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In this paper, we study complex symmetric C0-semigroups on the Bergman space A2(ℂ+) of the right half-plane ℂ+. In contrast to the classical case, we prove that the only involutive composition operator on A2(ℂ+) is the identity operator, and the class of J-symmetric composition operators does not coincide with the class of normal composition operators. In addition, we divide semigroups {ϕt} of linear fractional self-maps of ℂ+ into two classes. We show that the associated composition operator semigroup {Tt} is strongly continuous and identify its infinitesimal generator. As an application, we characterize Jσ-symmetric C0-semigroups of composition operators on A2(ℂ+).

In this paper, by use of matrix inversions, we establish a general $q$-expansion formula of arbitrary formal power series $F(z)$ with respect to the base $$\left\{z^n\frac{(az:q)_n}{(bz:q)_n}\bigg|n=0,1,2\cdots\right\}.$$ Some concrete expansion formulas and their applications to $q$-series identities are presented, including Carlitz's $q$-expansion formula, a new partial theta function identity and a coefficient identity of Ramanujan's ${}_1\psi_1$ summation formula as special cases., Comment: 18pages

We describe a new connection between the dilogarithm function and the solutions of Pell’s equation $$x^2-ny^2 = \pm 1$$ . For each solution x, y to Pell’s equation, we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $$x+y\sqrt{n} \in {\mathbb {Z}}[\sqrt{n}]$$ . We further show that Ramanujan’s dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $\theta$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb{C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb{C}^\ell \to \mathbb{C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture., Comment: 13 pages

This study discusses manner verb construction and reduplication of the Kedang language from the perspective of typology. This study is a descriptive qualitative study by describing the pattern of manner verb construction of the Kedang language. The data is analyzed by using distributional (agih) and identity (padan) methods through breaking down words and sentences. The choice and equivalent of sentences are adjusted with the typology rule of universal language by Dixon (2006); serial verb construction. The result of this study showed that the manner verb construction of the Kedang language is an action-manner with verb and reduplication. There are five patterns of manner verb construction; V1 Intransitive + V2 transitive, V1 Intransitive + V2 Intransitive, V1 Transitive + V1 Transitive, V1 transitive + V2 Intransitive, and V Tran/Inf +Adj. However, manner verb construction with reduplication is classified into three categories: V1 intransitive + total reduplication, V1 transitive + total reduplication, and V1 transitive/intransitive + total reduplication with sound change. The verbs that accompany the reduplication of the word are motion and action verb. The manner verb is always in second place to clarify the activity of the first verb.

In this paper, we introduce and study the notion of the maximal ideal graph of a commutative ring with identity. Let R be a commutative ring with identity. The maximal ideal graph of R, denoted by MG(R), is the undirected graph with vertex set, the set of non-trivial ideals of R, where two vertices I1 and I2 are adjacent if I1 I2 and I1+I2 is a maximal ideal of R. We explore some of the properties and characterizations of the graph.

In this paper, we seek to prove the equality of the $q$-graded fermionic sums conjectured by Hatayama et al. [ 14] in its full generality, by extending the results of Di Francesco and Kedem [ 9] to the non-simply laced case. To this end, we will derive explicit expressions for the quantum $Q$-system relations, which are quantum cluster mutations that correspond to the classical $Q$-system relations, and write the identity of the $q$-graded fermionic sums as a constant term identity. As an application, we will show that these quantum $Q$-system relations are consistent with the short exact sequence of the Feigin–Loktev fusion product of Kirillov–Reshetikhin modules obtained by Chari and Venkatesh [ 5].

In this paper, we prove two theta-function identities using modular equation of degree 3. Furthermore, as an application of this we establish combinatorial interpretations of colored partitions. 1. INTRODUCTION Ramanujan’s theta function f(x, y) is defined as f(x, y) := X∞ n=−∞ x n(n+1)/2 y n(n−1)/2 |xy| < 1. The function f(x, y) enjoys the well-known Jacobi’s triple-product identity [5, p. 35] given by f(x, y) = (−x; xy)∞(−y; xy)∞(xy; xy)∞ , where here and throughout the paper, we assume |q| < 1 and employ the standard notation (x; q)∞ := Y∞ n=0 (1 − xqn ).

Let R be a commutative ring with identity and X r 1 ( R ) the set of regular height one prime ideals of R. We will show that R is a Krull ring if and only if each regular prime ideal of R contains ...

The purpose of this paper is to analyze the relationship between the ownership structure and the investment in research and development (R&D) for a sample of listed Spanish companies.,Following the agency theory and the socioemotional wealth (SEW) perspective, the authors propose that R&D investment is affected by ownership structure, specifically by the identity of the controlling owner (family firms and firms with an institutional investor) and the level of contestability by other shareholders. In order to test these hypotheses, the authors build an original database identifying, at a 10% threshold, the ultimate shareholders of a sample of 96 Spanish firms listed during 2008–2018 (1,002 obs).,The results show that there is no significant relationship between the ownership concentration and the R&D investment. Only when the authors consider the nature of the main shareholder, the authors find that in family firms there is an inverted U relationship between ownership and R&D, so that at low levels of ownership, the R&D increases, while at high levels of ownership (that we compute around 54%) the R&D decreases. Also, when the main shareholder is an institutional investor, the greater its ownership, the higher the R&D investment. Finally, the authors test that, contrary to what mainstream suggests, contestability in family firms is higher when ownership in the hands of other family shareholders increases.,The work uses an original database to test a nonlinear relationship between ownership and R&D investment in family firms. Also, the study addresses a topic hardly ever discussed in the literature about R&D as it is the role of the contestability by other controlling shareholders.,El objetivo del presente trabajo es analizar la relacion existente entre la estructura de propiedad y la innovacion corporativa para una muestra de empresas cotizadas espanolas.,Utilizando los planteamientos de la teoria de la agencia y de la perspectiva de la riqueza socioemocional proponemos que la I+D empresarial esta relacionada con la estructura de propiedad, especificamente con la naturaleza del accionista de control (empresas familiares y empresas con un inversor institucional como principal accionista) y con el grado de contestabilidad por parte de otros accionistas significativos. A fin de testar nuestras hipotesis, construimos ad hoc una base de datos de propiedad original en la que identificamos, al umbral del 10% de propiedad, a los accionistas ultimos de una muestra de 96 empresas cotizadas espanolas para el periodo 2008–2018 (1.002 obs).,Nuestros resultados muestran que no existe relacion significativa entre la concentracion de propiedad y la inversion en I + D. Solo cuando consideramos la naturaleza del principal accionista encontramos que en las empresas familiares la relacion entre la propiedad de la familia y la innovacion corporativa adopta una forma de U invertida, tal que a bajos niveles de propiedad la I + D crece, mientras que a altos niveles de propiedad (que computamos en torno al 54% de propiedad) la inversion en I + D decrece. Asimismo, en las empresas con un inversor institucional como principal accionista, cuanto mayor es la propiedad de este inversor institucional, mayor es la I + D de la empresa. Finalmente testamos que, en contra de la corriente dominante, en las empresas familiares la propiedad en manos de otras familias incrementa el grado de contestabilidad a la familia controladora respecto a su inversion en I + D.,El trabajo utiliza una base de datos de propiedad original para testar una relacion no lineal entre concentracion de propiedad e innovacion corporativa en las empresas familiares. Asimismo, el estudio aborda un tema apenas analizado en la literatura de I + D como es el papel de la contestabilidad al accionista de control.

There is a local ring H of order 4, without identity for the multiplication, defined by generators and relations as H =〈a, b | 2a = 2b = 0, a 2 = 0, b 2 = b, ab = ba = 0〉. We classify self orthogonal codes of length n and size 2 n (called here quasi self-dual codes or QSD) up to the length n = 6. In particular, we classify quasi Type IV codes (a subclass of Type IV codes, viz QSD codes with even weights) up to n = 6.

In the setting of various complex Banach spaces we consider the questions of when a square of a Hermitian operator is also Hermitian, and when a Hermitian operator has a Hermitian square root. We also describe the structure of Hermitian projections and Hermitian square roots of the identity operator.

In this manuscript, firstly we introduce and study the concept of (α,m_1,m_2 )-Geometric-Arithmetically (GA) convex functions and some algebraic properties of such type functions. Then, we obtain Hermite-Hadamard type integral inequalities for the newly introduced class of functions by using an identity together with Hölder integral inequality, power-mean integral inequality and Hölder-İşcan integral inequality giving a better approach than Hölder integral inequality. Inequalities have been obtained with the help of Gamma function. In addition, results were obtained according to the special cases of α, m_1 and m_2.

This paper presents a case study of a Hong Kong university student’s experiences of learning English as a second language (L2) over a four-year period, with particular attention to the changes in her identities and beliefs across time and space. Drawing on a narrative inquiry approach, the study revealed that the student’s L2 identities appeared to be shaped by specific contextual conditions and agentic choices made by the student in response to different contexts, including consultation sessions with native English-speaking tutors, study abroad in the U.S., interactions with non-native English-speaking peers, and classroom interactions. It was also found that her L2 identities and beliefs not only varied over time in a complex and dynamic manner, but also appeared to be closely interconnected and interacted with each other in a reciprocal and bi-directional manner. The case study points to the need to pay more attention to the complex and dynamic interrelationship between identity and belief in L2 learning trajectories.

Let R be a commutative ring with a nonzero identity. Badawi et al. (Bull Korean Math Soc 51(4):1163–1173, 2014) defined the concept of 2-absorbing primary ideal as follows: a proper ideal P of R is said to be a 2-absorbing primary ideal if whenever $$xyz\in P$$ for some $$x,y,z\in R,$$ then either $$xy\in P$$ or $$xz\in \sqrt{P}$$ or $$yz\in \sqrt{P}.$$ It is clear that every primary ideal is also a 2-absorbing primary ideal. The author in this paper is to study rings in which every 2-absorbing primary ideal is primary. A ring R is said to be $$2-ABP$$ -ring if every 2-absorbing primary ideal is primary.

Let X be a T -module, T is a commutative ring with identity and K be a proper submodule of X. In this paper we introduce the concepts of 2-visible submodules and fully 2-visible modules as a generalizations of visible submodules and fully visible modules resp., where K is said to be 2-visible whenever K=I2 K for every nonzero ideal I of T and AT -module X is called fully 2-visible if for any proper submodule of it is 2-visible.Study some of the properties of these concepts also discuss the relationship 2-visible submodules and fully 2-visible modules with 2-pure submoules and other related submodules and modules resp. are given

In this paper, we first provide the generalized degenerate Gould-Hopper polynomials via thedegenerate exponential functions and then give various relations and formulas such as addition formula andexplicit identity. Moreover, we consider the generalized Gould-Hopper based degenerate central factorialnumbers of the second kind and present several identities and relationships. Furthermore, we introducethe generalized Gould-Hopper based fully degenerate central Bell polynomials and investigated multifariouscorrelations and formulas including summation formulas, derivation rule and correlations with the Stirlingnumbers of the first kind, the generalized Gould-Hopper based degenerate central factorial numbers of thesecond kind and the generalized degenerate Gould-Hopper polynomials. We then acquire some relations withthe degenerate Bernstein polynomials for the generalized Gould-Hopper based fully degenerate central Bellpolynomials. Finally, we consider the Gould-Hopper based fully degenerate Bernoulli, Euler and Genocchipolynomials and by utilizing these polynomials, we develop some representations for the generalized Gould-Hopper based fully degenerate central Bell polynomials.

We characterise the real extreme points of the unit ball of $$m^0_\Psi $$ , the complex extreme points of the unit ball of $$m_\Psi $$ and the real extreme and exposed points of the unit ball of $$(m_\Psi ^0)'$$ . Using these characterisations we show that, depending on the length of the extreme points, the multipliers of $$m^0_\Psi $$ are either constant multiple of the identity or diagonal operators.

In this paper, it is proved that for λ > 1, an additive map that strongly preserves the set of λ-scrambling matrices over the Boolean semiring B is a bijection. The general form of such a map over any antinegative commutative semiring with identity and without zero divisors is characterized. Bibliography: 20 titles.

Let R and S be commutative rings with identity. A proper (R,S)submodule P of M is called a left weakly jointly prime if for each element a and b in R and (R,S)-submodule K of M with abKS ⊆ P implies either aKS ⊆ P or bKS ⊆ P. In this paper, we present some properties of left weakly jointly prime (R,S)-submodule. We show some necessary and sufficient condition of left weakly jointly prime (R,S)-submodule. Moreover, we present that every left weakly jointly prime (R,S)-submodule contains a minimal left weakly jointly prime (R,S)submodule. At the end of this paper, we also show that in left multiplication (R,S)-module, every left weakly jointly prime (R,S)-submodule is equal to jointly prime (R,S)-submodules.

In the article, we establish a Simpson-type generalized identity containing multi-parameters and derive some new estimates for the generalized Simpson's quadrature rule via the Raina fractional integrals. As applications, we provide several inequalities for the f -divergence measures and probability density functions.

In this paper, the new version of the celebrated Montgomery identity is determined via quantum integral operators. By using it, certain quantum integral inequalities of Ostrowski type are established. Moreover, the relevant connection of the obtained results of this work with the derived results in previously published works is discussed.

In this paper, we use the elementary methods, the properties of the Gauss sums and the estimate for character sums to study the calculating problems of a certain cubic residues modulo p, and give some interesting identities and asymptotic formulas for their counting functions.

In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators A,B that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and B is the identity operator then one version of this quotient is the spectral radius of A. In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with the Collatz-Wielandt quotient for two nonnegative operators. In this paper we treat the two important cases: a pair of rectangular nonnegative matrices and a pair completely positive operators. We give a characterization of minimal optimal solutions and polynomially computable bounds on the Collatz-Wielandt quotient., Comment: 24 pages. To appear in Applications of Mathematics, ISSN 0862-7940

In 1996, A. Norton and D. Sullivan asked the following question: If $f:\mathbb{T}^2\rightarrow\mathbb{T}^2$ is a diffeomorphism, $h:\mathbb{T}^2\rightarrow\mathbb{T}^2$ is a continuous map homotopic to the identity, and $h f=T_{\rho} h$ where $\rho\in\mathbb{R}^2$ is a totally irrational vector and $T_{\rho}:\mathbb{T}^2\rightarrow\mathbb{T}^2,\, z\mapsto z+\rho$ is a translation, are there natural geometric conditions (e.g. smoothness) on $f$ that force $h$ to be a homeomorphism? In [ J. Wang and Z. Zhang, GAFA 2018 ], the first author and Z. Zhang gave a negative answer to the above question in the $C^{\infty}$ category: In general, not even the infinite smoothness condition can force $h$ to be a homeomorphism. In this article, we give a negative answer in the $C^{\omega}$ category: We construct a real-analytic conservative and minimal totally irrational pseudo-rotation of $\mathbb{T}^2$ that is semi-conjugate to a translation but not conjugate to a translation, which simultaneously answers a question raised in [ J. Wang and Z. Zhang, GAFA 2018 ].

Let [Formula: see text] be a field of characteristic zero, algebraically closed and [Formula: see text] be the unitary infinite-dimensional Grassmann [Formula: see text]-algebra. Our aim is to determine the minimal degree of an [Formula: see text]-polynomial satisfied by [Formula: see text]. Also, we find an explicit [Formula: see text]-identity of [Formula: see text].

An interesting identity for 3-point Gauss-Legendre quadrature rule using functions that are n-times differentiable. By applying the established identity, a sharp inequality which gives an error bound for 3-point Gauss-Legendre quadrature rule and some generalizations are derived. At the end, an application in numerical integration is given.

A commutative non-associative division closed lattice-ordered ring with identity that is not an f -ring is presented. More conditions are provided to ensure that an associative division closed latt...

In this paper, a new identity is established for differentiable mappings. By using the mathematical analysis techniques, some new integral inequalities of Hermite-Hadamard type for differentiable p...

We present new generalizations of the weighted Montgomery identity constructed by using the Hermite interpolating polynomial. The obtained identities are used to establish new generalizations of weighted Ostrowski type inequalities for differentiable functions of class $C^{n}$ C n . Also, we consider new bounds for the remainder of the obtained identities by using the Chebyshev functional and certain Grüss type inequalities for this functional. By applying those results we derive inequalities for the class of n-convex functions.

The (bivariate) Poisson distribution is the most common distribution for (bivariate) count random variables. The univariate Poisson distribution is characterized by the famous Stein–Chen identity. ...

In this paper, we research Noether’s theorems of fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives. First, the generalized Pfaff–Birkhoff principle and Birkhoff’s equations with classical and combined Caputo derivatives are given. Then, in the case of without and with transforming time, respectively, we obtain two kinds of Noether symmetry and their conserved quantities by the method of time re-parameterization. Finally, by taking case of generalized Birkhoffian systems, we study the relationship between Noether symmetry and Mei symmetry. It is concluded that the generalized Birkhoff equations, Noether identity and Noether conserved quantity obtained by using the generalized Pfaff–Birkhoff principle based on the action functional with dynamical functions after infinitesimal transformation are completely consistent with the criterion equation, structural equation and Mei conserved quantity of the fractional generalized Birkhoffian system, respectively.

Let R be associative; ring; with an identity and let D be unitary left R- module; . In this work we present semiannihilator; supplement submodule as a generalization of R-a- supplement submodule, Let U and V be submodules of an R-module D if D=U+V and whenever Y≤ V and D=U+Y, then annY≪R;. We also introduce the the concept of semiannihilator -supplemented ;modules and semiannihilator weak; supplemented modules, and we give some basic properties of this conseptes.

In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of [GK93] and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over Fp and the sum of the Fourier coefficients of the Siegel-Eisenstein series for Sp_4 of weight 2, which is independent of p (> 2). In addition, we will explain a description of the local intersection multiplicities of the special cycles over F_p on the supersingular locus of the `special fiber' of the Shimura varieties for GSpin(n; 2), n, 52 pages

Let $$\mathbb{D}$$ be the two-dimensional real algebra generated by 1 and by a hyperbolic unit k such that $$k^{2} = 1$$. This algebra is often referred to as the algebra of hyperbolic numbers. A function $$f : \mathbb{D} \rightarrow \mathbb{D}$$ is called $$\mathbb{D}$$-holomorphic in a domain $$\Omega \subset \mathbb{D}$$ if it admits derivative in the sense that $${\rm lim}_{h\rightarrow{0}}\frac{f({\mathfrak{z}_{0}+h)} -f{(\mathfrak{z}_{0})}} {h}$$ exists for every point $$\mathfrak{z}_0$$ in $$\Omega$$, and when h is only allowed to be an invertible hyperbolic number. In this paper we prove that $$\mathbb{D}$$-holomorphic functions satisfy an unexpected limited version of the identity theorem. We will offer two distinct proofs that shed some light on the geometry of $$\mathbb{D}$$. Since hyperbolic numbers are naturally embedded in the four-dimensional algebra of bicomplex numbers, we use our approach to state and prove an identity theorem for the bicomplex case as well.

In this note, we evaluate sums of partial theta functions. Our main tool is an application of an extended version of the Bailey transform to an identity of Gasper and Rahman on $q$-hypergeometric series.

The main result of this paper is an identity expressing the r-Stirling number of the first kind as a sum involving binomial coefficients and the Mobius function of the set-partition lattice. We provide three different proofs of this result: analytic, inductive, and combinatorial.

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove a Frech\'{e}t differentiability result for the eigenvalues, we compute the corresponding Hadamard formulas and we prove a continuity result for the eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem and we show how to deduce the Rellich-Pohozaev identity for the Finsler $p$-Laplacian from the Hadamard formula.

Social psychologists have noted that role identities contain both conventional (culturally influenced) and idiosyncratic (uniquely interpreted) meanings. To this point, this distinction has mostly been theoretical, and little empirical research exists that addresses how individuals define role identities in conventional or idiosyncratic ways. In this study, we create a scale that captures the conventional vs. idiosyncratic balance of role identity meanings and report the results of a study that examines the relationship between this balance and identity prominence and salience. We administer a survey to 934 study participants that measures the conventional/idiosyncratic balance, prominence, and salience of the student, worker, and friend role identities. Our results show that the more conventional one’s role identities, the more prominent and salient the identities.

The main motivation behind this paper is to study some structural properties of a non-associative structure as it hasn't attracted much attention compared to associative structures. In this paper, we introduce the concept of an ordered A*G**-groupoid and provide that this class is more generalized than an ordered AG-groupoid with left identity. We also define the generated left (right) ideals in an ordered A*G**-groupoid and characterize a (2; 2)-regular ordered A*G**-groupoid in terms of these ideals. We then study the structural properties of an ordered A*G**-groupoid in terms of its semilattices, (2; 2)-regular class and generated commutative monoids. Subsequently, compare -fuzzy left/right ideals of an ordered AG-groupoid and respective examples are provided. Relations between an -fuzzy idempotent subsets of an ordered A*G**-groupoid and its -fuzzybi-ideals are discussed. As an application of our results, we get characterizations of (2; 2)-regular ordered A*G**-groupoid in terms of semilattices and -fuzzy left (right) ideals. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.

An identity of degree 6 is given which is satisfied in an alternative monster and in a Skosyrskii algebra. It is proved that this identity is not a consequence of the alternativity identity, the commutativity identity, and an identity of nil-index 3. Also it is the only identity of degree at most 6, which is not a consequence of the identities mentioned. Therefore, the alternative monster and the Skosyrskii algebra are known to satisfy five independent identities.

An R d -valued Markov process X t ( x ) = ( X t 1 , x 1 , … , X t d , x d ) , t ≥ 0 , x ∈ R d is said to be multi-self-similar with index ( α 1 , … , α d ) ∈ [ 0 , ∞ ) d if the identity in law ( c i X t i , x i ∕ c i , t ≥ 0 ) 1 ≤ i ≤ d = ( d ) ( X c α t ( x ) , t ≥ 0 ) , where c α = ∏ i = 1 d c i α i , is satisfied for all c 1 , … , c d > 0 and all starting point x . Multi-self-similar Markov processes were introduced by Jacobsen and Yor (2003) in the aim of extending the Lamperti transformation of positive self-similar Markov processes to R + d -valued processes. This paper aims at giving a complete description of all R d -valued multi-self-similar Markov processes. We show that their state space is always a union of open orthants with 0 as the only absorbing state and that there is no finite entrance law at 0 for these processes. We give conditions for these processes to satisfy the Feller property. Then we show that a Lamperti-type representation is also valid for R d -valued multi-self-similar Markov processes. In particular, we obtain a one-to-one relationship between this set of processes and the set of Markov additive processes with values in { − 1 , 1 } d × R d . We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes.

Geranines were manufactured initially as textile dyes; they were made by coupling diazotized aromatic amines with sulfonated 1-naphthols. Most commonly encountered was geranine G, which for more than fifty years was thought to be derived from 1-naphthol monosulfonic acid, but later was considered to be derived from a 1-naphthol disulfonic acid. Currently, geranine G is thought to be a mixture of two isomers derived from 1-naphthol disulfonic acids. This species and others are described here by chemical structure and by other reference names and numbers where available. The occasional uses of geranines as biological stains are documented.

Let be a unitary left R-module on associative ring with identity. A submodule of is called -annihilator small if , where is a submodule of , implies that ann( )=0, where ann( ) indicates annihilator of in . In this paper, we introduce the concepts of -annihilator-coessential and - annihilator - coclosed submodules. We give many properties related with these types of submodules.

The usual homogeneous form of equality type in Martin-L\"of Type Theory contains identifications between elements of the same type. By contrast, the heterogeneous form of equality contains identifications between elements of possibly different types. This paper introduces a simple set of axioms for such types. The axioms are equivalent to the combination of systematic elimination rules for both forms of equality, albeit with typal (also known as "propositional") computation properties, together with Streicher's Axiom K, or equivalently, the principle of uniqueness of identity proofs., Comment: 13 pages

Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in R, then R contains one of them under various conditions.

The authors have proved an identity for a generalized integral operator via differentiable function. By applying the established identity and using the new and improved form of Holder’s integral inequality called Holder–Iscan integral inequality, some new generalized trapezoidal type integral inequalities have been obtain. Various special cases have been identified. Some applications to special means and error estimations of presented results have been analyzed.

Based on the work of Vershik (J Sov Math 59(5):1029–1040, 1992), we introduce two new combinatorial identities. We show how these identities can be used to prove a new hook-content identity. The main motivation for deriving this identity was a particular optimization problem in the field of quantum information processing.

Twist-structure representation theorems are established for De Morgan and Kleene lattices. While the former result relies essentially on the quasivariety of De Morgan lattices being finitely generated, the representation for Kleene lattices does not and can be extended to more general algebras. In particular, one can drop the double negation identity (involutivity). The resulting class of algebras, named semi-Kleene lattices by analogy with Sankappanavar’s semi-De Morgan lattices, is shown to be representable through a twist-structure construction inspired by the Cornish–Fowler duality for Kleene lattices. Quasi-Kleene lattices, a subvariety of semi-Kleene, are also defined and investigated, showing that they are precisely the implication-free subreducts of the recently introduced class of quasi-Nelson lattices.

Recently, Wang and Ma propose a conjecture associated with the possible generalization of Andrews-Warnaar identities. It is confirmed in this paper. As the applications of this conjecture, we prove that a family of series can be expressed by the partial theta functions and construct some new partial theta function identities.