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College student athletes are at risk for mental health difficulties. Athletic identity – the degree of strength and exclusivity to which a person identifies with the athletic role – is associated with health risk behaviors and mental health concerns broadly among student athletes. Given a variety of barriers to college athletes disclosing mental health difficulties, measuring athletic identity may be an indirect way to screen athletes for being at-risk for mental health concerns. The current study examined the factor structure of an increasingly popular measure of athletic identity (the AIMS-Plus) across two samples of self-identified athletes (Nstudy1 = 201; 60.2% male; 61.19% White; 21.39% Hispanic; NStudy2 = 175; 45.29% male; 70.86% White; 20.57% Hispanic) to offer a peer-reviewed validation of the measure. Results suggest that the factor structure of the AIMS-Plus was not supported; however, both a modified version of the AIMS-Plus (the AIMS-Plus-R) and the 7-item AIMS may be acceptable. Continued work refining measures of athletic identity is needed. [ABSTRACT FROM AUTHOR]

We study the functional identity G (x) f (x) = H (x) on a division ring D , where f : D → D is an additive map and G (X) ≠ 0 , H (X) are generalized polynomials in the variable X with coefficients in D. Precisely, it is proved that either D is finite-dimensional over its center or f is an elementary operator. Applying the result and its consequences, we prove that if D is a noncommutative division ring of characteristic not 2, then the only solution of additive maps f , g on D satisfying the identity f (x) = x n g (x − 1) with n ≠ 2 a positive integer is the trivial case, that is, f = 0 and g = 0. This extends Catalano and Merchán's result in 2023 to get a complete solution. [ABSTRACT FROM AUTHOR]

Here, we study the extension of p-trigonometric functions sinp and cosp family in complex domains and p-hyperbolic functions sinhp and the coshp family in hyperbolic complex domains. These functions satisfy analogous relations as their classical counterparts with some unknown properties. We show the relationship of these two classes of special functions viz. p-trigonometric and p-hyperbolic functions with imaginary arguments. We also show many properties and identities related to the analogy between these two groups of functions. Further, we extend the research bridging the concepts of hyperbolic and elliptical complex numbers to show the properties of logarithmic functions with complex arguments. [ABSTRACT FROM AUTHOR]

Combinatorial proofs of binomial identities involve establishing an identity by arguing that each side enumerates a certain set of outcomes. In this paper, we share results from interviews with experienced provers (mathematicians and upper-division undergraduate mathematics students) and examine one particular aspect of combinatorial proof, namely the kinds of contexts that experienced provers used to establish combinatorial proofs of binomial identities. Our findings show that overall, our participants used a variety of contexts in their work; we also demonstrate ways in which previous experiences influenced the contexts they chose, and we offer some instances in which features of a context supported their combinatorial proof production. We offer some theoretical implications of our work, and we conclude with a discussion of limitations and avenues for future work. [ABSTRACT FROM AUTHOR]

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained. [ABSTRACT FROM AUTHOR]

In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2) a1,b1 (p1, k), where a1, a2, b1, b2 are given complex numbers, a1, a2 ?= 0, and p1, p2 are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1,p2 (x1, x2). By using results for S(1,x2,p2) 1,x1 (p1, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case. [ABSTRACT FROM AUTHOR]

We study lattice paths by combinatorial methods on the positive lattice. We give some identity that produces the functional equations and generating functions to counting the lattice paths on or below the main diagonal. Also, we consider the subdiagonal lattice paths in relation to lower triangular arrays. This presents a Riordan array in conjunction with the columns of the matrix of the coefficients of certain formal power series by implying an infinite lower triangular matrix . We derive new combinatorial interpretations in terms of restricted lattice paths for some Riordan arrays. [ABSTRACT FROM AUTHOR]

This paper focused on Riemannian curvature tensor R of C9 −manifolds. The components of covariant derivative of R determined on the space of G −structure. There are fifteen non-zero of such components and the others components given by the symmetry property and Bianchi identity of R. According to these components, the conditions on curvature tensor R of C9 −manifolds to be has inheritance symmetry established. These conditions summarized by five equations that have common arbitrary scalar function ᴪ. [ABSTRACT FROM AUTHOR]

Organisms must adapt to environmental stresses to ensure their survival and prosperity. Different types of stresses, including thermal, mechanical, and hypoxic stresses, can alter the cellular state that accompanies changes in gene expression but not the cellular identity determined by a chromatin state that remains stable throughout life. Some tissues, such as adipose tissue, demonstrate remarkable plasticity and adaptability in response to environmental cues, enabling reversible cellular identity changes; however, the mechanisms underlying these changes are not well understood. We hypothesized that positive and/or negative "Integrators" sense environmental cues and coordinate the epigenetic and transcriptional pathways required for changes in cellular identity. Adverse environmental factors such as pollution disrupt the coordinated control contributing to disease development. Further research based on this hypothesis will reveal how organisms adapt to fluctuating environmental conditions, such as temperature, extracellular matrix stiffness, oxygen, cytokines, and hormonal cues by changing their cellular identities. [ABSTRACT FROM AUTHOR]

Let K be a finite field of characteristic 2, and U T 2 : = U T 2 (K) be the Lie algebra of 2 × 2 upper triangular matrices over K with the multiplication x ∘ y = x y + y x = x y − y x. In this paper, we exhibit a finite basis of graded identities for the variety of Lie algebras generated by U T 2 for any grading and show that it has the Specht property. It is important to highlight that the technique used in order to solve the Specht problem is independent of the characteristic of the field and also of its cardinality. [ABSTRACT FROM AUTHOR]

Modern glossarial practices remain rooted in a definitional project that suppresses the semantic relations and conflicts that exist within many early modern signifiers. In this essay, I challenge the premise that a responsible gloss must render apprehensible what might otherwise prove opaque to modern readers. Tracing out the intersectional histories, relations, and relational opacities that can be evinced by attending to the early modern "lowne," I demonstrate how conventional glossarial practices have limited our understanding not only of the glossed term and the texts within which it circulates, but of the relations between the intersectional identities and social affects that the term conjures in early modern England. My methodology therefore insists that we take seriously the historical, political, and pedagogical affordances of writing glosses that "confound" rather than clarify meaning: that reveal a mixture of elements precisely where we might otherwise expect a singular sense; and that aim to curate, rather than quell, the kinds of cognitive disorientation, emotional embarrassment, and astonishment that may attend such encounters. [ABSTRACT FROM AUTHOR]

Let D be an alternative division ring with characteristic different from two. The purpose of this paper is to characterize additive mappings f , g : D → D satisfying certain identities studied by Vukman, Brešar, and Catalano previously on an associative division ring. We also present some new identities concerning Lie and Jordan products, and provide a complete description of commuting maps on octonion algebras. [ABSTRACT FROM AUTHOR]

Let K be a field and let J n , k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n − k over K. Then one can consider the decomposition J n , k = B n − k ⊕ D k , where B n − k represents the corresponding Jordan algebra, denoted as B n − k = K ⊕ V. In this algebra, the restriction of b on the (n − k) -dimensional subspace V is non-degenerate, while D k accounts for the degenerate part of J n , k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for J n , k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T -ideal of polynomial identities of J n , k. Over a field of characteristic zero, we also prove that the ideal of identities of J n , k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z 2 -grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in [8] in case of infinite dimensional Jordan algebras with trace. [ABSTRACT FROM AUTHOR]

The liver is one of the few organs that retain the capability to regenerate in adult mammals. This regeneration process is mainly facilitated by the dynamic behavior of hepatocytes, which are the major functional constituents in the liver. In response to liver injury, hepatocytes undergo remarkable alterations, such as reprogramming, wherein they lose their original identity and acquire properties from other cells. This phenomenon of hepatocyte reprogramming, coupled with hepatocyte expansion, plays a central role in liver regeneration, and its underlying mechanisms are complex and multifaceted. Understanding the fate of reprogrammed hepatocytes and the mechanisms of their conversion has significant implications for the development of innovative therapeutics for liver diseases. Herein, we review the plasticity of hepatocytes in response to various forms of liver injury, with a focus on injury‐induced hepatocyte reprogramming. We provide a comprehensive summary of current knowledge on the molecular and cellular mechanisms governing hepatocyte reprogramming, specifically in the context of liver regeneration, providing insight into potential applications of this process in the treatment of liver disorders, including chronic liver diseases and liver cancer. [ABSTRACT FROM AUTHOR]

Harmonic numbers and generalized harmonic numbers have been studied in connection with combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. We introduce Lerch-harmonic numbers which generalize the harmonic numbers and the generalized harmonic numbers. The aim of this note is to derive some identities expressing certain finite sums as the infinite sums involving the Lerch-harmonic numbers. Then, by taking limits of such identities we obtain the corresponding infinite sums of the finite sums as the infinite sums involving the Lerch transcendents. [ABSTRACT FROM AUTHOR]

Dedekind-type DC sums and their properties are defined in terms of Euler functions. Ma et al. recently introduced poly-Dedekind-type DC sums and demonstrated that they satisfy a reciprocity relation. In this paper, we introduce the degenerate poly-Euler polynomials and numbers, and we also consider the reciprocity relations of degenerate poly-Dedekind-type DC sums. Equivalently, several properties and identities of degenerate poly-Euler functions are obtained. [ABSTRACT FROM AUTHOR]

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable. [ABSTRACT FROM AUTHOR]

Recently, Kim-Kim introduced the truncated degenerate Bell polynomials and numbers. In this paper, weintroduce the truncated Lah-Bell polynomials and numbers. We obtain some identities, recurrence relations and properties. Furthermore, we also introduce the truncated modified Lah-Bell polynomials and numbers, truncated degenerate Lah-Bell polynomials and numbers and we derive some properties and identities involving with those polynomials and numbers. [ABSTRACT FROM AUTHOR]

We introduce the generalized degenerate Euler-Genocchi polynomials as a degenerate version of the Euler-Genocchi polynomials. In addition, we introduce their higher-order version, namely the generalized degenerate Euler-Genocchi polynomials of order a, as a degenerate version of the generalized Euler-Genocchi polynomials of order a. The aim of this paper is to study certain properties and identities involving those polynomials, the generalized falling factorials, the degenerate Euler polynomials of order a, the degenerate Stirling numbers of the second kind, and the 'alternating degenerate power sum of integers'. [ABSTRACT FROM AUTHOR]

In this paper, we provide a generating function for mix type Apostol-Genocchi polynomials of order η associated with Bell polynomials. We also derive certain important identities of Apostol Genocchi polynomials of order η based on Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some correlation with Stirling numbers and their special instances. Moreover, we discover some symmetric identities and their related known results. [ABSTRACT FROM AUTHOR]

This document discusses recent advances in special functions and their applications across various scientific and engineering disciplines. It highlights the importance of special functions in fields such as mathematics, physics, engineering, statistics, astronomy, computer science, economics, chemistry, biology, medicine, materials science, and environmental science. The document provides summaries of 13 articles that explore topics such as Fibonacci polynomials, Appell and Horn functions, generalized hypergeometric functions, Lagrange-based hypergeometric Bernoulli polynomials, hypergeometric series, Humbert functions, integral formulas, twisted Catalan numbers, incomplete gamma function, existence theorems, finite sum identities, Euler polynomials, and q-generalized tangent-based Apostol-type Frobenius-Euler polynomials. These articles contribute to the understanding and application of special functions in various research realms. [Extracted from the article]

Recently, Kim-Kim introduced the degenerate Whitney numbers of the first and second kind involving the degenerate Dowling polynomials and numbers. In this paper, we introduce the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and derive some identities involving these polynomials. We also obtain the generating functions, expressions, and recurrence relations for the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and other special polynomials and numbers. [ABSTRACT FROM AUTHOR]

We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra 픸 for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to 픸 , which is also known as the Lorch derivative with respect to 픸 , and recently simply called 픸 -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity. [ABSTRACT FROM AUTHOR]

The aim of this paper is to further study some properties and identities on the degenerate Fubini and the degenerate Bell polynomials which are degenerate versions of the Fubini and the Bell polynomials, respectively. Especially, we find several expressions for the generating function of the sum of the values of the generalized falling factorials at positive consecutive integers. [ABSTRACT FROM AUTHOR]

As a powerful tool for models of quantum computing, q-calculus has drawn the attention of many researchers in the discipline of special functions. In this paper, we present new properties and characterize q-Bessel functions of the first kind using some identities of q-calculus. The results presented in this article help us to obtain new expression results related to q-special functions. New summation and integral representations for q-Bessel functions of the first kind are also established. A few examples are also provided to demonstrate the effectiveness of the proposed strategy. [ABSTRACT FROM AUTHOR]

In this paper, we first prove two new identities concerning the q-integers and then for these integers we derive several divisibility properties based on these identities and on properties of the q-integers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce type 2 degenerate poly-Euler polynomials and numbers, briefly called degenerate poly-Euler polynomials and numbers, by using the modified degenerate polyexponential function and derive several properties on these polynomials and numbers. In the last section, we also consider the type 2 degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities of these polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials. [ABSTRACT FROM AUTHOR]

We construct new degenerate (p, q)-Euler polynomials and find some properties and identities of these polynomials. It can be seen that the degenerate (p, q)-Euler polynomials which is related to Euler polynomials have various relation with other polynomials. [ABSTRACT FROM AUTHOR]

In deep neural networks, the activation function is an important component. The most popular activation functions at the moment are Sigmoid, Sin, rectified linear unit (ReLU), and some variants of ReLU. However, each of them has its own weakness. To improve the network fitting and generalization ability, a new activation function, TSin, is designed. The basic design idea for TSin function is to rotate the Sin function 45° counterclockwise and then finetune it to give it multiple better properties needed as an activation function, such as nonlinearity, global differentiability, unsaturated property, zero-centered property, monotonicity, quasi identity transformation property, and so on. The first is a theoretical derivation of TSin function by formulas. Then three experiments are designed for performance test. The results show that compared with some popular activation functions, TSin has advantages in terms of training stability, convergence speed, and convergence precision. The study of TSin not only provides a new choice of activation function in deep learning but also provides a new idea for activation function design in the future. [ABSTRACT FROM AUTHOR]

In the last letter to Hardy, Ramanujan [ Collected Papers , Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962] introduced seventeen functions defined by q-series convergent for |q|<1 with a complex variable q, and called these functions "mock theta functions". Subsequently, mock theta functions were widely studied in the literature. In the survey of B. Gordon and R. J. McIntosh [ A survey of classical mock theta functions , Partitions, q-series, and modular forms, Dev. Math., vol. 23, Springer, New York, 2012, pp. 95–144], they showed that the odd (resp. even) order mock theta functions are related to the function g_3(x,q) (resp. g_2(x,q)). These two functions are usually called "universal mock theta functions". D. R. Hickerson and E. T. Mortenson [Proc. Lond. Math. Soc. (3) 109 (2014), pp. 382–422] expressed all the classical mock theta functions and the two universal mock theta functions in terms of Appell–Lerch sums. In this paper, based on some q-series identities, we find four functions, and express them in terms of Appell–Lerch sums. For example, \begin{equation*} 1+(xq^{-1}-x^{-1}q)\sum _{n=0}^{\infty }\frac {(-1;q)_{2n}q^{n}}{(xq^{-1},x^{-1}q;q^2)_{n+1}}=2m(x,q^2,q). \end{equation*} Then we establish some identities related to these functions and the universal mock theta function g_2(x,q). These relations imply that all the classical mock theta functions can be expressed in terms of these four functions. Furthermore, by means of q-series identities and some properties of Appell–Lerch sums, we derive four radial limit results related to these functions. [ABSTRACT FROM AUTHOR]

College student athletes are at risk for mental health difficulties. Athletic identity – the degree of strength and exclusivity to which a person identifies with the athletic role – is associated with health risk behaviors and mental health concerns broadly among student athletes. Given a variety of barriers to college athletes disclosing mental health difficulties, measuring athletic identity may be an indirect way to screen athletes for being at-risk for mental health concerns. The current study examined the factor structure of an increasingly popular measure of athletic identity (the AIMS-Plus) across two samples of self-identified athletes (Nstudy1 = 201; 60.2% male; 61.19% White; 21.39% Hispanic; NStudy2 = 175; 45.29% male; 70.86% White; 20.57% Hispanic) to offer a peer-reviewed validation of the measure. Results suggest that the factor structure of the AIMS-Plus was not supported; however, both a modified version of the AIMS-Plus (the AIMS-Plus-R) and the 7-item AIMS may be acceptable. Continued work refining measures of athletic identity is needed. [ABSTRACT FROM AUTHOR]

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients. [ABSTRACT FROM AUTHOR]

The absence of discussions of identity that consider both subjective/personal and social aspects of identity is an important concern in mathematics education research. This paper proposes a framework to analyse the mathematical identities offered to and constructed by learners during peer interactions, by considering personal and social identities. To exemplify the framework, we discuss how two pairs of secondary school learners, who were engaged with similarly by their teachers and who performed similarly academically, interacted with each other and offered and constructed micro-identities during moments of a lesson, and macro-identities across lessons. We show that when learners were offered identities by a peer as higher, lower or equal status, they constructed their identities by affiliating with or resisting the identities offered by their peer. Their interactions and the identities that they constructed were informed by their personal identities, including their emotions when learning mathematics, and their social identities, including their choices to occupy certain roles in the classroom community. [ABSTRACT FROM AUTHOR]

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. [ABSTRACT FROM AUTHOR]

We study a critical Grushin-type equation by applying Lyapunov-Schmidt reduction method and obtain the existence of infinitely many positive multi-bubbling solutions with cylindrical symmetry. Particularly, since the concentration points here can be saddle points of some function related to the potential, we locate the concentration points of these solutions by local Pohozaev identities rather than estimating the derivatives of the reduced functional as usual. [ABSTRACT FROM AUTHOR]

In this paper we extend general Hardy's inequality by appropriately combining Montgomery's identity and Green functions. Related Grüss and Ostrowski-type inequalities are also derived. [ABSTRACT FROM AUTHOR]

The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certain trilinear relations that we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its SL(2, ℂ) covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemann ϑ-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers. [ABSTRACT FROM AUTHOR]

In this paper, based on a given parameterized identity that generates a quadrature rule family similar to Simpson's second formula, we establish some new Simpson-like type inequalities for functions with bounded as well as Lipchitzian derivatives from which we can deduce the famous 3/8-Simpson's inequality. The study concludes with an application example from management science. [ABSTRACT FROM AUTHOR]

The objective of this study is to identify novel quantum midpoint-type inequalities for twice  q -differentiable functions by utilizing Mercer's approach. We introduce a new auxiliary variant of the quantum Mercer midpoint-type identity related to twice  q -differentiable functions. By applying the theory of convex functions to this identity, we introduce new bounds using well-known inequalities, such as H"older's inequality and power-mean inequality. We provide explicit examples along with graphical demonstrations. The findings of this study explain previous studies on midpoint-type inequalities. Analytic inequalities of this type, as well as related strategies, have applications in various fields where symmetry plays an important role. [ABSTRACT FROM AUTHOR]

Let D be a noncommutative division ring. In a recent paper, Lee and Lin proved that if char D ≠ 2 , the only solution of additive maps f , g on D satisfying the identity f (x) = x n g (x − 1) on D ∖ { 0 } with n ≠ 2 a positive integer is the trivial case, that is, f = 0 and g = 0. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer n if char D = 2. [ABSTRACT FROM AUTHOR]

We provide a family of isolated tangent to the identity germs f:(C3,0)→(C3,0)$f:(\mathbb {C}^3,0) \rightarrow (\mathbb {C}^3,0)$ which possess only degenerate characteristic directions, and for which the lift of f$f$ to any modification (with suitable properties) has only degenerate characteristic directions. This is in sharp contrast with the situation in dimension 2, where any isolated tangent to the identity germ f$f$ admits a modification where the lift of f$f$ has a non‐degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of f$f$, showing that this phenomenon is not related to the non‐existence of complex separatrices for vector fields of Gomez‐Mont and Luengo. Finally, we describe the set of formal f$f$‐invariant curves, and the associated parabolic manifolds, using the techniques recently developed by López‐Hernanz et al. [ABSTRACT FROM AUTHOR]

Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a given differential form is invariant, with special attention paid to the case of the existence of invariant volumes. Particular attention is given to impact systems where the continuous dynamics are Lagrangian and subject to nonholonomic constraints. A celebrated result for volume-preserving dynamical systems is Poincaré recurrence. In order to be recurrent, trajectories need to exist for long periods of time, which can be controlled in continuous-time systems through e.g. compactness. For hybrid systems, an additional mechanism can occur which breaks long-time existence: Zeno (infinitely many discrete transitions in a finite amount of time). We demonstrate that the existence of a smooth invariant volume severely inhibits Zeno behavior; hybrid systems with the "boundary identity property" along with an invariant volume-form have almost no Zeno trajectories (although Zeno trajectories can still exist). This leads to the result that many billiards (e.g. the classical point, the rolling disk, and the rolling ball) are recurrent independent on the shape of the compact table-top. [ABSTRACT FROM AUTHOR]

This article is inspired by the reciprocal Catalan sums associated with problem 11765, proposed by David Beckwith and Sag Harbor. For this reason, partial derivative equations, the first-order linear differentiation equation and integral representations for series and generating functions for reciprocal Catalan-type sums containing the Catalan-type numbers are constructed. Some special values of these series and generating functions, which are given solutions of problem 11765, are found. Partial derivative equations of the generating function for the Catalan-type numbers are given. By using these equations, recurrence relations and derivative formulas involving these numbers are found. Finally, applying the p-adic Volkenborn integral to the Catalan-type polynomials, some combinatorial sums and identities involving the Bernoulli numbers, the Stirling numbers and the Catalan-type numbers are derived. [ABSTRACT FROM AUTHOR]

The main problem of Cartesian Dualism which turns it to an outdated view is mental causation; If mental and physical substance have no mutual property, how we could explain the causal relation between them. Jonathan Lowe reconsiders Cartesian view and introduces his Non-Cartesian Substance Dualism which maintains that I, as a subject of my own experiences, literally can have some physical properties of my body. However, I think his afford is not consequent; By introducing a distinction between negative and positive possibility, first I argue that he needs positive possibility for his view. But as he committed to this idea that the identity of a property depends, at least in part, on its substance, he cannot use the idea of having a property derivatively to show this possibility. However, by resorting to the idea of spatiality of mind, I think we can maintain his core idea and believe that mind literally could has some physical properties. At the end, I will also try to answer to Kim's critiques about the idea of spatiality of mind. [ABSTRACT FROM AUTHOR]

All multicellular life relies on differential gene expression, determined by regulatory DNA elements and DNA‐binding transcription factors that mediate activation and repression via cofactor recruitment. While activators have been extensively characterized, repressors are less well studied: the identities and properties of their repressive domains (RDs) are typically unknown and the specific co‐repressors (CoRs) they recruit have not been determined. Here, we develop a high‐throughput, next‐generation sequencing‐based screening method, repressive‐domain (RD)‐seq, to systematically identify RDs in complex DNA‐fragment libraries. Screening more than 200,000 fragments covering the coding sequences of all transcription‐related proteins in Drosophila melanogaster, we identify 195 RDs in known repressors and in proteins not previously associated with repression. Many RDs contain recurrent short peptide motifs, which are conserved between fly and human and are required for RD function, as demonstrated by motif mutagenesis. Moreover, we show that RDs that contain one of five distinct repressive motifs interact with and depend on different CoRs, such as Groucho, CtBP, Sin3A, or Smrter. These findings advance our understanding of repressors, their sequences, and the functional impact of sequence‐altering mutations and should provide a valuable resource for further studies. Synopsis: Transcriptional repressors and their respective repressive domains (RD) have, in contrast to activators, not been systematically characterized. Here, high‐throughput identification of RDs via a newly developed RD‐seq screening approach reveals that RDs contain short recurrent and conserved peptide motifs required for the recruitment of co‐repressors and repressive function. RD‐seq identifies known and novel transcriptionally repressive domains.RDs contain short peptide motifs that are required for the repressive function.Repressive peptide motifs can predict the interacting co‐repressor.RD function and repressive motifs are conserved throughout evolution. [ABSTRACT FROM AUTHOR]

Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. In this paper, we propose a new fractional identity for multiplicatively differentiable functions; based on this identity, we establish some new fractional multiplicative Bullen-type inequalities for multiplicative differentiable convex functions. Some applications of the obtained results are given. [ABSTRACT FROM AUTHOR]

In this paper, we relate some properties of non-comaximal graph of ideals of a commutative ring with identity with the properties of the ring. [ABSTRACT FROM AUTHOR]

Through an assessment of the data recorded in two books of habices (Span., libros de habices – inventories of goods from Islamic pious endowments) dated 1527 and 1530, this study examines the situation of Morisco women in the Alpujarra, a rural area of Granada, just three decades after the forced conversion of the Muslim population to Christianity. Various aspects of the economic and social position of these women are explored, paying particular attention to their participation in the legal framework related to property ownership and the transfer of their possessions in the form of bequests. Although the study focuses primarily on the Morisco period, its most immediate precedent, Islamic and Mudéjar Granada, is not forgotten. [ABSTRACT FROM AUTHOR]