Multiplication Theorem Research Articles

Euler and the Duplication Formula for the Gamma-Function

We show how the formulas in Euler’s paper "Variae considerationes circa serieshypergeometricas" [ 4] imply Legendre’s duplication formula for the Γ-function. Thispaper can be seen as an Addendum to [2].

Exploration on the Practice of "Project Based Learning" in the Field of Mathematics for Primary School Students

Project-based learning is a learning method that can implement the goal of educating people, can effectively promote the improvement of students' various abilities, and promote the cultivation of core literacy. Project based learning advocates that students are the main body, guide students to go out of the classroom, combine various learning methods to obtain knowledge, pay attention to students' actual ability, and develop ability. Project based learning has been supported by nearly 40 years of academic research and data, with rich theoretical research results, but there are few case studies on the practice of project-based learning in primary school mathematics. The 2022 new curriculum standard takes project-based learning as the implementation way to implement the goal of education, and provides some primary school mathematics project-based learning topics. In this paper, the author selected the second grade mathematics "Play with" Multiplication Formula "as the research topic, and carried out a practical study on the application of project-based learning in primary school mathematics teaching. According to the general process of "project-based learning", conduct practical research from the six steps of determining the project subject, analyzing project elements, designing driven problems, implementing project activities, publishing project results, and evaluating project learning, summarize the key links to carry out project-based learning, summarize the practical experience of developing project-based learning, and explore a practical path of project-based learning that conforms to the content of the discipline and the actual situation of the school. The implementation of project-based learning activities has improved students' interest in learning, developed their abilities, and yielded rich results in project-based learning. This study is expected to provide some reference and reference for the later development of the second classroom education of primary school mathematics.

On the joint universality for Dirichlet series connected to periodic zeta-functions

In the paper, we consider a collection of absolutely convergent Dirichlet series which in the mean are close to periodic and periodic Hurwitz zeta-functions, and prove that the shifts of mentioned Dirichlet series approximate simultaneously a wide class of analytic functions.

Multiplication formulas and isomorphism theorem of ıSchur superalgebras

We introduce the notion of ı Schur superalgebra, which can be regarded as a type B/C counterpart of the q -Schur superalgebra (of type A) formulated as centralizer algebras of certain signed q -permutation modules over Hecke algebras. Some multiplication formulas for ı Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the ı Scuhr superalgebras and the q -Schur superalgebras of type A, which helps us derive semisimplicity criteria of the ı Schur superalgebras.

Transfer-Learning-Assisted Multielement Calibration for Active Phased Antenna Arrays

A transfer-learning-based method for accelerating power-only calibration of phased array antennas by combining the conventional array theory with deep learning is presented in this communication. The existing power-only calibration methods either require a significant number of measurement cycles or have restrictive phase shifter resolution requirements. The proposed array calibration method uses a surrogate model to calibrate all the array elements in one pass without restricting phase resolution requirements. We developed a novel feature extraction scheme (FES) that picks out the most important power features resulting in reduced measurement cycles. The burden of data acquisition for model training is further reduced by relational knowledge transfer learning. The surrogate model acquires its general calibration capability from massive theoretical data, which is easily collected by the radiation multiplication theorem, and captures the detailed nonideal response from a small number of simulations. The proposed methodology has been demonstrated and tested on several arrays for validation. The effectiveness and performance of the method have been verified, and hence, it can serve as a complementary tool to accelerate the calibration process of phased antenna arrays.

Affine Hecke algebras and quantum symmetric pairs

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra K n c {\mathbf K}^{\mathfrak c}_n . We show that K n c {\mathbf K}^{\mathfrak c}_n is a coideal subalgebra of quantum affine algebra U ( g l ^ n ) {\mathbf {U}}(\widehat {\mathfrak {gl}}_n) , and ( U ( g l ^ n ) , K n c ) \big ({\mathbf {U}}(\widehat { \mathfrak {gl}}_n), {\mathbf K}^{\mathfrak c}_n) forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.

Side-Channel Resistant 2048-Bit RSA Implementation for Wireless Sensor Networks and Internet of Things

We present a practical realization of Rivest-Shamir-Adleman (RSA) with a 2048-bit key on MSP430, a widely used microcontroller in wireless sensor network and Internet of things applications, and show that 2048-bit RSA is feasible on a constrained microcontroller. We exploit several methods for acceleration, e.g. Montgomery modular multiplication, subtractive Karatsuba-Ofman and Chinese remainder theorem (CRT) based modular exponentiation, and achieve RSA encryption and decryption with a 2048-bit key on MSP430 in just 0.14 s and 7.56 s, respectively. Our implementation on the low-end MSP430 microcontroller achieves 2048-bit RSA significantly faster (×2.9 and ×2.4 for encryption and decryption) with respect to the existing implementation in the literature on the comparable ATmega128 microcontroller. While our implementation is secure against the brute force attack due to its 2048-bit key, and thus 112-bit security level, it also includes the necessary side-channel countermeasures, e. g. message and key blinding, to help mitigate implementation attacks such as simple power analysis and differential power analysis.

Fast change of level and applications to isogenies

Let $$(A, {{\mathscr {L}}}, \Theta _n)$$ be a dimension g abelian variety together with a level n theta structure over a field k of odd characteristic. We thus denote by $$(\theta ^{\Theta _{{\mathscr {L}}}}_{i})_{(\mathbb {Z}/ n \mathbb {Z})^g} \in \Gamma (A, {{\mathscr {L}}})$$ the associated standard basis. For a positive integer $$\ell $$ relatively prime to n and the characteristic of k, we study change of level algorithms which allow one to compute level $$\ell n$$ theta functions $$(\theta _i^{\Theta _{{{\mathscr {L}}^\ell }}}(x))_{i \in (\mathbb {Z}/ \ell n \mathbb {Z})^g}$$ from the knowledge of level n theta functions $$(\theta ^{\Theta _{{\mathscr {L}}}}_i(x))_{(\mathbb {Z}/n \mathbb {Z})^g}$$ or vice versa. The classical duplication formulas are an example of change of level algorithm to go from level n to level 2n. The main result of this paper states that there exists an algorithm to go from level n to level $$\ell n$$ in $$O(n^g \ell ^{2g})$$ operations in k. We derive an algorithm to compute an isogeny $$f : A \rightarrow B$$ from the knowledge of $$(A, {{\mathscr {L}}}, \Theta _n)$$ and $$K \subset A[\ell ]$$ isotropic for the Weil pairing which computes f(x) for $$x \in A(k)$$ in $$O((n\ell )^g)$$ operations in k. We remark that this isogeny computation algorithm is of quasi-linear complexity in the size of K.

Quantum cluster characters of Hall algebras revisited

Let Q be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of Q. As an application, we recover the surjective homomorphism defined in [12], which realizes the principal coefficient quantum cluster algebra \(\mathcal {A}\,_q(Q)\) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero–Chapoton formula, as well as some multiplication formulas between quantum Caldero–Chapoton characters.

On Data Augmentation for Models Involving Reciprocal Gamma Functions

In this article, we introduce a new and efficient data augmentation approach to the posterior inference of the models with shape parameters when the reciprocal gamma function appears in full conditional densities. Our approach is to approximate full conditional densities of shape parameters by using Gauss’s multiplication formula and Stirling’s formula for the gamma function, where the approximation error can be made arbitrarily small. We use the techniques to construct efficient Gibbs and Metropolis–Hastings algorithms for a variety of models that involve the gamma distribution, Student’s t-distribution, the Dirichlet distribution, the negative binomial distribution, and the Wishart distribution. The proposed sampling method is numerically demonstrated through simulation studies. Supplementary materials for this article are available online.

PERAN MODEL PEMBELAJARAN TUTOR SEBAYA TERHADAP HASIL BELAJAR MATEMATIKA SISWA KELAS 5 SDN 2 TAHUNAN

This study aims to determine the effect of the Tutor's Learning Method Peer to the mathematics learning outcomes of fifth graders of bar chart material in grade 5 SD N 2 Tahunan. In this study, the type of research used was a descriptive qualitative. Qualitative data analysis techniques were chosen in this study because the data analysis process did not involve or take the form of numbers. The results of the research from observations when it was carried out at SD N 2 Tahunan, several problems were found including, low student learning outcomes, lack of knowledge about the material being taught and weak memorization of multiplication formulas as well as errors in working on questions that occurreddifficulty in applying the formula to solving practice problems

TEGRAS: An Efficient Tegra Embedded GPU-Based RSA Acceleration Server

Industrial Internet of Things (IIoT) has strict requirements on the performance and security of devices. Public-key cryptography, as a kind of computing resource-consuming algorithm, is widely used in the digital signature, key exchange, and so on. The embedded graphics processing units (GPUs) are now rapidly achieving extraordinary computing power, such as NVIDIA Tegra K1/X1/X2/Xavier, which are also treated as edge computing devices. They are widely used in IIoT environments, such as intelligent manufacturing, smart cities, and vehicle-mounted systems. The performance advantages endow embedded GPUs with the possibility of accelerating cryptography that also requires high-density computing. This article implements an efficient Tegra-based embedded GPU RSA acceleration server-oriented IIoT, named TEGRAS. Various optimization methods are employed to promote efficiency, including multithreaded Montgomery multiplication and Chinese Remainder Theorem implementation on the resource-constricted embedded GPUs. With about 40–50 W of power consumption, TEGRAS can deliver 34 kops/s of RSA2048 signature generation and 1007 kops/s of RSA signature verification, which outperforms implementations in the desktop GPUs and embedded CPUs in the perspective of performance-to-power ratio. To evaluate TEGRAS in real-world scenarios, we additionally build a network stack to deliver digital signature services, which can provide more than 34 and 978 kops of signature generation and signature verification, respectively. In a word, based on the embedded GPU, we provide a high-throughput, low-latency, and ready-to-use RSA accelerator-oriented IIoT.

Reliability methods for delivery of primary energy resources at a power plant during a reliability audit of electric energy systems

Various approaches (nodal, system and estimation) to assessing the reliability of electric power systems, including the effectiveness of fuel supply, are compared and evaluated. A method of statistical testing (Monte Carlo method) is used to assess the reliability of electric power and gas supply systems. For calculating power shortages in electric power systems, the internal points method was used; in order to calculate minimum costs in gas supply systems, the Basaker-Gowan method was used. To calculate the reliability indicators of these systems, probability theory methods were employed, including calculation of probability distribution series, addition and multiplication theorems. Independent tests based on the Bernoulli formula, addition and multiplication theorems, as well as the composition method of distribution series, were used to calculate the distribution series of random states for fuel and electricity supply systems when examining the proposed approaches (nodal, system and estimation approaches). All approaches (nodal, system and estimation) to assessing the reliability of electric power systems, including reliable fuel supply to electric power plants, were evaluated using the following examples: "nodal" and "system" using nominal examples of fuel-supply and electric power systems, as well as "evaluation" using the computational schemes of gas supply and power systems of the North-West Federal District. It is recommended to apply approaches depending on the technological characteristics of the given systems and their connection structure for calculating the base-load provision reliability of energy systems, as well as their feasibility regarding the accuracy of the obtained results, time consumption and the complexity of the data search, preparation and presentation in the model.

On multiplication formulas of affine q-Schur algebras

We provide a simple and conceptual proof of Du-Fu's multiplication formula of affine q -Schur algebras via Lusztig's formula. We use the multiplication formulas to provide a proof of the existence of generic affine Schur algebras, in return, and a formula of the generators under comultiplication.

Multiplication theorems for self-conjugate partitions

In 2011, Han and Ji proved addition-multiplication theorems for integer partitions, from which they derived modular analogues of many classical identities involving hook-length. In the present paper, we prove addition-multiplication theorems for the subset of self-conjugate partitions. Although difficulties arise due to parity questions, we are almost always able to include the BG-rank introduced by Berkovich and Garvan. This gives us as consequences many self-conjugate modular versions of classical hook-lengths identities for partitions. Our tools are mainly based on fine properties of the Littlewood decomposition restricted to self-conjugate partitions.Mathematics Subject Classifications: 05A15, 05A17, 05A19, 05E05, 05E10, 11P81Keywords: Hook-length formulas, BGP-ranks, Integers partitions, Littlewood decomposition, core partitions

On the Left Connected Subalgebra of the Descent Algebra of a Coxeter Group of Classical Type

A Coxeter group of classical type A_n, B_n or D_n contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof, we use Solomon’s descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra.

Contribution to the Hadamard multiplication theorem

In this article we define a binary linear operator T for holomorphic functions in given open sets \(A\) and \(B\) in the complex plane under certain additional assumptions. It coincides with the classical Hadamard product of holomorphic functions in the case where \(A\) and \(B\) are the unit disk. We show that the operator T exists provided \(A\) and \(B\) are simply connected domains containing the origin. Moreover, T is determined explicitly by means of an integral form. To this aim we prove an alternative representation of the star product \(A*B\) of any sets \(A,B\subset\mathbb{C}\) containing the origin. We also touch the problem of holomorphic extensibility of Hadamard product.

DISCRETE UNIVERSALITY OF ABSOLUTELY CONVERGENT DIRICHLET SERIES

In the paper, an universality theorem of discrete type on the approximation of analytic functions by shifts of a special absolutely convergent Dirichlet series is obtained. These series is close in a certain sense to the periodic zeta-function and depends on a parameter.

Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<t<1, and 1<p<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.

Quantum queer supergroups via υ-differential operators

By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup Uυ(qn). The underlying space of this representation is a deformed polynomial superalgebra in 2n2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of Uυ(qn). A monomial basis M for Uυ(qn) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup Uυ(qn) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous components.The existence of the bases M and L and the new presentation show that the seminal construction of quantum gln established by Beilinson–Lusztig–MacPherson thirty years ago extends to this “queer” quantum supergroup via a completely different approach.