Parallelogram Law Research Articles

Higher Order Strongly Biconvex Functions and Biequilibrium Problems

In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and k-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the k-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.

Is It Possible to Measure the Amount of the SMAS Face Lift According to Facial Vectors?

Since the initial description of the importance of the superficial musculoaponeurotic system (SMAS) in the face-lift procedure, surgical treatment of the facial aging through SMAS face lift evolved to be less invasive, mainly because of potential temporary or permanent facial nerve damage1 with disturbance of the normal function of facial muscles, which could require reconstruction such as the transplantation of subunits of the latissimus dorsi muscle.2 When treatment of the SMAS involves plication, there is a consensus that the SMAS face lift must be performed superiorly and posteriorly; regardless of the technical choice, the concept of the facial vectors is well established.1 The main superior vectors are performed parallel to the body of the mandible, at the level of the projection of the zygomatic arch, and the posterior ones are performed along the projection of the ramus of the mandible. The convergence of these two vectors generally occurs at the preauricular area in the level of projection of the root of the helix, resulting in an inverted-L shape or a 7 shape with an oblique final vector.3 Considering all of the different types of faces and deformities, the distance between the two horizontal and vertical vectors and, consequently, the amount of SMAS lift, will vary according to each patient.1,3 Real et al. described a technique of frontotemporal minilifting using the concept of facial vectors and, based on anatomical points, quantified the amount of eyebrow tail suspension.4 For the past 10 years, the preferred technique for face lift in our group has been the plication of the SMAS, with its imbrication to return volume to the malar area. After skin dissection, the SMAS is marked as described above with two parallel 7-shaped figures, according to the maximum traction possible by both main vectors (Fig. 1).Fig. 1.: Vectors of the two parallel 7-shaped figures at the third middle of the face (arrow indicates the oblique final vector according to the maximum traction of the SMAS).Considering the theory of vector sum, it can be noted that the arrangement of the vectors as two 7-shaped figures enables calculation of the difference between the areas of each one, through the parallelogram law. The so-called parallelogram law gives the rule for vector addition of two or more vectors. For two vectors a and b, the vector sum a + b is obtained by placing them head to tail and drawing the vector from the free tail to the free head. In Cartesian coordinates, vector addition can be performed simply by adding the corresponding components of the vectors; therefore, A = (a1, a2, …, an) and B = (b1, b2, …, bn).5 Therefore, this rule could be applied when there is an overlapping of two geometric figures (Fig. 2). In theory, this numeric value could direct surgeons to estimate the amount of SMAS traction in each patient and then evaluate the effectiveness of the procedure. This idea has been developed and incorporated in the routine of our face-lifting procedures and the results will be presented soon.Fig. 2.: Overlapping of the two geometric figures according to the parallelogram law. The difference between their areas will predict the total amount of lifting.DISCLOSURE The authors declare that they have no commercial interest in the subject of study or in the source of any financial or material support. Marcus Vinicius J. Barbosa, M.D., Ph.D.School of MedicineMorphofunctional LaboratoryMorphofunctional Laboratory Tiago Bianchi, Ph.D.IBM Fabio X. Nahas, Ph.D., M.B.A.Lydia M. Ferreira, M.D., Ph.D.Department of SurgeryDivision of Plastic SurgeryFederal University of São PauloSão Paulo, Brazil

Further generalizations of the parallelogram law

Further generalizations of the parallelogram law

Further generalizations of the parallelogram law

In a recent work of Alessandro Fonda, a generalization of the parallelogram law in any dimension $N\geq 2$ was given by considering the ratio of the quadratic mean of the measures of the $(N-1)$-dimensional diagonals to the quadratic mean of the measures of the faces of a parallelotope. In this paper, we provide a further generalization considering not only $(N-1)$-dimensional diagonals and faces, but the $k$-dimensional ones for every $1\leq k\leq N-1$.

Some results around quadratic maps

This paper dedicated to study quadratic maps. We present some new operator equalities and inequalities by using quadratic map in the framework of B(H). Applications for particular case of interest are also provided. The parallelogram law is recovered and some other interesting operator equalities are established. Afterward, we get an extension of some well known inequalities such as, triangle inequality. Especially, Bohr's inequality is generalized to the context of quadratic map. Some results concerning this inequality are surveyed. We give an application of our results in the previous sections. We show that our results are a generalization of some well known works due to Fujii and Hirzallah.

The Delayed Doubly Stochastic Linear Quadratic Optimal Control Problem

In this paper, the delayed doubly stochastic linear quadratic optimal control problem is discussed. It deduces the expression of the optimal control for the general delayed doubly stochastic control system which contained time delay both in the state variable and in the control variable at the same time and proves its uniqueness by using the classical parallelogram rule. The paper is concerned with the generalized matrix value Riccati equation for a special delayed doubly stochastic linear quadratic control system and aims to give the expression of optimal control and value function by the solution of the Riccati equation.

A Role of Hilbert Space in Sampled Data to Reduced Error Accumulation by Over Sampling Then the Computational and Storage Cost Increase Using Signal Processing On 2-Sphere Dimension”

Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of "Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable. Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.

Higher order strongly general convex functions and variational inequalities

In this paper, we define and consider some new concepts of the higher order strongly general convex functions with respect to an arbitrary function. Some properties of the higher order strongly general convex functions are investigated under suitable conditions. It is shown that the optimality conditions of the higher order strongly general convex functions are characterized by a class of variational inequalities, which is called the higher order strongly variational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general variational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. It is shown that the parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine convex functions. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

Strongly Convex Functions of Higher Order Involving Bifunction

Some new concepts of the higher order strongly convex functions involving an arbitrary bifuction are considered in this paper. Some properties of the higher order strongly convex functions are investigated under suitable conditions. Some important special cases are discussed. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine convex functions as novel applications. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

Kant on Time I: The Kinematics of the Metaphysical Foundations of Natural Science

Abstract The theory of space-time developed in Kant’s Critique of Pure Reason and his (1786) Metaphysical Foundations of Natural Science is connected to Leonhard Euler’s proof of invariance under Galilean transformations in the “On Motion in General” of the latter’s 1736 Analytical Mechanics. It is argued that Kant, by using the Principle of Relativity that is the output of Euler’s proof as an input to his own proof of the kinematic parallelogram law, makes essential use of absolute simultaneity. This is why, in the Transcendental Aesthetic, he observes that “our theory of time explains as much a priori knowledge as the general theory of motion displays.” (KrV, B 67) In conclusion, it is shown that the same proof-method, under a different definition of simultaneity, leads to the parallelogram law of the “Kinematic Part” of Einstein’s 1905 “On the Electrodynamics of Moving Bodies”.

Encoding the electrodynamics in spatiotemporal boundaries

Maxwell’s equations represented by differential operators describe the local dependence between both electric and magnetic fields in every location of space and time. This description responds to the field concept proposed by Michael Faraday and formalized by James Clerk Maxwell. The discretized version in the form of the finite-difference time-domain (FDTD) technique relies on a proposal of local computation of the respective differential operators making use of the central difference approximation of the second derivative of functions through the Taylor series expansion. This article introduces a novel time-symmetric “non-local” technique from the mathematical formalism of electromagnetic potentials in the wave equation and its physical interpretation in the Minkowski spacetime. In the proposed case study, the “non-local” proposal is 1643 faster than FDTD, i.e., more than three orders of magnitude, and it uses a time-step that is 4096 times greater than the Courant–Friedrichs–Lewy limit without encountering stability problems. New electromagnetic potentials are calculated from the previous and distant ones located at vertices of spatiotemporal regions called causal diamonds that tessellates spacetime without the need for computations inside them. We show that the performance gain is proportional to the size of the spacetime tessellations because the ratio “domain to boundary” increases as the domain extension does. Consequently, our novel “non-local” approach provides a reduction in computational complexity and a more comprehensible explanation of their fundamental physical aspects, without contradicting the principles of the successful classical field theory. Program summaryProgram Title: SOME_modelsProgram Files doi:http://dx.doi.org/10.17632/7sgh6grm7g.1Licensing provisions: GPLv3Programming language: CNature of problem: The Courant–Friedrichs–Lewy (CFL) condition imposes a numerical bound to the size of the Yee cell in the FDTD approach. This restriction causes excessive oversampling, degrading the computation time of this technique.Solution method: We applied the parallelogram rule to solve the wave equation for the case of (1+1) time–space dimensions. With this rule, we proposed a novel algorithm that allows encoding potentials along the characteristic lines that tessellate the spacetime. The new technique produces higher performance gains than those achieved through traditional FDTD.

Micromagnetic simulations on demagnetization processes in anisotropic Nd2Fe14B magnets

Individual grains with diverse dimensional parameters were introduced to investigate the magnetization reversals in anisotropic Nd2Fe14B magnets. The micromagnetic simulations were carried out via Object Oriented MicroMagnetic Framework (OOMMF). With the same bottom area and height, analysis results show that the coercive fields for different bottom shapes are of similar values. Designed as a cubic grain, the coercive field presents descending tendency as grain volume ascends. Under constant grain volume, with aspect ratio increasing, the coercive field decreases in the beginning and increases soon. Based on the demagnetization field vector, the effects of bottom shape, grain volume and aspect ratio on the coercive field can be explained. The nucleation point is chosen to discuss. Its synthetic field and reversal field are calculated by parallelogram law and inverse external field equation, respectively. The synthetic field equal to the reversal field is defined as critical field, which always shows the same tendency as the coercive field for all cases of this study. It can be concluded that critical field is qualified to be a reference index to measure the magnitude of coercive field.

A generalization of the parallelogram law to higher dimensions

We propose a generalization of the parallelogram identity in any dimension N ≥ 2 , establishing the ratio of the quadratic mean of the diagonals to the quadratic mean of the faces of a parallelotope. The proof makes use of simple properties of the exterior product of vectors.

Influence of the Ply Angle Deviation on the Out-of-Plane Deformation of the Composite Space Mirror

In order to discuss the influence of the ply angle deviation, six symmetric quasi-isotropic layup sequences imposed the fixed or random angle deviation were studied based on the numerical analysis, such as Monte Carlo stochastic finite element method and statistic analysis. The out-of-plane deformation was observed in spite of the symmetric quasi-isotropic sequences subjected to a temperature change due to the ply angle deviation in the space mirror. The surface peak-to-valley (PV), which is one kind of out-of-plane deformation, is proportional to the magnitude of the fixed angle deviation, and the mean of PVs is also proportional to the standard deviation of the random angle deviation. The out-of-plane deformation resulting from two-ply angle deviation indirectly satisfies the parallelogram law, and if each ply angle deviation is random independently, the probability density of that satisfies the Rayleigh distribution. In this paper, it is found that the most effective method to improve the surface accuracy of the space mirror is decreasing the standard deviation of the angle deviation, next to decreasing ply thickness and optimizing layup sequence, which is useful for application in the space mirror.

Light Path

This paper focuses on the mathematisation of mechanics in the seventeenth century, specifically on how the representation of compounded rectilinear motions presented in the ancient Greek Mechanica found its way into Newton’s Principia almost two thousand years later. I aim to show that the path from the former to the latter was optical: the conceptualisation of geometrical lines as paths of reflection created a physical interpretation of dia­grammatic principles of geometrical point-motion, involving the kinematics and dynamics of light reflection. Upon the atomistic conception of light, the optical interpretation of such geometrical principles entailed their mechanical generalisation to local motion; rectilinear motion via the physico-mathemat­ics of reflection and the Mechanica’s parallelogram rule; circular motion via the physico-mathematics of reflection, the Archimedean squaring of the circle and the Mechanica’s extension of the parallelogram rule to centripetal motion. This appeal to the physico-mathematics of reflection forged a realist founda­tion for the mathematisation of motion. Whereas Aristotle’s physics rested on motions which had their source in the nature of the elements, early modern thinkers such as Harriot, Descartes, and Newton based their new principles of mechanical motion upon selected elements of the mechanics of light motion, projected upon the geometry of the parallelogram rule for rectilinear and, ultimately, circular motion.

Modeling Parametric Processes in Few-Mode Semiconductor Optical Amplifiers

Few-mode semiconductor optical amplifiers (FM SOAs) allow more than one transverse modes to propagate inside of the waveguide region. They are useful in optical signal processing of distinct spatial mode signals when applied to spatial division multiplexing in an optical fiber communication. A model for FM SOAs including carrier rate and coupled-mode equations, which govern linear and nonlinear effects, is derived for the first time. The model reveals two distinctions of parametric process in FM SOAs through carrier gratings. First, the carrier gratings with transverse variation patterns provide unique mode selectivity, which only allows efficient interaction between certain modes. This mode selectivity is summarized as a parallelogram rule, a simple method to find the interacting modes and categorize them. Second, degenerate four wave mixing (FWM) via static carrier grating diffraction distinguishes itself from other types of FWMs for the highest wave interaction efficiency. Theoretical analysis and numerical simulation show that optical signal processing based on the degenerate FWM features high power efficiency, broadband operation, phase sensitivity, and mode selectivity. The processing can be done either in the spatial mode domain or in the frequency domain, or by combing them together. These features bestowed by the introduction of spatial degree of freedom, make FM SOAs promising in optical signal processing.

Coordinating analytic and visual approaches: Math majors’ understanding of orthogonal Hermite polynomials in the inner product space [formula omitted] in a technology-assisted learning environment

The purpose of this research study was to understand how linear algebra students in a university in the United States make sense of the orthogonal Hermite polynomials as vectors of the inner product space ℙnℝ in a Dynamic Geometry Software (DGS) – MATLAB facilitated learning environment. Math majors came up with a diversity of innovative and creative ways in which they coordinated visual and analytic approaches (Zazkis, Dubinsky, & Dautermann, 1996) for visualizing inner products of Hermite 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. Research participants not only produced such creative inner product space visualizations of the Hermite polynomials with the induced improper integral inner product 〈f,g〉=∫−∞∞e−x2f(x)g(x)dx on the DGS, but they also verified their findings both analytically and visually in coordination. The paper concludes by offering pedagogical implications along with implications for mathematics teaching profession and recommendations for future research.

Coordinating representation registers: linear algebra students' understanding of orthogonal Legendre polynomials in the inner product space ℙn in a technology-assisted learning environment

The purpose of this research study was to understand how linear algebra students in a university in the United States make sense of the orthogonal Legendre polynomials as vectors of the inner product space ℙn in a DGS (Dynamic Geometry Software)-MATLAB facilitated learning environment. Math majors came up with a diversity of innovative and creative ways in which they coordinated semiotic registers (Duval, 1993, 2006) for understanding inner products of Legendre 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. Research participants not only produced such creative inner product space visualizations of the Legendre polynomials with the associated integral inner product 〈f,g〉 = ∫1-1 f(x)g(x)dx on the DGS, but they also verified their findings both analytically and visually in coordination. The paper concludes by offering pedagogical implications along with implications for mathematics teaching profession and recommendations for future research.

Fuzzy inner product spaces and fuzzy orthogonality

In this paper, we present two new fuzzy inner product spaces and investigate some basic properties of these spaces. Specially, we prove parallelogram law for two and three vectors. Also, we introduce a suitable notion of orthogonality.

What happens when a climber falls? Young climbers mathematise a climbing situation

Students in Norway and other countries experience vectors as a difficult topic. Four young skilled climbers, who all did well in mathematics at school, participated in the Vector Study (VS). They participated for free and each lesson lasted until the students decided it was over. The idea was to investigate how climbing may function as a basis for students’ development of a vector concept. The teaching goal was the parallelogram law of vector addition. The students investigated what happens when a climber falls. They discussed the situation, and they tested it out in practice. They also performed a supporting activity, a rope-pulling situation, which provides insight into what happens in the climbing situation. The students’ development is analysed by identifying a) Bishop’s six basic activities, b) the role of angles, and c) manipulation of mental objects. The analysis reveals that relations between a vector’s magnitude and direction was central in the students’ investigations. It is important that students develop two aspects of vectors before the parallelogram law of addition is introduced. These are a) relations between angles and vectors, and b) the zero vector.