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This paper investigates the way speakers construct their identities as representatives of their companies (institutional identity construction) in relation to the way they "project" an identity onto their audiences. The audience is "altercasted" (Weinstein and Deutschberger 1963) in the role of potential buyer of a product, thus evoking the standardized relational pair (Sacks 1972) of seller/buyer. The speaker then presents his company in the complementary role of seller of a product and as such a link is established between the identities of the speaker's company and the audience. This discursive co-construction of identities is crucial for the way both identities receive meaning. The two cases that are discussed here on the one hand show similarities in the general pattern of the two identity constructions and the way they are interwoven with one another, but on the other hand also demonstrate that there are many unique and diverging ways of constructing and linking these identities. ispartof: Pragmatics vol:18 issue:3 pages:491-509 status: published

This article describes how comedians and radio professionals in Corsica draw on a bilingual linguistic and metalinguistic cultural repertoire. In the context of Corsica’s history of language domination, language shift, and linguistic revitalization efforts, many of the products of language contact - mixed codes and compentences - are socially stigmatized. In the logic of dominant language ideologies, these mixed codes do not ‘count as’ language and depreciate individual speakers and collective identitities. Comic performances, it is argued, derive part of their tension and effect from the dominant view of languages as fixed and bounded codes which index single identities. Yet at the same time, performers make use of bilingual repertoires in ways that validate mixed language practices and identities. They do so by making maximal use of fluidity and indeterminacy in speaker stances towards mixed codes and identities. Bilingual comic performance also validates mixed codes and identities by evoking an ‘expert’ bilingual audience.

We study the subconvexity problem for GL 3 ( R ) L-functions in the t-aspect using integral representations by combining techniques employed by Michel–Venkatesh in their study of the corresponding problem for GL 2 with ideas from recent works of Munshi, Holowinsky–Nelson and Lin. Our main objective is to give – from the perspective of integral representations of L-functions and automorphic representation theory – a possible explanation of the origin of the “key identity” arising in the latter series of works.

We collect some interesting results for equations driven by the fractional relativistic Schrodinger operator ( − Δ + m 2 ) s with s ∈ ( 0 , 1 ) and m > 0 . More precisely, for the linear theory, we prove Holder-Schauder-Zygmund regularity results and a Kato's inequality. For the nonlinear theory, we obtain L ∞ -regularity, exponential decay, a Pohozaev-type identity, and a symmetry result for solutions of certain nonlinear fractional problems.

The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically Euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically Euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur type inequality valid in this non-compact setting which does not need restrictions on the Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.

In this article, we undertake a systematic, security-level analysis of momentum and reversal strategies in US Treasuries covering more than 40 years of data. We distinguish between what we call “market” and “self” time-series momentum (reversal) strategies and present an exact identity between these two time-series and the cross-sectional momentum (reversal) strategies. This identity helps us identify the sources of profitability of the various strategies and raises an interesting question regarding the contribution to the profitability of the first and second principal components of yield changes. We find that there exist look-back and investment periods for which momentum time series strategies (both “self” and “market”) give rise to statistically and economically significant positive Sharpe ratios; but we find that after adjusting for duration, the reversal cross-sectional strategy has an even larger Sharpe ratio and is profitable over a wider range of look-back and investment periods. We find an explanation for this finding in the mean-reverting properties of the yield-curve slope. Finally, we discover that the duration-adjusted reversal cross-sectional strategy can be successfully implemented in a long-only fashion.

A non-uniform (skewed) mixture of probability density functions occurs in various disciplines. One needs a measure of similarity to the respective constituents and its bounds. We introduce a skewed Jensen–Fisher divergence based on relative Fisher information, and provide some bounds in terms of the skewed Jensen–Shannon divergence and of the variational distance. The defined measure coincides with the definition from the skewed Jensen–Shannon divergence via the de Bruijn identity. Our results follow from applying the logarithmic Sobolev inequality and Poincaré inequality.

A new fractional integral identity is obtained involving n -th order differentiable functions and χ -Riemann–Liouville fractional integrals. Then, some associated estimates of upper bounds involving γ -preinvex functions are obtained. In order to relate some unrelated results, several special cases are discussed.

PageRank is an algorithm used in Internet search to score the importance of web pages. The aim of this paper is demonstrate some new results concerning the relationship between the concept of PageRank and automorphisms of a graph. In particular, we show that if vertices u and v are similar in a graph G (i.e., there is an automorphism mapping u to v), then u and v have the same PageRank score. More generally, we prove that if the PageRanks of all vertices in G are distinct, then the automorphism group of G consists of the identity alone. Finally, the PageRank entropy measure of several kinds of real-world networks and all trees of orders 10–13 and 22 is investigated.

Let R be a prime ring, Q the Utumi quotient ring of R and Ω the expansion closed set of products of automorphisms and skew derivations of Q. Let Λ ⊆ Ω ∖ Ω m , where m ≥ 0 , be such that for any w ∈ Λ ( g , h ) , where g , h are automorphisms of Q, w ( x y ) − g ( x ) w ( y ) − w ( x ) h ( y ) is a sum of terms with δ ( x ) and δ ′ ( y ) , where δ ∈ Λ | w | − 1 or δ ′ ∈ Λ | w | − 1 or δ , δ ′ ∈ Ω m ( Definition 3 ). Suppose that no monic linear identities with words in Λ ∪ Ω m involve words in Λ. We prove the following: (i) Λ extends to a basis of Ω ( Theorem 1 ). (ii) If a generalized polynomial φ with words in Λ ∪ Ω m is an identity, then so is the generalized polynomial obtained from φ by replacing every occurrence of ν ( x ) in φ by an arbitrary new variable y ν , x for ν ∈ Λ and variable x ( Theorem 2 ). Dedekind's Lemma asserts that distinct automorphisms σ i of a commutative field are independent maps over the field and hence, via linearization, σ i ( x j ) in distinct variables x j , considered as distinct maps for distinct ordered pair ( i , j ) , are transcendental over the field in the case of characteristic 0. (ii) above generalizes this to maps defined by w ∈ Λ with dependency and transcendency over the prime ring R.

Let R be an algebraic algebra over an infinite field and $$*$$ be an involution on R. We show that if the units of R, $${\mathcal {U}}(R)$$ , satisfy a $$*$$ -Laurent polynomial identity, then R satisfies a polynomial identity. Also, let G be a torsion group and F a field. As a generalization of Hartley’s Conjecture, in Broche et al. (Arch Math 111:353–367, 2018), it is shown that if $${\mathcal {U}}(FG)$$ satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra $$F[\alpha , \beta :\alpha ^2=\beta ^2=0]$$ , then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for $${\mathcal {U}}(FG)$$ to satisfy a Laurent polynomial identity.

The aim of this paper is to approximate the panjer distribution by the poisson and binomial distributions, where each bound is obtained by using the z-function and the Stein-Chen identity. For these bounds, it is indicated that a result of each of the Poisson and Binomial approximations yields a good approximation if both α and λ are small.

The purpose of this note is to characterize all the sequences of orthogonal polynomials $$(P_n)_{n\ge 0}$$ such that $$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$ where $$\,\mathrm {I}$$ is the identity operator, x defines a class of lattices with, generally, nonuniform step-size, and $$\triangle f(s)=f(s+1)-f(s)$$ . The proposed method can be applied to similar and to more general problems involving the mentioned operators, in order to obtain new characterization theorems for some specific families of classical orthogonal polynomials on lattices.

Around 1991, J. M. Borwein and P. B. Borwein introduced three cubic theta functions a(q), b(q) and c(q) and discovered many interesting identities associated with these functions. The cubic theta functions b(q) and c(q) have product representations and these representations were first established using the theory of modular forms. The first elementary proof of the product representation of b(q) was discovered in 1994 by the Borweins and F. G. Garvan using one of Euler’s identity. They then derived the product representation of c(q) using transformation formulas of Dedekind’s $$\eta (\tau )$$ and some elementary identities satisfied by a(q), b(q) and c(q). In this note, we present three proofs of the product representation of c(q) without the use of the transformation of Dedekind’s $$\eta $$ -function. We also discuss the connections between these proofs and the works of Baruah and Nath (Proc Am Math Soc 142:441–448, 2014) and Ye (Int J Number Theory 12(7):1791–1800, 2016). We also adopt the idea of the Borweins and Garvan to derive the product representation of Jacobi theta function $$\vartheta _4(0|\tau )$$ which leads to a proof of the Jacobi triple product identity.

"The authors have proved an identity for a generalized integral operator via di erentiable function with parameters. By applying the established identity, the generalized trapezium, midpoint and Simpson type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identi ed. Some applications of presented results to special means and new error estimates for the trapezium and midpoint quadrature formula have been analyzed. The ideas and techniques of this paper may stimulate further research in the eld of integral inequalities."

Let R be a commutative ring with identity and S⊈R a multiplicative subset. We define a proper ideal P of R disjoint from S to be S-primary if there exists an s∈S such that for all a,b∈R if ab∈P the...

UDC 517.5The authors have proved an identity with two parameters for differentiable function with respect to another function via generalized integral operator. By applying the established identity, the generalized trapezium, midpoint and Simpson type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means and new error estimates for the trapezium and midpoint quadrature formula have been analyzed. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities.

For more than a century, psychologists have been interested in how visual information can arouse emotions. Several studies have shown that rounded shapes evoke positive feelings due to their link with happy/baby-like expressions, compared with sharp angular shapes, usually associated with anger and threatening objects having negative valence. However, to date, no-one has investigated the preference to associate simple geometric shapes to personal identities, including one’s own, that of a close acquainted, or that of a stranger. Through 2 online surveys we asked participants to associate a geometric shape, chosen among a circle, a square and a triangle, to each of three identities, namely “you” (the self), “your best friend” or “a stranger”. We hypothesized that the circle would be more associated with the self, the square with the friend and the triangle with the stranger. Moreover, we investigated whether these associations are modulated by 3 personality traits: aggressivity, social fear and empathy. As predicted, we found that participants associate more often the circle with the self, both the circle and the square with the best friend, whereas they matched angular shapes (both the triangle and the square) to the stranger. On the other hand, the possibility that personality traits can modulate such associations was not confirmed. The study of how people associate geometric figures with the self or with other identities giving them an implicit socio-affective connotation, is interesting for all the disciplines interested in the automatic affective processes activated by visual stimuli.

This paper deals with the following nonlinear elliptic problem \begin{equation}\label{eq0.1} -\varepsilon^2\Delta u+\omega V(x)u=u^{p}+u^{2^{*}-1},\quad u> 0\quad\text{in}\ \R^N, \end{equation} where $\omega\in\R^{+}$, $N\geq 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/({N-2})$, $\varepsilon> 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (\ref{eq0.1}) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $\varepsilon$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.

We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.

We establish the nonlinear stability of N-soliton solutions of the modified Korteweg–de Vries (mKdV) equation. The N-soliton solutions are global solutions of mKdV behaving at (positive and negative) time infinity as sums of one-solitons with speeds 0 < c 1 c N . The proof relies on the variational characterization of N-solitons. We show that the N-solitons realize the local minimum of the (N + 1)th mKdV conserved quantity subject to fixed constraints on the N first conserved quantities. To this aim, we construct a functional for which N-solitons are critical points, we prove that the spectral properties of the linearization of this functional around an N-soliton are preserved on the extended timeline, and we analyze the spectrum at infinity of linearized operators around one-solitons. The main new ingredients in our analysis are a new operator identity based on a generalized Sylvester law of inertia and recursion operators for the mKdV equation.

We shall show a spectral mapping property of the generalized Fredholm spectrum in the more general context of Banach algebras. This is an extension of (Schmoeger in Demonstr Math XXXI(4):723–733, 1998, Theorem 3.3). More precisely, for $$a \in {\mathcal{A}}$$ which is a complex semisimple Banach algebra with identity, we shall prove that if g is a analytic function on a neighbourhood of the spectrum of a and if it has no zero in the generalized Fredholm spectrum of a then g(a) is a generalized Fredholm element in $${\mathcal{A}}$$ .

We prove a nonlinear analog of Picone’s identity for the ∆γ -Laplace operator. As an application, we give a Hardy-type inequality and the Sturmian comparison principle. We also establish strict monotonicity of the principal eigenvalue and degenerate elliptic system.

In the present paper, we propose a new method of derivation of number-theoretic identities which is applied to the proof of the multidimensional analogue of one of the Ramanujan identities. This method allows us to obtain new infinite series representations for the number $$\pi$$ , of the values of the Riemann zeta function, and of the $$L$$ -Dirichlet series at integer points.

The distributivity properties of fuzzy implication functions play an important role in fuzzy research. Making use of the solution of the autodistributivity functional equations, we give the necessary and sufficient conditions of two types of the distributivity property of fuzzy implications. It is essentially based on the weighted disjunctive operator, and new weighted and mean ( S , N ) -implication functions are introduced. Then using the pliant operators – where all the operators have a common generator function – we give a parametric form of the implication, namely the neutral value-dependent implication, and we show that some weakened properties of the fuzzy weighted implications are valid such as the identity principle, the ordering principle, the T-conditionality, the strong negation principle, etc. With this new concept, we use the fixed point of the negation as a threshold. We give a specific form of the new implication.

The van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature.

In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length n over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ . We give the MacWilliams type identity for the complete joint Jacobi polynomials of codes. We also introduce the concepts of the average Jacobi polynomial and the average complete joint Jacobi polynomial over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ . We give a representation of the average of the complete joint Jacobi polynomials of two linear codes of length n over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ in terms of the compositions of n and its distribution in the codes. Further we present a generalization of the representation for the average of the $$(g+1)$$ -fold complete joint Jacobi polynomials of codes over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ . Finally, we give the notion of the average Jacobi intersection number of two codes.

Let R be a commutative ring with identity element. The concept of 2-absorbing ideal was introduced by Badawi (Bull Aust Math Soc 75:417–429, 2007) as a generalization of the notion of prime ideal. Weakly prime ideals introduced by Anderson (Houst J Math 29(4):831–840 2003) are also generalizations of prime ideals. The concept of a weakly prime ideal was generalized to the concept of weakly 2-absorbing ideal in Badawi and Ece (J Korean Math Soc 52(1):97–111, 2015). Up till now research on these topics mainly concentrated on commutative rings. The concept of a 2-absorbing ideal in a non-commutative ring was introduced by Groenewald (J Algebra Number Theory Appl 40(5):855–867, 2018) as a generalization of the notion of prime ideal. A weakly prime ideal for non-commutative rings was introduced by Hirano et al. (Bull Korean Math Soc 47(5):1077–1087, 2010). In this note we introduce the notion of a weakly 2-absorbing ideal as a generalization of a weakly prime ideal in a non-commutative ring with identity and show that many of the results in commutative rings also hold in non-commutative rings with identity.

Combining McShane's identity on a hyperbolic punctured torus with Schmutz's work on the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent to the identity \begin{equation} \sum_{n=1}^\infty \, \left( 3- L_n \right) \, = \, 4 - \varphi - \sqrt 2 \end{equation} where $L_n$ is the $n$th Lagrange number and $\varphi=\frac{1+\sqrt5}2$ is the golden ratio., Comment: 5 pages, 2 figures, comments welcome!

Quantum private information retrieval (QPIR) is a protocol in which a user retrieves one of multiple files from $ \mathsf {n}$ non-communicating servers by downloading quantum systems without revealing which file is retrieved. As variants of QPIR with stronger security requirements, symmetric QPIR is a protocol in which no other files than the target file are leaked to the user, and $ \mathsf {t}$ -private QPIR is a protocol in which the identity of the target file is kept secret even if at most $ \mathsf {t}$ servers may collude to reveal the identity. The QPIR capacity is the maximum ratio of the file size to the size of downloaded quantum systems, and we prove that the symmetric $ \mathsf {t}$ -private QPIR capacity is $\min \{1,2( \mathsf {n}- \mathsf {t})/ \mathsf {n}\}$ for any $1\leq \mathsf {t} . We construct a capacity-achieving QPIR protocol by the stabilizer formalism and prove the optimality of our protocol. The proposed capacity is greater than the classical counterpart.

Let be an associative ring with identity and let be a unitary left -module. Let be a non-zero submodule of .We say that is a semi- - hollow module if for every submodule of such that is a semi- - small submodule ( ). In addition, we say that is a semi- - lifting module if for every submodule of , there exists a direct summand of and such that The main purpose of this work was to develop the properties of these classes of module.

Exploring teachers’ identities while teaching in the Fourth Industrial Revolution (4IR) is essential. Such assists teachers to reflect, improving their teaching through various networks. Identities are conscious thoughts interrogating subconscious thoughts that drive teachers to understand their personal needs. Eleven teachers, registered for a Master of Education degree at a university in South Africa, were purposively selected for this study. The objective of this study was to understand teachers’ identities while teaching mathematics in the 4IR. Focus-group discussions, reflective activities, and one-on-one semi-structured interviews, framed by a pragmatic case study, were used for data collection. The teachers’ identities revealed that the Three Tree Rings Theory with Currere was useful when applied as the teaching framework. Such application of the theories generated three categories of teaching identities within the 4IR. Categories were societal, professional, and personal identities. The university at which these participants were registered prescribed Moodle. However, the students mostly used social media sites such as Facebook and WhatsApp. As a result, students used digital technologies even during face-to-face classes. The study concluded that, although there were elements of both personal and professional identities, the societal identity was that which drove teaching. Consequently, this study recommends that teachers should learn to reflect on, in, and for their actions, in order to address the ‘what’, ‘how’, ‘who’, and ‘why’ questions of education, natural identity, and the 4IR/5IR.

Let $$\overline{ A}_\ell (n)$$ be the number of $$\ell $$ -regular overpartitions of n, i.e., overpartitions of n into parts not divisible by $$\ell $$ . Let $$\overline{ B}_\ell (n)$$ be the number of almost $$\ell $$ -regular overpartitions of n, i.e., overpartitions of n in which none of its overlined parts is divisible by $$\ell $$ . In this paper, we study the connections between $$\overline{ A}_3(n)$$ , respectively $$\overline{ B}_3(n)$$ , and the singular overpartition functions $$\overline{ C}_{12,5}(n)$$ and $$\overline{ C}_{12,1}(n)$$ which count the number of overpartitions into parts not divisible by 12 and in which only parts congruent to $$\pm 5 \pmod {12}$$ , respectively $$\pm 1 \pmod {12}$$ , may be overlined. We give a combinatorial proof for the the surprising identity $$\overline{ C}_{12,5}(n)=\overline{ C}_{12,1}(n-1)$$ . We also provide a linear homogeneous recurrence relation for $$\overline{ B}_3(n)$$ and give an alternate combinatorial interpretation for $$\overline{ B}_3(n)$$ .

Marginalization based on geographic-isolation has serious repercussions on construction of identities. Identities in geographically isolated borderlands in Jammu and Kashmir are doubly-marginalized...

We consider the operator Δ(α,β)−I, where Δ(α,β) is the three term recurrence relation for the Jacobi polynomial Rn(α,β)(x) normalized at the point x = 1 and I is the identity operator acting on N. ...

We construct infinite families of closed hyperbolic surfaces that are local maxima for the systole function on their respective moduli spaces. The systole takes values along a linearly divergent sequence $(L_n)_{n\geq 1}$ at these local maxima. The only surface corresponding to $L_1\approx 3.057$ is the Bolza surface in genus $2$. For every genus $g\geq 13$, we obtain either one or two local maxima in $\mathcal{M}_g$ whose systoles have length $L_2\approx 5.909$. For each $n\geq 3$, there is an arithmetic sequence of genera $(g_k)_{k\geq 1}$ such that the number of local maxima of the systole function in $\mathcal{M}_{g_k}$ at height $L_n$ grows super-exponentially in $g_k$. In particular, level sets of the systole function can have an arbitrarily large number of connected components. Many of the surfaces we construct have trivial automorphism group, and are the first examples of local maxima with this property., Comment: 39 pages, 19 figures

Let $${\mathcal {O}}_r(n)$$ be the set of r-regular partitions of n, $${\mathcal {D}}_r(n)$$ the set of partitions of n with parts repeated at most $$r-1$$ times, $${\mathcal {O}}_{1,r}(n)$$ the set of partitions with exactly one part (possibly repeated) divisible by r, and let $${\mathcal {D}}_{1,r}(n)$$ be the set of partitions in which exactly one part appears at least r times. If $$E_{r, t}(n)$$ is the excess in the number of parts congruent to $$t \pmod r$$ in all partitions in $${\mathcal {O}}_r(n)$$ over the number of different parts appearing at least t times in all partitions in $${\mathcal {D}}_r(n)$$ , then $$E_{r, t}(n) = |{\mathcal {O}}_{1,r}(n)| = |{\mathcal {D}}_{1,r}(n)|$$ . We prove this analytically and combinatorially using a bijection due to Xiong and Keith. As a corollary, we obtain the first Beck-type identity, i.e., the excess in the number of parts in all partitions in $$\mathcal {O}_r(n)$$ over the number of parts in all partitions in $$\mathcal {D}_r(n)$$ equals $$(r - 1)|\mathcal {O}_{1,r}(n)|$$ and also $$(r - 1)|\mathcal {D}_{1,r}(n)|$$ . Our work provides a new combinatorial proof of this result that does not use Glaisher’s bijection. We also give a new combinatorial proof based of the Xiong–Keith bijection for a second Beck-type identity that has been proved previously using Glaisher’s bijection.

A derivation (of order 1) satisfies the reduction formula $$f(x^k) = kx^{k-1}f(x)$$ for any integer k. In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation f of order n and every positive integer k we find an explicit formula for $$f(x^k)$$ as a linear combination of $$x^{k-1}f(x),x^{k-2}f(x^2), \ldots , x^{k-n}f(x^n)$$ . The proof hinges on the hypergeometric identity $$\begin{aligned} \sum _{k \ge 0} (-1)^k \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+d+1-k\\ n-j\end{array}}\right) \left( {\begin{array}{c}d+j-k\\ j\end{array}}\right) = \left( {\begin{array}{c}n\\ j\end{array}}\right) \end{aligned}$$ for any positive integer n, nonnegative integer d, and integer j satisfying $$0 \le j \le n$$ . We prove this identity via the WZ-method.

In this paper, the authors introduce the (p,q)-trapezoidal integral inequalities, which are the (p,q)-analogues of the recently introduced q-trapezoidal integral inequalities. We derive a new (p,q)-integral identity for twice (p,q)-differentiable function. Utilizing this as an auxiliary result, we establish several new (p,q)-trapezoidal type integral inequalities for the function whose absolute value of twice (p,q)-derivative is (η1,η2)-convex functions. Some special means of real numbers are also given. At the end, we give brief conclusion. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.

The generalization of McShane’s identity for the quotient surface $$\Sigma_q$$ of any Hecke group $$H_q$$ , where q is an integer greater than 3.

The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. Recall that an element x of R is an exact zero-divisor if there exists a non-zero element y of R such that $$Ann(x) = Ry$$ and $$Ann(y) = Rx$$ . As in Lalchandani (International J. Science Engineering and Management (IJSEM) 1(6): 14-17, 2016), for a ring R, we denote the set of all exact zero-divisors of R by EZ(R) and $$EZ(R)\backslash \{0\}$$ by $$EZ(R)^{*}$$ . Let R be a ring. In the above mentioned article, Lalchandani introduced and studied the properties of a graph denoted by $$E\Gamma (R)$$ , which is an undirected graph whose vertex set is $$EZ(R)^{*}$$ and distinct vertices x and y are adjacent in $$E\Gamma (R)$$ if and only if $$Ann(x) = Ry$$ and $$Ann(y) = Rx$$ . Let R be a reduced ring such that $$EZ(R)^{*}\ne \emptyset $$ . The aim of this article is to study the interplay between the graph-theoretic properties of $$E\Gamma (R)$$ and the ring-theoretic properties of R.

In the first part of the paper, we use states on $C^{*}$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^{*}$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^{*}$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^{*}$-module under the assumption that their inner product has a negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff–James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^{*}$-modules.

This paper introduces a natural deduction calculus for intuitionistic logic of belief$$\mathsf {IEL}^{-}$$ IEL - which is easily turned into a modal$$\lambda $$ λ -calculus giving a computational semantics for deductions in $$\mathsf {IEL}^{-}$$ IEL - . By using that interpretation, it is also proved that $$\mathsf {IEL}^{-}$$ IEL - has good proof-theoretic properties. The correspondence between deductions and typed terms is then extended to a categorical semantics for identity of proofs in $$\mathsf {IEL}^{-}$$ IEL - showing the general structure of such a modality for belief in an intuitionistic framework.

For the Fibonacci sequence the identity $$F_n^2+F_{n+1}^2 = F_{2n+1}$$ holds for all $$n \ge 0$$ . Let $${\mathcal {X}}:= (X_\ell )_{\ell \ge 1}$$ be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation $$X^2-dY^2=\pm 1$$ corresponding to a nonsquare integer $$d>1$$ . In this paper, we investigate all positive nonsquare integers d for which there are at least two positive integers X and $$X'$$ of $$\mathcal {X}$$ having a representation as the sum of xth powers of two consecutive terms of a Lucas sequence. Then we solve this problem for Fibonacci numbers.

In this paper, we have established some new integral identities connected with the left-hand side of Hermite–Hadamard inequality. By using this identity, we have obtained some new bounds for functions whose derivatives in absolute values are s-convex.

In this paper, we construct extensions of the differential field obtained from Halphen’s system of ordinary differential equations (ODEs) using quasi-modular forms of levels two, four, and six. In addition, we derive analogs of Jacobi’s derivative formula and Jacobi’s quartic identity using theta functions with rational characteristics. Moreover, we also treat representability as sums of squares, sums of triangular numbers and mixed sums of squares and triangular numbers.

We prove a Pohozaev-type identity for both the problem (−Δ+m2)su=f(u) in RN and its harmonic extension to R+N+1 when 0

In this paper, the author establishes a set of three new theta-function identities involving Rα, Rβ and Rm functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper we answer a open question of Srivastava et al [33], and established relations in terms of Rα, Rβ and Rm (for m = 1, 2, 3), and q-products identities. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities.

The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure A = 〈A, ·,→, 1, R〉, where (A, ·) is a commutative monoid with the identity 1 as the top element in this ordered monoid under a quasi-order R. The author introduced and analyzed the concepts of filters and implicative filters in this type of algebraic structures. In this article, the concept of weak implicative filters in a quasi-ordered residuated system is introduced as a continuation of previous researches. Also, some conditions for a filter of such system to be a weak implicative filter are listed.

In this paper, we use an identity connecting a modified [Formula: see text]-Bessel function and a [Formula: see text] function to give [Formula: see text]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [Formula: see text]-version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, [Formula: see text]-Lommel and [Formula: see text]-Bessel polynomials. We introduce and study [Formula: see text]-analogues of the spherical Bessel functions.