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In the development of Theory Radical of Rings, there are two kinds of radical constructions. The first radical construction is the lower radical construction and the second one is the upper radical construction. In fact, the class π of all prime rings forms a special class and the upper radical class of forms a radical class which is called the prime radical. An upper radical class which is generated by a special class of rings is called a special radical class. On the other hand, we also have the class of all semiprime rings which is weakly special class of rings. Moreover, we can construct a special class of modules by using a given special class of rings. This condition motivates the existence of the question how to construct weakly special class modules by using a given weakly special class of rings. This research is a qualitative research. The results of this research are derived from fundamental axioms and properties of radical class of rings especially on special and weakly special radical classes. In this paper, we introduce the notion of a weakly special class of modules, a generalization of the notion on a special class of modules based on the definition of semiprime modules. Furthermore, some properties and examples of weakly special classes of modules are given. The main results of this work are the definition of a weakly special class of modules and their properties.

In this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of p...

The semisimple subalgebras of the rank 2 symplectic Lie algebra sp ( 4 , C ) are well-known, and we recently classified its Levi decomposable subalgebras. In this article, we classify the solvable subalgebras of sp ( 4 , C ) , up to inner automorphism. This completes the classification of the subalgebras of sp ( 4 , C ) . More broadly speaking, in completing the classification of the subalgebras of sp ( 4 , C ) we have completed the classification of the subalgebras of the rank 2 semisimple Lie algebras.

We classify the solvable subalgebras, semisimple subalgebras, and Levi decomposable subalgebras of , up to inner automorphism. By Levi's Theorem, this is a full classification of the subalgebras of .

A classification of the semisimple subalgebras of the Lie algebra of traceless 3 × 3 matrices with complex entries, denoted A 2 , is well-known. We classify its nonsemisimple subalgebras, thus completing the classification of the subalgebras of A 2 .

We examine embeddings of an important class of Levi decomposable algebras into the classical Lie algebras. In particular, we classify the embeddings of A n C n + 1 , B n C 2 n + 1 , C n C 2 n , and D n C 2 n into the complex simple Lie algebras A n + 1 , B n + 1 , C n + 1 , and D n + 1 , respectively, up to inner automorphism.

Let $$\sigma $$ be an automorphism of an arbitrary algebra. In this paper, we introduce the notions of inner and extremal $$\sigma $$ -biderivations and of proper $$\sigma $$ -commuting maps. We prove that (under certain assumptions) every $$\sigma $$ -biderivation of a triangular algebra is the sum of an extremal $$\sigma $$ -biderivation and an inner $$\sigma $$ -biderivation; and provide sufficient conditions on a triangular algebra for all of its $$\sigma $$ -biderivations (respectively, $$\sigma $$ -commuting maps) to be inner (respectively, proper). We introduce and describe a new class of automorphisms of triangular algebras. We provide many classes of triangular algebras whose automorphisms can be determined.

An abelian extension of the special orthogonal Lie algebra D n is a nonsemisimple Lie algebra D n ⨭ V , where V is a finite-dimensional representation of D n , with the understanding that [ V , V ] = 0 . We determine all abelian extensions of D n that may be embedded into the exceptional Lie algebra E n + 1 , n = 5 , 6 , and 7 . We then classify these embeddings, up to inner automorphism. As an application, we also consider the restrictions of irreducible representations of E n + 1 to D n ⨭ V , and discuss which of these restrictions are or are not indecomposable.

It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive pairs of holomorphic representations which are bi-orthogonal duals of each other with respect to a simple Bargmann inner product. It is then shown that the dual of a standard holomorphic representation has an integral expression for its inner product, with a Bargmann measure and a simply-defined kernel, which is not restricted to discrete-series representations. Dual pairs of holomorphic representations also provide practical ways to construct orthonormal bases for unitary irreps which bypass the need for evaluating the integral expressions for their inner products. This leads to practical algorithms for the application of holomorphic representations to model problems with dynamical symmetries in physics.

The Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

The literature contains two different classifications of solvable Lie algebras of dimensions up to and including 4. This paper is devoted to comparing the two classifications and translating each into the other. In particular, we exhibit an isomorphism between each solvable Lie algebra of one classification and the corresponding algebra of the second. The first classification is provided by de Graaf, and the second classification is from a recent book by Snobl and Winternitz.

Many breeding systems have multiple mating, in which males or females mate with multiple partners. With the advent of molecular markers, it is now possible to detect multiple mating in nature. However, no model yet exists to effectively assess the frequency of multiple mating (f(mm))--the proportion of broods with at least two males (or females) genetically contributing--from limited genetic data. We present a single-sex model based on Bayes' rule that incorporates the numbers of loci, alleles, offspring, and genetic parents. Two genetic criteria for calculating f(mm) are considered: the proportion of broods with three or more paternal (or maternal) alleles at any one locus and the total number of haplotypes observed in each brood. The former criterion provides the most precise estimates of f(mm). The model enables the calculation of confidence intervals and allows mutations (or typing errors) to be incorporated into the calculation. Failure to account for mutations can result in overestimates of f(mm). The model can also utilize other biological data, such as behavioral observations during mating, thereby increasing the accuracy of the calculation as compared to previous models. For example, when two sires contribute equally to multiply mated broods, only three loci with five equally common alleles are required to provide estimates of f(mm) with high precision. We demonstrate the model with an example addressing the frequency of multiple paternity in small versus large clutches of the endangered Kemp's Ridley sea turtle (Lepidochelys kempi) and show that females that lay large clutches are more likely to have multiply mated.

The (real) GraviGUT algebra is an extension of the spin(11;3) algebra by a 64- dimensional Lie algebra, but there is some ambiguity in the literature about its definition. Recently, Lisi constructed an embedding of the GraviGUT algebra into the quaternionic real form of E8. We clarify the definition, showing that there is only one possibility, and then prove that the GraviGUT algebra cannot be embedded into any real form of E8. We then modify Lisi's construction to create true Lie algebra embeddings of the extended GraviGUT algebra into E8. We classify these embeddings up to inner automorphism.

We have recently presented models to estimate parentage in breeding systems with multiple mating and incomplete sampling of the candidate parents. Here we provide formulas to calculate the statistical confidence and the optimal trade-off between the number of loci and offspring. These calculations allow an understanding of the statistical significance of the parentage estimates as well as the appropriate sampling regime required to obtain a desired level of confidence. We show that the trade-off generally depends on the parentage of the putative parents. When parentage is low, sampling effort should concentrate on increasing the number of loci. Otherwise, there are similar benefits from increasing the number of loci or offspring. We demonstrate these methods using genetic data from a nest of the bluegill sunfish ( Lepomis macrochirus ).

Many breeding systems include 'multiple mating' in which males or females mate with multiple partners. We identify two forms of multiple mating: 'single-sex', where the next-generation individuals (NGIs) are the product of multiple mating by one sex; and 'two-sex', where the NGIs are the product of multiple mating by both sexes. For both mating systems we develop models that estimate the proportion of NGIs that is fathered (paternity) or mothered (maternity) by the putative parents. The models only require genetic data from the parent or parents in question and the sample of NGIs, as well as an estimate of population allele frequencies. The models provide unbiased estimates, can accommodate loci with many alleles and are robust to violations of their assumptions. They allow researchers to address intractable problems such as the parentage of seeds found on the ground, juvenile fish in a stream, and nestlings in a communal breeding bird. We demonstrate the models using genetic data from a nest of the bluegill sunfish Lepomis macrochirus, where the NGIs may be from multiple females that have spawned with multiple males from different life histories (cuckolder and parental).

We describe the construction of some finite dimensional nonunitary representations of E(2), the Lie group of Euclidean transformations in the plane. Some properties of these representations are also discussed, with emphasis on indecomposable representations.

In this paper we study reducibility of those representations of quasi-split unitary p p -adic groups which are parabolically induced from supercuspidal representations of general linear groups. For a supercuspidal representation associated via Howe’s construction to an admissible character, we show that in many cases a criterion of Goldberg for reducibility of the induced representation reduces to a simple condition on the admissible character.

The symplectic group Sp(n,R) is the group of linear canonical transformations of a real 2n-dimensional phase space and CM(n)⊂Sp(n,R) is a maximal parabolic subgroup. The symplectic groups are the fundamental dynamical groups of classical and quantal Hamiltonian mechanics. In particular, Sp(3,R) is the dynamical group of the spherical harmonic oscillator and its Weil (harmonic series) representations are important for the microscopic (shell model) description of the collective motions of many-particle systems. The subgroup CM(3)⊂Sp(3,R) also appears in the microscopic theory of nuclear collective motion as the dynamical group of a hydrodynamic model of quadrupole vibrations and rotations of a nucleus. Thus, the Sp(3,R)→CM(3) branching rules are needed in finding the embedding of the hydrodynamic collective model in the microscopic shell model. Some new developments are made in the vector-coherent-state theory of induced representations.

The conditional strategy is a theoretical framework that explains the existence within populations of individuals that express alternative behavioral, physical or life history tactics (phenotypes). An example is fighters and sneakers in many animal mating systems. In the conditional strategy the alternative tactics are chosen by individuals based on their state, for example large or small bodied. Since state is often heritable, due for example to additive genetic variance, the alternative tactics may also have inheritance. As the tactics do not have equal fitnesses, it is generally believed that any such inheritance would prevent the evolutionary stability of the conditional strategy. However, in previous work we introduced an Inheritance Theorem and were able to prove that a conditional strategy with tactic inheritances can have a unique equilibrium proportion of the tactics. We now prove a second property of our Inheritance Theorem, namely the stability of the equilibrium. This means that if the tactics are perturbed from their equilibrium proportions, they will return across generations to their equilibrium proportions. An example is provided in mites. We have therefore established an Inheritance Theorem which includes both the existence of an equilibrium and its stability for alternative tactics in a conditional strategy.Copyright 1998 Academic Press Limited

Analytical expressions for the matrices and an explicit algorithm for computing Clebsch-Gordan coupling coefficients are given forsu(4) in au(3)-coupled basis as an example of the construction for anysu(n) in au(n−1) basis. The results areinduced from the known results foru(3) by means of the vector-coherent-state (VCS) theory of induced representations. The important recent result that makes this possible is the discovery that a complete set of shift tensors for the finitedimensional representations of reductive Lie algebras can be induced, by VCS methods, from those of suitably defined subalgebras.

We present a set of tensors which are shift tensors (Wigner tensors) in accordance with the definitions of Biedenharn and Louck and satisfy the coherence conditions of Flath and Towber. Our tensors are defined for all connected compact Lie groups and for finite‐dimensional representations of connected reductive Lie groups. Thus, we have a realization of the coherent tensors in a rather general setting. Moreover, this realization enables us to confirm most of the conjectures of Flath and Towber concerning the properties of coherent tensors.

It is shown that the so‐called shift tensors of Biedenharn and Louck have a natural expression within the framework of vector coherent state (VCS) theory. VCS theory was developed to give a systematic construction of the representations of Lie groups and to derive the corresponding matrices representing the elements of their Lie algebras. In this paper there are explicit realizations of shift tensors within the framework of VCS theory for semisimple Lie groups in terms of shift tensors for suitably defined subgroups.

We investigate a natural generalization of the problem of the description of the eigenvalues of the sum of two Hermitian matrices both of whose eigenvalues are known. We describe more generally the convolution of the invariant probability measures supported on any two adjoint orbits of a compact Lie group. Our techniques utilize the convexity results of Guillemin and Sternberg and Kirwan on the one hand, and the character formulae of Weyl and Kirillov on the other. Applications to representation theory are discussed.

A general definition is given of vector‐coherent state (VCS) representation theory. It is shown that the theory is more general than suggested by previous applications and that it incorporates the standard theories of induced representations as special cases. The associated K‐matrix theory is also given a fuller treatment than hitherto and shown to provide a rather general algorithm both for projecting VCS representations from larger representations in which they are embedded and for determining the Hermitian form, with respect to which an isometric‐equivalent representation is, in fact, isometric.

The semisimple subalgebras of the symplectic algebra C2 are well known. In this article, we classify the Levi decomposable subalgebras of the symplectic algebra C2, up to inner automorphism. By Levi’s theorem, a full classification of the subalgebras of C2 would be complete with a classification of its solvable subalgebras.

It is shown that if H is a Hilbert space for a representation of a group G, then there are triplets of spaces F_H, H, F^H, in which F^H is a space of coherent state or vector coherent state wave functions and F_H is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps F_H -> H -> F^H which facilitates the construction of the corresponding inner products. After completion if necessary, the F_H, H, and F^H, become isomorphic Hilbert spaces. It is shown that the inner product for H is often easier to evaluate in F_H than F^H. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples., 33 pages, RevTex (Latex2.09) This paper is withdrawn because it contained errors that are being corrected

Coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model: (i) classical realizations which are the starting point for geometric quantization; (ii) induced unitary representations corresponding to prequantization; and (iii) irreducible unitary representations obtained in geometric quantization by choice of a polarization. These representations establish an intimate relation between coherent state theory and geometric quantization in the context of induced representations., Comment: 29 pages, part 1 of two papers, published version

We classify the embeddings of the Euclidean algebra e(2) into the rank 2 semisimple, classical Lie algebras. The classifications are up to inner automorphism. We also examine the finite-dimensional, irreducible representations of the rank 2 semisimple, classical Lie algebras restricted to e(2) with respect to various embeddings. All Lie algebras and representations are over the complex numbers.

We develop fractional allocation models and confidence statistics for parentage analysis in mating systems. The models can be used, for example, to estimate the paternities of candidate males when the genetic mother is known or to calculate the parentage of candidate parent pairs when neither is known. The models do not require two implicit assumptions made by previous models, assumptions that are potentially erroneous. First, we provide formulas to calculate the expected parentage, as opposed to using a maximum likelihood algorithm to calculate the most likely parentage. The expected parentage is superior as it does not assume a symmetrical probability distribution of parentage and therefore, unlike the most likely parentage, will be unbiased. Second, we provide a mathematical framework for incorporating additional biological data to estimate the prior probability distribution of parentage. This additional biological data might include behavioral observations during mating or morphological measurements known to correlate with parentage. The value of multiple sources of information is increased accuracy of the estimates. We show that when the prior probability of parentage is known, and the expected parentage is calculated, fractional allocation provides unbiased estimates of the variance in reproductive success, thereby correcting a problem that has previously plagued parentage analyses. We also develop formulas to calculate the confidence interval in the parentage estimates, thus enabling the assessment of precision. These confidence statistics have not previously been available for fractional models. We demonstrate our models with several biological examples based on data from two fish species that we study, coho salmon (Oncorhychus kisutch) and bluegill sunfish (Lepomis macrochirus). In coho, multiple males compete to fertilize a single female's eggs. We show how behavioral observations taken during spawning can be combined with genetic data to provide an accurate calculation of each male's paternity. In bluegill, multiple males and multiple females may mate in a single nest. For a nest, we calculate the fertilization success and the 95% confidence interval of each candidate parent pair.

We classify embeddings of the Poincare algebra p(3,1) into the rank 3 simple Lie algebras. Up to inner automorphism, we show that there are exactly two embeddings of p(3,1) into sl(4,C), which are, however, related by an outer automorphism of sl(4,C). Next, we show that there is a unique embedding of p(3,1) into so(7,C), up to inner automorphism, but no embeddings of p(3,1) into sp(6,C). All embeddings are explicitly described. As an application, we show that each irreducible highest weight module of sl(4,C) (not necessarily finite-dimensional) remains indecomposable when restricted to p(3,1), with respect to any embedding of p(3,1) into sl(4,C).

A proof is presented to show that, when fitnesses from alternative tactics within a population depend on both their frequency and the phenotypic condition of individuals, there will be a unique ESS switchpoint s* that determines both the condition at which an individual will switch between tactics and the resulting frequency of the tactics in the population. For an individual at the ESS s*, the fitnesses of the alternative tactics will be equal. When fitness is averaged over the population, however, the average fitnesses of the alternative tactics will not be equal.

The Diamond Lie algebra D is a four-dimensional, solvable Lie algebra. It is a central extension of the Poincare Lie algebra in two dimensions. We classify the embeddings of D into each of the classical Lie algebras of rank 2, namely, sl(3,C) and sp(4,C), up to conjugation by elements of SL(3,C) or Sp(4,C), respectively. We then show that the irreducible representations of sl(3,C) or sp(4,C) remain indecomposable upon restriction to D under any embedding.

The Euclidean group E(3) is the noncompact, semidirect product group E(3) ≊ SO(3) 3. It is the Lie group of orientation-preserving isometries of 3-dimensional Euclidean space. The Euclidean algebra (3) is the complexification of the Lie algebra of E(3). We embed (3) into the 10-dimensional symplectic algebra (4, ), the simple Lie algebra of type C2. We show that, up to conjugation by an element of Sp(4, ), there is only one embedding of (3) into (4, ), and then prove that the irreducible representations of (4, ) remain indecomposable upon restriction to (3), thus creating a new class of indecomposable (3)-representations.

The Euclidean group E(3) is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. It is the noncompact, semidirect product group E(3)≅SO(3)⋉R3. The Euclidean algebra e(3) is the complexification of the Lie algebra of E(3). We classify the embeddings of the Euclidean algebra e(3) into the simple Lie algebra sl(4,C) and, as an application of this classification, discuss the restriction to various embeddings of e(3) of certain irreducible representations of sl(4,C). In particular, we consider which of these restrictions are e(3)-indecomposable.

In applying symmetry concepts to the description of a physical system, the following situation arises. One is given a Hilbert space ℍ which carries a reducible representation of some dynamical symmetry group G. The observables of the system are expressible as tensors under the group G and the basic problem is to carry out the following steps: (i) Construct a basis for ℍ which reduces G and its relevant subgroups. (ii) Calculate the matrix representations of the observables in this basis. (iii) Diagonalize the Hamiltonian for the system, calculate the observable properties of the eigenstates and compare them with experiment.

Consider a semidirect product group G=H⋉V, where H is reductive and V is a vector group. Two irreps π1 and π2 of H can be “assembled” into a representation of G if it is possible to construct an indecomposable representation Π of G whose restriction to H is π1⊕π2. It is shown that this is equivalent to the existence of a tensor operator from π2 to π1 carrying a representation of H which is equivalent to a nontrivial quotient of the representation which defines the semidirect product. This provides a systematic method for deciding whether two irreps can be assembled, and, if so, in how many inequivalent ways. The method is applied in many of the standard examples that arise in physical questions.

Shalika germs at the identity in p-adic GL(n) will blow up at other singular points. Waldspurger has developed a technique for describing this behaviour at intermediate singular points, giving a «germ expansion for germs». In this paper we discuss which germs occur in such an expansion and fint that the answer is more complicated than expected

For an elliptic torus in SL(n), explicit formulae are given for the germs which are associated to the regular unipotent conjugacy classes. Using them, a formula is found for the germ associated to the "subregular" class, the class whose Jordan canonical form contains a 1 x 1 and an (n - 1) x (n - 1) block.

Let G be a type I separable locally compact group. By studying a representation of G × G × G G \times G \times G we show that a measure class λ \lambda on G × G × G G \times G \times G which describes the decompositions of tensor products is invariant under permutations, and that the multiplicity n ( π 1 , π 2 , π 3 ) n({\pi _1},{\pi _2},{\pi _3}) of π ¯ 3 {\bar \pi _3} in π 1 ⊗ π 2 {\pi _1} \otimes {\pi _2} is a symmetric function of its variables up to a λ \lambda null set.

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.