Mackey's Theory Research Articles

Analisis Buku Ajar Bahasa Arab: Telaah Buku Arabiyyah Linnasyiin Dalam Perspektif Teori William Frances Mackey

Interesting to study is the Arabic textbook Arabiyyah Linnasyiin This book will be further examined using William Francis Mackey's theory, which explains that a good textbook must fulfill three aspects: selection, gradation, presentation, and repetition. To achieve maximum research outcomes, this study employs a descriptive analysis method with a library research approach, where data sources are gathered from library materials such as books, journals, seminar proceedings, newspapers, and similar resources. The results of this study show that the Arabiyyah Linnasyiin textbook consists of four language skills: speaking (Muhadatsah), reading (maharah Qiroah), writing (maharah Kitabah), and grammatical analysis in the form of exercises (Ta'bir Syafahiyah). When viewed from the level of student learning, Arabiyyah li Nasyiin has six levels, consisting of Volume I, Volume II, Volume III, Volume IV, Volume V, and Volume VI. These levels are based on learning Arabic at the beginner, intermediate, and advanced levels. This book includes indicators and basic competency standards, though there is no set time limit. Secondly, in the Arabiyyah Linnasyiin textbooks, from the first to the sixth volume, the levels are categorized by themes, vocabulary, grammatical analysis, and the exercises provided. Thirdly, the Arabiyyah Linnasyiin textbook is comprehensive in terms of covering the four aspects of Arabic language skills, with content that includes maharah kalam, maharah qiroah, maharah kitabah, and grammatical analysis in the form of exercises. In terms of procedure, the Arabiyyah Linnasyiin textbook uses a pictorial procedure, a teaching method that employs images. Fourth, regarding speaking, from Volume I to Volume VI, the speaking-related material is extensive and complete, corresponding to the levels and themes provided. For reading, the reading themes maharah qiroah in Volumes I to VI are dense and thorough, appropriate to the students' level of proficiency. In terms of writing, the writing-related material in Arabiyyah Linnasyiin is only provided in Volume I, and this is tailored to the students' proficiency level in learning Arabic.

Analisis penyajian materi kitab An-Nahwu Al-Wadhih berdasarkan teori mackey

Analysis of material presentation is an activity in determining the validity and suitability of a textbook for educational institutions. In essence, textbooks are one of the factors that support the success of learning. However, another fact was found that textbooks can also be a factor causing poor learning progress. This study aims to provide an overview for educational institutions related to the textbooks studied which can be used as a reference for selecting textbooks. This research uses a qualitative approach with descriptive research type. The object of research is the book Nahwu Wadhih. Data collection techniques were carried out using library research and documentation. Data analysis was carried out using content analysis techniques. Based on the results of this study, from the selection aspect of the book Nahwu Wadhih it aims to familiarize beginners in learning Arabic through an effective model, it is common for educational institutions in Indonesia to complete learning the book Nahwu Wadih every volume for 1 academic year. Judging from the aspect of gradation, the book Nahwu Wadhih uses Rotary Gradation, the type of linguistic gradation is grammatical gradation. Judging from the presentation aspect, this book can be taught by teachers using all existing procedures. Judging from the repetition aspect, this book contains various tests in each chapter. Selection of inappropriate textbooks can cause difficulties for students and teachers. This study has limitations in the literature review because there are few studies using the same subject. For future researchers, it is advisable to carry out an analysis based on other theories as well because there is a possibility that this problem will change. Keywords: textbooks, nahwu wadhih, mackey theory

Content Analysis Kitab “Nubdzatul Bayan” Sebagai Buku Ajar Akselerasi Maharah Qiraah Di Ponpes Nurul Jadid

The research departs from the existence of the accelerated book 'Nubdzatul Bayan' which is used as an acceleration book for Qiraah proficiency at the Nurul Jadid Situbondo Islamic boarding school. This book with 6 volumes of Qawaid Arabic science content has made several Islamic boarding school students able to read Arabic texts in a short time. This research will focus on dissecting the contents of the six volumes of this book with the analysis knife of Willian Francis Mackey's learning theory with four principles, namely Selection, Gradation, Presentation, and Presentation. The research method that the writer uses is literature studies with the primary data being the book. Secondary data was obtained from several research journals and books. The conclusion from the content analysis of this textbook provides information about the purpose of the textbook, as well as the use of this Nubdzatul Bayan book in accordance with its learning outcomes, namely as a Qiraah proficiency acceleration book. The material presented helps determine the relevance of the textbook with the desired learning objectives. This study also provides information about the quality of the presentation of material that is systematic, clear, and easily understood by students. In conclusion, the content analysis of this textbook provides useful information for evaluating the effectiveness and efficiency of textbooks as learning aids. Thus, the analysis of the contents of textbooks becomes important in selecting and using the appropriate textbooks in accordance with the goals and needs of the students' Qiraah proficiency acceleration

Review and concrete description of the irreducible unitary representations of the universal cover of the complexified Poincaré group

In this paper, we give a pedagogical presentation of the irreducible unitary representations of [Formula: see text], that is, of the universal cover of the complexified Poincaré group [Formula: see text]. These representations were first investigated by Roffman in 1967. We provide a modern formulation of his results together with some facts from the general Wigner–Mackey theory which are relevant in this context. Moreover, we discuss different ways to realize these representations and, in the case of a non-zero “complex mass”, we give a detailed construction of a more explicit realization. This explicit realization parallels and extends the one used in the classical Wigner case of [Formula: see text]. Our analysis is motivated by the interest in the Euclidean formulation of Fermionic theories.

ANALISIS BUKU BAHASA ARAB KELAS 10 DIREKTORAT KURIKULUM SARANA KELEMBAGAAN DAN KESISWAAN (KSKK) TINJAUAN KONTEN BERDASARKAN TEORI MACKEY

<p style="text-align: justify;"><em>of KSKK (content review based on Mackey theory) aims to determine the feasibility of textbooks based on Mackey's theory so that teaching materials can be used in learning. This article uses a qualitative approach with a library research model. As for the data collection is to use documentation and data analysis techniques using content analysis. In analyzing teaching materials, Mackey's theory provides a good overview of aspects that need to be considered in analyzing teaching materials, including 1) selection; 2) gradation; 3) presentation; 4) reps. From these four aspects, it can be concluded that the aspect of the selection of textbooks is still general, but the material's content tends to be more complex. In contrast, from the aspect of gradation, it has fulfilled the aspects of grouping and grading. As for the presentation aspect, it has fulfilled the differential, potential, pictorial, and context procedures. Meanwhile, the repetition aspect has been fulfilled in the form of various evaluations.</em></p>

ANALISIS MATERI NAHWU DALAM KITAB AUDLOHUL MANAHIJ FI MU’JAM QOWA’IDUL LUGHOH AL ‘AROBIYAH

This article is aim to analyze the appropriated of Nahwu Book to educated someone who want to learn arabic language easily. This book created by Agus Shohib Khoironi, Lc in 2001 by using Mustaqilli method for the teaching arabic language. In analizing this book, this research is a library research then analyzing by nahwu material analysis, in the other side, this research also analyzing with conducting four principles of William F. Mackey’s theory which includes: repetition, presentation, gradation, and selection. From the results of the analysis of the nahwu material in the book Audhohul Manahij Fi Mu'jam Qowaidul Lughoh Al-Arabiyah, it was found that the preparation of this book was in accordance with Mackey's theory of gradation, selection, presentation and repetition. The nahwu material is arranged in a coherent way from a discussion of the basic nahwu qoidah to more depth, starting with the kalam chapter to the Jama 'Taksir chapter in the first part of this book, which consists of 14 discussion chapters.

Analysis of Arabic Textbook "Takallam Bil 'Arabiyyah Volume 6" Based on Mickey's Theory

Textbooks have an important role in achieving learning success. learning will run effectively if the teaching materials used are in accordance with the learning objectives. it is necessary to do an in-depth study of the textbooks used in schools. This study aims to analyze the textbook "Takallam Bil 'Arabiyyah Volume 6" which is used in Arabic learning based on Mackey's theory. This study uses the literature study method. Sources of data were taken from articles, books, research reports, and textbooks "Takallam Bil 'Arabiyyah Volume 6" with data analysis techniques using descriptive analysis. Finally, the data is presented using the deductive method, which describes general phenomena to the findings of the analysis. This research reveals that the textbook "Takallam Bil 'Arabiyyah Volume 6" was compiled based on the theory of William Francis Mackey. This can be seen from the presentation of the material which contains the concepts of selection, gradation, presentation, and repetition. The development of the textbook "Takallam Bil 'Arabiyyah Volume 6" needs to be done by adding more factual material for student life in the 21st-century era.

Multiplicity formulas for representations of transformation groupoids

Abstract We study the representations of transitive transformation groupoids with the aim of generalizing the Mackey theory. Using the Mackey theory and a bijective correspondence between the imprimitivity systems and the representations of a transformation groupoid we derive the irreducibility theory. Then we derive the direct sum decomposition for representations of a groupoid together with the formula for the multiplicity of subrepresentations. We discuss a physical interpretation of this formula. Finally, we prove the claim analogous to the Peter-Weyl theorem for a noncompact transformation groupoid. We show that the representation theory of a transitive transformation groupoids is closely related to the representation theory of a compact groups.

Factorial Unitary Representations of the Translational Group, Invariant Pure States and the Super Symmetric Graviton

Unitary Representations corresponding to local shifts in reference frames are a key topic for progress in Quantum Gravity. The fibre bundle construct defined in our previous work in which quantum fields become liftings of; or sections through; a fibre bundle with base space curved space-time, continues to be the context for this paper. We investigate in more depth the subgroup T of the Poincare group consisting of translations of space-time as a gauge group of automorphisms. We define a representation of T as a group of automorphisms of the local fibre algebra A(x) which we assume to be isomorphic to a von Neumann algebra with trivial centre acting on a separable Hilbert space. Provided this group representation α is weakly measurable, then we have previously proved that it is also norm continuous, and is implemented by a norm, hence weakly and strongly, continuous unitary representation, by unitaries in the algebra. We discuss the dependence of this key result on the strange properties of Stonean spaces and extend our exploitation of Mackey Theory to give a more straightforward group theoretic proof. From such unitary representations a minimal form of Supersymmetry naturally emerges, which predicts the existence of both the mass zero graviton, and its gravitino partner.

A Commentary on Single-photon Wave Function Advocated by Bia\l ynicki-Birula

We present in this paper how the single-photon wave function for transversal photons (with the direct sum of ordinary unitary representations of helicity 1 and -1 acting on it) is subsumed within the formalism of Gupta-Bleuler for the quantized free electromagnetic field. Rigorous Gupta-Bleuler quantization of the free electromagnetic field is based on our generalization (published formerly) of the Mackey theory of induced representations which includes representations preserving the indefinite Krein inner-product given by the Gupta-Bleuler operator. In particular it follows that the results of Bia{\l}ynicki-Birula on the single-photon wave function may be reconciled with the causal perturbative approach to QED.

Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups

Let \({\mathbb{G}}\) denote a higher-rank \({\mathbb{R}}\)-split simple Lie group of the following type: \({SL(n,\mathbb{R})}\), SOo(m, m), E6(6), E7(7) and E8(8), m ≥ 4. For any unitary representation \({(\pi,\mathcal{H})}\) of \({\mathbb{G}}\) without non-trivial \({\mathbb{G}}\)-invariant vectors, we study smooth solutions of the cohomological equation \({\mathfrak{u}f=g}\), where \({\mathfrak{u}}\) is a vector in the root space of \({\mathfrak{G}}\), the Lie algebra of \({\mathbb{G}}\) and g is a given vector in \({\mathcal{H}}\). We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for f. We also study common solutions to (the infinitesimal version of) the cocycle equation \({\mathfrak{u}h=\mathfrak{v}g}\), where \({\mathfrak{u}}\) and \({\mathfrak{v}}\) are commutative vectors in different root spaces of \({\mathfrak{G}}\) and g and h are given vectors in \({\mathcal{H}}\). We give the sufficient condition under which the cocycle equation has a common solution: (∗) if \({\mathfrak{u}}\) and \({\mathfrak{v}}\) embed in \({\mathfrak{sl}(2,\mathbb{R})\times \mathbb{R}}\), then the common solution exists. In fact, condition (∗) is also necessary when \({\mathbb{G}=SL(n,\mathbb{R})}\). We show counter examples in each \({SL(n,\mathbb{R})}\), n ≥ 3 if condition (∗) is not satisfied. Especially, the cocycle rigidity always fails for \({SL(3,\mathbb{R})}\). As an application, we obtain smooth cocycle rigidity for higher rank parabolic actions over \({\mathbb{G}/\Gamma}\) if the Lie algebra of the acting parabolic subgroup contains a pair \({\mathfrak{u}}\) and \({\mathfrak{v}}\) satisfying property (∗). The main new ingredient in the proof is making use of unitary duals of \({SL(2,\mathbb{R})\ltimes\mathbb{R}^2}\) and \({(SL(2,\mathbb{R})\ltimes\mathbb{R}^2)\ltimes\mathbb{R}^3}\) obtained by Mackey theory.

Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks

A counterpart of the Mackey–Arens Theorem for the class of locally quasi-convex topological Abelian groups (LQC-groups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasi-convex groups, stem from the following fact: The supremum of all compatible locally quasi-convex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasi-convex compact subsets of the dual group $$G^\wedge $$ . Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQC-groups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Diaz Nieto and Martin-Peinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasi-convex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasi-convex groups. We point out some general questions which are still open, under the name of Problems.

Primary spaces, Mackey’s obstruction, and the generalized barycentric decomposition

We call a hamiltonian N-space \emph{primary} if its moment map is onto a single coadjoint orbit. The question has long been open whether such spaces always split as (homogeneous) x (trivial), as an analogy with representation theory might suggest. For instance, Souriau's \emph{barycentric decomposition theorem} asserts just this when N is a Heisenberg group. For general N, we give explicit examples which do not split, and show instead that primary spaces are always flat bundles over the coadjoint orbit. This provides the missing piece for a full "Mackey theory" of hamiltonian G-spaces, where G is an overgroup in which N is normal.

Group valued null sequences and metrizable non-Mackey groups

Abstract. For a topological abelian group X we topologize the group c 0 ( X ) $c_0(X)$ of all X-valued null sequences in a way such that when X = ℝ ${X={\mathbb {R}}}$ the topology of c 0 ( ℝ ) $c_0({\mathbb {R}})$ coincides with the usual Banach space topology of the classical Banach space c0 . If X is a non-trivial compact connected metrizable group, we prove that c 0 ( X ) $c_0(X)$ is a non-compact Polish locally quasi-convex group with countable dual group c 0 ( X ) ∧ $c_0(X)^{\wedge }$ . Surprisingly, for a compact metrizable X, countability of c 0 ( X ) ∧ $c_0(X)^{\wedge }$ leads to connectedness of X. Our principal application of the above results is to the class of locally quasi-convex Mackey groups ( LQC $\rm {LQC}$ -Mackey groups). A topological group ( G , μ ) $(G,\mu )$ from a class 𝒢 $\mathcal {G}$ of topological abelian groups will be called a Mackey group in 𝒢 $\mathcal {G}$ or a 𝒢 $\mathcal {G}$ -Mackey group if it has the following property: if ν is a group topology in G such that ( G , ν ) ∈ 𝒢 ${(G,\nu )\in \mathcal {G}}$ and ( G , ν ) $(G,\nu )$ has the same character group as ( G , μ ) $(G,\mu )$ , then ν ≤ μ ${\nu \le \mu }$ . Based upon the results obtained for c 0 ( X ) $c_0(X)$ , we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC $\rm {LQC}$ -Mackey. Namely, we show that for a connected compact metrizable group X ≠ { 0 } ${X\ne \lbrace 0\rbrace }$ , the group c 0 ( X ) $c_0(X)$ , endowed with the topology induced from the product topology on X ℕ $X^{{\mathbb {N}}}$ , is a metrizable precompact connected group which is not a Mackey group in LQC. Since metrizable locally convex spaces always carry the Mackey topology – a well-known fact from Functional Analysis –, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role.

Unbounded Induced Representations of ∗-Algebras

Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra \({{\mathcal{A}}}\) onto a unital *-subalgebra \({{\mathcal{B}}}\) are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras \({{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g\) for which the *-subalgebra \({{\mathcal{B}}}:={{\mathcal{A}}}_e\) is commutative. Then the canonical projection \(p:{{\mathcal{A}}}\to{{\mathcal{B}}}\) is a conditional expectation and there is a partial action of the group G on the set \({{\mathcal{B}}}p\) of all characters of \({{\mathcal{B}}}\) which are nonnegative on the cone \(\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.\) The complete Mackey theory is developed for *-representations of \({{\mathcal{A}}}\) which are induced from characters of \({{\widehat{{{\mathcal{B}}}}^+}}.\) Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras \({{\mathcal{A}}}\) is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.

Induced representations and Mackey theory

This is an introduction to the theory of induced representations of finite groups with emphasis on the contributions of George Mackey. Some applications to the representation theory of the symmetric group are also presented.

Hausdorff–Young Inequalities for Group Extensions

AbstractThis paper studies Hausdorff–Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies in particular to yield a Hausdorff–Young inequality for nonunimodular groups.

Mackey Theory for p–adic Lie Groups

This paper gives a p–adic analogue of the Mackey theory, which relates representations of a group of type G = H × t A to systems of imprimitivity.

Clifford and Mackey theory for Weil representations of symplectic groups

Clifford and Mackey theory for Weil representations of symplectic groups

Representations of the canonical group, (the semidirect product of the unitary and Weyl$ndash$Heisenberg groups), acting as a dynamical group on noncommutative extended phase space

The unitary irreducible representations of the covering group of the Poincare group define the framework for much of particle physics on the physical Minkowski space = /, where is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a (3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically {t, e, qi, pi} → {t, e, pi, −qi} where {t, e, qi, pi} are the time, energy, position and momentum degrees of freedom. This implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b, where b is a universal constant. The appropriate group of dynamical symmetries that embodies this is the canonical group (1, 3) = (1, 3) ⊗s (1, 3) = (1, 3) ⊗s (1, 3) and in this theory the non-commuting space = (1, 3)/(1, 3) is the physical quantum space endowed with a metric that is the second Casimir invariant of the canonical group, T2 + E2/c2b2 − Q2/c2 − P2/b2 + 2I/bc (Y/bc − 2) where {T, E, Qi, Pi, I, Y} are the generators of the algebra of (1, 3) = (1) ⊗s (1, 3). The idea is to study the representations of the canonical dynamical group using Mackey's theory to determine whether the representations can encompass the spectrum of particle states. The unitary irreducible representations of the canonical group contain a direct product term that is a representation of (1, 3) that Kalman has studied as a dynamical group for hadrons. The (1, 3) representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of (3) (finite dimensional) or (2) (with degenerate (1) ⊗ (2) finite-dimensional representations) corresponding to the rest or null frames.