Algebraic Identities Research Articles

Cutkosky rules and 1-loop κ -deformed amplitudes

In this paper, we show that the Cutkosky cutting rules are still valid term by term in the expansion in powers of κ of the κ-deformed 1-loop correction to the propagator of the κ-deformed complex scalar field. We first present a general argument which relates each term in the expansion to a nondeformed amplitude containing additional propagators with mass M>κ. We then show the same thing more pragmatically, by reducing the singularity structure of the coefficients in the expansion of the κ-deformed amplitude, to the singularity structure of nondeformed loop amplitudes, by using algebraic and analytic identities. We will explicitly show this up to second order in 1/κ, but the technique can be generalized to higher orders in 1/κ. Both the abstract and the more direct approach easily generalize to different deformed theories. We will then compute the full imaginary part of the κ-deformed 1-loop correction to the propagator in a specific model, up to second order in the expansion in 1/κ, highlighting the usefulness of the approach for the phenomenology of deformed models. This explicitly confirms previous qualitative arguments concerning the behavior of the decay width of unstable particles in the considered model. Published by the American Physical Society 2024

Some commutativity criteria of a factor ring R/P using semiderivations of R

Let [Formula: see text] be a ring and [Formula: see text] a prime ideal of [Formula: see text] This paper establishes a connection between the commutativity of the factor ring [Formula: see text] and semiderivations of [Formula: see text] satisfying some algebraic identities involving the prime ideal [Formula: see text] Moreover, we extended some results for derivations and semiderivations of prime rings to semiprime rings.

Algebraic Identities on Generalized Derivations in Prime Rings with Involution

This research paper aims to investigate the commutativity properties of prime rings in relation to generalized derivations and a left multiplier that fulfil specific algebraic conditions involving involution. Additionally, we present various examples studies to illustrate that the constraints imposed in our theorems are indeed necessary and cannot be omitted without compromising the validity of our results.Introduction: Consider a ring S that satisfies the associative property, and let Z(S) represent its center. An involution denoted by *, which is an additive function mapping S to itself. This involution has specific properties: for any α and β in S, applying the involution twice returns the original element ((α^* )^*=α), it distributes over addition ((α+β)^*=α^*+β^*), and it reverses the order of multiplication ((αβ)^*=β^* α^*). We categorize elements as hermitian when they remain unchanged under the involution (α^*=α), and as skew-hermitian when they change sign (α^*=-α). We use H(S) to represent the collection of all hermitian elements in S, and S(S) for all skew-hermitian elements. The involution is classified as first kind if H(S) is a subset of the center of S, denoted as Z(S). If this is not the case, it’s considered second kind, and in this scenario, the intersection of H(S) and Z(S) contains more than just the zero element. We also define several types of mappings on ring S. A left multiplier, △, is an additive map where △(υω)=△(υ)ω, ∀υ,ω∈S. A derivation, ψ, is an additive mapping that satisfies ψ(υω)=ψ(υ)ω+υψ(ω), ∀υ,ω∈S. Extending this concept, we define a generalized derivation, Γ, which is linked to a derivation ψ. This function satisfies Γ(υω)=Γ(υ)ω+υψ(ω), ∀υ,ω∈S. It’s worth noting that any derivation can be considered a generalized derivation associated with itself.Objectives: In this study, we intend to examine the commutativity of a prime ring S by utilizing generalized derivations Γ_1, Γ_2, and a left multiplier △, while adhering to specific algebraic identities that involve involution. Specifically, we will delve into the commutativity of rings S that fulfill the following algebraic conditions: • [Γ_1 (υ),Γ_2 (υ^*)]+△([υ,υ^*])∈Z(S), for all υ∈S; • Γ_1 (υ)∘Γ_2 (υ^*)+△(υ∘υ^*)∈Z(S), for all υ∈S; • [Γ_1 (υ),Γ_2 (υ^*)]+△(υ∘υ^*)∈Z(S), for all υ∈S; • Γ_1 (υ)∘Γ_2 (υ^*)+△([υ,υ^*])∈Z(S), for all υ∈S; • Γ(υυ^*)±Γ(υ)Γ(υ^*)+△([υ,υ^*])∈Z(S), for all υ∈S; • Γ(υυ^*)±Γ(υ^*)Γ(υ)+△(υ∘υ^*)∈Z(S), for all υ∈S. Lastly, we offer examples to illustrate that the constraints applied to our hypotheses are necessary and not redundant.Results: Building on the work of Nejjar ([8], Theorems 3.5, 3.8), who demonstrated that a 2-torsion-free prime ring with involution and a derivation ψ satisfying certain conditions must be commutative, we explore broader generalizations of these conditions. Specifically, Nejjar showed that if the derivation ψ meets either of the following criteria: [ψ(υ),ψ(υ)]±υ∘υ∈Z(S), ∀υ∈S or ψ(υ)∘ψ(υ)±υ∘υ∈Z(S), ∀υ∈S, then S is necessarily commutative. Our work extends these findings by introducing new identities for pairs of generalized derivations that are connected to a left multiplier △. Finally, we offer examples to illustrate that the constraints applied to our hypotheses are necessary and not redundant.Conclusions: In this research, we investigate the commutativity of prime rings S admitting an involution and generalized derivations satisfying some algebraic identities. We can conclude our paper with an open question.Open question: are these results correct if we replace the generalized derivation by the generalized (α,β)-derivation, where α and β are automorphisms of ring S?

IDENTITIES. PEDAGOGICAL METHODS FOR TEACHING SHORT MULTIPLICATION FORMULAS

The effective teaching of short multiplication formulas, which play a crucial role in algebra, requires innovative pedagogical methods. This paper explores how different teaching strategies impact students' understanding and retention of these algebraic identities. The study employs a mixed-methods approach, combining quantitative analysis of student performance with qualitative feedback from both students and teachers. Results suggest that interactive methods, such as problem-based learning and visual aids, significantly enhance comprehension. These findings have important implications for improving the quality of algebra education in secondary and tertiary settings.

Nonexistence of Majorana fermions in Kerr-Newman type spacetimes with nontrivial charge* *Supported by the National Natural Science Foundation of China (12326602)

We show that the Dirac equation is separated into four differential equations for time-periodic Majorana fermions in Kerr-Newman and Kerr-Newman-(A)dS spacetimes. Although they cannot be transformed into radial and angular equations, the four differential equations yield two algebraic identities. When the electric or magnetic charge is nonzero, they conclude that there is no differentiable time-periodic Majorana fermions outside the event horizon in Kerr-Newman and Kerr-Newman-AdS spacetimes, or between the event horizon and the cosmological horizon in Kerr-Newman-dS spacetime.

A simple linear algebra identity to optimize large-scale neural network quantum states

Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105 parameters, achieving state-of-the-art ground-state energy in the J1–J2 Heisenberg model at J2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.

Lepowsky's and Wakimoto's product formulas for the affine Lie algebras [formula omitted]

In this paper, we recall Lepowsky's and Wakimoto's product character formulas formulated in a new way by using arrays of specialized weighted crystals of negative roots for affine Lie algebras of type Cl(1), Dl+1(2) and A2l(2). Lepowsky–Wakimoto's infinite periodic products appear as one side of (conjectured) Rogers–Ramanujan-type combinatorial identities for affine Lie algebras of type Cl(1).

Ideals in Bipolar Quantum Linear Algebra

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c∈(0,1] such that I=(−x,y)∈R2|cx≤y≤xcandx,y≥0. This shows that its shape has to be symmetric with the graph y=−x.

Inverse conductivity problem with one measurement: uniqueness of multi-layer structures

In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.

Ghost problem, spectrum identities and various constraints on brane-localized gravity

This paper investigates the brane-localized interactions, including DGP gravity and higher derivative (HD) gravity localized on the brane. We derive the effective action on the brane, which suggests the brane-localized HD gravity suffers the ghost problem generally. Besides, we obtain novel algebraic identities of the mass spectrum, which reveal the global nature and can characterize the phase transformation of the mass spectrum. We get a powerful ghost-free condition from the spectrum identities, which rules out one type of brane-localized HD gravity. We further prove the mass spectrum is real and non-negative m2 ≥ 0 under the ghost-free condition.Furthermore, we discuss various constraints on parameters of brane-localized gravity in AdS/BCFT and wedge holography, respectively. They include the ghost-free condition of Kaluza-Klein and brane-bending modes, the positive definiteness of boundary central charges, and entanglement entropy. The ghost-free condition imposes strict constraint, which requires non-negative couplings for pure DGP gravity and Gauss-Bonnet gravity on the brane. It also rules out one class of brane-localized HD gravity. Thus, such HD gravity should be understood as a low-energy effective theory on the brane under the ghost energy scale. Finally, we briefly discuss the applications of our results.

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Algebraic identities and generalized derivations in prime rings

Let R be a prime ring with center Z(R). A map F from R to R is called a multiplicative generalized derivation associated with d (not necessarily additive) if F(xy)= F(x)y + xd(y) for all x and y in R. In this paper, our intention is to study the commutativity of R, using a multiplicative generalized derivation that satisfies some algebraic identities.

Lie Modules of Banach Space Nest Algebras

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let A be a Banach space nest algebra and L be a weakly closed Lie A-module. We show that there exist a weakly closed A-bimodule K, a weakly closed subalgebra DK of A, and a largest weakly closed A-bimodule J contained in L,such that J⊆L⊆K+DK, with [K,A]⊆L. The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras.

Skew-Symmetric Identities of Finitely Generated Alternative Algebras

Skew-Symmetric Identities of Finitely Generated Alternative Algebras

Images of locally nilpotent derivations acting on ideals of polynomial algebras

Images of locally nilpotent derivations acting on ideals of polynomial algebras

Connecting ideals in evolution algebras with hereditary subsets of its associated graph

AbstractIn this article, we introduce a relation including ideals of an evolution algebra A and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and ideals having the absorption property of an evolution algebra in terms of its associated graph. In particular, the maximal ideals can be determined through maximal hereditary subsets of vertices except for those containing $$A^2$$ A 2 . We also define a couple of order-preserving maps, one from the sets of ideals of an evolution algebra to that of hereditary subsets of the corresponding graph, and the other in the reverse direction. Conveniently restricted to the set of absorption ideals and to the set of hereditary saturated subsets, this is a monotone Galois connection. According to the graph, we characterize arbitrary dimensional finitely-generated (as algebras) evolution algebras under certain restrictions of its graph. Furthermore, the simplicity of finitely-generated perfect evolution algebras is described on the basis of the simplicity of the graph.

Pedagogical Processes to Help Children Learn Algebraic Identities in Mathematics

This paper focuses on refining the pedagogical processes instrumental in the mastering of algebraic identities, employing a meticulous approach to instruction. Emphasizing the utility of resources specifically tailored for visualization, the paper primarily seeks to deepen understanding. Its aim is to instill and foster algebraic thinking among children, encouraging active engagement in the exploration of patterns and problem-solving within the realm of algebra. A significant facet of the research lies in its nuanced acknowledgment and targeted resolution of common challenges inherent in comprehending patterns and algebraic thinking, providing a thoughtful intervention in educational practices. Moreover, the research advocates for the exploration of diverse representations of identical algebraic concepts, acknowledging the multifaceted nature of learning, drawing from other subjects, the idea of memorization of patterns in order to internalize the concepts such that their application can be made irrespective of circumstances such as in grammar where the base rules for the constructions of sentences have to memorized after which their rules can be applied in varied circumstances. The outlined learning objectives for students extend beyond rote memorization, encompassing the identification and recall of common algebraic identities, graphical representation of their meanings, application in simplifying expressions, and the cultivation of critical thinking and problem-solving skills. The paper's holistic approach is further highlighted by its emphasis on effective communication in mathematics, stressing the importance of utilizing appropriate language. This well-rounded strategy, integrating theoretical understanding, practical skills, and critical thinking, is meticulously designed to optimize learning outcomes in the intricate domain of algebraic identities, offering a comprehensive foundation for students to build upon.

The Ptolemy-Euler Theorem via Reflection

Summary Although there are several generalizations of the Ptolemy-Euler theorem in plane geometry, its higher dimensional cases are still interesting enough to deserve attention. In this note, we give a new proof of the Ptolemy-Euler theorem in R n , via a modified matrix version (reflection transformation) of a classical algebraic identity.

On multiplicative (generalized)- ( α , β ) -derivations in prime rings

UDC 512.5 We discuss some algebraic identities related to multiplicative (generalized)-derivations and multiplicative (generalized)- ( α , β ) -derivations on appropriate subsets in prime rings.

A Real Approximation to Riemann Hypothesis

This work present the requisites to define ζ(s) like an analytic function on real numbers; as well as a numerical evaluation made computationally to obtain criteria of convergence. By the other hand its discusses some algebraic identities to establish where the function zeta Riemann gives place to ζ(s) = 0