In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics $(\mathscr{A},\mathscr{B})$, with $\mathscr{A}$ smooth or singular and $\mathscr{B}$ smooth, in a fixed pencil of conics will admit a triangle or a tetragon inscribed in $\mathscr{A}$ and circumscribed about $\mathscr{B}$. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic; effectively allowing the case where our conic pairs intersect non-transversally.

Twisted group algebra of dihedral groups over finite fields

We investigate the coupled dynamics of concentration and charge in asymmetric 1:1 electrolytes, focusing on the interplay between diffusion asymmetry and external electric fields. Using Brownian dynamics simulations and linearized stochastic density functional theory (SDFT), we analyze the transient response of charge and number currents to inhomogeneous electric fields, as well as the steady-state spatio-temporal fluctuations under uniform fields. Our results reveal that asymmetry in ionic diffusion coefficients introduces a non-trivial coupling between charge and number transport, which modifies the two relaxation modes already present in symmetric electrolytes-a fast one associated with charge relaxation and a slow one linked to ambipolar diffusion. The dynamics are further modulated by the applied field, which enhances diffusion, alters screening lengths, and induces oscillatory behavior in the relaxation modes. The SDFT framework provides closed-form expressions for the intermediate scattering matrix, capturing the dynamics of density fluctuations and cross-correlations between number and charge. These predictions are validated by simulations, demonstrating excellent agreement across a wide range of wave vectors, both at equilibrium and under a finite electric field. Our findings highlight the critical role of diffusion asymmetry and external fields in tuning the transport properties of electrolytes, with implications for nanofluidic devices, energy harvesting, and iontronic circuits. This study bridges theoretical insights with practical applications, offering a robust framework for understanding and controlling electrolyte dynamics in asymmetric systems.

Abstract Let L be a field of positive characteristic p with a fixed algebraic closure $$\overline{L}$$ L ¯ , and let $$\alpha _1,\alpha _2,\beta \in L$$ α 1 , α 2 , β ∈ L . For an integer $$d\ge 2$$ d ≥ 2 , we consider the family of polynomials $$f_{\lambda }(z):= z^d+\lambda $$ f λ ( z ) : = z d + λ , parameterized by $$\lambda \in \overline{L}$$ λ ∈ L ¯ . Define $$C(\alpha _1,\alpha _2;\beta )$$ C ( α 1 , α 2 ; β ) to be the set of all $$\lambda \in \overline{L}$$ λ ∈ L ¯ for which there exist $$m,n\in {\mathbb {N}}$$ m , n ∈ N such that $$f_{\lambda }^m(\alpha _1)=f_{\lambda }^n(\alpha _2)=\beta $$ f λ m ( α 1 ) = f λ n ( α 2 ) = β . In other words, $$C(\alpha _1,\alpha _2;\beta )$$ C ( α 1 , α 2 ; β ) consists of all $$\lambda \in \overline{L}$$ λ ∈ L ¯ with the property that the orbit of $$\alpha _1$$ α 1 collides with the orbit of $$\alpha _2$$ α 2 under the same polynomial $$f_{\lambda }$$ f λ precisely at the point $$\beta $$ β . Assuming $$\alpha _1,\alpha _2,\beta $$ α 1 , α 2 , β are not all contained in a finite subfield of L , we provide explicit necessary and sufficient conditions under which $$C(\alpha _1,\alpha _2;\beta )$$ C ( α 1 , α 2 ; β ) is infinite. We also discuss the remaining case where $$\alpha _1,\alpha _2,\beta \in \overline{\mathbb {F}}_p$$ α 1 , α 2 , β ∈ F ¯ p and provide ample computational data that suggest a somewhat surprising conjecture. Our problem fits into a long series of questions in the area of unlikely intersections in arithmetic dynamics, which have been primarily studied over fields of characteristic 0. Working in characteristic p adds significant difficulties, but also reveals the subtlety of our problem, especially when some of the points lie in a finite field or when d is a power of p .

This work presents a secure data transmission process in the Internet of Things (IoT). Initially, the required data are collected and given to the Adaptive and Sparse Attention-based Dense Long Short-Term Memory (ASA-DLSTM) network for intrusion detection. The adaptive nature of the model allows for optimizing the parameters using the Sorted Fitness-based Addax Optimization Algorithm (SF-AOA). Once intrusions are detected, the data is used for the data transmission phase. It is performed using Optimal Key-based Elliptic Galois Cryptography (OK-EGC). By combining Elliptic with Galois fields and an optimal key management strategy, the proposed OK-EGC method enhances both encryption efficiency and security. Moreover, the integration of optimal key-based management using the same SF-AOA ensures that cryptographic keys are dynamically optimized based on the network's security requirements. Then, the effectiveness of the model is compared with existing systems. The accuracy of the implemented SF-AOA-ASA-DLSTM technique is 95.97%, which is higher than the conventional techniques, such as DNN (83.77%), SVM (83.19%), 1DCNN (90.26%), and ASA-DLSTM (93.6%) for the batch size value 64. Thus, the results display that the designed model addresses the critical challenges of IoT data security by providing both robust intrusion detection and secure communication.

Snevily conjecture on vector spaces over finite fields

Spaces of triangularizable matrices (II): Finite fields with odd characteristic

Duality theorem over finite fields and applications to Brauer groups

Corrigendum to “Divisibility of the multiplicative order modulo monic irreducible polynomials over finite fields” [J. Number Theory 277 (2025) 105–123

HighlightsWhat are the main findings?Develops and optimizes rubber granule concrete (12% content) as a stress-release layer.Validates field performance: ~64% stress reduction in high-ground-stress soft rock roadways.Achieves dual benefits of geotechnical stability and waste tire diversion.What are the implications of the main findings?Proposes a high-volume, high-value pathway for waste tire valorization in geotechics.Provides a complete case study from material design to field deployment.Offers a sustainable solution for waste management and deep roadway stability.Waste tire disposal and high-ground-stress soft rock roadway instability are pressing global challenges. This study develops sustainable rubber granule concrete (RGC) using waste tire rubber as a key component, aiming to realize waste valorization and floor heave control. RGC’s mechanical properties (uniaxial/triaxial compression, compressibility, ductility) were systematically tested, and its pressure relief mechanism was validated via finite element analysis (ABAQUS/FLAC) and 60-day field monitoring. Results show that RGC with optimal parameters (12% rubber content, 3–4 GPa elastic modulus, 250–350 mm thickness) achieves 64% bottom stress reduction and >40% displacement control. The material’s excellent energy absorption and flexibility address the brittleness of conventional concrete, ensuring stable support in high-stress environments. This work provides a sustainable, cost-effective concrete modification strategy, bridging waste recycling and geotechnical engineering, with broad implications for low-intensity, high-toughness material applications.

Let f be a monic quadratic polynomial over a finite field of odd characteristic. In 2012, Boston and Jones constructed a Markov process based on the post-critical orbit of f and conjectured that its limiting distribution explains the factorization of large iterates of f . Later, Xia, Boston, and the author performed extensive Magma computations and found some exceptional families of quadratics that do not seem to follow the original Markov model conjectured by Boston and Jones. They discovered this by empirically observing that certain factorization patterns predicted by the Boston–Jones model never seem to occur for these polynomials, and they suggested a multi-step Markov model that accounts for these missing factorization patterns. In this note, we provide proofs for all these missing factorization patterns. These are the first results that explain why the original conjecture of Boston and Jones does not hold for all monic quadratic polynomials.

Let F q \mathbb F_q be the finite field of characteristic p p with q = p a q=p^a elements. Let X X be a set of n n variables, and let A A be a nonempty subset of { 1 , … , n } \{1,\ldots ,n\} . Let N \mathbb {N} denote the set of nonnegative integers, and let σ ( d ) \sigma (d) be the digital sum of d ∈ N d\in \mathbb N in base p p . For f ( X ) = ∑ j ∈ J a j X j f(X)=\sum _{j\in J}a_{j}X^{j} with J J being a finite subset of N n \mathbb {N}^n , a j ∈ F q ∖ { 0 } a_j\in \mathbb {F}_q \setminus \{0\} and X j = x 1 j 1 ⋯ x n j n X^j=x_1^{j_1}\cdots x_n^{j_n} , we define deg A ⁡ ( f ) = max j ∈ J { ∑ i ∈ A j i } \deg _{A}(f)=\max _{j\in J}\{\sum _{i\in A}j_i\} and w p , A ( f ) = max j ∈ J { ∑ i ∈ A σ ( j i ) } w_{p,A}(f)= \max _{j\in J}\{\sum _{i\in A}\sigma (j_i)\} . Let { X 1 , … , X m } \{X_1,\ldots ,X_m\} be a partition of X X , S k S_k be the index set of X k X_k , and A k A_k be any nonempty subset of S k S_k for 1 ≤ k ≤ m 1\le k\le m . Let N ( V ) N(V) be the number of common zeros of the polynomials f i ( X ) = f i 1 ( X 1 ) + ⋯ + f i m ( X m ) , i = 1 , … , r , \begin{equation*} f_i(X)=f_{i1}(X_1)+\cdots +f_{im}(X_m),\ i=1,\ldots , r, \end{equation*} where f i k ( X k ) ∈ F q [ X k ] f_{ik}(X_k)\in \mathbb {F}_q[X_k] for all 1 ≤ i ≤ r 1\le i\le r and 1 ≤ k ≤ m 1\le k\le m . By using the Newton polyhedron introduced by Adolphson and Sperber [Ann. Sci. École Norm. Sup. (4) 20 (1987), 545–556], and the prime field reduction of Moreno and Moreno [Amer. J. Math. 117 (1995), 241–244], we show that if f 1 , … , f r f_1,\ldots ,f_r are not all polynomials in some proper subset of X X , then o r d q N ( V ) {ord}_qN(V) is no less than max { ⌈ ∑ k = 1 m | A k | max 1 ≤ i ≤ r { deg A k ⁡ ( f i k ) } ⌉ , 1 a ⌈ ∑ k = 1 m a | A k | max 1 ≤ i ≤ r { w p , A k ( f i k ) } ⌉ } − r . \begin{align*} \max \bigg \{\bigg \lceil \sum _{k=1}^m\frac {|A_k|} {\max \limits _{1\le i\le r}\{\deg _{A_k}(f_{ik})\}}\bigg \rceil , \frac {1}{a}\bigg \lceil \sum _{k=1}^m\frac {a|A_k|} {\max \limits _{1\le i\le r}\{w_{p,A_k}(f_{ik})\}}\bigg \rceil \bigg \}-r. \end{align*} This result extends previous work on diagonal systems to separated-variable systems and establishes a partial generalization of the Ax-Katz-MorenoMoreno theorem. Furthermore, by adding different variables, we extend the study to a more general case.

Let [Formula: see text] denote the finite field with [Formula: see text] elements. While an LCD code [Formula: see text] satisfies the duality criteria [Formula: see text] with [Formula: see text] being the dual code with respect to [Formula: see text], each constituent code [Formula: see text] of a [Formula: see text]-ary [Formula: see text]-Direct code is an LCD code in the sense that [Formula: see text] is dual to [Formula: see text] with respect to [Formula: see text], [Formula: see text]. In this paper, a concatenation procedure for the class of [Formula: see text]-ary [Formula: see text]-Direct codes is presented to obtain codes of larger lengths. Further, as the constituent codes of [Formula: see text]-ary [Formula: see text]-Direct codes are known to provide an optimal coding solution for the two-user binary adder channel, the constructed concatenated [Formula: see text]-Direct codes are employed over the [Formula: see text]-user [Formula: see text]-adder channel to harness the usefulness of [Formula: see text]-Direct codes in multi-user coding. The coding and decoding procedures for the constructed class of concatenated codes over the noiseless as well as noisy [Formula: see text]-user [Formula: see text]-adder channel also presented. The fact that [Formula: see text] allows and facilitates the required error correction for [Formula: see text]-Direct codes. With suitable parameters of concatenated [Formula: see text]-Direct codes [Formula: see text], the concatenation and coding procedures are illustrated through examples.

Let [Formula: see text] with [Formula: see text] an odd prime power, and let [Formula: see text] denote the degree-[Formula: see text] extension of the finite field [Formula: see text]. In this paper, we study the existence of an element [Formula: see text] for which both [Formula: see text] and [Formula: see text] are simultaneously primitive and normal over [Formula: see text]. Furthermore, when [Formula: see text] for some [Formula: see text], such a pair is guaranteed except for exact three choices of [Formula: see text].

Anti-toppling helical piles exhibit superior load-bearing performance due to enhanced interaction between the helices and the underlying soil; however, rigorous theoretical frameworks for their compressive analysis remain scarce. To address this limitation, this study proposes a computationally efficient analytical model utilizing the Modified Cam-Clay (MCC) constitutive framework to calibrate plane strain elements for pile–soil interaction simulations. Wedge-shaped and bulb-shaped fictitious soil pile models are introduced to accurately capture vertical capacity mobilization beneath the helix and pile tip, respectively. After successfully validating the framework against 3D finite element simulations and field test data, extensive parametric analyses were conducted. The key findings reveal that (1) unlike conventional piles, skin friction for anti-toppling helical piles increases monotonically with depth; (2) an optimal helix-to-pile diameter ratio of approximately 1.5 maximizes coordinated bearing capacity; (3) increasing pile length below a fixed helix depth provides negligible additional capacity; and (4) the critical state parameter M strictly controls the ultimate bearing threshold.

Abstract This work focuses on developing an efficient numerical method to solve the relativistic hydrogen-like atom in a finite magnetic field. To this end, we derive and implement an algorithm based on Gaussian-type orbitals that exploits fermionic symmetry to accelerate the calculations and to distinguish states of different character. This is then used to investigate a novel type of mixing regime between internal and external magnetic interactions. Finally, we assess the implications for the core region of heavy elements, where spin-orbit coupling is much stronger than the effects of external magnetic fields, and the consequences for quantum chemical calculations on heavy atoms in the vicinity of white dwarfs.

Abstract We prove that a Kakeya set in a vector space over a finite field of size always supports a probability measure, whose Fourier transform is bounded by for all non‐zero frequencies. We show that this bound is sharp in all dimensions at least 2. In particular, this provides a Fourier analytic proof that a Kakeya set in dimension 2 has size at least (which is asymptotically sharp). We also establish analogous results for sets containing ‐planes in a given set of orientations.

Shi et al. (2021) and Cheng (2024) constructed LCD double Toeplitz codes from tridiagonal symmetric and skew-symmetric Toeplitz matrices by factorizing Dickson polynomials. This paper extends their work to linear codes generated by anti-tridiagonal Hankel matrices with zero on the sub-diagonal, denoted H2n+1(b, 0, c). Using a permutation involution, it is shown that H2n+1(b, 0, c) is similar to a symmetric tridiagonal 2-Toeplitz matrix with zero main diagonal, allowing the factorization of the Hankel matrix’s characteristic polynomial via Dickson polynomials. Based on this factorization, necessary and sufficient conditions for the code C2n+1(b, 0, c) = [I2n+1,H2n+1(b, 0, c)] to be LCD are derived over finite fields of both even and odd characteristic. For even characteristic, the LCD condition is expressed in terms of roots of unity and parameters b, c, including an extension when gcd(p, n+1) ̸= 1. For odd characteristic, conditions involve primitive 2(n+1)-th roots of unity and the element μ with μ2 = −1. The results are further generalized to the case pr ∥ (n + 1) by reducing to primitive (m + 1)-th or 2(m + 1)-th roots. These constructions provide new families of LCD codes from Hankel matrices, complementing existing Toeplitz-based constructions and offering flexible parameter choices for side-channel and fault-injection attack resistant cryptography. An example over F4 is given.

This paper presents a systematic algebraic construction of noncyclic generalizations of BCH and Srivastava codes over Galois rings [Formula: see text] The proposed codes are defined via parity-check matrices whose entries are carefully chosen from the Galois ring, leading to determinants of the Alternant type. This structure, when combined with careful selection of ring elements to ensure key determinants are units, allows us to derive a rigorous lower bound on the minimum distance, providing a theoretically guaranteed error-correcting capability. We explicitly construct these codes and compute their core parameters. A comparative analysis with classical constructions over finite fields shows that for the same length [Formula: see text] and designed distance d, the ring-based construction achieves the same dimension k but with symbols drawn from a larger alphabet of size [Formula: see text] This yields a codebook of size [Formula: see text] representing an increase in information density (bits per codeword) compared to the field-based codebook of size [Formula: see text] The increased information rate comes at the cost of greater algebraic complexity in implementation, while the guaranteed minimum distance remains unchanged. This work establishes a foundational framework for applying advanced algebraic structures in noncyclic coding theory, with implications for modern communication systems requiring robust error control.

Recent works have combined random linear network coding (RLNC) with guessing random additive noise decoding (GRAND) to leverage RLNC packets to partially correct bit errors prior to RLNC decoding, so as to reduce the packet erasure rates in wireless broadcast networks. However, existing schemes are restricted to scalar RLNC over the finite field GF(2L). In this paper, we first formulate a general GRAND-assisted decoding framework for vector RLNC over the vector space GF(2)L, and further propose a design rule for vector RLNC schemes such that estimated error vectors can be efficiently obtained without incurring any additional computational overhead. Necessary and sufficient conditions for the correctness of every efficiently obtained estimated error vector are characterized. Two explicit vector RLNC schemes satisfying the proposed design rule are constructed. The first scheme is designed based on the matrix representation of GF(2L), and analytical results show that it achieves the same completion delay performance as the counterpart scalar RLNC scheme over GF(2L), while achieving up to a 37.3% reduction in coding computational complexity compared with the scalar one. The second scheme is designed based on sparse coding coefficient matrices. It further reduces computational complexity by up to 33.6% compared with the first scheme, at the cost of a slight degradation in completion delay performance.