Notion Of Orthogonality Research Articles

Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curves

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

Isosceles constant in Banach spaces

The rectangular constant in Banach spaces was introduced in a paper by N. Gastinel and J.L. Joly in 1970 and has also received attention recently. To define such a constant, the notion of orthogonality, according to Birkhoff and James, is used. Here, we introduce and study a similar constant, but based on isosceles orthogonality. We indicate several properties of the new constant as a characterization of Hilbert spaces and of uniformly non-square spaces. These characterizations are similar to those proven for the old constant. Some illustrative examples complete the paper.

Local orthogonal maps and rigidity of holomorphic mappings between real hyperquadrics

We propose a coordinate-free approach to study the holomorphic maps between the real hyperquadrics in complex projective spaces. It is based on a notion of orthogonality on the projective spaces induced by the Hermitian structures that define the hyperquadrics. There are various kinds of special linear subspaces associated to this orthogonality which are well respected by the relevant holomorphic maps and we obtain rigidity theorems by analyzing the images of these linear subspaces, together with techniques in projective geometry. Our method allows us to recover and generalize a number of well-known results in the field with simpler arguments.

Discrete Isothermic Nets Based on Checkerboard Patterns

This paper studies the discrete differential geometry of the checkerboard pattern inscribed in a quadrilateral net by connecting edge midpoints. It turns out to be a versatile tool which allows us to consistently define principal nets, Koenigs nets and eventually isothermic nets as a combination of both. Principal nets are based on the notions of orthogonality and conjugacy and can be identified with sphere congruences that are entities of Möbius geometry. Discrete Koenigs nets are defined via the existence of the so-called conic of Koenigs. We find several interesting properties of Koenigs nets, including their being dualizable and having equal Laplace invariants. Isothermic nets can be defined as Koenigs nets that are also principal nets. We prove that the class of isothermic nets is invariant under both dualization and Möbius transformations. Among other things, this allows a natural construction of discrete minimal surfaces and their Goursat transformations.

A necessary and sufficient condition for a direct sum of modules to be distributive

Let R be an associative ring with unity. A unital left R-module M is said to be distributive if for every submodules S, T and U of M, the equality S ∩ ( T + U ) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for a direct sum of left R-modules to be distributive. This condition is given by the notion of splitting of submodules of the direct sum and the proof uses the notion of orthogonality, where both notions are discussed and revisited.

On the Iterative Multivalued ⊥-Preserving Mappings and an Application to Fractional Differential Equation

In this paper, we introduce orthogonal multivalued contractions, which are based on the recently introduced notion of orthogonality in the metric spaces. We construct numerous fixed point theorems for these contractions. We show how these fixed point theorems aid in the generalization of a number of recently published findings. Additionally, we offer a theorem that establishes the existence of a fractional differential equation’s solution.

Hessian geometry of nonequilibrium chemical reaction networks and entropy production decompositions

We derive the Hessian geometric structure of nonequilibrium chemical reaction networks on the flux and force spaces induced by the Legendre duality of convex dissipation functions and characterize their dynamics as a generalized flow. With this geometric structure, we can extend theories of nonequilibrium systems with quadratic dissipation functions to more general cases with nonquadratic ones, which are pivotal for studying chemical reaction networks. By applying generalized notions of orthogonality in Hessian geometry to chemical reaction networks, two generalized decompositions of the entropy production rate are obtained, each of which captures gradient-flow and minimum-dissipation aspects in nonequilibrium dynamics.

Some fixed point theorems on orthogonal metric spaces via extensions of orthogonal contractions

Orthogonal metric space is a considerable generalization of a usual metric space obtained by establishing a perpendicular relation on a set. Very recently, the notions of orthogonality of the set and orthogonality of the metric space are described and notable fixed point theorems are given in orthogonal metric spaces. Some fixed point theorems for the generalizations of contraction principle via altering distance functions on orthogonal metric spaces are presented and proved in this paper. Furthermore, an example is presented to clarify these theorems.

Orthogonal Semiderivations and Symmetric Bi-semiderivations in Semiprime Rings

In this paper, orthogonality for symmetric bi-semiderivations is defined and some results are obtained when two symmetric bi-semiderivations are orthogonal. Also, this paper gives the notion of orthogonality between semiderivations and symmetric bi-semiderivations of a 2-torsion free semiprime ring and offers some results of orthogonality.In this paper, orthogonality for symmetric bi-semiderivations is defined and some results are obtained when two symmetric bi-semiderivations are orthogonal. Also, this paper gives the notion of orthogonality between semiderivations and symmetric bi-semiderivations of a 2-torsion free semiprime ring and offers some results of orthogonality.

Nonlinear steepest descent approach to orthogonality on elliptic curves

We consider the recently introduced notion of denominators of Padé-like approximation problems on a Riemann surface. These denominators are related as in the classical case to the notion of orthogonality over a contour. We investigate a specific setup where the Riemann surface is a real elliptic curve with two components and the measure of orthogonality is supported on one of the two real ovals. Using a characterization in terms of a Riemann–Hilbert problem, we determine the strong asymptotic behaviour of the corresponding orthogonal functions for large degree. The theory of vector bundles and the non-abelian Cauchy kernel play a prominent role even in this simplified setting, indicating the new challenges that the steepest descent method on a Riemann surface has to overcome.

Quantum combinatorial designs and k-uniform states

Goyeneche et al [2018 Phys. Rev. A 97 062326] introduced several classes of quantum combinatorial designs, namely quantum Latin squares, quantum Latin cubes, and the notion of orthogonality on them. They also showed that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Moreover, they established a relationship between quantum combinatorial designs and a remarkable class of entangled states called k-uniform states, i.e. multipartite pure states such that every reduction to k parties is maximally mixed. In this article, we put forward the notions of incomplete quantum Latin squares and orthogonality on them and present construction methods for mutually orthogonal quantum Latin squares and mutually orthogonal quantum Latin cubes. Furthermore, we introduce the notions of generalized mutually orthogonal quantum Latin squares and generalized mutually orthogonal quantum Latin cubes, which are equivalent to quantum orthogonal arrays of size d 2 and d 3, respectively, and thus naturally provide two- and three-uniform states.

A new notion of orthogonality involving area and length

Abstract In this paper, a space called geometric space, involving both the notions of area and length, is introduced in general setting. The interplay, between these two ideas, is studied. As a result, a new notion of orthogonality, called area-length orthogonality or A-L orthogonality, is demonstrated. It is shown that A-L orthogonality coincides with the standard notion of orthogonality for inner product spaces. Finally, it is proved that A-L orthogonality implies Birkhoff orthogonality, but not conversely.

Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc

The symmetrized bidisc G=def{(z+w,zw):|z|<1,|w|<1},\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} G {\\mathop {=}\\limits ^\\mathrm{{def}}}\\{(z+w,zw):|z|<1,\\quad |w|<1\\}, \\end{aligned}$$\\end{document}under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, G does not admit a natural notion of angle, but we nevertheless show that there is a notion of orthogonality. The complex tangent bundle TG splits naturally into the direct sum of two line bundles, which we call the sharp and flat bundles, and which are geometrically defined and therefore covariant under automorphisms of G. Through every point of G, there is a unique complex geodesic of G in the flat direction, having the form Fβ=def{(β+β¯z,z):z∈D}\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} F^\\beta {\\mathop {=}\\limits ^\\mathrm{{def}}}\\{(\\beta +{\\bar{\\beta }} z,z)\\ : z\\in \\mathbb {D}\\} \\end{aligned}$$\\end{document}for some beta in mathbb {D}, and called a flat geodesic. We say that a complex geodesic Dis orthogonal to a flat geodesic F if D meets F at a point lambda and the complex tangent space T_lambda D at lambda is in the sharp direction at lambda . We prove that a geodesic D has the closest point property with respect to a flat geodesic F if and only if D is orthogonal to F in the above sense. Moreover, G is foliated by the geodesics in G that are orthogonal to a fixed flat geodesic F.

Partial isometries in an absolute order unit space

In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.

On the orthogonality and convolution orthogonality via the Kontorovich–Lebedev transform

Notions of orthogonality and convolution orthogonality are explored with the use of the Kontorovich–Lebedev transform and its convolution. New classes of the corresponding orthogonal polynomials and functions are investigated. Integral representations, orthogonality relations and explicit expressions are established.

Orthogonality: Developing a structural/perspectival approach for improving theoretical models

AbstractIn order to create more useful theories we must attempt to choose better concepts (variables). Although we may strive to identify “independent” variables, that effort is problematic because we live in a world where everything is connected; also, attempting to obtain even nominally independent variables may require sophisticated analytical methodologies available only to well‐funded researchers and practitioners. To help resolve those problems, and so provide an improved path for developing more useful theories, we explore the notion of orthogonality as a tool for rethinking concepts in a theory, from a structural perspective, in a way that clarifies research opportunities (from a data perspective) and clarifies situational/theoretical perspectives (from a relevance perspective).

Birkhoff Orthogonality and Different Particular Cases of Carlsson's Orthogonality on Normed Linear Spaces

Let x, y ∈ X, where X is an inner-product space. We say x is orthogonal to y if ⟨x, y⟩ = 0. When we move to general normed spaces there are many possibilities of extending the notion of orthogonality. Since 1934, different types of orthogonality relations in normed spaces have been introduced and studied. In this study, we enlist some properties of Birkhoff's orthogonality and Carlsson's orthogonality along with it we introduce two new particular cases of Carlsson's orthogonality and check some properties of othogonality in relation to these particular cases in normed spaces.

Relation of Pythagorean and Isosceles Orthogonality with Best approximations in Normed Linear Space

In an arbitrary normed space, though the norm not necessarily coming from the inner product space, the notion of orthogonality may be introduced in various ways as suggested by the mathematicians like R.C. James, B.D. Roberts, G. Birkhoff and S.O. Carlsson. We aim to explore the application of orthogonality in normed linear spaces in the best approximation. Hence it has already been proved that Birkhoff orthogonality implies best approximation and best approximation implies Birkhoff orthogonality. Additionally, it has been proved that in the case of ε -orthogonality, ε -best approximation implies ε -orthogonality and vice-versa. In this article we established relation between Pythagorean orthogonality and best approximation as well as isosceles orthogonality and ε -best approximation in normed space.

Some characterizations of inner product spaces based on angle

A problem in functional analysis that arises naturally is about finding necessary and sufficient conditions for a normed space to be an inner product space. By answering this question, mathematicians try to understand the inner product and normed spaces features. In this note, we have discussed this issue and we prove some results concerned with it. We introduce a notion of angle between two vectors in a normed space, denoted by $A_\theta(.,.)$ where $\theta\neq{k\pi\over2}$. We also speak about a notion of orthogonality concerning it, we call it $\theta$-orthogonality.

On Orthogonality of Elementary Operators in Norm-attainable Classes

Various notions of orthogonality of elementary operators have been characterized by many mathematicians in different classes. In this paper, we characterize orthogonality of these operators in norm-attainable classes. We first give necessary and sufficient conditions for norm-attainability of Hilbert space operators then we give results on orthogonality of the range and the kernel of elementary operators when they are implemented by norm-attainable operators in norm-attainable classes.