Commutation Properties Research Articles

A note on second order Riesz transforms in 3-dimensional Lie groups

We prove explicit $$L^p$$ bounds for second order Riesz transforms of the sub-Laplacian and of the Laplacian in the Lie groups $${\mathbb {H}}$$ , $$\mathbb {SU}(2)$$ , and $$\widetilde{\mathbb {SL}}(2)$$ . Our proof makes use of martingale transform techniques and specific commutation properties between the complex gradient and the sub-Laplacian in those Lie groups.

Derivative bounds and continuity of maximal commutators

We study the regularity properties of maximal commutators acting on $W^{1,1}$-functions. More precisely, let $\mathfrak {M}_{b}$ be the one-dimensional maximal commutator with symbol $b$. Under the condition that $b\in W^{1,1}(\mathbb {R})$ and $\|b’\|_{L

Interval‐Valued q‐Rung Orthopair Fuzzy QUALIFLEX Decision Analysis Method with Dombi Operators

As a new and useful decision tool, interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS), has strong ability to address uncertain or imprecision information in practical decision‐making processes. This paper develops a new interval‐valued q‐rung orthopair fuzzy QUALIFLEX decision analysis method with Dombi operators for handling the group decision‐making problems with IVq‐ROFSs data. We also conduct a case study on the health management of hypertensive patients for demonstrating the feasibility and superiority of the developed method. The main contributions of this paper are as follows: (1) we first propose interval‐valued q‐rung orthopair fuzzy Dombi operational laws and investigate their desirable properties; (2) using the developed Dombi operational laws, we develop an interval‐valued q‐rung orthopair fuzzy Dombi weighted average (IVq‐ROFDWA) operator, and we prove that the IVq‐ROFDWA operator satisfies the properties of idempotency, commutativity, boundedness, and monotonicity; (3) we introduce the concept of interval‐valued q‐rung orthopair fuzzy ideal group opinions with the help of IVq‐ROFDWA operator and propose a new weight‐determining method for identifying the weights of experts; (4) we develop a new interval‐valued q‐rung orthopair fuzzy distance measure. With the help of the developed distance measure, we calculate interval‐valued q‐rung orthopair fuzzy concordance/discordance index under each permutation, and then a new interval‐valued q‐rung orthopair fuzzy QUALIFLEX decision analysis method is developed for addressing the complex multiattribute group decision‐making problems with IVq‐ROFSs.

Geometrical approach on set theoretical solutions of Yang-Baxter equation in Lie algebras

In this paper, we handle set-theoretical solutions of Yang-Baxter equation and Lyubashenko set theoretical solutions in Lie algebras. We present a new commutative binary operation on these structures, and we obtain new set theoretical solutions including this operation by using property of commutativity of it. Also, we show that some set theoretical solutions of Yang-Baxter equation corresponds to the Lyubashenko set theoretical solutions on these structures. Additionally, we give some relations to verify set theoretical solution of Yang-Baxter equation. Moreover, we put an interpretation for these solutions from the point of geometrical view in Euclidean space, Minkowski space and differentiable manifolds by using Lie algebras.

Counting in Cycles

Counting in terms of cycles allows modeling many processes of Nature. We make use of a slight numerical incongruence within the numbering system to find a translational mechanism which connects sequential ↔ commutative properties of assemblies. The algorithms allow picturing the logical syntax Nature uses when reading the DNA. Ordering a collection on two different properties of its members will impose two differing sequences on the members. The coordinates of a point on a plane of which the axes are the two sorting orders sidestep the logical contradiction arising from the different linear assignments. During a reorder, elements aggregate into cycles. Using an etalon collection of simple logical symbols (which are pairs of natural numbers), which we reorder, we see typical movement patterns along the path of the string of elements that are members of the same cycle. We split the {value, position} descriptions of a natural number and observe the places the unit occupies at specific instances of time among its peers, while being a member of a cycle, under different orders prevailing. Which places elements occupy among their peers, under specific order conditions, is determined by numeric properties of the natural numbers that make up the element. The model presented is therefore tautologic, and can be compared to a hybrid of sudokus with the ultimate form of Rubik‘s cube. It is necessary for the reader to do some self-education on the matter of cycles as a prerequisite to understanding the working principle of the model. The contraption being a new development, it can’t have references in literature. One cannot lose a bet on the idea that sorting and ordering a collection of elementary logical elements will turn up typical patterns and that these archetypes of patterns will be of interest to Theoretical Physics, Chemistry, Biology, Information Theory, and some other fields, too.

Generalized commutative quaternions of the Fibonacci type

Quaternions are a four-dimensional hypercomplex number system discovered by Hamilton in 1843 and next intensively applied in mathematics, modern physics, computer graphics and other fields. After the discovery of quaternions, modified quaternions were also defined in such a way that commutative property in multiplication is possible. That number system called as commutative quaternions is intensively studied and used for example in signal processing. In this paper we define generalized commutative quaternions and next based on them we define and explore Fibonacci type generalized commutative quaternions.

МАТЕМАТИЧЕСКИЙ ФОРМАЛИЗМ ДЛЯ РАСЧЕТОВ СЕЧЕНИЯ УПРУГОГО р7Ве- РАССЕЯНИЯ В РАМКАХ ТЕОРИИ ГЛАУБЕРА

In this paper, we solve the problem of constructing a commutative algebra of quaternions and octonions. A proof of the theorem is given that the commutativity of quaternions can be ensured by specifying a set of sign coefficients of the directions of reference of the angles between the radius vectors in the coordinate planes of the vector part of the coordinate system of the quaternion space. The method proposed in the development of quaternions possessing the commutative properties of multiplication is used further to construct a commutative octonion algebra. The results obtained on improving the algebra of quaternions and octonions can be used in the development of new hypercomplex numbers with division over the field of real numbers, and can also find application for solving a number of scientific and technical problems in the areas of field theory, physical electronics, robotics, and digital processing of multidimensional signals.

بناء مصفوفات تبديلية من مراتب عليا وتطبيقاتها في التشفير

نقدّم في هذا البحث طريقة لبناء مصفوفات تحقّق الخاصة التبديلية للضرب انطلاقا من مصفوفات دائريّة، حيث يمكن تجزئة كلّ مصفوفة من هذه المصفوفات إلى أربع مصفوفات دائريّة، وكذلك يمكن أن نبني مصفوفات تحقق الخاصة التبديلية للضرب من مراتب عليا ولا تُجزّأ بالضرورة إلى مصفوفات دائريّة. وباستخدام هذه المصفوفات نقدّم طريقة لتبادل مفتاح سري للتشفير عبارة عن مصفوفة مربّعة بشكل آمن عبر قنوات اتّصال مفتوحة، ثم نستخدم هذا المفتاح في تبادل رسائل مشفّرة بين جهتين أو طرفين. الأكثر من ذلك نقدّم أيضاً باستخدام هذه المصفوفات طريقة تشفير ذات مفتاح معلن، بحيث يتبادل الطرفان رسائل مشفّرة دون الاتفاق مسبقاً على مفتاحٍ سرّيٍّ مشتركٍ بينهما.

Regularity of Commutators of Maximal Operators with Lipschitz Symbols

This paper is devoted to studying Sobolev regularity properties of commutators of Hardy–Littlewood maximal operator and its fractional case with Lipschitz symbols, both in the global and local case. Some new pointwise estimates for the weak gradients of the above commutators will be established. As applications, some bounds for the above commutators on the Sobolev spaces will be obtained.

Aczel–Alsina aggregation operators and their application to intuitionistic fuzzy multiple attribute decision making

This paper describes the new intuitionistic fuzzy aggregation operators in consequence of Aczel–Alsina operations that possess certain advantages in cases of solving real life problems. We first present some new operations of intuitionistic fuzzy sets (IFSs), for example, Aczel–Alsina sum, Aczel–Alsina product, and Aczel–Alsina scalar multiplication. At that point, we create some IF aggregation operators, for example, the IF Aczel–Alsina weighted averaging operator, the IF Aczel–Alsina ordered weighted averaging operator and IF Aczel–Alsina hybrid averaging operator. We set up different properties of these operators. It is demonstrated that suggested averaging operators have the properties of idempotency, boundary, monotonicity, and commutativity. Then, we design new techniques dependent on these operators to fix multiattribute decision making issues. We present an example of human resource selection to elaborate on the performance of our proposed approach. The outcome shows the practicality and viability of the new technique. Eventually, an organized comparison between the prevailing techniques and the suggested technique has been given.

The Effects of a CRA-I Intervention on Students’ Number Sense and Understanding of Addition

As students develop understanding and fluency in single-digit operations such as addition, they develop sophisticated strategies and number sense (magnitude, number order, and composition). Deficits in number sense and reliance on inefficient approaches can lead to struggle in mathematics. Intervention research in this area reported effects on students’ automaticity. Research reported observational data regarding strategy-use, but not growth in number sense and understanding of operations. This study investigated the effects of an intervention using the concrete-representational-abstract-integrated (CRA-I) sequence. The purpose of this study was to assess the effects of CRA-I on students’ number knowledge (magnitude, place value, and flexibility in strategy-use) and understanding of addition (commutative property and relation to subtraction). The researchers used a multiple probe across participants design. In addition, the researchers collected descriptive data on students’ automaticity. There was a functional relation between CRA-I and students’ number sense and understanding of addition, and students’ automaticity increased.

Characterizations of Double Commutant Property on B H

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

Corrections and Extensions in Left and Right Almost Semigroups

In this paper we elaborated the concept that on what conditions left almost semigroup (LA-Semigroup), right almost semigroup (RA-Semigroup) and a groupoid become commutative and further extended these results on medial, LA-Group and RA-Group. We proved that the relation of LA-Semigroup with left double displacement semigroup (LDD-semigroup), RA-Semigroup with left double displacement semigroup (RDD-semigroup) is only commutative property. We highlighted the errors in the recently developed results on LA-Semigroup and semigroup [17, 1, 18] and proved that example discussed in [18] is semigroup with left identity but not paramedial. We extended results on locally associative LA-Semigroup explained in [20, 21] towards LA-Semigroup and RA-Semigroup with left zero and right zero respectively. We also discussed results on n-dimensional LA-Semigroup, n-dimensional RASemigroup, non commutative finite medials with three or more than three left or right identities and finite as well as infinite commutative idempotent medials not studied in literature.

Secure and Efficient High Throughput Medium Access Control for Vehicular Ad-Hoc Network.

The evolution of the internet has led to the growth of smart application requirements on the go in the vehicular ad hoc network (). enables vehicles to communicate smartly among themselves wirelessly. Increasing usage of wireless technology induces many security vulnerabilities. Therefore, effective security and authentication mechanism is needed to prevent an intruder. However, authentication may breach user privacy such as location or identity. Cryptography-based approach aids in preserving the privacy of the user. However, the existing security models incur communication and key management overhead since they are designed considering a third-party server. To overcome the research issue, this work presents an efficient security model namely secure performance enriched channel allocation () by using commutative . This work further presents the commutative property of the proposed security scheme. Experiments conducted to evaluate the performance of the proposed over state-of-the-art models show significant improvement. The outcome shows that minimizes collision and maximizes system throughput considering different radio propagation environments.

PENGUATAN PENERAPAN NILAI-NILAI KARAKTER SIFAT KOMUTATIF, ASOSIATIF, DAN DISTRIBUTIF OPERASI HITUNG BILANGAN CACAH TERHADAP ASPEK AFEKTIF SISWA SEKOLAH DASAR

The study aims to (1) explain the conditions of learning, (2) determining strategies for strengthening; and (3) explain the results of strengthening the application of character values of the commutative, associative, and distributive traits of counting whole numbers in improving the affective aspects of the 6th-grade students of SD 2 and 3 Temulus, Mejobo, Kudus. This research is a qualitative research with the type of field research in 2 locations, in SD 2 and 3 Temulus, Mejobo, Kudus. The research method used is multi-case qualitative with the type of field research in two locations, namely SD 2 and 3 Temulus, Mejobo, Kudus. The data collection technique was carried out using observation, interviews, and documentation. Data analysis used individual case data analysis and cross-case data analysis. The validity of the research data was tested by credibility test, transferability test, dependability test, and confirmability test. The results of this study are (1) learning conditions in strengthening the application of character values to the commutative, associative, and distributive traits of counting whole numbers are generally good. The learning atmosphere is more dynamic and students show interest and motivation to understand character values extracted from mathematical notations on formulas of the commutative, associative, and distributive properties of numeric arithmetic operations; (2) strategies to strengthen the application of character values including mapping of character values, preparation of learning strategies, habituation of character values, and teachers as role models in practicing character values; and (3) the results of strengthening the application of character values shown students have an understanding of the character values that appeared.

Graded Medial n-Ary Algebras and Polyadic Tensor Categories

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.

On the Commutation Properties of Finite Convolution and Differential Operators I: Commutation.

The commutation relation $KL = LK$ between finite convolution integral operator $K$ and differential operator $L$ has implications for spectral properties of $K$. We characterize all operators $K$ admitting this commutation relation. Our analysis places no symmetry constraints on the kernel of $K$ extending the well-known results of Morrison for real self-adjoint finite convolution integral operators.

Designing a Novel Star Topology using Operad Linear Differential Theory

In general, topology design is the most important framework in network communication. In this research, a star topology is adapted as an application. Normally, the star topologies are freely connected to the closed-loop systems and effectively created for all network applications. However, the failure of the central node leads to function error and system collapse. Once the network system gets collapsed then attaining the original states becomes difficult. To overcome this problem, a novel star topology is designed using the operad linear differential theory. This proposed mathematical star design efficiently reconstructs the network, when the collapse occurs. Moreover, the linear differential theory is mostly used to create a pertinent computational replica, and operad is perturbed with ideal algebras including associative and commutative properties. Furthermore, the proposed star topology design is applicable for Storage Area Network (SAN). Henceforth, the stability of the proposed method is determined using proper eigenvalues with specific theorems and conditions. The implementation of this research is done in the MATLAB platform. Thus, the proposed differential star topology is validated in the SAN application for data transmission. Besides, the proposed model is validated with other existing models using different key metrics to make the comparison assessment. Finally, the comparison results proved that the proposed linear differential theory is sufficient to be applied for SAN application with different network conditions.

Subsystems and Fuzzy Relation Equations of Fuzzy Automata Based on Generalized Residuated Lattice

The objective of this paper is to establish the concepts of subsystems, reverse subsystems and double subsystems of a fuzzy automata based on generalized residuated lattices where the fuzzy sets are equipped with generalized residuated lattice in which the commutative property does not hold and characterizes them in terms of fuzzy relation inequalities and equations. Also, the opening and closure systems in the lattices of fuzzy subsets of [Formula: see text] can be discussed for these six subsystems.

Towards Green Computing Oriented Security: A Lightweight Postquantum Signature for IoE.

Postquantum cryptography for elevating security against attacks by quantum computers in the Internet of Everything (IoE) is still in its infancy. Most postquantum based cryptosystems have longer keys and signature sizes and require more computations that span several orders of magnitude in energy consumption and computation time, hence the sizes of the keys and signature are considered as another aspect of security by green design. To address these issues, the security solutions should migrate to the advanced and potent methods for protection against quantum attacks and offer energy efficient and faster cryptocomputations. In this context, a novel security framework Lightweight Postquantum ID-based Signature (LPQS) for secure communication in the IoE environment is presented. The proposed LPQS framework incorporates a supersingular isogeny curve to present a digital signature with small key sizes which is quantum-resistant. To reduce the size of the keys, compressed curves are used and the validation of the signature depends on the commutative property of the curves. The unforgeability of LPQS under an adaptively chosen message attack is proved. Security analysis and the experimental validation of LPQS are performed under a realistic software simulation environment to assess its lightweight performance considering embedded nodes. It is evident that the size of keys and the signature of LPQS is smaller than that of existing signature-based postquantum security techniques for IoE. It is robust in the postquantum environment and efficient in terms of energy and computations.