Search Results (133)

We develop a comparative synthesis of quantum-information geometry beyond complex numbers, with emphasis on what different algebraic frameworks contribute to information-processing structure rather than on their formal novelty alone. The organizing idea is a layer-by-layer test of the standard complex Hilbert-space formalism: each non-complex or deformed framework modifies the scalar field, phase group, projective state space, Born-probability semantics, composition rule, measurement geometry, symmetry algebra or representation category. The central thesis is that such frameworks are physically meaningful when they identify which assumptions make complex quantum mechanics operationally stable: positive probabilities, associative multipartite composition, reversible dynamics, experimentally testable phases, locality constraints, informationally complete measurements, error bases and clear operational semantics. Real quantum theory probes the necessity of complex phases and local tomography; quaternionic quantum mechanics probes non-Abelian phase while retaining associativity and admitting complex embeddings; octonionic proposals probe the boundary where exceptional geometry survives but generic circuit composition is obstructed by non-associativity; Jordan algebras test ordered probabilistic state spaces; Clifford algebras and Bott periodicity provide the spinorial and topological grammar connecting gates, Hopf maps and periodic dimensions; and quantum-group or q-deformed constructions probe coproducts, braiding and representation categories rather than scalar amplitudes. We distinguish three roles that are often conflated: genuine hypercomplex kinematics, Hopf-fibration coordinates for ordinary complex multipartite entanglement, and deformed algebraic or categorical structures. The resulting map separates established equivalence and experimental-constraint results from useful representation tools and speculative programs, while identifying concrete open problems for non-complex quantum information. Full article

This paper proposes a spatial analog of the compass rose, interpreted as a discrete analog of cylindrical coordinates and considered as a basis for direction-based command filtering in Unmanned Aerial Vehicle (UAV) groups. The initial formulation is the problem of determining the direction to a radio signal source using data obtained by a group of four UAVs located at different altitudes. It is shown that, under conditions where the distance to the signal source significantly exceeds the characteristic size of the UAV spatial configuration, the direction to the source is determined much more reliably than the range to it. The results of Monte Carlo simulations confirm that the angular component of the solution remains meaningful under Time Difference of Arrival (TDoA) noise, whereas range reconstruction is substantially less stable. On this basis, a transition from a continuous description to a discrete sector representation of directions is proposed. The spatial compass rose is defined as a partition of the cylinder’s surface into a finite number of elements differing in azimuth and altitude. It is shown that this representation admits a natural algebraization: discrete directions can be one-to-one mapped to elements of finite fields and, therefore, interpreted in terms of multivalued logic. The obtained result creates the basis for simplifying computational procedures related to direction-sector classification and command processing in the on-board systems of UAV groups, provided that the method is interpreted as directional classification rather than complete three-dimensional localization. Full article

This paper proposes a spatial analog of the compass rose, constructed using finite fields and the discrete logarithm operation. The basic idea is to match the geometric elements of regular and semiregular polyhedra with elements of Galois fields (GF), which allows for the introduction of discrete spherical coordinates defined in algebraic form. The icosahedron is considered as a basic example. It is shown that using the icosahedron faces and the GF(41) field results in a 20-directed spatial structure that can be interpreted through discrete analogs of polar and azimuthal coordinates. Next, a variant based on the icosahedron edges and the GF(31) field is investigated, in which the number of directions increases to 30 while maintaining the regularity of the construction. A further generalization to the case of a truncated icosahedron, associated with the GF(181) field, is also considered, demonstrating the possibility of increasing the angular resolution without abandoning the algebraic organization of the set of directions. The obtained results demonstrate that the spatial rose of compass points can be represented as a finite system of directions with an explicit internal structure, convenient for coding, enumeration, and algorithmic processing. The proposed approach is of interest for problems of discrete description of rotations, construction of finite coordinate systems, and development of sectoral control algorithms, including those applicable to UAVs and their groups. The proposed formalism may also be considered as a sector-level coding layer for command-and-control architectures in which it is sufficient to identify a spatial sector rather than reconstruct full continuous coordinates. Full article

Tanner quasi-cyclic low-density parity-check (QC-LDPC) codes form an important family of structured LDPC codes with favorable girth properties. This paper studies the girth of Tanner $(2, L)$-regular QC-LDPC codes (referred to as Tanner QC-LDPC cycle codes) for arbitrary integers $L>2$ and develops a novel algebraic number theoretic method to determine the girth for all sufficiently large primes p with $p\equiv 1\left(\mathrm{mod}2L\right)$. We first analyze the case $L=3$ and prove that the girth is 12 for every prime $p\equiv 1\left(\mathrm{mod}6\right)$ through exhaustive resultant computations. We then extend the method to arbitrary L and obtain a clear classification: when L is even, the girth is exactly 8 for all admissible primes; when L is odd, the girth attains the maximum value 12 for all sufficiently large admissible primes. The proof transforms cycle existence conditions into polynomial equations and applies resultant theory. This approach converts the infinite task of checking all primes into a finite set of algebraic checks. Numerical simulations show that the Tanner $(2, 5)$-regular non-binary code over $\mathrm{GF}\left(64\right)$ achieves a coding gain of approximately $0.2$ dB over the 5G LDPC code of equivalent binary length. Full article

This paper proposes a recursive method for constructing intra-resolvable balanced incomplete block designs (BIBDs). The approach exploits the algebraic and geometric structure of finite projective geometries over Galois fields to generate resolvable designs with improved efficiency in terms of the number of blocks and treatment replications. The recursive procedure produces symmetric and uniform designs that are particularly suitable for high-dimensional settings. By systematically nesting resolvable blocks, we derive a new class of balanced n-ary designs that are both economical and scalable. These designs hold significant value for the statistical community, offering broad applicability in resource-constrained experimental environments such as precision agriculture, high-throughput drug screening, and computer-based simulation studies. We provide theoretical foundations through explicit constructions and comparative evaluations, demonstrating the advantages of our method over classical approaches. Full article

Let $0,1\ne m\in \mathbb{Z}$ and $\alpha =\sqrt[8]{m}$. According to the results of Gaál and El Fadil, $\alpha$ generates a power integral basis in $K=\mathbb{Q}\left(\alpha \right)$, if and only if m is square-free and $m\not\equiv 1\left(\mathrm{mod}4\right)$. In the present paper, we consider totally complex pure octic fields, as in the case $m<0$, with m satisfying the above property. In this case, $(1,\alpha ,{\alpha }^2,\dots ,{\alpha }^7)$ is an integral basis. Our purpose is to investigate whether K admits any other generators of power integral bases, inequivalent to $\alpha$. We present an efficient method to calculate generators of power integral bases in this type of fields with coefficients $<{10}^{200}$ in the above integral basis. We report on the results of our calculation for this type of field with $0>m>-5000$, which yields 2024 fields. Full article

Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and fading wiretap channels. This article offers a comprehensive survey of lattice-based wiretap coding, covering foundational concepts in algebraic number theory, Construction A over number fields, and the structure of modular and unimodular lattice families. We review key secrecy metrics, including secrecy gain, flatness factor, and equivocation, and consolidate classical and recent results to provide a unified perspective that links wireless-channel models with their underlying algebraic lattice structures. In addition, we review a newly proposed family of p-modular lattices in Khodaiemehr, H., 2018 constructed from cyclotomic fields $\mathbb{Q}\left({\zeta }_p\right)$ for primes $p\equiv 1\left(\mathrm{mod}4\right)$ via a generalized Construction A framework. We characterize their algebraic and geometric properties and establish a non-existence theorem showing that such constructions cannot be extended to prime-power cyclotomic fields $\mathbb{Q}\left({\zeta }_{p^n}\right)$ with $n>1$. Finally, motivated by the fact that these p-modular lattices naturally yield mixed-signature structures for which classical theta series diverge, we integrate recent advances on indefinite theta series and modular completions. Drawing on Vignéras’ differential framework and generalized error functions, we outline how modularly completed indefinite theta series provide a principled analytic foundation for defining secrecy-relevant quantities in the indefinite setting. Overall, this work serves both as a survey of algebraic lattice techniques for PLS and as a source of new design insights for secure wireless communication systems. Full article

This paper presents a set of survey-style notes linking core themes of pure algebra with central topics in algebraic and analytic number theory. We begin with finite extensions of $\mathbb{Q}$ and describe algebraic number fields through their realization as finite-dimensional $\mathbb{Q}$-algebras (via multiplication operators and matrix representations), leading naturally to the arithmetic invariants—trace, norm, and discriminant—and to the ring of integers, ideals, Dedekind domains, and the ideal class group. We then develop the classical theory of cyclotomic fields, emphasizing their Galois structure and their role in abelian extensions of $\mathbb{Q}$. Next, we discuss ramification in general extensions, including decomposition and inertia groups, the Frobenius element, and the Chebotarev density theorem. The exposition continues with a concise algebraic introduction to elliptic curves and their L-functions, and it places key conjectural links (including Birch and Swinnerton-Dyer) in context. Finally, a collection of examples highlights a common operational language between fractional calculus and number theory: Laplace and Mellin transforms turn convolution-type operators into multiplication, clarifying the appearance of $\Gamma$-factors, Dirichlet series, and zeta- and L-function structures in both settings. Full article

Colour algebras are noncommutative Jordan algebras and closely related to octonion algebras. They initially emerged in physics since the multiplication of a colour algebra over the real or complex numbers describes the colour symmetry of the Gell–Mann quark model. Over fields of characteristic not equal to two, their structure is now well-known. We initiate the study of colour algebras over a unital commutative base ring R where two is an invertible element, and show when colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, their automorphism group and their derivations. We show that there is again a close connection between the colour algebras obtained from hermitian forms and certain types of octonion algebras. Full article

Let p be a prime number and r a positive integer. This paper investigates the construction and classification of finite commutative rings of order $p^{6r}$ in which the set of zero-divisors J forms an ideal satisfying the conditions $J^3=0$, $J^2\ne 0$ with $J^2$ being principal. Under these conditions, the rings considered are precisely the Frobenius local rings. A Frobenius local (completely primary) ring R with these properties is referred to as a ring with property (P). These rings naturally divide into three classes according to their characteristic: p, $p^2$, or $p^3$. In the case of characteristic $p^2$, a further distinction is made depending on whether p lies in $J^2$ or in $J\setminus J^2$, where J denotes the Jacobson radical of R. The classification is achieved by associating to each ring a canonical matrix corresponding to a bilinear form and then applying matrix congruence techniques to reduce the problem to linear algebra over finite fields. This yields a complete and explicit description of all Frobenius local rings with property (P) of order $p^{6r}$, including their algebraic structure and enumeration. Full article

Let $\mathcal{A}$ be a unital ∗-algebra with the unit I over the complex field $\mathbb{C}$ and let $\eta \ne 0,\pm 1$ be a complex number. For any $A,B\in \mathcal{A}$, $A{\diamond }_{\eta }B=AB+\eta B{A}^{*}$ is referred to as the $\eta$-Jordan ∗-product. Suppose that $n\ge 3$ is a fixed positive integer. In this study, it is shown that if a map $\phi :\mathcal{A}\to \mathcal{A}$ satisfies $\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n)={\sum }_{k=1}^n{A}_1{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\cdots {\diamond }_{\eta }{A}_n$ for all $A_1,A_2\cdots A_{n-3}\in \{I,iI\}$ and $A_{n-2},A_{n-1},A_n\in \mathcal{A}$, then $\phi$ is an additive ∗-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$, where i is the imaginary unit. In application, characterizations of prime ∗-algebras, von Neumann algebras with no central summands of type $I_1$ and factor von Neumann algebras are obtained. Full article

The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time. Full article

DNA recombination is a fundamental biological process that encodes genetic information for organism development and function. In this study, we construct left-symmetric algebras arising from DNA insertion operations. That is, we define a modified insertion operation by weighting the simplified insertion. It generalizes the left-symmetric algebra constructed from the simplified DNA insertion operation. We prove that the algebra $\mathbb{F}\left(R\right)$ (over a field $\mathbb{F}$ of characteristic 0, with R being an infinite free semigroup generated by DNA nucleotides $\{A,G,C,T\}$) forms a left-symmetric algebra if and only if the function f satisfies a certain multiplicative condition for all positive integers m, n, and p. A key example of such a function is $f(m,n)=exp\left\{g\right(m,n\left)\right\}$, where $g(m,n)=k·mn,$ and k is a fixed positive number, which effectively models length-dependent DNA insertion dynamics. This work contributes an algebraic framework that may be useful for quantitative modeling of DNA recombination processes. Full article

In this paper we study the existence of Killing vector fields for right-invariant metrics on five-dimensional Lie groups. We begin by providing some explanation of the classification lists of the low-dimensional Lie algebras. Then we review some of the known results about Killing vector fields on Lie groups. We take as our invariant metric the sum of the squares of the right-invariant Maurer–Cartan one-forms, starting from a coordinate representation. A number of such metrics are uncovered that have one or more extra Killing vector fields, besides the left-invariant vector fields that are automatically Killing for a right-invariant metric. In each case the corresponding Lie algebra of Killing vector fields is found and identified to the extent possible on a standard list. The computations are facilitated by use of the symbolic manipulation package MAPLE. Full article

The Satisfiability Problem (SAT), a fundamental NP-complete problem, is widely applied in integrated circuit verification, artificial intelligence planning, and other fields, where the growing scale and complexity of practical problems demand higher solving efficiency. Due to redundant search paths, serialized reasoning steps, and inefficient pure literal detection, traditional serial SAT solvers require efficient parallelization of the pure literal rule. This paper adopts a parallel solving algorithm for the pure literal rule based on matrix representation. The algorithm can solve the shortcomings of poor universality, insufficient parallel collaborative mechanisms, and clause reduction. We first introduce a Clause-Numerical Incidence Matrix (CNIM) representation to provide a unified mathematical model for parallel operations. Second, we design a Column Vectors Pure Literal Parallel Topological Detection (CVPLPTD) algorithm that achieves pure literal detection with $O(mn/p)$ time complexity (p being the number of parallel threads) within the coefficient range $[1.0\times mn/p, 1.2\times mn/p]$. Finally, we adopt a dynamic matrix reduction strategy that compresses the matrix scale through row and column deletion after each pure literal assignment to reduce computational load. These innovations integrate matrix algebra and parallel computing, effectively breaking through the efficiency limitations of solving large-scale SAT problems while ensuring good universality across different computing platforms. Full article