Properties Of Quotients Research Articles

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the q-series coming from the Hall-Littlewood polynomials, $$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$ These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence $$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$

On some properties of quotients of homogeneous C(K) spaces

On some properties of quotients of homogeneous C(K) spaces

Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

Linear flags and Koszul filtrations

We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Gr\"obner basis with respect to the reverse lexicographic order, and show that this linear quotient property for algebras defined by binomial edge ideals characterizes closed graphs. Furthermore, for algebras defined by binomial edge ideals attached to a closed graph and for join-meet rings attached to a finite distributive lattice we present explicit Koszul filtrations.

Some remarks on representations of Yang-Mills algebras

In this article, we present some new properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine Kac-Moody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of Yang-Mills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the Yang-Mills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.

Dimensions of irreducible modules over W-algebras and Goldie ranks

The main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple Lie algebra. The first one, very classical, is the Goldie rank of an ideal. The second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite W-algebra. We concentrate on the integral central character case. We prove, modulo a conjecture, that in this case the two are equal. Our conjecture asserts that there is a one-dimensional module over the W-algebra with certain additional properties. The conjecture is proved for the classical types. Also, modulo the same conjecture, we compute certain scale factors introduced by Joseph, this allows to compute the Goldie ranks of the algebras of locally finite endomorphisms of simples in the BGG category $$\mathcal {O}$$ . This completes a program of computing Goldie ranks proposed by Joseph in the 80’s (for integral central characters and modulo our conjecture). We also provide an essentially Kazhdan–Lusztig type formula for computing the characters of the irreducibles in the Brundan–Goodwin–Kleshchev category $$\mathcal {O}$$ for a W-algebra again under the assumption that the central character is integral. In particular, this allows to compute the dimensions of the finite dimensional irreducible modules. The formula is based on a certain functor from an appropriate parabolic category $$\mathcal {O}$$ to the W-algebra category $$\mathcal {O}$$ . This functor can be regarded as a generalization of functors previously constructed by Soergel and by Brundan-Kleshchev. We prove a number of properties of this functor including the quotient property and the double centralizer property. We develop several side topics related to our generalized Soergel functor. For example, we discuss its analog for the category of Harish–Chandra modules. We also discuss generalizations to the case of categories $$\mathcal {O}$$ over Dixmier algebras. The most interesting example of this situation comes from the theory of quantum groups: we prove that an algebra that is a mild quotient of Luszitg’s form of a quantum group at a root of unity is a Dixmier algebra. For this we check that the quantum Frobenius epimorphism splits.

Soft congruence relations over rings.

Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft congruence relations by using the soft set theory. The notions of soft quotient rings, generalized soft ideals and generalized soft quotient rings, are introduced, and several related properties are investigated. Also, we obtain a one-to-one correspondence between soft congruence relations and idealistic soft rings and a one-to-one correspondence between soft congruence relations and soft ideals. In particular, the first, second, and third soft isomorphism theorems are established, respectively.

Intersection Theory on Abelian-Quotient V-Surfaces and Q-Resolutions

In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct embedded $\mathbf{Q}$-resolutions of plane curve singularities and abstract $\mathbf{Q}$-resolutions of surfaces.

Existence and properties of geometric quotients

In this paper, we study quotients of groupoids and coarse moduli spaces of stacks in a general setting. Geometric quotients are not always categorical, but we present a natural topological condition under which a geometric quotient is categorical. We also show the existence of geometric quotients of finite flat groupoids and give explicit local descriptions. Exploiting similar methods, we give an easy proof of the existence of quotients of flat groupoids with finite stabilizers. As the proofs do not use Noetherian methods and are valid for general algebraic spaces and algebraic stacks, we obtain a slightly improved version of Keel and Mori’s theorem.

Stability and robustness of [formula omitted]-minimizations with Weibull matrices and redundant dictionaries

We investigate the recovery of almost s-sparse vectors x∈CN from undersampled and inaccurate data y=Ax+e∈Cm by means of minimizing ‖z‖1 subject to the equality constraints Az=y. If m≍sln(N/s) and if Gaussian random matrices A∈Rm×N are used, this equality-constrained ℓ1-minimization is known to be stable with respect to sparsity defects and robust with respect to measurement errors. If m≍sln(N/s) and if Weibull random matrices are used, we prove here that the equality-constrained ℓ1-minimization remains stable and robust. The arguments are based on two key ingredients, namely the robust null space property and the quotient property. The robust null space property relies on a variant of the classical restricted isometry property where the inner norm is replaced by the ℓ1-norm and the outer norm is replaced by a norm comparable to the ℓ2-norm. For the ℓ1-minimization subject to inequality constraints, this yields stability and robustness results that are also valid when considering sparsity relative to a redundant dictionary. As for the quotient property, it relies on lower estimates for the tail probability of sums of independent Weibull random variables.

Individual differences in empathizing and systemizing in Japanese children: Psychometric properties of the children's versions of the Empathy Quotient (EQ) and Systemizing Quotient (SQ)

In the present study, children's versions of the Empathy Quotient (EQ-C) and the Systemizing Quotient (SQ-C) were administered to 626 Japanese children (313 boys, 313 girls) aged 6–15 years. The psychometric properties of both quotients were found to be satisfactory. Sex differences were found in the scores for both quotients. The girls scored higher than the boys on the EQ-C, while the boys scored higher than the girls on the SQ-C. Proportions of cognitive styles, which were determined by differences between the standardized scores of the EQ-C and SQ-C, showed clear sex differences. The proportion of participants whose cognitive style was classified as empathizing-dominant was greater for the girls, while the proportion of participants classified as systemizing-dominant was greater for the boys. No developmental changes were found in either the scores for the EQ-C and SQ-C or the proportions of the cognitive styles.

Compatibility and Schur complements of operators on Hilbert C*-module

Let E be a Hilbert C*-module, and S be an orthogonally complemented closed submodule of E. The authors generalize the definitions of S-complementability and S-compatibility for general (adjointable) operators from Hilbert space to Hilbert C*-module, and discuss the relationship between each other. Several equivalent statements about S-complementability and S-compatibility, and several representations of Schur complements of S-complementable operators (especially, of S-compatible operators and of positive S-compatible operators) on a Hilbert C*-module are obtained. In addition, the quotient property for Schur complements of matrices is generalized to the quotient property for Schur complements of S-complementable operators and S*-complementable operators on a Hilbert C*-module.

Cohomological properties of invariant quotients of affine spaces

We compute the Chow and the cohomology rings of varieties that are obtained as the GIT-quotient of affine spaces by actions of reductive groups. We prove the vanishing of the higher cohomology groups of numerically effective line bundles. As an application we prove a numerical effectiveness result for direct images of numerically effective line bundles.

Projectivity of analytic Hilbert and Kähler quotients

We investigate algebraicity properties of quotients of complex spaces by complex reductive Lie groups G. We obtain a projectivity result for compact momentum map quotients of algebraic G-varieties. Furthermore, we prove equivariant versions of Kodaira's Embedding Theorem and Chow's Theorem relative to an analytic Hilbert quotient. Combining these results we derive an equivariant algebraisation theorem for complex spaces with projective quotient.

On discrete and continuous quotient Riesz spaces

Order properties of quotient Riesz spaces E/N(f) by null ideals N(f) are investigated. We show relationships between properties of a Riesz space E and its order dual E~ and properties of quotients E/N(f) where f runs over some subspaces of E~. A characterization of metrizable locally convex topological Riesz spaces whose all quotients (by proper closed ideals) are discrete is also given.

Quotients of projective spaces and spheres

Abstract In this paper we prove results on topological properties of quotients of manifolds, in particular of projective spaces and spheres. Specifically, under suitable conditions we exhibit results on homology and homotopy groups of these quotients. As an application we generalize the characterization of topological translation planes obtained by Löwen, Steinke, and Van Maldeghem in [Adv. Geom.: S59–S74, 2003] to all possible (finite) dimensions. Our approach also yields a more unified proof for this characterization result. Most of the results in this paper are obtained by means of standard results from algebraic topology.

Network quotients: Structural skeletons of complex systems

A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article we utilize inherent network symmetry to collapse all redundant information from a network, resulting in a coarse graining which we show to carry the essential structural information of the "parent" network. In the context of algebraic combinatorics, this coarse-graining is known as the "quotient." We systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of "real-world" quotients with respect to those of their parent networks. In particular, we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity (heterogeneity and hub vertices) and communication (diameter and mean geodesic distance), suggesting that quotients constitute the essential structural skeletons of their parent networks. We summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms.

Evolution of soil microbial biomass in restoration process of Robinia pseudoacacia plantations in an eroded environment

Vegetation recovery is a key measure to improve ecosystems in the Loess Plateau in China. To understand the evolution of soil microorganisms in forest plantations in the hilly areas of the Loess Plateau, the soil microbial biomass, microbial respiration and physical and chemical properties of the soil of Robinia pseudoacacia plantations were studied. In this study, eight forest soils of different age classes were used to study the evolution of soil microbial biomass, while a farmland and a native forest community of Platycladus orientalis L. were chosen as controls. By measuring soil microbial biomass, metabolic quotient, and physical and chemical properties, it can be concluded that soil quality was improved steadily after planting. Soil microbial biomass of C, N and P (SMBC, SMBN and SMBP) increased significantly after 10 to 15 years of afforestation and vegetation recovery. A relatively stable state of soil microbial biomass was maintained in near-mature or mature plantations. There was an increase of soil microbial biomass appearing at the end of the mature stage. After 50 years of afforestation and vegetation recovery, compared with those in farmland, the soil microbial biomass of C, N and P increased by 213%, 201% and 83% respectively, but only accounting for 51%, 55% and 61% of the increase in P. orientalis forest. Microbial soil respiration was enhanced in the early stages, and then weakened in the later stage after restoration, which was different from the change of soil organic carbon. The metabolic quotient (qCO2) was significantly higher in the soils of the P. orientalis forest than that in farmland at the early restoration stage and then decreased rapidly. After 25 years of afforestation and vegetation recovery, qCO2 in soils of the R. pseudoacacia forest was lower than that in the farmland soil, and reached a minimum after 50 years, which was close to that of the P. orientalis forest. A significant relationship was found among soil microbial biomass, qCO2 and physical and chemical properties and restoration duration. Therefore, we conclude that it is possible to artificially improve the ecological environment and soil quality in the hilly area of the Loess Plateau; a long time, even more than 100 years, is needed to reach the climax of the present natural forest.

NUMBER THEORETIC PROPERTIES OF WRONSKIANS OF ANDREWS–GORDON SERIES

For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews–Gordon q-series [Formula: see text] This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of [Formula: see text] Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if Pb(a;n) denotes the number of partitions of n into parts which are not congruent to 0, ±a ( mod b), then for every positive integer n, we have [Formula: see text] We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k + 1 = p, where p ≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of supersingular j-invariants in characteristic p.

Compactly generated domain theory

We propose compactly generated monotone convergence spaces as a well-behaved topological generalisation of directed-complete partial orders (dcpos). The category of such spaces enjoys the usual properties of categories of ‘predomains’ in denotational semantics. Moreover, such properties are retained if one restricts to spaces with a countable pseudobase in the sense of E. Michael, a fact that permits connections to be made with computability theory, realizability semantics and recent work on the closure properties of topological quotients of countably based spaces (qcb spaces). We compare the standard domain-theoretic constructions of products and function spaces on dcpos with their compactly generated counterparts, showing that these agree in important cases, though not in general.