Modular Case Research Articles

Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions

Using the rings of Lipschitz and Hurwitz integers $$\mathbb {H}(\mathbb {Z})$$ and $$\mathbb {H}ur(\mathbb {Z})$$ in the quaternion division algebra $$\mathbb {H}$$ , we define several Kleinian discrete subgroups of $$PSL(2,\mathbb {H})$$ . We define first a Kleinian subgroup $$PSL(2,\mathfrak {L})$$ of $$PSL(2,\mathbb {H}(\mathbb {Z}))$$ . This group is a generalization of the modular group $$PSL(2,\mathbb {Z})$$ . Next we define a discrete subgroup $$PSL(2,\mathfrak {H})$$ of $$PSL(2,\mathbb {H})$$ which is obtained by using Hurwitz integers. It contains as a subgroup $$PSL(2,\mathfrak {L})$$ . In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic quaternionic half space. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients $$\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {L})$$ and $$\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {H})$$ which are quaternionic versions of the classical modular orbifold and they are of finite volume. Finally we give a thorough study of their descriptions by Lorentz transformations in the Lorentz–Minkowski model of hyperbolic 4-space.

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Improved Two-Level Voltage Source Converter for High-Voltage Direct Current Transmission Systems

This paper presents an improved two-level voltage source converter for dc transmission systems with relatively low rated power and dc operating voltage. Unlike conventional two-level converter, the presented converter employs two distributed cell capacitors per three phase; thus, do not contribute any current when converter is blocked during dc short-circuit fault as in modular multilevel converter case. The use of three-phase cells is proven to be beneficial because the arm currents do not contain second-order harmonic currents, and cell capacitors tend to be small as they only experience high-order harmonic current associated with the switching frequency. For the same rated dc-link voltage and switching devices, the rated power of the improved two-level converter will be twice than that of the conventional two-level converter. Averaged, switching function, and electromagnetic transient simulation models of the improved two-level converter are discussed and validated against detailed switch model. The viability of the improved two-level converter for high-voltage dc applications is examined, considering dc and ac short-circuit faults. Besides, reduced complexity of the control and power circuit of the improved two-level converter, it has been found that its transient responses to ac and dc faults are similar to that of the modular multilevel converter.

English

We investigate the invariant rings of two classes of finite groups G ≤ GL(n, F q) which are generated by a number of generalized transvections with an invariant subspace H over a finite field F q in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings.

Classical weight one forms in Hida families: Hilbert modular case

The purpose of this paper is to investigate the number of classical weight one specializations of a non-CM ordinary Hida family of parallel weight Hilbert cusp forms. It is known that a specialization of a primitive ordinary Hida family at any arithmetic points of weight at least two is a classical holomorphic Hilbert cusp form. However, this is not always the case for weight one specializations. Balasubramanyam, Ghate and Vatsal proved that such a Hida family admits infinitely many classical weight one specializations if and only if it is of CM type. We give an explicit upper bound on the number of classical weight one specializations of a non-CM primitive ordinary Hida family.

The asymptotic behavior of Frobenius direct images of rings of invariants

We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; we also calculate the Frobenius limit for the restriction of these to the ring of invariants. As an application, we generalize the description of the generalized F-signature of a ring of invariants by the second author and Nakajima to the modular case.

Counterexample to the Generalized Belfiore–Solé Secrecy Function Conjecture for <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math> </inline-formula>-Modular Lattices

In this paper, we show that the secrecy function conjecture that states that the maximum of the secrecy function of an $l$ -modular lattice occurs at $1/\sqrt {l}$ is false, by proving that the four-modular lattice $ C^{(4)}= \mathbb {Z}\oplus \sqrt {2} \mathbb {Z}\oplus 2 \mathbb {Z}$ fails to satisfy this conjecture. After this, we indicate how the secrecy function must be modified in the $l$ -modular case to provide a more meaningful comparison for $l$ -modular lattices, and show that this new secrecy function indeed has a maximum at $1/\sqrt {l}$ for various two-modular lattices and for $ C^{(4)}$ . We conjecture that this must hold true for all $l$ -modular lattices.

Three-dimensional isolated quotient singularities in odd characteristic

Let a finite group act linearly on a finite-dimensional vector space over an algebraically closed field of characteristic . Suppose that the quotient space has an isolated singularity only. The isolated singularities of the form are completely classified in the case when does not divide the order of , and their classification reduces to Vincent's classification of isolated quotient singularities over . In the present paper we show that, if , then the classification of isolated quotient singularities reduces to Vincent's classification in the modular case as well (when divides ). Some remarks on quotient singularities in other dimensions and in even characteristic are also given. Bibliography: 14 titles.

Multivariable codes in principal ideal polynomial quotient rings

Multivariable codes over a finite field are a natural generalization of several classes of codes, including cyclic, negacyclic, constancyclic, polycyclic and abelian codes. Since these particular families have been also considered in the context of codes over a finite chain ring, we proposed constructions of multivariable codes over such a class of finite rings. As in the case of traditional cyclic codes over finite fields the modular case (i.e., codes with repeated roots) is much more difficult to handle than the semisimple case (i.e., codes with non-repeated roots). In this sense, different authors have dedicated their efforts to provide a better understanding of the properties of cyclic, negacyclic, constancylic and polycyclic modular codes over a finite chain ring. Among these codes, those contained in an ambient space which is a principal ideal ring admit a relatively simple description, quite close to that of semisimple. This feature has been recently used in the description of abelian codes over a finite field, and in the description of modular additive cyclic codes over F4. As a natural continuation of these works, in this paper we consider the structure of multivariable modular codes in an ambient space which is a principal ideal ring.

Coordinating cross-sector partnerships for water provision in Malawi

Subject areaPublic sector management, stakeholder management, collaboration and strategy.Study level/applicabilityUndergraduate (final-year) or master's-level students (Master in Public Administration, Master in Management). Designed for courses in nonprofit management, public administration and/or international development. Can also be used in any course, such as strategic management, sustainable development or corporate social responsibility, that covers stakeholder theory, or stakeholder management as a topic.Case overviewDecentralization has changed the way core services are delivered to local populations in sub-Saharan Africa. This in turn has forced nongovernmental organizations, international aid agencies, corporations and other development partners to change the way they engage with government in their shared efforts to help improve the living conditions of people living under the threshold of poverty in this and other parts of the world. This modular ethnographic teaching case uses the specific example of the water sector in Malawi to help highlight the complexity of multiple stakeholder relations in an international development context.Expected learning outcomesUpon completion of this case, students should be able to: identify and understand the different goals and issues that individual stakeholders in cross-sector partnerships are dealing with; identify and understand the power/control dynamics at play in these relationships; analyse the advantages and disadvantages associated with different ways of coordinating multi-stakeholder partnerships; and develop recommendations for structuring multi-stakeholder relationships in developing and emerging markets that balance efficient service provision with concern for individual stakeholder priorities.Supplementary materialsTeaching Notes are available for educators only. Please contact your library to gain login details or emailsupport@emeraldinsight.comto request teaching notes.

Two-weight codes, graphs and orthogonal arrays

We investigate properties of two-weight codes over finite Frobenius rings, giving constructions for the modular case. A \(\delta \)-modular code (in: Honold T, Honold in Proceedings of the fifth international workshop on optimal codes and related topics, White Lagoon, Bulgaria, 2007) is characterized as having a generator matrix where each column \(g\) appears with multiplicity \(\delta |gR^\times |\) for some \(\delta \in \mathbb {Q}\). Generalizing (Delsarte in Discret Math 3:47–64, 1972) and (Byrne et al in Finite Fields Appl 18(4):711–727, 2012), we show that the additive group of a two-weight code satisfying certain constraint equations (and in particular a modular code) has a strongly regular Cayley graph and derive existence conditions on its parameters. We provide a construction for an infinite family of modular two-weight codes arising from unions of submodules with pairwise trivial intersection. The corresponding strongly regular graphs are isomorphic to graphs from orthogonal arrays.

Modular case of Levinson's theorem

We evaluate the integral mollified second moment of L-functions of primitive cusp forms and we obtain, for such L-function, an explicit positive proportion of zeros which lie on the critical line.

Double integrals on a weighted projective plane and Hilbert modular functions for Q(\sqrt 5)

The aim of this paper is to give an explicit extension of the classical elliptic integrals to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We study a family of Kummer surfaces corresponding to the Humbert surface of invariant $5$ with two complex parameters. Our Kummer surface is given by a double covering of the weighted projective space $\mathbb{P}(1:1:2)$ branched along a parabola and a quintic curve. The period mapping for our family is given by double integrals of an algebraic function on chambers coming from an arrangement of a parabola and a quintic curve in $\mathbb{C}^2$.

The structure of polynomial operations associated with smooth digraphs

With every digraph we associate an algebra whose fundamental operations are the polymorphisms of the digraph. In a 2012 paper, the second and third authors proved that the digraph of endomorphisms of any finite connected reflexive digraph is connected, provided that the algebra associated with the digraph lies in a congruence join-semidistributive over modular variety. In the same paper, this connectivity result led to a proof of the statement that if the algebra associated with a finite reflexive digraph generates a congruence modular variety, then the digraph has a near-unanimity polymorphism. A digraph is smooth if it has no sinks and no sources. Smooth digraphs of algebraic length 1 are a broad generalization of reflexive digraphs. In a 2009 paper, Barto et al. proved that every finite smooth digraph of algebraic length 1 whose associated algebra lies in a congruence meet-semidistributive over modular variety has a loop edge. This is a powerful theorem that has nice applications in algebra and computer science. In the present paper, we prove that the digraph of unary polynomial operations of the algebra associated with a finite smooth connected digraph of algebraic length 1 is connected, provided that the algebra lies in a congruence join-semidistributive over modular variety. This generalizes our connectivity result mentioned above and implies a restricted version of the result of Barto et al. in the congruence join-semidistributive over modular case. We also give a characterization of locally finite idempotent congruence join-semidistributive over modular varieties via smooth compatible digraphs of algebraic length 1. It remains as an open question whether the congruence modularity of the variety generated by the algebra associated with a finite smooth digraph of algebraic length 1 implies the existence of a near-unanimity polymorphism of the digraph.

On Designerization of Media Culture in the Age of Software

On Designerization of Media Culture in the Age of Software

Un scindage du morphisme de Frobenius quantique

We show that the quantum Frobenius morphism constructed by Lusztig in the setting of the quantum enveloping algebra specialized at a root of unity admits a multiplicative splitting (non unital). We also find a basis of the toral part of the small quantum algebra consisting of pairwise orthogonal idempotents summing up to 1, and likewise in the modular case of the algebra of distributions for a semisimple algebraic group.

On generalized invariants of cyclic groups

In this paper, we extend the definition of generalized invariants for any finite group. Further, we describe the structure of generalized invariants of a cyclic group for lower dimensional indecomposable representations in the modular case, where the characteristic of the ground field divides the group order. Finally, an analog of a conjecture about Hilbert ideals associated with invariant rings has been proven for the case of generalized invariants.

Yang–Baxter deformations and rack cohomology

In his study of quantum groups, Drinfeld suggested considering set-theoretic solutions of the Yang–Baxter equation as a discrete analogon. As a typical example, every conjugacy class in a group or, more generally, every rack Q Q provides such a Yang–Baxter operator c Q : x ⊗ y ↦ y ⊗ x y c_Q \colon x \otimes y \mapsto y \otimes x^y . In this article we study deformations of c Q c_Q within the space of Yang–Baxter operators over some complete ring. Infinitesimally these deformations are classified by Yang–Baxter cohomology. We show that the Yang–Baxter cochain complex of c Q c_Q homotopy-retracts to a much smaller subcomplex, called quasi-diagonal. This greatly simplifies the deformation theory of c Q c_Q , including the modular case which had previously been left in suspense, by establishing that every deformation of c Q c_Q is gauge equivalent to a quasi-diagonal one. In a quasi-diagonal deformation only behaviourally equivalent elements of Q Q interact; if all elements of Q Q are behaviourally distinct, then the Yang–Baxter cohomology of c Q c_Q collapses to its diagonal part, which we identify with rack cohomology. This establishes a strong relationship between the classical deformation theory following Gerstenhaber and the more recent cohomology theory of racks, both of which have numerous applications in knot theory.

Zero-separating invariants for finite groups

We fix a field k of characteristic p. For a finite group G denote by δ(G) and σ(G) respectively the minimal number d, such that for every finite dimensional representation V of G over k and every v∈VG∖{0} or v∈V∖{0} respectively, there exists a homogeneous invariant f∈k[V]G of positive degree at most d such that f(v)≠0. Let P be a Sylow-p-subgroup of G (which we take to be trivial if the group order is not divisible by p). We show that δ(G)=|P|. If NG(P)/P is cyclic, we show σ(G)≥|NG(P)|. If G is p-nilpotent and P is not normal in G, we show σ(G)≤|G|l, where l is the smallest prime divisor of |G|. These results extend known results in the non-modular case to the modular case.

On additive modular bivariate codes over [formula omitted

Additive multivariate codes over F4 (the Galois field with 4 elements) are a natural extension of additive cyclic and abelian codes. A complete description of such codes when the length is odd was presented in [11]. In this paper we study some properties of this family of codes in the case when the length is even (modular case) and the number of variables is two (bivariate codes).