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Recently Vasuki and Veeresha were able to prove Dedekind η-function identities of level 14 conjectured by Somos. In this paper, we show how these Somos's identities are the consequence of Ramanujan's modular equations of degree 7. Also, in their paper authors are using a known multiplier to prove Somos's identities and we explained the method of obtaining that multiplier in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree 0 and rank n on a compact Riemann surface X of genus g. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Invent. Math. 139 (2000), 541–549], and admit a projective symplectic resolution if and only if g = 1 or (g, n) = (2, 2). These results are an application of a recent paper by Bellamy and Schedler [ArXiv e-print (2016)] via the so-called Isosingularity Theorem. [ABSTRACT FROM AUTHOR]

This paper presents a high-frequency modular medium-voltage AC (4160 VAC or 13.8 kVAC) to low-voltage DC (400 VDC) system that is scalable in order to be used for different scale microgrids. A 15 kW, 500 kHz DC/DC converter is demonstrated as the most important stage of the system overall, which can be scalable to a 225 kW 4160 VAC to 400 VDC system. Motivation of a CLLC resonant converter and its design parameters determination is carefully illustrated. The high-frequency transformer is the key element for the DC/DC converter. Then, the paper focuses on high-frequency transformer design to realize high-voltage insulation, high efficiency, and high density at the same time. Based on a split winding UU core transformer structure, transformer insulation materials and dimension parameters are determined referring to the IEEE insulation standard. The transformer magnetic loss model is reviewed based on which loss design tradeoff is carefully analyzed. Different working frequency impact on transformer design over three different core materials is also presented. Finally, a 500 kHz transformer prototype has been developed and demonstrated with the IEEE standard required insulation tests. A whole CLLC resonant converter is also present with experiments results. The converter holds outstanding 98% efficiency and 2.9 kW/L power density. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, some new martingale inequalities in the framework of Orlicz‐Karamata spaces are provided. More precisely, we establish modular martingale inequalities associated with concave functions and slowly varying functions. [ABSTRACT FROM AUTHOR]

Abstract: We consider non‐trivial irreducible tensor products of modular representations of a symmetric group Σ n in characteristic 2 for even n completing the proof of a classification conjecture of Gow and Kleshchev about such products. [ABSTRACT FROM AUTHOR]

Given integers k and m with k ≥ 2 and m ≥ 2 , let P be an empty simplex of dimension ( 2 k − 1 ) whose δ -polynomial is of the form 1 + ( m − 1 ) t k . In the present paper, the necessary and sufficient condition for the k th dilation k P of P to have a regular unimodular triangulation will be presented. [ABSTRACT FROM AUTHOR]

We prove that for any valuation ring R of Krull dimension ≤1 or any commutative local ring R with Krull dimension 0 and n≥3, the special linear group SLn(R[x]) coincides with the group of elementary matrices En(R[x]), and for an arbitrary arithmetical ring R of Krull dimension ≤1 and n≥3, the group . These give partial solutions to two conjectures of Yengui. [ABSTRACT FROM AUTHOR]

We investigate a method for finding small integer solutions of a univariate modular equation, that was introduced by Coppersmith (Proceedings of Eurocrypt 1996, LNCS, vol 1070, pp 155-165, 1996) and extended by May (New RSA vulnerabilities using lattice reduction methods, Ph.D. thesis, University of Paderborn, 2003). We will refer this method as the Coppersmith technique. This paper provides a way to analyze a general limitations of the lattice construction for the Coppersmith technique. Our analysis upper bounds the possible range of U that is asymptotically equal to the bound given by the original result of Coppersmith and May. This means that they have already given the best lattice construction. In addition, we investigate the optimality for the bivariate equation to solve the small inverse problem, which was inspired by Kunihiro’s (LNCS 7483:55-69, 2012) argument. In particular, we show the optimality for the Boneh-Durfee’s equation (Proceedings of Eurocrypt 1999, LNCS, vol 1592, pp 389-401, 1999) used for RSA cryptoanalysis, To show our results, we establish framework for the technique by following the relation of Howgrave-Graham (Proceedings of cryptography and coding, LNCS, vol 1355, pp 131-142, 1997), and then concretely define the conditions in which the technique succeed and fails. We then provide a way to analyze the range of U that satisfies these conditions. Technically, we show that the original result of Coppersmith achieves the optimal bound for U when constructing a lattice in the standard way. We then provide evidence which indicates that constructing a non-standard lattice is generally difficult. [ABSTRACT FROM AUTHOR]

We obtain new constraints for the modular energy of general states by using the monotonicity property of relative entropy. In some cases, modular energy can be related to the energy density of states and these constraints lead to interesting relations between energy and entropy. In particular, we derive new quantum energy inequalities that improve some previous bounds for the energy density of states in a conformal field theory. Additionally, the inequalities derived in this manner also lead us to conclude that the entropy of the state further restricts the possible amount of negative energy allowed by the theory. [ABSTRACT FROM AUTHOR]

Trace formulas are investigated in non-commutative integration theory. The main result is to evaluate the standard trace of Takesaki duals and, for this, we introduce the notion of interpolator and accompanied boundary objects. The formula is then applied to explore a variation of Haagerup's trace formula. [ABSTRACT FROM AUTHOR]

Cloud computing gathers a lot of computing resources together, and provides cheap, efficient and convenient services. With the development of cloud computing, the importance of security issues is increasing. Modular exponentiations are one of the basic and expensive operations of public key cryptography. We consider the single untrusted programed model of modular exponentiations outsourcing algorithm in this paper. A new secure outsourcing of modular exponentiations scheme is proposed under this model. In this scheme, a new logical split method about modular exponentiation is put forward. The modular exponentiation is divided into several parts and a random value is kept for verification. The exponent and the base of modular exponentiations are keeping privacy in the process of outsourcing. The computational burden for users in the process of outsourcing is small, and the users can verify the results with high checkability. [ABSTRACT FROM AUTHOR]

In the present paper, we study the problem of approximation to a function by means of positive linear operators in modular spaces in the sense of power series method. Indeed, in order to get stronger results than the classical cases, we use the power series method which also includes both Abel and Borel methods. An application that satisfies our theorem is also provided. [ABSTRACT FROM AUTHOR]

In this paper, we prove existence result of renormalized solutions in the setting of Musielak-Orlicz spaces W10 Lφ(Ω) for the following strongly nonlinear Dirichlet problem A(u) + g(x, u,∇u) = f in Ω, where A is a Leray-Lions operator acting from its domain D(A) ⊂ W10 Lφ(Ω) into its dual, while g(x, u,∇u) is a nonlinear term having a growth conditions with respect only to ∇u, and does not satisfy any sign condition. The right-hand side f belongs to L1(Ω). A modular-inequality of Poincaré type in this setting is also proved (see Lemma 2.5). [ABSTRACT FROM AUTHOR]

Let be the number of -regular partitions of . Recently, Hou et al. established several infinite families of congruences for modulo , where and . In this paper, using the vanishing property given by Hou et al., we prove an infinite family of congruence for modulo . Moreover, for and , we obtain three infinite families of congruences for modulo and respectively using the theory of Hecke eigenforms. [ABSTRACT FROM AUTHOR]

The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sums for the positive integers , and with lcm. We reprove the known formulas for the number of representations of a positive integer by each of the quadratic forms as an application of the new identities proved in this paper. [ABSTRACT FROM AUTHOR]

The modular properties of generalized theta-functions with characteristics are used to obtain the formulas for the number of representations of positive integers by the diagonal quadratic forms in nine variables with coefficients that are ones and twos. [ABSTRACT FROM AUTHOR]

Let F be a non-Archimedean locally compact field, q be the cardinality of its residue field, and R be an algebraically closed field of characteristic ℓ not dividing q. We classify all irreducible smooth R-representations of GLn(F) having a non-zero GLn-1(F)-invariant linear form, when q is not congruent to 1 mod ℓ. Partial results in the case when q is 1 mod ℓ show that, unlike the complex case, the space of GLn-1(F)-invariant linear forms has dimension 2 for certain irreducible representations. [ABSTRACT FROM AUTHOR]

If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures gi's, we say that g is simply assembled from the gi's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the gi's, one obtains a Lie algebra assembled in two steps from gi's, and so on. We describe the process of modular disassembling of a Lie algebra into a unimodular and a non-unimodular part. We then study two inverse questions: which Lie algebras can be assembled from a given family of Lie algebras, and from which Lie algebras can a given Lie algebra be assembled. We develop some basic assembling and disassembling techniques that constitute the elements of a newapproach to the general theory of Lie algebras. The main result of our theory is that any finite-dimensional Lie algebra over an algebraically closed field of characteristic zero or over ℝ can be assembled in a finite number of steps from two elementary constituents, which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra, while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As an example, we describe constructions of classical Lie algebras from triadons. [ABSTRACT FROM AUTHOR]

We study the arithmetic properties of two partition functions p ω ( n ) , p ν ( n ) and a smallest parts function spt ‾ ω ( n ) . The first two functions were introduced recently by Andrews, Dixit and Yee, and the smallest parts function by Andrews, Dixit, Schultz and Yee. These functions are closely related to the Ramanujan/Watson mock theta functions ω ( q ) and ν ( q ) . We find many new congruences satisfied by these functions modulo 11 and powers of 2 or 3. [ABSTRACT FROM AUTHOR]

Modular covariance of torus one-point functions constrains the three point function coefficients of a two dimensional CFT. This leads to an asymptotic formula for the average value of light-heavy-heavy three point coefficients, generalizing Cardy's formula for the high energy density of states. The derivation uses certain asymptotic properties of one-point conformal blocks on the torus. Our asymptotic formula matches a dual AdS computation of one point functions in a black hole background. This is evidence that the BTZ black hole geometry emerges upon course-graining over a suitable family of heavy microstates. [ABSTRACT FROM AUTHOR]

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin topological quantum field theories at low energy. We formulate a 16-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of PSU(2)4m+2 with an eye towards a classification of the low-rank cases. [ABSTRACT FROM AUTHOR]

Consider a finite dimensional vector space V over a finite field F q . We give a minimal generating set for the ring of invariants F q [ V ⊕ V ⁎ ] GL ( V ) , and show that this ring is a Gorenstein ring but is not a complete intersection. These results confirm a conjecture of Bonnafé and Kemper [3, Conjecture 3.1] . [ABSTRACT FROM AUTHOR]

Linear equalities, disequalities and inequalities on fixed-width bit-vectors, collectively called linear modular constraints, form an important fragment of the theory of fixed-width bit-vectors. We present a practically efficient and bit-precise algorithm for quantifier elimination from conjunctions of linear modular constraints. Our algorithm uses a layered approach, whereby sound but incomplete and cheaper layers are invoked first, and expensive but complete layers are called only when required. We then extend this algorithm to work with arbitrary Boolean combinations of linear modular constraints as well. Experiments on an extensive set of benchmarks demonstrate that our techniques significantly outperform alternative quantifier elimination techniques based on bit-blasting and linear integer arithmetic. [ABSTRACT FROM AUTHOR]

We describe here a special subdirect decomposition of algebras with modular congruence lattice. Such a decomposition (called a star-decomposition) is based on the properties of the congruence lattices of algebras. We consider four properties of lattices: atomic, atomless, locally uniform and anti-uniform. In effect, we describe a star-decomposition of a given algebra with modular congruence lattice into two or three parts associated to these properties. [ABSTRACT FROM AUTHOR]

In this paper we compute the restricted 0, 1-, and 2-dimensional cohomology groups with trivial coefficients of the modular Witt algebras W for any prime p > 3, and give explicit description of all one-dimensional restricted central extensions of W. [ABSTRACT FROM AUTHOR]

We give simple explicit formulas for deformation quantization of Poisson–Lie groups and of similar Poisson manifolds which can be represented as moduli spaces of flat connections on surfaces. The star products depend on a choice of Drinfeld associator and are obtained by applying certain monoidal functors (fusion and reduction) to commutative algebras in Drinfeld categories. From a geometric point of view, this construction can be understood as a quantization of the quasi-Poisson structures on moduli spaces of flat connections. [ABSTRACT FROM AUTHOR]

By using the q-analog of van der Corput's method, we study the divisor function in an arithmetic progression to modulus q. We show that the expected asymptotic formula holds for a larger range of q than was previously known, provided that q has a certain factorization. [ABSTRACT FROM AUTHOR]

GalSim is a collaborative, open-source project aimed at providing an image simulation tool of enduring benefit to the astronomical community. It provides a software library for generating images of astronomical objects such as stars and galaxies in a variety of ways, efficiently handling image transformations and operations such as convolution and rendering at high precision. We describe the GalSim software and its capabilities, including necessary theoretical background. We demonstrate that the performance of GalSim meets the stringent requirements of high precision image analysis applications such as weak gravitational lensing, for current datasets and for the Stage IV dark energy surveys of the Large Synoptic Survey Telescope, ESA’s Euclid mission, and NASA’s WFIRST-AFTA mission. The GalSim project repository is public and includes the full code history, all open and closed issues, installation instructions, documentation, and wiki pages (including a Frequently Asked Questions section). The GalSim repository can be found at https://github.com/GalSim-developers/GalSim . [ABSTRACT FROM AUTHOR]

Modularity is a mandatory principle to apply Petri nets to real world-sized systems. Modular extensions of Petri nets allow to create complex models by combining smaller entities. They facilitate the modeling and verification of large systems by applying a divide and conquer approach and promoting reuse. Modularity includes a wide range of notions such as encapsulation, hierarchy and instantiation. Over the years, Petri nets have been extended to include these mechanisms in many different ways. The heterogeneity of such extensions and their definitions makes it difficult to reason about their common features at a general level. We propose in this article an approach to standardize the semantics of modular Petri nets formalisms, with the objective of gathering even the most complex modular features from the literature. This is achieved with a new Petri nets formalism, called the LLAMAS Language for Advanced Modular Algebraic Nets (LLAMAS). We focus principally on the composition mechanism of LLAMAS, while introducing the rest of the language with an example. The composition mechanism is introduced both informally and with formal definitions. Our approach has two positive outcomes. First, the definition of new formalisms is facilitated, by providing common ground for the definition of their semantics. Second, it is possible to reason at a general level on the most advanced verification techniques, such as the recent advances in the domain of decision diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we give a criterion for the dual Selmer group of an elliptic curve which has either good ordinary reduction or multiplicative reduction at every prime above p to satisfy the 픐H(G)-conjecture. As a by-product of our calculations, we are able to define the Akashi series of the dual Selmer groups assuming the conjectures of Mazur and Schneider. Previously, the Akashi series are defined under the stronger assumption that the dual Selmer group satisfies the 픐H(G)-conjecture. We then establish a criterion for the vanishing of the dual Selmer groups using the Akashi series. We will apply this criterion to prove some results on the characteristic elements of the dual Selmer groups. Our methods in this paper are inspired by the work of Coates-Schneider-Sujatha and can be extended to the Greenberg Selmer groups attached to other ordinary representations, for instance, those coming from a p-ordinary modular form. [ABSTRACT FROM AUTHOR]

In this paper, we study the Hyers-Ulam-Rassias stability of the quadratic functional equation f(x + y) + f(x - y) = 2f(x) + 2f(y), x⊥y in which ⊥ is orthogonality in the sens of Rätz in modular spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that two main fixed point theorems for commuting mappings in modular metric spaces. Our main results are more general than from those of Jungck [17] and Fisher [16]. [ABSTRACT FROM AUTHOR]

We present here a general framework to design algorithms that compute H-join. For a given bipartite graph H, we say that a graph G admits a H-join decomposition or simply a H-join, if the vertices of G can be partitioned in | H| parts connected as in H. This graph H is a kind of pattern, that we want to discover in G. This framework allows us to present fastest known algorithms for the computation of P-join (aka N-join), P-join (aka W-join), C-join (aka 6-join). We also generalize this method to find a homogeneous pair (also known as 2-module), a pair { M, M} such that for every vertex x∉( M∪ M) and i∈{1,2}, x is either adjacent to all vertices in M or to none of them. First used in the context of perfect graphs (Chvátal and Sbihi in Graphs Comb. 3:127-139, ), it is a generalization of splits (a.k.a. 1-joins) and of modules. The algorithmics to compute them appears quite involved. In this paper, we describe an O( mn)-time algorithm computing all maximal homogeneous pairs of a graph, which not only improves a previous bound of O( mn) for finding only one pair (Everett et al. in Discrete Appl. Math. 72:209-218, ), but also uses a nice structural property of homogenous pairs, allowing to compute a canonical decomposition tree for sesquiprime graphs (i.e., graphs G having no module and such that for every vertex v∈ G, G− v also has no module). [ABSTRACT FROM AUTHOR]

In this paper we defined the ... (Δumn, A, φ)μη defined by a modulus and exhibit some general properties of the space with an four dimensional infinite regular matrix. [ABSTRACT FROM AUTHOR]

By expressing infinite series in terms of definite integrals, we shall establish four formulae of BBP-type modulo 6 with an extra free parameter. Several new BBP-type formulae are consequently deduced. [ABSTRACT FROM AUTHOR]

Abstract: Twisted Calabi–Yau algebras are a generalisation of Ginzburg's notion of Calabi–Yau algebras. Such algebras A come equipped with a modular automorphism , the case being precisely the original class of Calabi–Yau algebras. The aim of this paper is to give a concise proof of the fact that every twisted Calabi–Yau algebra may be extended to a Calabi–Yau algebra. More precisely, we show that if A is a twisted Calabi–Yau algebra with modular automorphism σ, then the smash (semidirect) product algebras and are Calabi–Yau. [Copyright &y& Elsevier]

We construct the R-operator—the solution of the Yang–Baxter equation acting in the tensor product of two infinite-dimensional representations of Faddeev’s modular double. This R-operator intertwines the product of two L-operators associated with the modular double and it is built from three basic operators generating the permutation group of four parameters . [ABSTRACT FROM AUTHOR]

Abstract: In 1954, A.O.L. Atkin and H.P.F. Swinnerton-Dyer established the generating functions for rank differences modulo 5 and 7 for partition functions. In this paper, we derive formulas for the generating functions of ranks of partitions modulo 10 and some inequalities between them. [Copyright &y& Elsevier]

Abstract: We show that a classical uniformly distributed phenomenon for an element and its inverse in also exists in . A analogy of the uniform distribution on modular hyperbolas has also been considered. [Copyright &y& Elsevier]

In this note, we prove a conjecture of Steinberg [10] about modular invariants of some linear groups over finite commutative local rings. [ABSTRACT FROM PUBLISHER]

A ring R is quasipolar if for every a ∈ R there exists p2 = p ∈ R such that, a + p ∈ U(R) and ap ∈ Rqnil; the element p is called a spectral idempotent of a. Strongly π-regular rings are quasipolar and quasipolar rings are strongly clean. In this paper, the relationship among strongly regular rings, strongly π-regular rings and quasipolar rings are investigated. Moreover, we provide several equivalent characterizations on quasipolar elements in rings. Consequently, it is shown that any quasipolar element in a ring is strongly clean. It is also proved that for a module M, α ∈end(M) is quasipolar if and only if there exist strongly α-invariant submodules P and Q such that M = P ⊕ Q, α|Pis isomorphic and α|Qis quasinilpotent, which is shown to be a natural generalization of Fitting endomorphisms. [ABSTRACT FROM PUBLISHER]

In this paper, we establish the Hyers-Ulam stability of the orthogonal quadratic functional equation of Pexiderized type f(x + y) + f(x - y) = 2g(x) + 2h(y), x ⊥ y in which ⊥ is the orthogonality in the sense of Rätz in modular spaces. [ABSTRACT FROM AUTHOR]

We review our recent results in the vertex algebra approach in construction of identities for modular forms. [ABSTRACT FROM AUTHOR]

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we show that the probability that a uniformly random k × n matrix over can be extended to an n × n invertible matrix is . Connections with Dirichlet’s density theorem on co-prime integers and its various generalizations are also presented. [Copyright &y& Elsevier]

Let π(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρf. Let K be a totally complex quadratic extension of F embedding in B. Using families of CM points on towers of Shimura curves attached to B and K, we construct an Euler system for ρf. We prove that it extends to p-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory. [ABSTRACT FROM AUTHOR]

In this paper we give a sufficient condition on the class numbers of distinct quadratic imaginary fields so that on a given CM elliptic curve over Q with fixed modular parameterization, the Heegner points associated to (the maximal orders of) these quadratic imaginary fields are linearly independent. This extends results of Rosen and Silverman from the non-CM to the CM case. [ABSTRACT FROM AUTHOR]

Abstract: We compare the dimensions of the irreducible -modules over a field of characteristic constructed by Gow (1998) with the dimensions of the irreducible -modules that appear in the first approximation to representations of mapping class groups of surfaces in integral topological quantum field theory . For this purpose, we derive a trigonometric formula for the dimensions of Gow’s representations. This formula is equivalent to a special case of a formula contained in unpublished work of Foulle (2004) . Our direct proof is simpler than the proof of Foulle’s more general result, and is modeled on the proof of the Verlinde formula in TQFT. [Copyright &y& Elsevier]

We study modular Galois representations mod pm. We show that there are three progressively weaker notions of modularity for a Galois representation mod pm: We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a level-lowering result ("stripping-of-powers of p away from the level"): A mod pm strongly modular representation of some level Npr is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod pm corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod pm to any "dc-weak" eigenform, and hence to any eigenform mod pm in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general. [ABSTRACT FROM AUTHOR]

We present an example of a variety of algebras which is congruence modular but not congruence distributive or congruence n-permutable for each n ≥ 2. [ABSTRACT FROM AUTHOR]