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This paper studies a question of Bourin on noncommutative $ L^p $ L p spaces. Some inequalities due to Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin II. Int J Math. 2021;32:2150043] and Hayajneh et al. [Norm inequalities for positive semidefinite matrices and a question of Bourin III. Positivity. 2022;26:23] are extended to the case of operators in noncommutative $ L^p $ L p spaces associated with a semifinite von Neumann algebra. We give a partial answer to a question of Bourin on noncommutative $ L^p $ L p spaces. Some related submajorization inequalities are also given. [ABSTRACT FROM AUTHOR]

Quaternion algebra ℍ is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space ℍ M with M ≥ 2 . Write ℂ η = { ξ : ξ = ξ 0 + β ⁢ η , ξ 0 , β ∈ ℝ } for η ∈ { i , j , k } . We remark that not only ℂ η M -vectors cannot allow traditional conjugate phase retrieval in ℍ M , but also ℂ i M ∪ ℂ j M -complex vectors cannot allow phase retrieval in ℍ M . We are devoted to conjugate phase retrieval of ℂ i M ∪ ℂ j M -complex vectors in ℍ M , where "conjugate" is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory. [ABSTRACT FROM AUTHOR]

If A is an associative algebra, then we can define the adjoint Lie algebra A(−) and Jordan algebra A(+). It is easy to see that any associative Rota–Baxter (RB) operator on A induces a Lie and Jordan RB operator on A(−) and A(+), respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra H4, which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra H4(−). [ABSTRACT FROM AUTHOR]

Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing. [ABSTRACT FROM AUTHOR]

It is evident that skew polynomials offer promising directions for developing cryptographic schemes. This paper focuses on exploring skew polynomials and studying their properties with the aim of exploring their potential applications in fields such as cryptography and combinatorics. We begin by deriving the concept of resultants for bivariate skew polynomials. Then, we employ the derived resultant to incrementally eliminate indeterminates in skew polynomial systems, utilising both direct and modular approaches. Finally, we discuss some applications of the derived resultant, including cryptographic schemes (such as Diffie–Hellman) and combinatorial identities (such as Pascal's identity). We start by considering a bivariate skew polynomial system with two indeterminates; our intention is to isolate and eliminate one of the indeterminates to reduce the system to a simpler form (that is, relying only on one indeterminate in this case). The methodology is composed of two main techniques; in the first technique, we apply our definition of a (bivariate) resultant via a Sylvester-style matrix directly from the polynomials' coefficients, while the second is based on modular methods where we compute the resultant by using evaluation and interpolation approaches. The idea of this second technique is that instead of computing the resultant directly from the coefficients, we propose to evaluate the polynomials at a set of valid points to compute its corresponding set of partial resultants first; then, we can deduce the original resultant by combining all these partial resultants using an interpolation technique by utilising a theorem we have established. [ABSTRACT FROM AUTHOR]

In this paper, we study the derivations of group algebras of some important groups, namely, Dihedral (D2n), Dicyclic (T4n) and Semi-dihedral (SD8n). First, we explicitly classify all inner derivations of a group algebra 픽G of a finite group G over an arbitrary field 픽. Then we classify all 픽-derivations of the group algebras 픽D2n, 픽T4n and 픽(SD8n) when 픽 is a field of characteristic 0 or an odd rational prime p by giving the dimension and an explicit basis of these derivation algebras. We explicitly describe all inner derivations of these group algebras over an arbitrary field. Finally, we classify all derivations of the above group algebras when 픽 is an algebraic extension of a prime field. [ABSTRACT FROM AUTHOR]

In the present work, an extended description of the covariant noncommutative space is presented, which accommodates the Fuzzy Gravity model constructed previously. It is based on the historical lesson that the use of larger algebras containing all generators of the isometry of the continuous one helped in formulating a fuzzy covariant noncommutative space. Specifically a further enlargement of the isometry group leads the authors, in addition to the construction of the covariant noncommutative space, also to the suggestion of the group that should be gauged on such a space in order to construct a Fuzzy Gravity theory. As a result, two Fuzzy Gravity models are obtained, one in de Sitter and one in anti‐de Sitter space, depending on the extension of the isometry group, and their spontaneous symmetry breaking leading to fuzzy versions of the noncommutative SO(1,3)$SO(1,3)$ gravity are discussed. In addition, how to introduce fermions in the fuzzy gravity is discussed for the first time, and even more importantly, how to unify the constructed noncommutative‐fuzzy gravity with internal interactions based on SO(10)$SO(10)$ or SU(5)$SU(5)$ as grand unified theories. [ABSTRACT FROM AUTHOR]

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra A. The algebra A is associative, noncommutative, and infinite-dimensional. It is defined by two generators A , B and two relations called the down-up relations. In the present paper, we introduce the Z 3 -symmetric down-up algebra A. We define A by generators and relations. There are three generators A , B , C and any two of these satisfy the down-up relations. We describe how A is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras sl 2 and sl 3 , the sl 3 loop algebra, the Kac-Moody Lie algebra A 2 (1) , the q -Weyl algebra, the quantized enveloping algebra U q (sl 2) , and the quantized enveloping algebra U q (A 2 (1)). We give some open problems and conjectures. [ABSTRACT FROM AUTHOR]

It is well-known that a formal deformation of a commutative algebra A leads to a Poisson bracket on A and that the classical limit of a derivation on the deformation leads to a derivation on A , which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A . The deformation leads in this case to a Poisson algebra structure on Π (A) : = Z (A) × (A / Z (A)) and to the structure of a Π (A) -Poisson module on A . The limiting derivations are then still derivations of A , but with the Hamiltonian belong to Π (A) , rather than to A . We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra. [ABSTRACT FROM AUTHOR]

This paper uses noncommutative resolutions of non-Gorenstein singularities to construct classical deformation spaces by recovering the Artin component of the deformation space of a cyclic surface singularity using only the quiver of the corresponding reconstruction algebra. The relations of the reconstruction algebra are then deformed, and the deformed relations together with variation of the GIT quotient achieve the simultaneous resolution. This extends the work of Brieskorn, Kronheimer, Grothendieck, Cassens-Slodowy, and Crawley-Boevey-Holland into the setting of singularities C²/H with H ≤ GL(2, C) and furthermore gives a prediction for what is true more generally. [ABSTRACT FROM AUTHOR]

In this paper, we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as inductive limits of spectral triples on rotation algebras. In the final section, we prove a noncommutative version of Wiener's lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups. [ABSTRACT FROM AUTHOR]

The given document is a research paper titled "Quasiregular Radicals of Nonassociative Algebras" written by A. Yu. Golubkov. The paper discusses various mathematical concepts related to Lie algebras, Jordan linear pairs, and linear triple systems. It explores the construction of quasiregular radicals and their applications in the Amitsur-Procesi theorem. The document also discusses the properties and relationships of different radicals in these algebraic structures. The author acknowledges funding support and provides their contact information for further inquiries. [Extracted from the article]

Based on the position and momentum of noncommutative relations with a noncanonical map, we study the Dirac equation and analyze its parity and time reversal symmetries in a noncommutative phase space. Noncommutative parameters can be endowed with the Planck length and cosmological constant such that the noncommutative effect can be interpreted as an effective gauge potential or a (0 , 2) -type curvature associated with the Plank constant and cosmological constant. This provides a natural coupling between dynamics and spacetime geometry. We find that a free Dirac particle carries an intrinsic velocity and force induced by the noncommutative algebra. These properties provide a novel insight into the Zitterbewegung oscillation and the physical scenario of dark energy. Using perturbation theory, we derive the perturbed and nonrelativistic solutions of the Dirac equation. The asymmetric vacuum gaps of particles and antiparticles reveal the particle–antiparticle symmetry breaking in the noncommutative phase space, which provides a clue to understanding the physical mechanisms of particle–antiparticle asymmetry and quantum decoherence through quantum spacetime fluctuation. [ABSTRACT FROM AUTHOR]

In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or 'rounded', onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group |$\mathrm{SE}(3)$|⁠ , the group of rigid motions of |$\mathbb{R}^{3}$|⁠. We based our method on embedding |$\mathrm{SE}(3)$| into the algebra of dual quaternions, which has deep algebraic connections with the group |$\mathrm{SE}(3)$|⁠. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral |$\mathrm{SE}(3)$| synchronization, which uses a matrix embedding of |$\mathrm{SE}(3)$|⁠. We show by numerical experiments that our approach yields comparable results with the current state of the art in |$\mathrm{SE}(3)$| synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of |$\mathrm{SE}(3)$| while yielding estimators of similar quality. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of solutions of nonlinear fractional coupled system of $\varphi $ -Hilfer type in the frame of Banach spaces. We improve a property of a measure of noncompactness in a suitably selected Banach space. Darbo's fixed point theorem is applied to obtain a new existence result. Finally, the validity of our result is illustrated through an example. [ABSTRACT FROM AUTHOR]

In this article, we construct some examples of noncommutative projective Calabi–Yau schemes by using noncommutative Segre products and quantum weighted hypersurfaces. We also compare our constructions with commutative Calabi–Yau varieties and examples constructed in Kanazawa (2015, Journal of Pure and Applied Algebra 219, 2771–2780). In particular, we show that some of our constructions are essentially new examples of noncommutative projective Calabi–Yau schemes. [ABSTRACT FROM AUTHOR]

We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on. [ABSTRACT FROM AUTHOR]

Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra A is a 2-cluster category of Dynkin type A. Here we show that, if A has an admissible action by the group G with two elements, then the stable Cohen-Macaulay category of the skew group algebra AG is a 2-cluster category of Dynkin type D. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible G -action on the path algebra of type A the resulting skew group algebra is of type D. Moreover, we provide a geometric model of the syzygy category of AG in terms of a punctured polygon P with a checkerboard pattern in its interior, such that the 2-arcs in P correspond to indecomposable syzygies in AG and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type A and D respectively. We also provide examples of types E 6 , E 7 , and E 8. [ABSTRACT FROM AUTHOR]

We establish some \Phi-moment inequalities for noncommutative differentially subordinate martingales. Let \Phi be a p-convex and q-concave Orlicz function with 1

We describe the geometric and probabilistic properties of a noncommutative -torus in a magnetic field. We study the volume invariance, integrated scalar curvature, and the volume form by using the operator method of perturbation by an inner derivation of the magnetic Laplacian operator on the noncommutative -torus. We then analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative -torus, and discuss the related main properties. [ABSTRACT FROM AUTHOR]

Let G be a group with a metric invariant under left and right translations, and let 픻r be the ball of radius r around the identity. A (k,r)-metric approximate subgroup is a symmetric subset X of G such that the pairwise product set XX is covered by at most k translates of X픻r. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of X finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on X replacing finiteness. In particular, if G has bounded exponent, we show that any (k,r)-metric approximate subgroup is close to a (1,r′)-metric approximate subgroup for an appropriate r′. [ABSTRACT FROM AUTHOR]

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has a radical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremum being finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J³ = 0 and without considering that rad² (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case. [ABSTRACT FROM AUTHOR]

Let 퐷 be the nonsplit quaternion algebra over Q p . We prove that a class of admissible unitary Banach space representations of D × of global origin are topologically of finite length. [ABSTRACT FROM AUTHOR]

This paper presents a modified version of Leibman's group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring's problem for general discrete Heisenberg groups in a sequel to this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring D [ x , σ , δ ]. We extend results of the first author and Vishkautsan on polynomial dynamics in D [ x ]. In particular, we show that if a ∈ D and f ∈ D [ x , σ , δ ] satisfy f (a) = a , then f ∘ n (a) = a for every formal power of f. More generally, we give a sufficient condition for a point a to be r -periodic with respect to a polynomial f. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy. [ABSTRACT FROM AUTHOR]

Let 퐷 be the nonsplit quaternion algebra over Q p . We prove that a class of admissible unitary Banach space representations of D × of global origin are topologically of finite length. [ABSTRACT FROM AUTHOR]

This article considers arrowhead and diagonal-plus-rank-one matrices in F n × n where F ∈ { R , C , H } and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O (n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language. [ABSTRACT FROM AUTHOR]

A set of position-dependent noncommutative algebra in two dimension (2D) that describes the space near the Planck scale had been introduced [J. Phys. A: Math. Theor. 53 (2020) 115303]. This algebra predicted the existence of maximal length of graviton measurable at low energy. From this algebra, we deduce in the present paper, a new noncommutative algebra that is compatible with the deformed algebra proposed by Costa Filho et al. [Phys. Rev. A84 (2011) 050102] to describe the Position-Dependent Mass (PDM) system in 1D. To this aim, we derive from the momentum operators, the Schrödinger-like equation which describes PDM system in null quantum well potential. The spectrum of this system is asymetrically deformed and exhibits different behaviours from the one obtained by Costa Filho et al. Thus, we observe that by increasing the PDM, the energy decreases and this decrease is more pronounced as the quantum number increases. Finally, we evaluate the thermodynamic quantities within the canonical ensemble and we show that these results are consistent with the ones recently obtained by Bensalem and Bouaziz [Phys. A Stat. Mech. Appl.523 (2019) 583–592]. [ABSTRACT FROM AUTHOR]

The description of (commutative and noncommutative) Lp$L_p$‐isometries has been studied thoroughly since the seminal work of Banach. In the present paper, we provide a complete description for the limiting case, isometries on noncommutative L0$L_0$‐spaces, which extends the Banach–Stone theorem and Kadison's theorem for isometries of von Neumann algebras. The result is new even in the commutative setting. [ABSTRACT FROM AUTHOR]

We construct the noncommutative Poisson boundaries of tracial von Neumann algebras through the ultraproducts of von Neumann algebras. As an application of this result, we complete the proof of Kaimanovich-Vershik's fundamental theorems regarding noncommutative entropy. We also prove the Amenability-Trivial Boundary equivalence and Choquet-Deny-Type I equivalence for tracial von Neumann algebras. [ABSTRACT FROM AUTHOR]

In this paper, we will solve an important form of hidden discrete logarithm problem (HDLP) and a generalized form of HDLP (GHDLP) over non-commutative associative algebras (FNAAs). We will reduce them to discrete logarithm problem (DLP) in a finite field through analyzing the eigenvalues of the representation matrix. Through the analysis of computational complexity, we will show that HDLP and GHDLP are not good improvements of DLP. With all the instruments in hand, we will break a series of corresponding schemes. Thus, we can conclude that all ideas of constructing cryptographic schemes based on the two solved problems are of no practical significance. [ABSTRACT FROM AUTHOR]

Let I be the commutative non-unital ring of order 4 defined by generators and relations. I = a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0. Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6 , and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods. We find that when n = 8 there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions. [ABSTRACT FROM AUTHOR]

In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure. [ABSTRACT FROM AUTHOR]

We present the following constructions: extension of a trivial quandle by a group with a nontrivial abelian subgroup, a quandle over a noncommutative group by an arbitrary antiautomorphism, a quandle presenting from a generalized Alexander quandle with the replacement of the automorphism by a central antiautomorphism, and a quandle over an -dimensional module depending on parameters with in a commutative unital ring. [ABSTRACT FROM AUTHOR]

In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A ∘ act as an appropriate depiction of the noncommutative maximal spectrum of A ; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces. [ABSTRACT FROM AUTHOR]

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R [ t ] / (t 2) . Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q , R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F q [ t ] / (t r) . We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation. [ABSTRACT FROM AUTHOR]

We introduce the ozone group of a noncommutative algebra |$A$|⁠ , defined as the group of automorphisms of |$A$|⁠ , which fix every element of its center. In order to initiate the study of ozone groups, we study polynomial identity (PI) skew polynomial rings, which have long proved to be a fertile testing ground in noncommutative algebra. Using the ozone group and other invariants defined herein, we give explicit conditions for the center of a PI skew polynomial ring to be Gorenstein (resp. regular) in low dimension. [ABSTRACT FROM AUTHOR]

We investigate the notion of subsystem in the framework of spectral triple as a generalized notion of noncommutative submanifold. In the case of manifolds, we consider several conditions on Dirac operators which turn embedded submanifolds into isometric submanifolds. We then suggest a definition of spectral subtriple based on the notion of submanifold algebra and the already existing notions of Riemannian, isometric, and totally geodesic morphisms. We have shown that our definitions work at least in some relevant almost commutative examples. [ABSTRACT FROM AUTHOR]

In light of the significance of non-commutative quaternionic algebra in modern physics, this study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing quaternionic wave function, we rewrite the Klein–Gordon equation in extended quaternionic form that includes scalar and the vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein–Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle's kinetic energy is less than 0 − m 0 c 2 . Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero. [ABSTRACT FROM AUTHOR]

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely, E p = a , b | p a = p b = 0 , a 2 = a , b 2 = b , a b = a , b a = b , where a ≠ b and p is any odd prime. We also give a classification of self-orthogonal codes and self-dual codes over E p , where p = 3 , 5 , and 7, in short lengths. [ABSTRACT FROM AUTHOR]

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order 4, to generate quasi self-dual codes. In the present paper, we introduce three such propagation rules to generate self-orthogonal, self-dual and quasi self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes completely in length at most 3, and partially in lengths 4 and 5, up to monomial equivalence. [ABSTRACT FROM AUTHOR]

A class of left bialgebroids whose underlying algebra A ♯ H is a smash product of a bialgebra H with a braided commutative Yetter–Drinfeld H -algebra A has recently been studied in relation to models of field theories on noncommutative spaces. In Borowiec and Pachoł (2017 J. Phys. A: Math. Theor. 50 055205) a proof has been presented that the bialgebroid A F ♯ H F where HF and AF are the twists of H and A by a Drinfeld 2-cocycle F = ∑ F 1 ⊗ F 2 is isomorphic to the twist of bialgebroid A ♯ H by the bialgebroid 2-cocycle ∑ 1 ♯ F 1 ⊗ 1 ♯ F 2 induced by F. They assume H is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter–Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with a general coaction and the correct prebraiding, and even without the assumption of quasitriangularity. [ABSTRACT FROM AUTHOR]

In this short note, we formulate a stabilizer formalism in the language of noncommutative graphs. The classes of noncommutative graphs we consider are obtained via unitary representations of finite groups and suitably chosen operators on finite-dimensional Hilbert spaces. This type of construction exhibits all the correctable errors by the stabilizer codes. Furthermore, in this framework, we generalize previous results in this area for determining when such noncommutative graphs have anticliques. [ABSTRACT FROM AUTHOR]

A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z 2 -graded algebra E (θ) , which corresponds to a noncommutative quadric hypersurface E ! / (z) for some central regular element z ∈ E 2 !. It turns out that the bounded derived category D b (gr Z 2 E (θ)) is equivalent to the stable category of the maximal Cohen-Macaulay modules over E ! / (z) provided that E ! is noetherian. As a consequence, E ! / (z) is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E (θ) is a semisimple Z 2 -graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces. As an application, we recover Knörrer's periodicity theorem without using matrix factorizations. [ABSTRACT FROM AUTHOR]

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C*-algebras that are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities for the |$K_0$| -group, the |$K_1$| -group, and the tracial state space of such an algebra. We show that these C*-algebras satisfy the Universal Coefficient Theorem, which is new even for the already known example of an exact C*-algebra nonisomorphic to its opposite algebra produced in an earlier work. [ABSTRACT FROM AUTHOR]

This article focuses on a recently developed formulation based on the noncommutative branch‐cut cosmology, the Wheeler‐DeWitt (WdW) equation, the Hořava–Lifshitz quantum gravity, chaotic and the coupling of the corresponding Lagrangian approach with the inflaton scalar field. Assuming a mini‐superspace of variables obeying the noncommutative Poisson algebra, we examine the impact of the inflaton scalar field on the evolutionary dynamics of the branch‐cut Universe scale factor, characterized by the dimensionless helix‐like function ln−1[β(t)]$$ {\ln}^{-1}\left[\beta (t)\right] $$. This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the primordial singularities. We take the Hořava–Lifshitz action modeled by branch‐cut quantum gravity as our starting point, which depends on the scalar curvature of the branched Universe and its derivatives and which preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner formalism. We then investigate the sensitivity of the scale factor of the branch‐cut Universe's dynamics. [ABSTRACT FROM AUTHOR]

This article focuses on the implications of a noncommutative formulation of branch‐cut quantum gravity. Based on a mini‐superspace structure that obeys the noncommutative Poisson algebra, combined with the Wheeler–DeWitt equation and Hořava–Lifshitz quantum gravity, we explore the impact of a scalar field of the inflaton‐type in the evolution of the Universe's wave function. Taking as a starting point the Hořava–Lifshitz action, which depends on the scalar curvature of the branched Universe and its derivatives, the corresponding wave equations are derived and solved. The noncommutative quantum gravity approach adopted preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner Formalism. In this work we delve deeper into a mini‐superspace of noncommutative variables, incorporating scalar inflaton fields and exploring inflationary models, particularly chaotic and nonchaotic scenarios. We obtained solutions to the wave equations without resorting to numerical approximations. The results indicate that the noncommutative algebraic space captures low and high spacetime scales, driving the exponential acceleration of the Universe. [ABSTRACT FROM AUTHOR]

To extend the existing results of differential systems of integer order, we have considered a class of Caputo fractional linear differential systems with multiple delays. By applying the multinomial theorem for nonpermutable matrices without a commutativity assumption on the matrix coefficients, the representations of exact solutions for the studied systems were obtained. The results show that Hyers-Ulam stability in finite time holds for this class of systems. [ABSTRACT FROM AUTHOR]