depending only on s and q. Consequently, we successfully resolve the minimum modulus problem for \mathbb {F}_q[x] and disprove a conjecture by Azlin [ Covering Systems of Polynomial Rings Over Finite Fields , University of Mississippi, Electronic Theses and Dissertations. 39, 2011]. [ABSTRACT FROM AUTHOR]

Residual stress (RS) significantly impacts the mechanical performance of components. Measurement of RS often provides incomplete data in terms of components of stress and spatial density. Employing such fields in finite element simulations results in significant modification of the field to achieve equilibrium and compatibility among strains. To overcome this, an iterative stress reconstruction algorithm (ISRA) is developed to estimate 3D RS fields that satisfy equilibrium, are stress component-wise complete, and represent the characterized data sampled. An Al 7075-T651 plate and an additively manufactured (AM) A36 steel wall are considered for RS reconstruction using measurement data from the literature. A maximum variation of ~2.5 MPa in the Al plate, and ~10 MPa in the steel wall are observed between the reconstructed and measured stresses. Furthermore, unknown stress components emerge and reach significant magnitudes (upto ~2.3 MPa in the Al plate and ~45 MPa in the AM wall) during ISRA. Indeed, it is found that minor errors in measurement or data processing are eliminated through the physical requirements during ISRA. Employing a reconstructed RS field is hence not just more accurate given its compatibility, but it additionally corrects for minor errors in measurement. Furthermore, it is found that spatially dense measurement data result in convergence with fewer iterations. Finally, although ISRA yields a nonunique solution dependent on boundary conditions, measurement errors, fitting errors, and mesh density, it accommodates for uncertainties and inaccuracies in measurement, as opposed to failing to reach a physically realistic converged solution. [ABSTRACT FROM AUTHOR]

Let 픽 q d be the d-dimensional vector space over the finite field 픽 q with q elements. For each non-zero r in 픽 q and E ⊂ 픽 q d , we define W ⁢ (r) as the number of quadruples (x , y , z , w) ∈ E 4 such that Q ⁢ (x - y) Q ⁢ (z - w) = r , where Q is a non-degenerate quadratic form in d variables over 픽 q . When Q ⁢ (α) = ∑ i = 1 d α i 2 with α = (α 1 , ... , α d) ∈ 픽 q d , Pham (2022) recently used the machinery of group actions and proved that if E ⊂ 픽 q 2 with q ≡ 3 ⁢ (mod ⁡ 4) and | E | ≥ C ⁢ q , then we have W ⁢ (r) ≥ c ⁢ | E | 4 q for any non-zero square number r ∈ 픽 q , where C is a sufficiently large constant, c is some number between 0 and 1, and | E | denotes the cardinality of the set E. In this article, we improve and extend Pham's result in two dimensions to arbitrary dimensions with general non-degenerate quadratic distances. As a corollary of our results, we also generalize the sharp results on the Falconer-type problem for the quotient set of distance set due to the first two authors and Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. The key new ingredient is to relate the estimate of the W ⁢ (r) to a quadratic homogeneous variety in 2 ⁢ d -dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums which are more handleable than the Kloosterman sums appearing in the standard distance-type problems. [ABSTRACT FROM AUTHOR]

Proxy signature schemes can be used in all the scenarios where a signer is not able to sign an important message. In some cases which involve a group of proxy signers, the anonymity of the proxy signer is required. We propose the first multivariate anonymous proxy signature scheme over finite field, and discuss all its security properties such as anonymity, unforgeability, verifiability, distinguishability, and undeniability. Additionally, we give a toy example to demonstrate the mathematical functioning of the proposed scheme. We compare our scheme with the existing post-quantum anonymous proxy signature scheme. [ABSTRACT FROM AUTHOR]

Fatigue‐induced clip fractures are the most common type of damage in railway systems, and they are exacerbated by rail corrugation. This research investigated the development of rail corrugation through long‐term measurements and examined the impact of rail corrugation on the fatigue life of e‐clips using a finite element model. The examination included different corrugation wavelengths, namely, short‐pitch corrugation (SPC), long‐pitch corrugation (LPC), and a new form of corrugation called superposed SPC and LPC. The results showed that increasing corrugation amplitude reduces an e‐clip's fatigue life and alters the region in the clip where fatigue damage occurs. A short corrugation wavelength shortens fatigue life. On railway tracks with superposed SPC and LPC, a significant increase in LPC amplitude sharply reduces a clip's fatigue life, even under minuscule SPC amplitude. Effective measures are proposed to prolong the fatigue life of e‐clips subjected to the effects of rail corrugation. Highlights: Studying the impact of three types of rail corrugation on e‐clip's fatigue life.Long‐term tracking investigation on rail corrugation and fastening system failures.Two measures for prolonging e‐clip's fatigue life were proposed.Establishment and verification of the whole solid‐element wheel–track system model. [ABSTRACT FROM AUTHOR]

Let G be a symplectic or orthogonal group defined over a finite field with odd characteristic and let D ≤ G be a Sylow 2-subgroup. In this paper, we classify the essential 2-subgroups and determine the essential 2-rank of the Frobenius category ℱD(G). Together with the results of An-Dietrich and Cao–An–Zeng, this completes the work of essential subgroups and essential ranks of classical groups. [ABSTRACT FROM AUTHOR]

Let A be the Steenrod algebra over the finite field k : = F 2 and G(q) be the general linear group of rank q over k. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, Ext A q , ∗ (k , k) , for all homological degrees q ⩾ 0. The Singer algebraic transfer of rank q, formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of G(q) to Ext A q , ∗ (k , k). Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all q ⩾ 4. This paper establishes Singer's conjecture for rank four in the generic degrees n = 2 s + t + 1 + 2 s + 1 - 3 whenever t ≠ 3 and s ⩾ 1 , and n = 2 s + t + 2 s - 2 whenever t ≠ 2 , 3 , 4 and s ⩾ 1. In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four. [ABSTRACT FROM AUTHOR]

Let Fq be the finite field of q elements, and Fqn its extension of degree n. A normal basis of Fq n over Fq is a basis of the form {α, αq, · · ·, αqn−1 }. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial x n − 1 in Fq, while the latter is closely related to the cyclotomic polynomials. Denote by δ(xn − 1) the set of all distinct monic irreducible factors of xn − 1 in Fq. The criteria for [δ(xn-1)] ≤2 have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for [δ(xn-1)]=s where s is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for [δ(xn-1)]=3,4,5 [ABSTRACT FROM AUTHOR]

A scheme for the Fourier–Galois spectrum analyzer for the field GF(31) is proposed. It is shown that this analyzer allows for solving a wide enough range of problems related to image processing, in particular those arising in the course of experimental studies in the field of physical chemistry. Such images allow digital processing when divided into a relatively small number of pixels, which creates an opportunity to use Galois fields of relatively small size. The choice of field GF(31) is due to the fact that the number 31 is a Mersenne prime number, which considerably simplifies the algorithm of calculating the Fourier–Galois transform in this field. The proposed scheme of the spectrum analyzer is focused on the use of threshold sensors, at the output of which signals corresponding to binary logic are formed. Due to this fact, further simplification of the proposed analyzer scheme is achieved. The constructiveness of the proposed approach is proven using digital modeling of electronic circuits. It is concluded that when solving applied problems in which an image can be divided into a relatively small number of pixels, it is important to take into account the specificity of particular Galois fields used for their digital processing. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. We review the characterization of p-valued Maiorana-McFarland bent functions with the Reed-Muller and the Reed-Muller-Fourier Transforms. We show that both transforms generate the same characterization and, based on the orthogonality of vectors, we prove why this is possible. Following the Language Philosopher Ludwig von Wittgenstein we consider this as a case of Family Resemblance. [ABSTRACT FROM AUTHOR]

Many modern cryptographic primitives for hashing and (authenticated) encryption make use of constructions that are instantiated with an iterated cryptographic permutation that operates on a fixed-width state consisting of an array of bits. Often, such permutations are the repeated application of a relatively simple round function consisting of a linear layer and a non-linear layer. These constructions do not require that the underlying function is a permutation and they can plausibly be based on a non-invertible transformation. Recently, Grassi proposed the use of non-invertible mappings operating on arrays of digits that are elements of a finite field of odd characteristic for so-called MPC-/FHE-/ZK-friendly symmetric cryptographic primitives. In this work, we consider a mapping that we call γ that has a simple expression and is based on squaring. We discuss, for the first time, the differential and linear propagation properties of γ and observe that these follow the same rules up to a relabeling of the digits. This is an intriguing property that, as far as we know, only exists for γ and the binary mapping χ 3 that is used in the cryptographic permutation Xoodoo. Moreover, we study the implications of its non-invertibility on differentials with zero output difference and on biases at the output of the γ mapping and show that they are as small as they can possibly be. [ABSTRACT FROM AUTHOR]

AbstractLet o be a compact discrete valuation ring with maximal ideal ℘ such that the finite residue field o/℘ has characteristic p. For r≥2 and p=2, we obtain the branching rules for the restriction of a regular representation of GL2(o/℘r) to SL2(o/℘r). These results have different behavior than that of the known case of p≠2. [ABSTRACT FROM AUTHOR]

We prove sum representations of Appell-Lauricella functions over a finite field using confluent hypergeometric functions over the finite field. As an application, we also prove transformation formulas, summation formulas and reduction formulas for Appell-Lauricella functions over the finite field. [ABSTRACT FROM AUTHOR]

The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle’s spacetime coordinates, 〈xμ〉, becomes uncertain. There exists a specific range of coordinates where the accessibility to 〈(xμ)2〉 is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value 〈xμ〉. This expectation value is dependent on 〈(xμ)2〉, which is itself limited by a minimum measurable length Δxmin2. A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between Δxmin and −〈g〉μν is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between Δxmin and the Planck length ℓp is established with a proportionality factor of −〈g〉μνβ2, where β2 denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle’s wavefunction ψ(x,t), leading to an unrealistic Δp. It is also noted that the lower limit of 〈(pμ)2〉 is directly proportional to the fundamental tensor. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and q = pr, r ≥ 1. For positive integers n, let nGn[⋯]q denote McCarthy’s p-adic hypergeometric function. In this paper, we prove an identity expressing a 4G4[⋯]q hypergeometric function as a sum of two 2G2[⋯]q hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transformation that relates n+2Gn+2[⋯]q and nGn[⋯]q hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of 4G4[⋯]q and 6G6[⋯]q hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes. [ABSTRACT FROM AUTHOR]

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set 픽qm and connection set the nonzero (qℓ + 1)th powers in 픽qm, as well as their complements. We explicitly compute the spectrum and the energy of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular, of pseudo or negative Latin square type. Using the spectral information we compute several invariants of these graphs. We exhibit infinitely many pairs of integral equienergetic non-isospectral graphs. As applications, on the one hand we solve Waring’s problem over 픽qm for the exponents qℓ + 1, for each q and for infinitely many values of ℓ and m. We obtain that the Waring number g(qℓ + 1,qm) = 1 or 2, depending on m and ℓ, thus solving some open cases. On the other hand, we construct infinite towers of integral Ramanujan graphs in all characteristics. Finally, we give the Ihara zeta functions of these graphs. [ABSTRACT FROM AUTHOR]

Let 픽q be an arbitrary finite field and n ≥ 2 be an integer with gcd(n,q) = 1. We study all 픽q-linear additive cyclic codes over 픽q3 of length n systematically. This is much more complicated compared to the same task over 픽q2. We obtain a canonical unique representation. We explicitly obtain the dual codes in the canonical form under the Euclidean and trace the Galois inner products. We characterize and construct large classes of complementary dual codes among 픽q-linear additive cyclic codes over 픽q3 of length n under the trace Euclidean and the trace Galois inner products. We obtain interesting differences depending on the canonical representation and also on the inner products. We also study subfield subcodes and trace (onto 픽q) codes. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic different from 2, and let L = F(b + 2ca) be a tower of two quadratic field extensions. We construct a filtration on the relative Brauer group Br(L/F) and compute its factors in terms of certain norm groups. The answer depends on whether the extension L/F is biquadratic or not. As a consequence we obtain a description of the group F∗2N E/FE∗ for a Galois field extension E/F with dihedral Galois group. Also, we prove that for any 2-torsion abelian group G and any biquadratic extension L/F there exists a field extension K/F linearly disjoint with L/F such that Br(LK/K)/2Br(LK/K) ≃ G. [ABSTRACT FROM AUTHOR]

Given a |$g$| -dimensional abelian variety |$A$| over a finite field |$\mathbf{F}_{q}$|⁠ , the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most |$g$|⁠. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for |$g \le 3$|⁠. We also give a partial classification for simple ordinary abelian varieties of prime dimension |$g\geq 3$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, our main aim is to study periodic and potent elements of a ring. We specially study periodic elements of graded rings and generalize some classical results related to idempotent of polynomial rings. We show that a (von Neumann) quasi-inverse of a potent element is a root of unity. We study the isomorphism of potent elements and analyze some closure properties of the set Pot(R) of potent elements of a ring R. The potent elements of the endomorphism ring of a Fitting module are described, and we apply this to matrices over division rings. In the case of matrices over finite fields, we connect features of potent elements with the exponent of their minimal polynomials. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we make explicit an application of the wreath product construction to the Galois groups of field extensions. More precisely, given a tower of fields F⊆K⊆L with L/F finite and separable, we explicitly construct an embedding of the Galois group Gal(Lc/F) into the regular wreath product Gal(Lc/Kc)≀rGal(Kc/F) . Here Lc (resp. Kc ) denotes the Galois closure of L/F (resp. K/F). Similarly, we also construct an explicit embedding of the Galois group Gal(Lc/F) into the smaller sized wreath product Gal(Lc/K)≀ΩGal(Kc/F) , where Ω=HomF(K,Kc) is acted on by composition of automorphisms in Gal(Kc/F) . Moreover, when L/K is a Kummer extension we prove a sharper embedding, that is, that Gal(Lc/F) embeds into the wreath product Gal(L/K)≀ΩGal(Kc/F) . As corollaries we obtain embedding theorems when L/K is cyclic and when it is quadratic with char(F)≠2 . We also provide examples of these embeddings and as an illustration of the usefulness of these embedding theorems, we survey some recent applications of these types of results in field theory, arithmetic statistics, number theory and arithmetic geometry. [ABSTRACT FROM AUTHOR]

In this paper, we study which field F can be represented as sums of two proper subfields. We prove that a field F is a sum of two proper subfields if and only if the transcendence degree of F over its prime subfield is infinite. This answers a question posed in the paper [M. Kȩpczyk and R. Mazurek, On the representation of fields as finite sums of proper subfields, Results Math. 75(2) (2020) 65]. [ABSTRACT FROM AUTHOR]

In this article, we find several novel and efficient quantum error-correcting codes ( Q ecc) by studying the structure of constacyclic ( C cc ), cyclic ( C c ), and negacyclic codes (N C c ) over the ring A k = Z p r 1 , r 2 , ⋯ , r k / ⟨ (r b (m b + 1) - r b) , r l r b = r b r l = 0 , b ≠ l ⟩ , where p = q m for m, m b ∈ N , m b | - 1 + q ∀ b , l ∈ 1 to k , q ≥ 3 is a prime, Z p is a finite field. We define distance-preserving gray map δ k . Moreover, we determine the quantum singleton defect (Q SD) of Q ecc, which indicates their overall quality. We compare our codes with existing codes in recent publications. The rings discussed by Kong et al. (EPJ Quantum Technol 10:1–16, 2023), Suprijanto et al. (Quantum codes constructed from cyclic codes over the ring F q + vF q + v 2 F q + v 3 F q + v 4 F q , pp 1–14, 2021. arXiv: 2112.13488v2 [cs.IT]), and Dinh et al. (IEEE Access 8:194082–194091, 2020) are specific cases of our work. Furthermore, we construct several novel and optimum linear complementary dual (Lcd) codes over A k. [ABSTRACT FROM AUTHOR]

A well-known difficulty of perturbative approaches to quantum field theory at finite temperature is the necessity to address theoretical constraints that are not present in the vacuum theory. In this work, we use lattice simulations of scalar correlation functions in massive ϕ4 theory to analyse the extent to which these constraints affect the perturbative predictions. We find that the standard perturbative predictions deteriorate even in the absence of infrared divergences at relatively low temperatures, and that this is directly connected to the analytic structure of the propagators used in the expansion. This suggests that the incorporation of non-perturbative thermal effects in the propagators is essential for a consistent perturbative formulation of scalar quantum field theories at finite temperature. By utilising the spectral constraints imposed on finite-temperature correlation functions, we explore how these effects manifest themselves in the lattice data, and discuss why the presence of distinct thermoparticle excitations provides a potential resolution to these issues. [ABSTRACT FROM AUTHOR]

We continue the formalization of field theory in Mizar [2], [3], [4]. We introduce separability of polynomials and field extensions: a polynomial is separable, if it has no multiple roots in its splitting field; an algebraic extension E of F is separable, if the minimal polynomial of each a ∈ E is separable. We prove among others that a polynomial q(X) is separable if and only if the gcd of q(X) and its (formal) derivation equals 1 – and that a irreducible polynomial q(X) is separable if and only if its derivation is not 0 – and that q(X) is separable if and only if the number of q(X)'s roots in some field extension equals the degree of q(X). A field F is called perfect if all irreducible polynomials over F are separable, and as a consequence every algebraic extension of F is separable. Every field with characteristic 0 is perfect [13]. To also consider separability in fields with prime characteristic p we define the rings Rp = { ap| a ∈ R} and the polynomials Xn− a for a ∈ R. Then we show that a field F with prime characteristic p is separable if and only if F = F p and that finite fields are perfect. Finally we prove that for fields F ⊆ K ⊆ E where E is a separable extension of F both E is separable over K and K is separable over F. [ABSTRACT FROM AUTHOR]