Parallelogram Law Research Articles

Functional Inequalities in the Framework of Banach Spaces

AbstractThis paper aims to illustrate the usefulness of majorization theory and higher-order convexity in generalizing several classical inequalities as functional inequalities. This includes results due to Ball et al. (Invent Math 115:463-482, 1994), Hanner (Ark Mat 3:239-244, 1956), Popoviciu (Analele ştiinţifice Univ “Al. I. Cuza” Iasi, Secţia I-a Mat 11:155-164, 1965) and Zhang (Matrix Theory: Basic Results and Techniques, Springer, New York, 2011). Also, based on a substitute for the parallelogram law due to Clarkson, a quadrilateral inequality established by Schötz (in Quadruple inequalities: Between Cauchy-Schwarz and triangle. ArXiv preprint, arXiv:2307.01361 v.3, 2024) in the context of Hilbert spaces is extended to the general framework of Banach spaces.

Optimal Motion Planning for Heterogeneous Multi-USV Systems Using Hexagonal Grid-Based Neural Networks and Parallelogram Law Under Ocean Currents

Optimal Motion Planning for Heterogeneous Multi-USV Systems Using Hexagonal Grid-Based Neural Networks and Parallelogram Law Under Ocean Currents

An integrative socio-hydrological resilience assessment and management implications for oasis sustainability in arid regions, Northwest China

Study regionThe oases of the lower reaches of inland river basin in south Xinjiang, Northwest China Study focusThe increase of man-made water demands has led to water shortage and water pollution issues worldwide. It is important to determine the contributions of oasis change to socio-hydrological system resilience for oasis sustainability in arid regions. This study proposes an integrative socio-hydrological resilience assessment framework to examine the relationships between oasis evolution and system resilience at spatial-temporal scales. Hooke's Law and parallelogram law are adopted to quantitatively evaluate the resilience of drought, water resource utilization and water pollution, and comprehensive resilience of socio-hydrological system. Meanwhile, the impact of oasis expansion on socio-hydrological system resilience is quantified. New hydrological insights for the regionThe research and application of oases in Hotan prefecture of northwest China have proved that the proposed framework is very promising for assessing the relationships between oasis evolution and socio-hydrological system resilience in arid regions. During the study period from 2009 to 2018, The oases show “expansion-contraction-expansion” trends. There is significant negative correlation between oasis change and system resilience. The population size, socioeconomic development level and structure of cultivated land are the main factors affecting the oasis area and system resilience. The expanded agricultural irrigation reduces the resilience of socio-hydrological system. The water pollution has gradually affected oasis sustainability and system resilience. This study also provides a management implication concerning the impact of water pollution on the oasis sustainability in the future work.

Research on the revitalization of Huizhou traditional villages based on the PAF model

ABSTRACT The protection and development of traditional villages have become a pressing issue. This study deconstructed the driving elements of village revitalization into two major systems: the kernel and the regulation system, using the parallelogram law, Analytic Hierarchy Process (AHP), and fuzzy membership functions integrated to form the PAF model of traditional village revitalization. The village revitalization can be classified into two types of primary modes and six types of secondary modes: core dominance and regulation dominance. Combining the characteristics of each model and the development status quadrant chart, two village revitalization paths were determined: 1) First, strengthen the core driving force of villages with low development in the third quadrant, promoting their independent development in the fourth quadrant, and optimizing their external development to achieve a state of high-quality development in the first quadrant; 2) optimize the external development environment first, promote third quadrant villages to enter the externally led development state of the second quadrant, and then enhance their competitiveness to reach a better development level in the first quadrant. Finally, it outlines a revitalization path that is consistent with village reality, including the continuation of the traditional style, protection of traditional buildings, and inheritance of excellent culture.

General biconvex functions and bivariational inequalities

<p style='text-indent:20px;'>In this paper, we define and introduce some new concepts of the higher order strongly general biconvex functions involving the arbitrary bifunction and a function. Some new relationships among various concepts of higher order strongly general biconvex functions have been established. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine general biconvex functions, which is itself an novel application. It is proved that the optimality conditions of the higher order strongly general biconvex functions are characterized by a class of variational inequalities, which is called the higher order strongly general bivariational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general bivariational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.</p>

Multi-vector management model of decentralization program management in Ukraine

Multi-vector management model of decentralization program management in Ukraine

New type parallelogram laws in Banach spaces and geodesic spaces with curvature bounded above

The parallelogram law characterizes a part of the structure of Hilbert spaces. To hold the parallelogram law by the norm, reasonably good conditions are required for Banach spaces. This paper proposes new type parallelogram laws with bifunctions on Banach spaces and geodesic spaces, respectively.

Visualizing Laguerre polynomials as a complete orthonormal set for the inner product space ℙ n

This linear algebra note offers teaching and learning ideas in the treatment of the orthonormal Laguerre polynomials as vectors of the inner product space with the associated improper integral inner product in a DGS (Dynamic Geometry Software)-MATLAB facilitated learning environment. The proposed learning module demonstrates creative ways of integrating DGS-MATLAB technology into various properties of the integral inner products of Laguerre polynomials along with other notions inherent in the inner product space, such as Triangle Inequality, Pythagorean Theorem, Parallelogram Law, Orthogonality and Orthonormality, Coordinates Relative to an Orthonormal Basis, and Parseval’s Identity.

Vector semi-inner products

We formalize the notion of vector semi-inner products and introduce a class of vector seminorms which are built from these maps. The classical Pythagorean theorem and parallelogram law are then generalized to vector seminorms that have a geometric mean closed vector lattice for codomain. In the special case that this codomain is a square root closed, semiprime $f$-algebra, we provide a sharpening of the triangle inequality as well as a condition for equality.

An Introduction to Multi Inner Product Space

In this paper, for the first time, notion of multi complex numbers and multi complex number valued inner product is introduced in multi linear (vector) space. Starting from the definition, some basic properties of multi inner product spaces are studied along with examples. Multi number valued parallelogram law and polarization identity are established in multi inner product space.

Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces

It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets F(ℝn) and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, ‖u‖q = where 1 ≤ q ≤ ∞, we stated that (F(ℝn), ‖u‖q) is a complete normed space. Furthermore, we introduced an inner product in this space for the case q = 2. The inner product must be in the form. For u, v ∈ F(ℝn). We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of F(ℝn). Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.

On the geometry of the multiplier space of ℓp A

Abstract For p ∈ (1, ∞) \ {2}, some properties of the space ℳ p of multipliers on ℓp A are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for ℳ p . It is also shown that the extremal multipliers on the ℓp A spaces are exactly the monomials, in stark contrast to the p = 2 case.

Vector Current Measurement Using Doppler Scatterometry with Optimally Selected Observation Azimuths

The Doppler scatterometer is a new style of remote sensing tool that can provide current measurements over a wide swath for rapid global coverage. The existing current estimation method for Doppler scatterometry uses the maximum likelihood method to jointly derive the wind and current fields but shows high computational complexity. Moreover, the current radial speeds measured along two arbitrary observation azimuths are used to derive the vector current according to the parallelogram rule, which is not applicable for the case where two observation azimuths are not perpendicular. In this paper, a vector current velocity inversion method using an optimally selected observation azimuth combination—as well as a general current velocity calculation method—is proposed for Doppler scatterometry. Firstly, current radial speeds along several different observation azimuths are estimated using an interferometric phase difference matching method with low computational complexity. Then, two current radial components of each point are arbitrarily selected to estimate a preliminary current direction using the proposed vector current velocity derivation method. Finally, two observation azimuths that have the smallest intersection angles with the preliminarily estimated current direction are selected for vector current velocity determination. With the Ocean Surface Current Analyses Real-time (OSCAR) data as current input, vector current estimation experiments were conducted based on simulation analysis using an instrument conceptual design model for a pencil-beam scatterometer. The results show that the standard deviation of the estimated current velocity magnitude is 0.06 m/s. Compared with the reported results obtained by the existing method, the inversion accuracy of velocity magnitude is improved by 67%.

On New Moduli Related to the Generalization of the Parallelogram Law

On New Moduli Related to the Generalization of the Parallelogram Law

Some aspects of generalized Zbăganu and James constant in Banach spaces

Abstract We shall introduce a new geometric constant C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias. First, it is shown that, for a Banach space, C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) is equal to 1 if and only if the norm is induced by an inner product. Next, a characterization of uniformly non-square is given, that is, X X has the fixed point property. Also, a sufficient condition which implies weak normal structure is presented. Moreover, a generalized James constant J ( λ , X ) J\left(\lambda ,X) is also introduced. Finally, some basic properties of this new coefficient are presented.

Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

Operator equalities and Characterizations of Orthogonality in Pre-Hilbert C*-Modules

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

Strongly log-biconvex Functions and Applications

In this paper, we consider some new classes of log-biconvex functions. Several properties of the log-biconvex functions are studied. We also discuss their relations with convex functions. Several interesting results characterizing the log-biconvex functions are obtained. New parallelogram laws are obtained as applications of the strongly log-biconvex functions. Optimality conditions of differentiable strongly log-biconvex are characterized by a class of bivariational inequalities. Results obtained in this paper can be viewed as significant improvement of previously known results.

Wolfram-технологии в реализации принципа наглядности при обучении математическим дисциплинам в высшей экономической школе

The article focuses on the substantive and methodological features of the implementation of the principle of clarity by means of Wolfram technologies. The need and methodological feasibility of applying Wolfram technologies in the practice of teaching mathematical disciplines at an economic university is justified. Special attention is paid to the management of educational and cognitive activities of students when working with the created base of visualizations of mathematical concepts and objects, as well as the results of economic and mathematical modeling. Recommendations are presented for teachers of mathematical disciplines, compliance with which contributes to a deeper penetration of undergraduate students into the essence of mathematical methods and a meaningful interpretation of the results obtained on their basis. Disassembled and methodically characterized visualizations affect educational situations that arise when studying various mathematical disciplines: Linear Algebra, Higher Mathematics, Probability Theory and Mathematical Statistics, Theory of Optimal Control, Game Theory, etc. Among them is the visualization of the probability of a random variable falling into the interval under the condition of a normal distribution of a random variable, as well as under the condition that a random variable has a Weibul distribution; visualization of probability density of hypergeometric distribution; visualization of a fragment of a function graph on a period length segment; visualization of parallelogram rule for addition of two vectors in three-dimensional space; visualization of the direction field and integral curves when solving the ordinary differential equation of the first order; visualization of the direction field when solving a system of linear differential equations of the first order. The main results of the study of the new instrumental implementation of the classical didactic principle of clarity described in this article can be useful for improving the professional training system of the future bachelor of economics, as well as for designing the content of new professionally significant disciplines in the educational field «Mathematics and mathematical modeling».

Type synthesis of plane-symmetric deployable grasping parallel mechanisms using constraint force parallelogram law

Type synthesis methods of parallel mechanisms or metamorphic mechanisms have drawn rising interest for mechanism designers. However, existing synthesis approaches cannot synthesize both parallel mechanisms and metamorphic mechanisms. This paper develops a synthesis method of metamorphic mechanisms, which enables it to construct a family of 2-degree-of-freedom (DOF) deployable grasping parallel mechanisms. Namely, the originality of this paper is that it explores a common synthesis method for both parallel mechanisms and metamorphic mechanisms. First, a new type of grasping sub-mechanisms is designed. Then, a kinematic sub-chain is constructed after assembly of a base, a platform, and the grasping sub-mechanism, whose constraint-screw system is obtained using reciprocal screw theory. This constraint-screw system is further applied to derive the desired constraints of auxiliary sub-mechanisms using the constraint force parallelogram law. Subsequently, auxiliary sub-mechanisms are type synthesized based on such desired constraints. Finally, a class of parallel mechanisms is constructed by assembling the auxiliary and the kinematic sub-chain. Thus, this paper and related work generate fresh insight into synthesizing both single-DOF metamorphic mechanisms and 2-DOF parallel mechanisms.