This paper introduces the concept of Fg-metric Space. We generalize the concept of G-metric space. Some supporting examples are given.

In this paper, we present the concept of conventional $F_G$-contraction and prove the results of a new coincidence point for multi-valued in b-metric spaces endowed with a digraph $G.$

We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping ${\bf r}$ from a two dimensional parameter space $M$ to the three dimensional Euclidean space ${\bf R}^3$. The metric variable $g_{ab}$, which is always fixed to the Euclidean metric $\delta_{ab}$, can be extended to a more general non-Euclidean metric on $M$ in the continuous model. The problem we focus on in this paper is whether such an extension is well-defined or not in the discrete model. We find that a discrete surface model with nontrivial metric becomes well-defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in $M$ depends on the direction. It is also shown that the discrete FG model is orientation assymetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some assymetries for orientation changing transformations.

An actions Φ on compact metric G – space M are shown to have G – inverse shadowing property with respect to a class of d – methods which are represented by continuous mappings. We proved that fgg_j has G – inverse shadowing property if f_g has this property and the converse also true for all j ∈ ℕ. And f_g ο h_g, f_g × k_g, f_g + h_g, f_g − h_g, and f_g·h_g also have G – inverse shadowing property if f_g and h_g have G – inverse shadowing property. Finally, we showed that this property is deserved in the topologically conjugate with respect to the classes of homeomorphisms methods : if there exists a homeomorphism h_g on M such that f_g ο h_g = h_g ο k_g then f_g has G – inverse shadowing property with respect to the classes of homeomorphism methods if k_g has G – inverse shadowing property with respect to the classes of homeomorphism methods.

In this article, we have attained some interesting results on existence of fixed points for a newly developed larger class of self mappings called FG -Kannan defined on a metric space equipped with a special type of graph called ʊ-orbitally connected graph. This extended class is a merger of the most recently developed Kannan type mapping called F-Kannan contractions defined in a metric space and G-Kannan mapping defined in a metric space with an underlying graph. It is highlighted through example that such a graph condition is sufficient to study fixed points of FG -Kannan mappings. Several interesting examples are illustrated which justify that our obtained results are more general and further, many previously developed fixed point results related to Kannan mappings are encompassed in our main result. The article closes by raising some open problems of this work.

This paper extends parts of the results from [P.W.Michor and D. Mumford, \emph{Appl. Comput. Harmon. Anal.,} 23 (2007), pp. 74--113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of $\mathbb R^n$ of type $M$, or the orbifold of immersions from $M$ to $\mathbb R^n$ modulo the group of diffeomorphisms of $M$. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: $$ G_f(h,k) = \int_{M} \Phi(\on{Vol}(f),\operatorname{Tr}(L))\g(h, k) \operatorname{vol}(f^*\g),$$ where $\g$ is the Euclidean metric on $\mathbb R^n$, $f^*\g$ is the induced metric on $M$, $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$ to the space of embeddings or immersions, where $\Phi:\mathbb R^2\to \mathbb R_{>0}$ is a suitable smooth function, $\operatorname{Vol}(f) = \int_M\operatorname{vol}(f^*\g)$ is the total hypersurface volume of $f(M)$, and the trace $\operatorname{Tr}(L)$ of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of $\Phi$ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.

In this paper we introduced a new notation G – fg – contraction of Caristi type and a new edge preserving property. With help of these we proved a some coupled fixed point results for four maps endowed with a graph in a complete metric space. Also we gave an application to integral equations.

It is shown that the set F of bijective mappings on a metric space which preserve the metric, forms a group under composition. A metric don Rⁿ is developed using continuous bijective functions g₁... ,g_non R, it is shown that dinduces the usual topology on Rⁿ. In R¹ with the metric d,the group F_g of metric preserving bijective mappings is shown to be isomorphic to the group F₍ of bijective mappings in R which preserve the usual metric or a subgroup of F₁ if the function g₁ is not surjective. A method of generating metrics from an existing metric is introduced. It is shown that these metrics induce the same topology on Rⁿ and the group of metric preserving bijective mappings is isomorphic to the group F_n of metric preserving mappings in Rⁿ with the same topology. If the metric on Rⁿ is induced by some norm, then it is shown that the group of bijective mappings which preserve the above metric is isomorphic to a subgroup of F_n. It is seen that Ulam's conjecture ‘if dis a metric on Rⁿ which induces the usual topology on Rⁿ, then the set of bijective mappings in Rⁿ which preserve dis a group under composition and is isomorphic to a subgroup of the group of mappings which preserve the usual metric’, holds in the cases studied above.

The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim $\sigma$-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by considering a triplet of Lie groups $F \supset G \supset H$. We show that for every symmetric space $F/G$, the generalized sine-Gordon models can be derived from the $G/H$ WZW action, plus a potential term that is algebraically specified. Thus, the symmetric space sine-Gordon models describe certain integrable perturbations of coset conformal field theories at the classical level. We also briefly discuss their vacuum structure, Backlund transformations, and soliton solutions.

In continuation of [3] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^p\Phi_i(\Vol(f)) \g((P_i)_fh,k) \vol(f^*\g) $$ on the space of immersions $\Imm(M,N)$ and on shape space $B_i(M,N)=\Imm(M,N)/\on{Diff}(M)$. Here $(N,\g)$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\g$ is the induced Riemannian metric on $M$, $\vol(f^*\g)$ is the induced volume density on $M$, $\Vol(f)=\int_M\vol(f^*\g)$, $\Phi_i$ are positive real-valued functions, and $(P_i)_f$ are operators like some power of the Laplacian $\Delta^{f^*\g}$. We derive the geodesic equations for these metrics and show that they are sometimes well-posed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights $\Ph_i(\Vol(f))$ which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail.

We define a class of space-times that we call asymptotically locally Schroedinger space-times. We consider these space-times in 3 dimensions, in which case they are also known as null warped AdS. The boundary conditions are formulated in terms of a specific frame field decomposition of the metric which contains two parts: an asymptotically locally AdS metric and a product of a lightlike frame field with itself. Asymptotically we say that the lightlike frame field is proportional to the particle number generator N regardless of whether N is an asymptotic Killing vector or not. We consider 3-dimensional AlSch space-times that are solutions of the massive vector model. We show that there is no universal Fefferman-Graham (FG) type expansion for the most general solution to the equations of motion. We show that this is intimately connected with the special role played by particle number. Fefferman-Graham type expansions are recovered if we supplement the equations of motion with suitably chosen constraints. We consider three examples. 1). The massive vector field is null everywhere. The solution in this case is exact as the FG series terminates and has N as a null Killing vector. 2). N is a Killing vector (but not necessarily null). 3). N is null everywhere (but not necessarily Killing). The latter case contains the first examples of solutions that break particle number, either on the boundary directly or only in the bulk. Finally, we comment on the implications for the problem of holographic renormalization for asymptotically locally Schroedinger space-times.

Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \g(P^f h, k)\, \vol(f^*\g)$$ where $\g$ is some fixed metric on $N$, $f^*\g$ is the induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where $\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\bar g$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.Comment: 52 pages, final version as it will appea

Using the invariance of Quadratic Gravity in FLRW metric under the group of diffeomorphisms of the time coordinate, we rewrite the action A of the theory in terms of the invariant dynamical variable g(τ).We propose to consider the path integrals ∫F(g) exp {−A}dg as the integrals over the functional measure μ(g) = exp {−A₂}dg, where A₂ is the part of the action A quadratic in R. The rest part of the action in the exponent stands in the integrand as the “interaction” term. We prove the measure μ(g) to be equivalent to the Wiener measure, and, as an example, calculate the averaged scale factor in the first nontrivial perturbative order.

The optical system of the James Webb Space Telescope (JWST) is split between two of the Observatory's element, the Optical Telescope Element (OTE) and the Integrated Science Instrument Module (ISIM). The OTE optical design consists of an 18-hexagonal segmented primary mirror (25m² clear aperture), a secondary mirror, a tertiary mirror, and a flat fine steering mirror used for fine guidance control. All optical components are made of beryllium. The primary and secondary mirror elements have hexapod actuation that provides six degrees of freedom rigid body adjustment. The optical components are mounted to a very stable truss structure made of composite materials. The OTE structure also supports the ISIM. The ISIM contains the Science Instruments (SIs) and Fine Guidance Sensor (FGS) needed for acquiring mission science data and for Observatory pointing and control and provides mechanical support for the SIs and FGS. The optical performance of the telescope is a key performance metric for the success of JWST. To ensure proper performance, the JWST optical verification program is a comprehensive, incremental, end-to-end verification program which includes multiple, independent, cross checks of key optical performance metrics to reduce risk of an on-orbit telescope performance issues. This paper discusses the verification testing and analysis necessary to verify the Observatory's image quality and sensitivity requirements. This verification starts with component level verification and ends with the Observatory level verification at Johnson Space Flight Center. The optical verification of JWST is a comprehensive, incremental, end-to-end optical verification program which includes both test and analysis.

The concept of $FG$- coupled fixed point introduced recently is a generalization of coupled fixed point introduced by Guo and Lakshmikantham. A point $(x,y)\in X\times X$ is said to be a coupled fixed point of the mapping $F: X\times X \rightarrow X$ if $F(x,y)=x$ and $F(y,x)=y$, where $X$ is a non empty set. In this paper, we introduce $FG$- coupled fixed point in cone metric spaces for the mappings $F:X\times Y \rightarrow X$ and $G:Y\times X\rightarrow Y$ and establish some $FG$- coupled fixed point theorems for various mappings such as contraction type mappings, Kannan type mappings and Chatterjea type mappings. All the theorems assure the uniqueness of $FG$- coupled fixed point. Our results generalize several results in literature, mainly the coupled fixed point theorems established by Sabetghadam et al. for various contraction type mappings. An example is provided to substantiate the main theorem.

The usefulness of the hit-miss transform (HMT) and related transforms for pattern matching in document image applications is examined. Although the HMT is sensitive to the types of noise found in scanned images, including both boundary and random noise, a simple extension, the blur HMT, is relatively robust. The noise immunity of the blur HMT derives from its ability to treat both types of noise together, and to remove them by appropriate dilations. In analogy with the Hausdorff metric for the distance between two sets, metric generalizations for special cases of the blur HMT are derived. Whereas Hausdorff uses both directions of the directed distances between two sets, a metric derived from a special case of the blur HMT uses just one direction of the directed distances between foreground and background components of two sets. For both foreground and background, the template is always the first of the directed sets. A less restrictive metric generalization, where the disjoint foreground and background components of the template need not be set complements, is also derived. For images with a random component of noise, the blur HMT is sensitive only to the size of the noise, whereas Hausdorff matching is sensitive to its location. It is also shown how these metric functions can be derived from the distance functions of the foreground and background of an image, using dilation by the appropriate templates. The blur HMT is implemented efficiently with Boolean image operations. The FG and BG images are dilated with structuring elements that depend on image noise and pattern variability, and the results are then eroded with templates derived from patterns to be matched. Subsampling the patterns on a regular grid can improve speed and maintain match quality, and examples are given that indicate how to explore the parameter space. The blur HMT can be used as a fast heuristic to avoid more expensive integer-based matching techniques. Truncated matches give the same result as full erosions and are much faster.

This is a review paper based on a recent article on FG- coupled fixed points [17], in which the authors established FG- coupled fixed point theorems in partially ordered complete S∗ metric space. The results were illustrated by suitable examples, too. An S∗ metric is an n-tuple metric from n-product of a set to the non negative reals. The theorems in [17] generalizes the main results of Gnana Bhaskar and Lakshmikantham [5].

In this paper, we introduce a geometry called F-geometry on a statistical manifold S using an embedding F of S into the space R_X of random variables. Amari’s α-geometry is a special case of F-geometry. Then using the embedding F and a positive smooth function G, we introduce (F,G)-metric and (F,G)-connections that enable one to consider weighted Fisher information metric and weighted connections. The necessary and sufficient condition for two (F,G)-connections to be dual with respect to the (F,G)-metric is obtained. Then we show that Amari’s 0-connection is the only self dual F-connection with respect to the Fisher information metric. Invariance properties of the geometric structures are discussed, which proved that Amari’s α-connections are the only F-connections that are invariant under smooth one-to-one transformations of the random variables.

In this work, our intention is to introduce the notion of rational (?-?-FG)-contraction mapping in b-metric-like spaces, and produce relevant fixed point and periodic point results for weakly ?-admissible mappings. Ulam-Hyers stability of this problem is also investigated. To illustrate our results, we give throughout the paper some examples, in particular in order to justify the use of rational terms. As an application, we obtain sufficient conditions for the existence of solutions for Cantilever Beam Problem.

Algebras and Lattices We continue with our study of B(X), the space of bounded real-valued functions on a set X. As we have seen, B(X) is a Banach space when supplied with the norm ∥f∥∞ = sup x∈X |f(x)|. Moreover, convergence in B(X) is the same as uniform convergence. Of course, if X is a metric space, we will also be interested in C(X), the space of continuous real-valued functions on X, and its cousin Cb (X) = C(X) ∩ B(X), the closed subspace of bounded continuous functions in B(X). Finally, if X is a compact metric space, recall that Cb (X) = C(X). But now we want to add a few more ingredients to the recipe: It's time we made use of the algebraic and lattice structures of B(X). In this chapter we will make formal our earlier informal discussions of algebras and lattices. In particular, we will see how this additional structure leads to a generalization of the Weierstrass approximation theorem in C(X), where X is a compact metric space. To begin, an algebra is a vector space A on which there is defined a multiplication (f, g) ↦ fg (from A × A into A) satisfying (i) (fg)h = f(gh), for all f, g, h ∈ A; (ii) f(g + h) = fg + fh, (f + g)h = fh + gh, for all f, g, h ∈ A; (iii) α(fg) = (αf)g = f(αg), for all scalars α and all f, g ∈ A. […]

We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping $\mathbf{r}$ from a two-dimensional parameter space M to the three-dimensional Euclidean space ${\mathbf{R}}^3$. The metric variable $g_{ab}$, which is always fixed to the Euclidean metric ${\delta }_{ab}$, can be extended to a more general non-Euclidean metric on M in the continuous model. The problem we focus on in this paper is whether such an extension is well defined or not in the discrete model. We find that a discrete surface model with a nontrivial metric becomes well defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in M depends on the direction. It is also shown that the discrete FG model is orientation asymmetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some asymmetries for orientation-changing transformations.

In this paper we prove FG-coupled fixed point theorems for different contractive mappings and generalized quasi- contractive mappings in partially ordered complete metric spaces. We prove the existence of FG-coupled fixed points of continuous as well as discontinuous mappings. We give some examples to illustrate the results.

This introductory chapter begins with a brief definition of conformal geometry. Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. In [FG], the authors outlined a construction of a nondegenerate Lorentz metric in n+2 dimensions associated to an n-dimensional conformal manifold, which they called the ambient metric. This association enables one to construct conformal invariants in n dimensions from pseudo-Riemannian invariants in n+2 dimensions, and in particular shows that conformal invariants are plentiful. The formal theory outlined in [FG] did not provide details. This book provides these details. An overview of the subsequent chapters is also presented.

In this paper, we give a complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ having positive isotropic curvature. We prove that for a pair $(f, \kappa)$ of a nontrivial smooth function $f: M \to {\Bbb R}$ and a nonnegative real number $\kappa$, if $(M, g)$ having positive isotropic curvature satisfies $$ Ddf - (\Delta f)g - f{\rm Ric} = \kappa g, $$ then $(M, g)$ is isometric to a geodesic ball in ${\Bbb S}^n$ when $\kappa >0$, and either $M$ isometric to ${\Bbb S}^n_+$, or the product $I \times {\Bbb S}^{n-1}$, up to finite cover when $\kappa =0$.

The purpose of this document is to investigate the universe in f(G) gravity. A wgeneral static plane symmetric space-time is chosen and exact solutions are explored using a viable f(G) gravity model. In particular, power and exponential law solutions are discussed. In addition, the physical relevance of the solutions with Taub’s metric and anti-deSitter space-time is shown. Graphical analysis of energy density and pressure of the universe is done to substantiate the study.

Functional trait and biological integrity approaches in stream ecology enable the determination and prediction of aquatic community responses to a variety of environmental stressors, such as chemical pollution, habitat disturbance, and biological invasion. Here, we used multi-trait based functional groups (FGs) to predict the functional responses of fish assemblages to the physicochemical and ecological health gradients in a temperate stream. The multi-metric biological integrity model (mIBI model) was used to evaluate stream ecological health. The FGs were derived from the distance matrix of trophic, tolerance, and physical habitat traits among fish species. The leading water quality indicators (conductivity [EC], total suspended solids [TSS], and chlorophyll-a [CHL-a]) varied conspicuously with the stream gradient and anthropogenic pollution. The multi-metric water-pollution index (mWPI) showed differences in chemical health from upstream to downstream. Monsoon precipitation may have affected the variations in fish species and associated changes linked to irregular chemical health. The fish FGs varied more by space (longitudinal) than by season (premonsoon and postmonsoon). Functional metrics, which reflected trophic and tolerance traits, as well as vertical position preference, were strongly correlated with water quality degradation downstream. Changes were evident in FG (II, III, and IV) combinations from the upstream to downstream reaches. Furthermore, the structure of the fish assemblages from FG-II and FG-III was significantly correlated with chemical (R² = 0.43 and 0.35, p < 0.001) and ecological health (R² = 0.69 and 0.66, p < 0.001), as well as the metrics of mWPI. In conclusion, the results indicate significant variations in both trait-based FGs and biological integrity among stream-fish communities, influenced by chemical water quality gradients. The causes included longitudinal zones and intensifying degradation of water quality downstream. Therefore, multi-trait based FGs can facilitate ecological health assessment and develop the mIBI model based on fish assemblages by reflecting the prevailing chemical health status of streams and rivers.

We set up the existence of a symmetric outcome of a system of simultaneous nonlinear fractional integral equations, that arises in motion of water wave on smooth surface, with the help of a common fixed point theorem satisfying a generalized FG-contractive condition. To accomplish this, we introduce first the concept of generalized FG-contractive condition for two pairs of self-mappings in a complete metric space and then we establish requisites for common fixed point results for weakly compatible mappings followed by a suitable example.