Multiplication Theorem Research Articles

Motivic Cluster Multiplication Formulas in 2-Calabi–Yau Categories

Abstract We introduce a notion of motivic cluster characters via virtual Poincaré polynomials, and prove a motivic version of multiplication formulas obtained by Chen–Xiao–Xu for weighted quantum cluster characters associated to a 2-Calabi–Yau triangulated category ${{\mathcal{C}}}$ with a cluster tilting object. Furthermore, a refined form of this formula is also given. When ${{\mathcal{C}}}$ is the cluster category of an acyclic quiver, our certain refined multiplication formula is a motivic version of the multiplication formula in SIGMA 19 (2023), 097-157.

IDENTITIES. PEDAGOGICAL METHODS FOR TEACHING SHORT MULTIPLICATION FORMULAS

The effective teaching of short multiplication formulas, which play a crucial role in algebra, requires innovative pedagogical methods. This paper explores how different teaching strategies impact students' understanding and retention of these algebraic identities. The study employs a mixed-methods approach, combining quantitative analysis of student performance with qualitative feedback from both students and teachers. Results suggest that interactive methods, such as problem-based learning and visual aids, significantly enhance comprehension. These findings have important implications for improving the quality of algebra education in secondary and tertiary settings.

Correction of ‘J. Laderman, V. Pan, X.–H. Sha, On practical Algorithms for Accelerated Matrix Multiplication, Linear Algebra and its Applications. Vol. 162-164 (1992) pp. 557-588’

In this article, we corrected the trilinear formula for triple disjoint matrix multiplication given in the article ‘J. Laderman, V. Pan, X. H. Sha, On practical Algorithms for Accelerated Matrix Multiplication, Linear Algebra and its Applications. Vol. 162-164 (1992) pp. 557-588’, which is incorrect for matrix dimensions equal to two or greater. That formula is a base of two algorithms, for disjoint and single matrix multiplication. The necessary correction made the amount of non–scalar products rise, slightly increasing the algorithm time complexity. We also corrected explicit formulas, in the bilinear form, for triple disjoint matrix multiplication. They are explicit, and thus convenient for the practical use of the fast matrix multiplication algorithms in question.

On the Hadamard multiplication theorem in ℂ n

The Hadamard product of two holomorphic functions given by Taylor series ∑ α a α z α and ∑ α b α z α , defined on neighborhoods Ω 1 and Ω 2 of the origin, is the function whose Taylor series is ∑ α a α b α z α . For n = 1 , it is known that the Hadamard product extends to a holomorphic function on the so-called *-product of Ω 1 and Ω 2 . In this paper, we generalize this result to the case of several complex variables.

Stable envelopes for slices of the affine Grassmannian

The affine Grassmannian associated to a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}$$\\end{document} is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of Maulik and Okounkov (Astérisque 408:ix+209, 2019) in this family. We construct an explicit recursive relation for the stable envelopes in the G=PSL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}= \ extbf{PSL}_{2}$$\\end{document} case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.

The automorphism group of finite 2-groups associated to the Macdonald group

We consider the Macdonald group [Formula: see text] and its Sylow 2-subgroup [Formula: see text], where [Formula: see text] and [Formula: see text] is odd. Then [Formula: see text] has order [Formula: see text], and nilpotency class 5 if [Formula: see text] and 3 if [Formula: see text]. We determine the automorphism group of the 2-groups [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. Explicit multiplication, power, and commutator formulas for [Formula: see text], [Formula: see text] and [Formula: see text] are given, and used in the calculation of [Formula: see text], [Formula: see text] and [Formula: see text].

나눗셈 도입 단원의 곱셈과 나눗셈의 관계 지도 내용 분석

Multiplication and division can cancel each other's calculation results, and they are in the relationsheep of inverse operations. Division is mathematically defined based on multiplication. This means that division assumes a multiplication structure. It is also possible to explain the inverse operation relationship between multiplication and division in the way that division utilizes the multiplication structure. In particular, when expressed as a multiplicative expression, division corresponds to the problem of finding the number to be multiplied or multiplied with a given product. This is the opposite of the calculation direction of multiplication, and the number to be found can vary depending on the context of division. Based on the analysis of the inverse operation relationship between multiplication and division, the division introduction unit of 11 elementary school textbooks were analyzed. Korea's elementary school mathematics textbooks teach division and division calculation methods based on the context of the inverse operation relationship between multiplication, and the contexs of division. As a result of the textbook analysis, it was found that the textbooks taught the relationship between multiplication and division as the quotient of division could be obtained using the related multiplication formula, and the number to divide the related multiplication formula was fixed as the number to be multiplied. Accordingly, the position of the quotient is at the multiplier in the multiplication expression, and this means that the context of the division is measurement division. As a result of adhering to this division quotient calculation method, problems arise when Korean elementary school mathematics textbooks explain the division calculation process in the context of the equal division system using multiplication formulas, and there are cases where the division quotient calculation method is inappropriate for the multiplication structure. Based on the textbook analysis results, some ideas were proposed to improve the teaching of the relationship between multiplication and division, and division.

The Livšic function of a homogeneous symmetric operator and the multiplication theorem

The Livšic function of a homogeneous symmetric operator and the multiplication theorem

A DISCRETE VERSION OF THE MISHOU THEOREM RELATED TO PERIODIC ZETA-FUNCTIONS

In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts $\zeta_{u_N}(s+ikh_1; \ga)$ and $\zeta_{u_N}(s+ikh_2, \alpha; \gb)$ of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence $\ga$, and the periodic Hurwitz zeta-function, respectively. We suppose that $u_N\to\infty$ and $u_N\ll N^2$ as $N\to\infty$, and the set $\{(h_1\log p:\! p\in\! \PP), (h_2\log(m+\alpha): m\in \NN_0), 2\pi\}$ is linearly independent over $\QQ$.

A Comparative Study of Junior Middle School Mathematics Example Exercises between People’s Education Edition and Zhejiang Education Edition—Taking Multiplication Formula as an Example

A Comparative Study of Junior Middle School Mathematics Example Exercises between People’s Education Edition and Zhejiang Education Edition—Taking Multiplication Formula as an Example

A generalized quantum cluster algebra of Kronecker type

<abstract><p>The notion of generalized quantum cluster algebras was introduced as a natural generalization of Berenstein and Zelevinsky's quantum cluster algebras as well as Chekhov and Shapiro's generalized cluster algebras. In this paper, we focus on a generalized quantum cluster algebra of Kronecker type which possesses infinitely many cluster variables. We obtain the cluster multiplication formulas for this algebra. As an application of these formulas, a positive bar-invariant basis is explicitly constructed. Both results generalize those known for the Kronecker cluster algebra and quantum cluster algebra.</p></abstract>

The Multiplication Formulas of Weighted Quantum Cluster Functions

By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2-Calabi-Yau triangulated categories with cluster-tilting objects.

A New Generalization of the Alternating Harmonic Series

Kilmer and Zheng (2021) recently introduced a generalized version of the alternating harmonic series. In this paper, we introduce a new generalization of the alternating harmonic series. A special case of our generalization converges to the Kilmer-Zheng series. Then we investigate several interesting and useful properties of this generalized, such as a summation formula related to the Hurwitz -Lerch Zeta function, a duplication formula, an integral representation, derivatives, and the recurrence relationship. Some important special cases of the main results are also discussed.

Somatic embryo formation, plantlet vigorous and greenhouse preliminary test of garlic bulblet obtained through somatic embryogenesis

Garlic production in Indonesia is still low due to limited availability of seedling. Approximately 10% of garlic yield are used for the next cultivation. Therefore about 90 % of garlic consumption are still imported from China. In addition to conventional seed production, alternative routes are needed, such as through somatic embryogenesis. Research consists of three activities, namely 1) induction and multiplication of callus, 2) embryo somatic and plantlet formation and 3) preliminary test of bulblet in vitro in the greenhouse. Parameters observed in the laboratory activities were time of callus emergence, callus weight and morphology, time of somatic embryo emergence, embryo diameter and weight per bottle, plantlet number and vigorous and bulblet number per bottle, while in the greenhouse, parameter observed were survival of bulblet number, plant height, leaf number per plant, and clove number and weight. The result showed that application of 2,4-D 3 mg/l with casein hydrolysate acid was suitable for callus induction of garlic var. Lumbu Kuning, while BAP 2 mg/l was the most suitable formula for callus multiplication, the addition of paclobutrazol 1 ppm and sucrose 20% in MS medium was the most suitable treatment for plantlet vigorous, while the bulblet formation could be developed by using jasmonic acid with sucrose of 12%. The best medium for growjng media of in vitro bulblet in the greenhouse was the combination of sandy soil, cocopeat and manure.

Two-dimensional (p,q)-heat polynomials of Gould–Hopper type

We introduce a new class of holomorphic polynomials extending the classical Gould–Hopper to two complex variables. The considered polynomials include the 1-D and 2-D holomorphic and polyanalytic Itô–Hermite polynomials as particular cases. We emphasize studying their operational representation, various generating functions, and recurrence relations. We also establish some special identities including multiplication, Runge addition and Nielson types formulas. Higher-order partial differential equations are analyzed and the connection to Gould-Hopper polynomials and hypergeometric functions are investigated.

Generalized sines and duplication formulae

We derive explicit Weierstrassian expressions and duplication formulae for a variety of generalized sine functions sinp,q.

The Cluster Multiplication Theorem for Acyclic Quantum Cluster Algebras

Abstract Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can be viewed as the quantum version of the cluster multiplication theorem in the classical cluster algebra proved by Caldero–Keller for finite type, Hubery for affine type, and Xiao–Xu for acyclic quivers.

Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

The Recursive Structures of Manin Symbols over Q, Cusps and Elliptic Points on X0 (N)

Firstly, we present a more explicit formulation of the complete system D(N) of representatives of Manin’s symbols over Q, which was initially given by Shimura. Then, we establish a bijection between D(M)×D(N) and D(MN) for (M,N)=1, which reveals a recursive structure between Manin’s symbols of different levels. Based on Manin’s complete system Π(N) of representatives of cusps on X0(N) and Cremona’s characterization of the equivalence between cusps, we establish a bijection between a subset C(N) of D(N) and Π(N), and then establish a bijection between C(M)×C(N) and C(MN) for (M,N)=1. We also provide a recursive structure for elliptical points on X0(N). Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over Q, cusps, and elliptical points on X0(N). This may give rise to more efficient algorithms for modular elliptic curves. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and prove constructively formulas of the numbers of D(N), cusps, and elliptic points on X0(N).

Matrix analog of the four‐parameter Mittag‐Leffler function

In this paper, we address a matrix analog of the four‐parameter Mittag‐Leffler function and demonstrate its convergence on . Some properties, such as integral representations, derivative formulae, and finite summation formulae, have been determined. Additionally, we provide the addition and multiplication formulas for the addition and multiplication of two arguments of the four‐parameter Mittag‐Leffler matrix function. Further, we discuss the composition of generalized fractional calculus operators with respect to a function with the four‐parameter Mittag‐Leffler matrix function.