Unity Element Research Articles

IDEMPOTENT ELEMENTS IN MATRIX RING OF ORDER 2 OVER POLYNOMIAL RING $\mathbb{Z}_{p^2q}[x

An idempotent element in the algebraic structure of a ring is an element that, when multiplied by itself, yields an outcome that remains unchanged and identical to the original element. Any ring with a unity element generally has two idempotent elements, 0 and 1, these particular idempotent elements are commonly referred to as the trivial idempotent elements However, in the case of rings $\mathbb{Z}_n$ and $\mathbb{Z}_n[x]$ it is possible to have non-trivial idempotent elements. In this paper, we will investigate the idempotent elements in the polynomial ring $\mathbb{Z}_{p^2q}[x]$ with $p,q$ different primes. Furthermore, the form and characteristics of non-trivial idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$ will be investigated. The results showed that there are 4 idempotent elements in $\mathbb{Z}_{p^2q}[x]$ and 7 idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$.

A reflection of the Ottoman Empire in the poetry of Ahmad Shawki

Egypt is the most important region of the east-west route, the Mediterranean basin, which opens to the west. Because of its location as a gate, it has drawn the interest of major states in every century and has been the center of attention. Because of Egypt’s strategic importance in the 19th century, France and England pursued colonial interests in the nation. In this country, both European governments’ colonial actions in Egypt, as well as Egypt’s isolation from the Ottoman Empire, have an impact. It also influenced the creation of diverse thinking currents. Among them were Egyptian intellectuals and their views on the caliphate, as well as the formation of distinct understandings of nationality. The influence of Islamism and nationalism was deeply felt on Egypt and the intellectuals living in this country. We picked Ahmad Shawki, one of Egypt’s thinkers, since he is an Arab of Turkish descent who dedicated himself to the Ottoman empire during the time of Abdul-Hamid II. The notion of nationalism was seen in Egypt in a variety of ways throughout this time. “Egyptian Nationalism Related to the Ottoman Caliphate” was Ahmad Shawki’s definition of nationalism. Because he is devoted to the caliphate, he has the notion of Islamic Union; Ahmad Shawki is a nationalist thinker who loves Egypt, and the idea of Islamic unity and nationalist element is the major focus of this research. &nbsp

Near-Modul Kuat Faktor

Near-ring is a generalization of ring. In ring theory, let R be a ring over addition and multiplication operation with unity element and G be a commutative group under addition operation. Then G together with scalar multiplication of R and holds several axioms called module over ring. Let N be a near-ring, here we can construct the module over near-ring called near-module. We have three definitions of module over near-ring, such that, near-module, modified near-module, and strong near-module. Then we showed that strong near-module can be constructed into strong near-module factor as well as module factor in G. Furthermore, we generalized the fundamental theorem of homomorphism in module over ring to strong near-module over strong near-ring.

ON CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS

The Yangian Y (𝔤) of a simple Lie algebra 𝔤 can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra of the current algebra \( U\left(\mathfrak{g}\left[t\right]\right) \) and the coordinate ring of the first congruence subgroup \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right). \) Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Y (𝔤).Bethe subalgebras B(C) in Y (𝔤) form a natural family of commutative subalgebras depending on a group element C of the adjoint group G. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain.We describe the associated graded of Bethe subalgebras in the Yangian Y (𝔤) of a simple Lie algebra 𝔤 as subalgebras in \( U\left(\mathfrak{g}\left[t\right]\right) \) and in \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right) \) for all semisimple C ∈ G. In particular, we show that the associated graded in \( U\left(\mathfrak{g}\left[t\right]\right) \) of the Bethe subalgebra B(E) assigned to the unity element of G is the universal Gaudin subalgebra of \( U\left(\mathfrak{g}\left[t\right]\right) \) obtained from the center of the corresponding affine Kac–Moody algebra \( \hat{\mathfrak{g}} \) at the critical level. This generalizes Talalaev’s formula for generators of the universal Gaudin subalgebra to 𝔤 of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin–Frenkel center at the critical level.Using our general result on the associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple C ∈ G as C goes to an irregular semisimple group element C0. We show that this limit is the product of the smaller Bethe subalgebra B(C0) and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of C0 in 𝔤. This generalizes the Nazarov–Olshansky solution of Vinberg’s problem on quantization of the (Mishchenko–Fomenko) shift of argument subalgebras.

A Partition of unity finite element method for valuation American option under Black-Scholes model

Abstract In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.

Memorias de una Colombia dividida

La herencia colonial está presente en la configuración jurídica e institucional de los estados independizados. Este fenómeno es especialmente perceptible en América Latina donde lo hispano ha sido empleado como elemento de identificación, pertenencia y unión nacional. La Constitución colombiana de 1886 es una muestra de la fuerte influencia de las instituciones coloniales españolas y del especial rol que la religión católica ha jugado en Latinoamérica.

ROLE OF WALI, ANCIENT MOSQUE AND SACRED TOMB IN ISLAM SPREADING DYNAMICS IN CIKAKAK

AbstractThis qualitative research was conducted in the Islamic communities who base all its activities on Aboge calendar calculations in the village of Cikakak,Wangon, District of Banyumas. Data collected by interview, observation, and documentation. This study found an association between the role of the Wali, the ancient mosque and shrine in the spread of Islam in the village Cikakak. Jaro Rajab ritual or grave pilgrimage or nyadran held every 26th of Rajab as a symbol of respect to KyaI Mustolih who spread Islam in the 17th century. Kyai Mustolih built Saka Tunggal Mosque in 1288 AD. Social practices of mutual assistance and cooperation is reflected in Jaro Rajab ritual performed in the tomb and mosque complex. Jaro Rajab ritual functioning as a unity element for communities and faiths.

Deformation quantization and boundary value problems

We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.

Microstructural enrichment functions based on stochastic Wang tilings

This paper presents an approach to constructing microstructural enrichment functions to local fields in non-periodic heterogeneous materials with applications in the partition of unity and hybrid finite element schemes. It is based on a concept of aperiodic tilings by the Wang tiles, designed to produce microstructures morphologically similar to original media and enrichment functions that satisfy the underlying governing equations. An appealing feature of this approach is that the enrichment functions are defined only on a small set of square tiles and extended to larger domains by an inexpensive stochastic tiling algorithm in a non-periodic manner. The feasibility of the proposed methodology is demonstrated on constructions of stress enrichment functions for two-dimensional mono-disperse particulate media.

A Ring Construction Using Finite Directed Graphs

Abstarct In this paper we discuss an interesting class of noncommutative rings which can be constructed using finite directed graphs. This construction also creates a vector space. These structures provide undergraduate students connections between ring theory and graph theory and, among other things, allow them to see a ring unity element that looks quite different from the ones they are accustomed to using.

Because Judges Are Not Angels Either:1 Limiting Judicial Discretion by Introducing Objectivity into Piercing Doctrine

Ordinarily, individual shareholders are immune from liability arising from a corporation’s activities through the doctrine of limited liability. That is, absent a personal breach of duty either in contract or in tort, an individual shareholder is only financially exposed to judgments against, or debts of, the corporation up to the shareholder’s investment. All rules, of course, have exceptions. The most frequently litigated of those exceptions is the doctrine of piercing the corporate veil. Notwithstanding the frequent litigation surrounding the doctrine and the “bright-line rules” courts have created to cabin the doctrine, disparate results frequently occur both internally within a jurisdiction and externally between jurisdictions, as, in most cases, the courts use a factually specific factors analysis to determine when piercing is proper. These disparate results have led to concerns by corporate scholars, as the lack of predictability and consistency in the doctrine’s application is concerning as a matter of justice, as a matter of logical consistency, and, perhaps most practically, as a matter of counseling clients to avoid this unruly beast. This Article argues that piercing doctrine could be made predictable and consistent through adoption of a conjunctive test requiring objective criteria for the elements of unity, injustice, causation, and insolvency. Part II of this Article briefly discusses the historical roots of the piercing doctrine, indicating its origination and the typical categorizations and classifications of its various incarnations. This section further reviews each state’s classifications of piercing doctrine, ultimately finding the traditional classifications are generally lip-service used to mask an unguided, and inconsistently applied, factors test. Part III of this Article discusses four remedies other scholars have advocated to alleviate the doctrine’s predictability concerns. This Part further discusses the deficiencies of each of those proposals. Finally, Part IV argues the doctrine can be cabined through the introduction of a four-part conjunctive test. First, the proposed test requires an injustice prong, compelling plaintiffs to demonstrate the corporation engaged in fraud, engaged in misrepresentation, or was undercapitalized. Special emphasis is provided in this section to the element of undercapitalization and a novel method of generating an objective test for determining undercapitalization. Ultimately, the Article advocates adopting the tort doctrine of custom as a mechanism to contain the discretion otherwise rampant in undercapitalization determinations. Second, the proposed test requires an objective unity element that is premised upon control over the decision-making process. This element dispenses with the notion of seeking to determine which shareholder has corporate unity and instead focuses piercing analysis on which shareholder(s) have unity with the decision. Third, the proposed test requires a causal element, requiring plaintiffs to demonstrate the inequitable conduct gave rise to the plaintiff’s harm. This section incorporates but-for causation and proximate causation into the piercing test. Finally, the proposed test requires an insolvency prong, requiring plaintiffs to prove the corporation is insolvent to pay the plaintiff’s debt.

Structures and Representations of Generalized Path Algebras

It is shown that an algebra $\Lambda $ can be lifted with nilpotent Jacobson radical $r = r(\Lambda)$ and has a generalized matrix unit $\{e_{ii}\}_I$ with each $\bar e_{ii} $ in the center of $\bar \Lambda = \Lambda /r$ iff $\Lambda $ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, $\Lambda $ is a finite algebra with non-zero unity element over perfect field $k$ (e.g. a field with characteristic zero or a finite field) iff $\Lambda $ is isomorphic to a generalized path algebra $k (D, \Omega, \rho)$ of finite directed graph with weak relations and $dim {\} \Omega < \infty $; $\Lambda $ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents iff $\Lambda $ is isomorphic to a path algebra with relations.

Group Homomorphism Generated Near-Rings and Rings: A Unity Not Fixing Each Element of the Group

Let (G, +) be a group, not necessarily abelian, and let K be a nontrivial subgroup of G. Let ℋ = ℋ(G, K) be the additive group generated by Hom (G, K). Then (ℋ(G, K), +, ○) is a d.g. near-ring. If K ≠ G, then ℋ(G, K) cannot contain the unity element of ℰ(G), the near-ring generated by End G. Surprisingly, examples exist which show it may indeed have a two-sided unity element. Conditions are developed involving G and K¯, the additive subgroup generated by ∪ {h(G) : h ∈ ℋ}, which characterize when ℋ(G, K) contains a one-sided or two-sided unity element. The cases when K¯ is abelian or an E-group are considered. As a consequence of this theory, connections between ℰ(G) and ℰ(K¯), via ℋ(G, K), are established. Numerous illustrative examples are given.

LOCALLY CONVEX SPACES OF VECTOR-VALUED DISTRIBUTIONS WITH MULTIPLICATIVE STRUCTURES

We construct an infinite-dimensional linear space [Formula: see text] of vector-valued distributions (generalized functions), or sequences, f*(x)=(fn(x)) finite from the left (i.e. fn(x)=0 for n&lt;n0(f*)) whose components fn(x) belong to the linear span [Formula: see text] generated by the distributions δ(m-1)(x-ck), P((x-ck)-m), xm-1, where m=1, 2, …, ck ∈ ℝ, k = 1, …, s. The space of distributions [Formula: see text] can be realized as a subspace in [Formula: see text] This linear space [Formula: see text] has the structure of an associative and commutative algebra containing a unity element and free of zero divizors. The Schwartz counterexample does not hold in the algebra [Formula: see text]. Unlike the Colombeau algebra, whose elements have no explicit functional interpretation, elements of the algebra [Formula: see text] are infinite-dimensional Schwartz vector-valued distributions. This construction can be considered as a next step and a "model" on the way of constructing a nonlinear theory of distributions similar to that developed by L. Schwartz. The obtained results can be considerably generalized.

Unitized Jordan algebras and dispersionless KdV equations

Multicomponent KdV systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied. Recursion laws for conserved densities are given under the assumption that the algebra possesses a unity element. Sufficient conditions are given for the unitized counterpart of a diagonalizable non-unital system to be diagonalizable. Hamiltonian structure is discussed within the context of DN Jordan algebras and N scattering problems.

Jordan Manifolds and Dispersionless KdV Equations

Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied, under the assumptions that the Jordan algebra has a unity element and a compatible non-degenerate inner product. Much of this structure may be encoded in a so-called Jordan manifold, akin to a Frobenius manifold. In particular the Hamiltonian properties of these systems are investigated.

Unbiased Separation and Enhancement of Sources by Using a Zero Tracking Algorithm with Post Processing Adaptive Noise Reduction

This paper proposes and investigates a new structure for the unbiased separation, tracking, and enhancement of uncorrelated sources by a linear array. The structure consists of a matrix preprocessing beamformer followed by an adaptive linear combiner and a postprocessing beamformer. The weights of the adaptive linear combiner are updated by using the LMS algorithm and, together with the preprocessor, implement an adaptive array whose response zeros are adjusted to minimize the output power with unity element weight vector norm constraint. By properly designing the preprocessor, first-order movements in the response zeros can be individually and proportionally controlled by first order changes in the weights in the linear combiner, and the preprocessor outputs will be due to individual sources and receiver noise in the steady state. By measuring the powers of the preprocessor outputs, it will be possible to determine if a particular preprocessor output is due to an actual source (if the associated zero is tracking a source) or merely receiver noise (if the associated zero is not tracking any source). Further SNR enhancement is then carried out by making use of those preprocessor outputs due to receiver noise to remove the correlated receiver noise components in the other preprocessor outputs tracking actual sources through the use of a matrix postprocessor. Apart from the implementation of the preprocessor and postprocessor which will be inevitable in any source separation system and which have to be designed only occasionally and whenever the response zeros have been found to have changed significantly, the algorithm has an implementation complexity which is proportional to the array size. However, as the preprocessor outputs are due primarily to individual sources in the steady state, the algorithm has only one single asymptotic time constant controlled by the designed misadjustment level and has fast tracking performance.

G-algebras, Clifford theory, and the Green correspondence

Let G be a finite group with subgroup H. Loosely speaking, (as developed by E. C. Dade and others) is a tool for constructing reduction arguments in the case where H is normal in G. The Green correspondence is a tool for constructing reduction arguments in the case where H contains the normaliser in G of a given p-subgroup, or more generally, where H contains the normaliser of a given local pointed group. Here, we cover both of these situations by considering the case where No(H) contains the normaliser in G of a given local pointed group. A brief discussion in [1] indicates how Puig's theory subsumes the Green correspondence and aspects of Clifford theory. Here, this theme is given a more systematic treatment. We shall see, in particular, that defect pointed subgroups, the Green correspondence, and Brauer's extended first main theorem are all manifestations of a more general mechanism. Let @ be a complete local commutative noetherian ring with an algebraically closed residue field k of prime characteristic p. Throughout, we fix a G-algebra A. (All algebras are deemed to possess a unity element; all algebras and modules are to be free over ~, or over k, and of finite rank.) For the rest of this section, let us fix points /3, a of H, G, respectively, on A, and suppose that Ht3, G,~ have a common defect pointed subgroup Pr such that Nc(P ~) <_ NG(H). It is easy to show--see Section 4 t h a t if P~. is another defect pointed subgroup of Ht3, then P~, is also a defect pointed subgroup of G,~, and NG(P~,) < NG(H). It t ranspires--again see Section 4 t h a t Ht3 _< G,~ if and only if a c TrGH(AH/3AH); when these equivalent

Distributive monoid algebras

All rings are assumed to be associative with a nonzero unity element, and modulles and subrings are assumed to be unital. A distributive module is a module with a distributive lattice of submodules. Right distributive monoid rings were studied in [1-6] under !various additional conditions. A monoid (ring) G is called regular if for any element f there exists an element g such that f = fgf. Our main results are Theorems 1 and 2.

A theorem of generalized Cauchy-Pompeiu type on finite-dimensional associative algebras

The author takes up the following problem: suppose given a class of regular hypercomplex functions, i.e., with domains and ranges in a finite-dimensional associative algebra (over the reals) with unity element (so that it is not, in general, a division algebra). Does there nevertheless exist for such functions an integral formula of Cauchy's type, or perhaps of Cauchy-Pompeiu tipe? The author obtains a formula of Cauchy-Pompeiu type (wich, in the commutative case, is similar to a Green identity).