Search Results (18)

Hyers and Ulam considered the problem of whether there is a true isometry that approximates the $\epsilon$-isometry defined on a Hilbert space with a stability constant $10\epsilon$. Subsequently, Fickett considered the same question on a bounded subset of the n-dimensional Euclidean space ${\mathbb{R}}^n$ with a stability constant of $27{\epsilon }^{1/2^n}$. And Vestfrid gave a stability constant of $27n\epsilon$ as the answer for bounded subsets. In this paper, by applying singular value decomposition, we improve the previous stability constants by $C\sqrt{n}\epsilon$ for bounded subsets, where the constant C depends on the approximate linearity parameter K, which is defined later. Full article

A study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach–Stone theorem to surjective and nonsurjective cases. Full article

Buckled rings, also known as pressurized elastic circles, can be described as critical points for a variational problem, namely the integral of a quadratic polynomial in the geodesic curvature of a curve. Thus, they are a generalization of elastic curves, and they are solitary wave solutions to a flow in a (three-dimensional) filament hierarchy. An example of such a curve is the Kiepert Trefoil, which has three leaves meeting at a central singular point. Such a variational problem can be considered for curves in other surfaces. In particular, researchers have found many examples of such curves in a unit sphere. In this article, we consider a new family of such curves, having a discrete dihedral symmetry about a central singular point. That is, these are spherical analogues of the Kiepert curve. We determine such curves explicitly using the notion of a Killing field, which is a vector field along a curve that is the restriction of an isometry of the sphere. The curvature k of each such curve is given explicitly by an elliptic function. If the curve is centered at the south pole of the sphere and has minimum value $\rho$, then $k-\rho$ is linear in the height above the pole. Full article

In this paper, we establish the stability of the almost-surjective coarse isometries of Banach spaces by means of the weak stability condition. In addition, we also discuss the existence of coarse left-inverse operators in classical Banach spaces. Making use of them, we generalize several known results related to $\epsilon$-isometries. Full article

Complexity is one of the most important variables in how the brain performs decision making based on esthetic values. Multiple definitions of perceptual complexity have been proposed, with one of the most fruitful being the Normalized Shannon Entropy one. However, the Normalized Shannon Entropy definition has theoretical gaps that we address in this article. Focusing on visual perception, we first address whether normalization fully corrects for the effects of measurement resolution on entropy. The answer is negative, but the remaining effects are minor, and we propose alternate definitions of complexity, correcting this problem. Related to resolution, we discuss the ideal spatial range in the computation of spatial complexity. The results show that this range must be small but not too small. Furthermore, it is suggested by the analysis of this range that perceptual spatial complexity is based solely on translational isometry. Finally, we study how the complexities of distinct visual variables interact. We argue that the complexities of the variables of interest to the brain’s visual system may not interact linearly because of interclass correlation. But the interaction would be linear if the brain weighed complexities as in Kempthorne’s λ-Bayes-based compromise problem. We finish by listing several experimental tests of these theoretical ideas on complexity. Full article

In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for $k\times 2k$ matrices M over R satisfying the conditions that its rows are linearly independent over R and $M{M}^{\top }={\mathit{\alpha }}^{\top }\mathit{\alpha }$ for an R-linearly independent vector $\mathit{\alpha }\in R^k.$ Let $\mathcal{C}$ be a linear code generated by such a matrix $M.$ We prove that the dual code ${\mathcal{C}}^{\perp }$ of $\mathcal{C}$ is also a free linear code with dimension k, as well as $\mathcal{C}/Hull\left(\mathcal{C}\right)$ and ${\mathcal{C}}^{\perp }/Hull\left(\mathcal{C}\right)$ are one-dimensional free R-modules, where $Hull\left(\mathcal{C}\right)$ represents the hull of $\mathcal{C}$. Based on these facts, an isometry from $R\mathit{x}+R\mathit{y}$ onto $R^2$ is established, assuming that $\mathit{x}+Hull\left(\mathcal{C}\right)$ and $\mathit{y}+Hull\left(\mathcal{C}\right)$ are bases for $\mathcal{C}/Hull\left(\mathcal{C}\right)$ and ${\mathcal{C}}^{\perp }/Hull\left(\mathcal{C}\right)$ over R, respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix $M{M}^{\top }$ takes the form of ${\mathit{\alpha }}^{\top }\mathit{\alpha }$ with $\mathit{\alpha }$ being a linearly independent vector in $R^k$, a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy $M{M}^{\top }=0$. The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach. Full article

Let $X,Y$ be two Banach spaces and $f:X\to Y$ be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when Y is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when Y is a $L_p(1<p<\infty )$ space, or both X and Y are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces. Full article

If T is a bounded linear operator on a Hilbert space $\mathcal{H}$ and V is a given linear isometry on a Hilbert space $\mathcal{K}$, we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry $\pi :\mathcal{H}\to \mathcal{K}$ such that $\pi T=V^{*}\pi$ (i.e., $(\pi ,V^{*})$ extends T). We parametrize the set of all solutions $\pi$ of this equation. We show, for example, that for a given unitary operator U on a Hilbert space $\mathcal{E}$ and for the multiplication operator by the independent variable $M_z$ on the Hardy space $H_{\mathcal{D}}^2\left(\mathbb{D}\right)$, there exists an isometric operator $\pi :\mathcal{H}\to \mathcal{E}\oplus H_{\mathcal{D}}^2\left(\mathbb{D}\right)$ such that $(\pi ,{(U\oplus M_z)}^{*})$ extends T if and only if T is a contraction, the defect index ${\delta }_T\le dim\mathcal{D}$ and, for some $Y:{\mathcal{A}}_T\to \mathcal{E}$, $(Y,U^{*})$ extends the isometric operator $A_T^{1/2}h\mapsto A_T^{1/2}Th$ on the space ${\mathcal{A}}_T=\overline{A_T\mathcal{H}}$, where $A_T$ is the asymptotic limit associated with T. We also prove that if T is isometric and V is unitary, there exists an isometric operator $\pi :\mathcal{H}\to \mathcal{K}$ such that $(\pi ,V)$ extends T if and only if (a) the spectral measures of the unitary part of T (in its Wold decomposition) and the restriction of V to one of its reducing subspaces ${\mathcal{K}}_0$ possess identical multiplicity functions and (b) $dim(ker{T}^{*})=dim({\mathcal{K}}_1\ominus V{\mathcal{K}}_1)$ for a certain subspace ${\mathcal{K}}_1$ of $\mathcal{K}$ that contains ${\mathcal{K}}_0$ and is invariant under V. The precise form of $\pi$, in each situation, and characterizations of the minimality conditions are also included. Several examples are given for illustrative purposes. Full article

In diverse branches of mathematics, several inequalities have been studied and applied. In this article, we improve Furuta’s inequality. Subsequently, we apply this improvement to obtain new radius inequalities that not been reported in the current literature. Numerical examples illustrate the main findings. Full article

We revisit universal features of duality in linear and nonlinear relativistic scalar and Abelian 1-form theories with single or multiple fields, which exhibit ordinary or generalized global symmetries. We show that such global symmetries can be interpreted as generalized Killing isometries on a suitable, possibly graded, target space of fields or its jet space when the theory contains higher derivatives. This is realized via a generalized sigma model perspective motivated from the fact that higher spin particles can be Nambu–Goldstone bosons of spontaneously broken generalized global symmetries. We work out in detail the 2D examples of a compact scalar and the massless Heisenberg pion fireball model and the 4D examples of Maxwell, Born–Infeld, and ModMax electrodynamics. In all cases we identify the ’t Hooft anomaly that obstructs the simultaneous gauging of both global symmetries and confirm the anomaly matching under duality. These results readily generalize to higher gauge theories for p-forms. For multifield theories, we discuss the transformation of couplings under duality as two sets of Buscher rules for even or odd differential forms. Full article

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms. Full article

Let $\mathrm{Lip}\left(\right[0,1\left]\right)$ be the Banach space of all Lipschitz complex-valued functions f on $[0,1]$, equipped with one of the norms: ${∥f∥}_{\sigma }=\left|f\left(0\right)\right|+{∥f^{\prime }∥}_{L^{\infty }}$ or ${∥f∥}_m=max\left\{\left|f\left(0\right)\right|,{∥f^{\prime }∥}_{L^{\infty }}\right\}$, where ${∥·∥}_{L^{\infty }}$ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of $\mathrm{Lip}\left(\right[0,1\left]\right)$. Namely, we prove that every approximate local isometry of $\mathrm{Lip}\left(\right[0,1\left]\right)$ can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of $\mathrm{Lip}\left(\right[0,1\left]\right)$. Additionally, some complete characterizations of such reflections and projections are stated. Full article

We will prove the generalized Hyers–Ulam stability of isometries, with a focus on the stability for restricted domains. More precisely, we prove the generalized Hyers–Ulam stability of the orthogonality equation and we use this result to prove the stability of the equations $\parallel f\left(x\right)-f\left(y\right)\parallel =\parallel x-y\parallel$ and ${\parallel f\left(x\right)-f\left(y\right)\parallel }^2={\parallel x-y\parallel }^2$ on the restricted domains. As we can easily see, these functional equations are symmetric in the sense that they become the same equations even if the roles of variables x and y are exchanged. Full article

We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures with almost contact metric homogeneous structures. This relation will have consequences in the class of the almost contact manifold. Indeed, if we choose a pseudo-Kähler homogeneous structure of linear type, then the reduced, almost contact homogeneous structure is of linear type and the reduced manifold is of type ${\mathcal{C}}_5\oplus {\mathcal{C}}_6\oplus {\mathcal{C}}_{12}$ of Chinea-González classification. Full article