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This research examines the Hyers–Ulam stability of norm-additive functional equations expressed as ∥ ξ (g h − 1) ∥ = ∥ ξ (g) − ξ (h) ∥ , ∥ ξ (g h) ∥ = ∥ ξ (g) + ξ (h) ∥ , through the utilization of (δ , ϵ) -isometries. In this context, ξ : G → X represents a surjective (δ-surjective) mapping, where G denotes noncommutative group (arbitrary group) and X signifies a Banach space (or a real Banach space). [ABSTRACT FROM AUTHOR]

In this research article, we focus on analyzing the generalized stability of norm-additive functional equations (FEs) ‖ ξ (g h) ‖ = ‖ ξ (g) + ξ (h) ‖ , and ‖ ξ (g h - 1) ‖ = ‖ ξ (g) - ξ (h) ‖ for every g , h ∈ G , where (G , ·) represents an arbitrary (noncommutative) group and B is a p-uniformly convex space, assuming that the mapping ξ is (δ , θ) -surjective. This proposed work generalizes and extends the findings of Y. Dong (2015), L. Sun (2023) and Y. Sun (2023) by applying (δ , θ , ϵ) -isometry. [ABSTRACT FROM AUTHOR]

This paper provides computational tools for the modeling and design of quad mesh mechanisms, which are meshes allowing continuous flexions under the assumption of rigid faces and hinges in the edges. We combine methods and results from different areas, namely differential geometry of surfaces, rigidity and flexibility of bar and joint frameworks, algebraic geometry, and optimization. The basic idea to achieve a time-continuous flexion is time-discretization justified by an algebraic degree argument. We are able to prove computationally feasible bounds on the number of required time instances we need to incorporate in our optimization. For optimization to succeed, an informed initialization is crucial. We present two computational pipelines to achieve that: one based on remeshing isometric surface pairs, another one based on iterative refinement. A third manner of initialization proved very effective: We interactively design meshes which are close to a narrow known class of flexible meshes, but not contained in it. Having enjoyed sufficiently many degrees of freedom during design, we afterwards optimize towards flexibility. [ABSTRACT FROM AUTHOR]

We show that if X is a complete metric space with uniform relative normal structure and G is a subgroup of the isometry group of X with bounded orbits, then there is a point in X fixed by every isometry in G. As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if L 1 ⁢ (μ) is an essential Banach L 1 ⁢ (G) -bimodule, then any continuous derivation δ : L 1 ⁢ (G) → L ∞ ⁢ (μ) is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra L 1 ⁢ (G) is weakly amenable if G is a locally compact group. [ABSTRACT FROM AUTHOR]

The normal curve is a space curve that plays an important role in the field of differential geometry. This research focuses on analyzing the properties of normal curves on smooth immersed surfaces, considering their invariance under isometric transformations. The primary contribution of this article is to explore the requirements for the image of a normal curve that preserves its invariance under isometric transformations. In this article, we investigate the invariant condition for the component of the position vector of the normal curves under isometry and compute the expression for the normal and geodesic curvature of such curves. Moreover, it has been investigated that the geodesic curvature and Christoffel symbols remain unchanged under the isometry of surfaces in R3{{\mathbb{R}}}^{3}.

An example of two noncongruent isometric analytic compact surfaces is given as a positive answer to M. Berger's question about the existence of such surfaces. [ABSTRACT FROM AUTHOR]

This paper examines characterizations of closed curves in a geodesic metric space with curvature bounded below, including closed geodesic polygons and closed spherical curves bounding surfaces isometric to convex polygons and circles in the model space. [ABSTRACT FROM AUTHOR]

The versatility of the shark body form is suggested to be one of the key factors underlying their evolutionary success and persistence. Nevertheless, sharks exhibit a huge diversity of body forms and morphological adaptations. More subtly, it is increasingly evident that in many species, morphology varies through ontogeny. Multiple competing hypotheses exist explaining both the function of specific morphological structures and the interspecific distribution of these ontogenetic morphological shifts. However, existing studies are restricted to a small number of mostly large‐bodied species. In this study, we report allometric scaling relationships from functionally important morphological structures in the spadenose shark (Scoliodon laticaudus). We find that a mosaic of isometric and allometric growth underlies the scaling trends in this species and that cases of allometry are consistent with an ontogenetic shift in diet. Moreover, our results refute suggestions that small‐bodied sharks grow isometrically. Given the small number of existing studies of ontogenetic morphometry in sharks and the life‐history/ecological characteristics of S. laticaudus, this study is a valuable contribution to our understanding of the adaptive value of ontogenetic morphological shifts in elasmobranchs. [ABSTRACT FROM AUTHOR]

In this paper, we study CR maps between hyperquadrics and Winkelmann hypersurfaces. Based on a previous study on the CR Ahlfors derivative of Lamel–Son and a recent result of Huang–Lu–Tang–Xiao on CR maps between hyperquadrics, we prove that a transversal CR map from a hyperquadric into a hyperquadric or a Winkelmann hypersurface extends to a local holomorphic isometric embedding with respect to certain Kähler metrics if and only if the Hermitian part of its CR Ahlfors derivative vanishes on an open set of the source. Our proof is based on relating the geometric rank of a CR map into a hyperquadric and its CR Ahlfors derivative. [ABSTRACT FROM AUTHOR]

Let H be a complex separable Hilbert space with $\dim H \geq 2$. Let $\mathcal {N}$ be a nest on H such that $E_+ \neq E$ for any $E \neq H, E \in \mathcal {N}$. We prove that every 2-local isometry of $\operatorname {Alg}\mathcal {N}$ is a surjective linear isometry. [ABSTRACT FROM AUTHOR]

For a locally compact Hausdorff space L, we denote by C 0 (L , R) the Banach space of all continuous real-valued functions on L vanishing at infinity equipped with the supremum norm. We prove that every surjective phase-isometry T : C 0 + (X , R) → C 0 + (Y , R) between the positive cones of C 0 (X , R) and C 0 (Y , R) is a composition operator induced by a homeomorphism between X and Y. Furthermore, we show that any surjective phase-isometry T : C 0 + (X , R) → C 0 + (Y , R) extends to a surjective linear isometry from C 0 (X , R) onto C 0 (Y , R) . [ABSTRACT FROM AUTHOR]

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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 × 2 k matrices M over R satisfying the conditions that its rows are linearly independent over R and M M ⊤ = α ⊤ α for an R-linearly independent vector α ∈ R k. Let C be a linear code generated by such a matrix M. We prove that the dual code C ⊥ of C is also a free linear code with dimension k, as well as C / H u l l (C) and C ⊥ / H u l l (C) are one-dimensional free R-modules, where H u l l (C) represents the hull of C. Based on these facts, an isometry from R x + R y onto R 2 is established, assuming that x + H u l l (C) and y + H u l l (C) are bases for C / H u l l (C) and C ⊥ / H u l l (C) 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 ⊤ takes the form of α ⊤ α with α 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 ⊤ = 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. [ABSTRACT FROM AUTHOR]

Skeletal ontogeny of xenarthrans is poorly known, especially because of the paucity of study specimens from distinct developmental stages. Here, we investigate morphometric aspects of the mandible ontogeny in the two-toed sloths, Choloepus spp. We examined mandibles of infant, juveniles and subadult sloths that were present in kill assemblages of harpy eagles, Harpia harpyja, and complemented our study with adult museum specimens. We carried out uni- and multivariate linear morphometric analyzes to assess the growth pattern of the mandible. Harpy eagles did not prey on adult two-toed sloths, preferring younger individuals. We found an overall strong correlation between the total length of the mandible and other mandibular measurements across age classes, with some of them scaling isometrically, and others presenting allometric growth. Also, morphometric data correlated with patterns of symphysial fusion across ontogenetic stages, rendering the latter a reliable indicator of the animal's age category. Although it was necessary to complement our sample with museum material, individuals obtained from the harpy eagle kill assemblage proved to be a valuable complementary source of specimens to be studied. [ABSTRACT FROM AUTHOR]

In this paper, we study composition and weighted composition operators that are close to isometries on H 2 but not necessarily isometric. We also obtain a Wold type decomposition for such operators. [ABSTRACT FROM AUTHOR]

In 1982, Fickett attempted to prove the Hyers-Ulam stability of isometries defined on a bounded subset of $ \mathbb{R}^n $. In this paper, we applied an intuitive and efficient approach to prove the Hyers-Ulam stability of isometries defined on the bounded subset of $ \mathbb{R}^4 $, and we significantly improved Fickett's theorem for the four-dimensional case.

The question of whether there is a true isometry that approximates the ε -isometry defined on a bounded set has long interested mathematicians. The first paper on this topic was published by Fickett, whose result was subsequently greatly improved by Alestalo et al., Väisälä and Vestfrid. Recently, the authors published some papers improving the previous results. The main purpose of this paper is to improve all of the abovementioned results by utilizing the properties of the norm and inner product for Euclidean space. [ABSTRACT FROM AUTHOR]

An open problem posed by G. Maksa and Z. Páles is to find the general solution of the functional equation { ‖ f (x) - β f (y) ‖ : β ∈ T n } = { ‖ x - β y ‖ : β ∈ T n } (x , y ∈ H) where f : H → K is between two complex normed spaces and T n : = { e i 2 k π n : k = 1 , ⋯ , n } is the set of the nth roots of unity. With the aid of the celebrated Wigner's unitary-antiunitary theorem, we show that if n ≥ 3 and H and K are complex inner product spaces, then f satisfies the above equation if and only if there exists a phase function σ : H → T n such that σ · f is a linear or anti-linear isometry. Moreover, if the solution f is continuous, then f is a linear or anti-linear isometry. [ABSTRACT FROM AUTHOR]

A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure lattice deviations from higher-symmetry neighbours. The geometric methods extend the past work of Delone, Conway, and Sloane. [ABSTRACT FROM AUTHOR]

We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of the operator in terms of the geometry of the concerned spaces. In particular, it is shown that such a study is related to the preservation of k-smoothness and the extremal properties of the unit ball of a Banach space. As an application of the results obtained in this direction, we obtain a refinement of the well-known Blanco-Koldobsky-Turnsek characterization of isometries on some polyhedral Banach spaces, including ℓ ∞ n , ℓ 1 n. [ABSTRACT FROM AUTHOR]

Let G L n (C) be the complex general linear group of degree n. In this paper, we present the general form of all continuous endomorphisms of G L 2 (C) with respect to the Jordan triple product. These are the continuous maps φ : G L 2 (C) → G L 2 (C) which satisfy φ (A B A) = φ (A) φ (B) φ (A) , A , B ∈ G L 2 (C). As a result, we present the general form of all continuous homomorphisms and automorphisms of G L 2 (C) . [ABSTRACT FROM AUTHOR]

In this paper, the group structure of the -adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations and on a ball and sphere, respectively, and prove that these sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere and show that the trajectory of any initial point that is not a fixed point is not convergent. We study ergodicity of this -adic dynamical system with respect to normalized Haar measure reduced on the sphere. For we prove that the dynamical systems are not ergodic. But for under some conditions the dynamical system may be ergodic. [ABSTRACT FROM AUTHOR]

ABSTRACT The versatility of the shark body form is suggested to be one of the key factors underlying their evolutionary success and persistence. Nevertheless, sharks exhibit a huge diversity of body forms and morphological adaptations. More subtly, it is increasingly evident that in many species, morphology varies through ontogeny. Multiple competing hypotheses exist explaining both the function of specific morphological structures and the interspecific distribution of these ontogenetic morphological shifts. However, existing studies are restricted to a small number of mostly large‐bodied species. In this study, we report allometric scaling relationships from functionally important morphological structures in the spadenose shark (Scoliodon laticaudus). We find that a mosaic of isometric and allometric growth underlies the scaling trends in this species and that cases of allometry are consistent with an ontogenetic shift in diet. Moreover, our results refute suggestions that small‐bodied sharks grow isometrically. Given the small number of existing studies of ontogenetic morphometry in sharks and the life‐history/ecological characteristics of S. laticaudus, this study is a valuable contribution to our understanding of the adaptive value of ontogenetic morphological shifts in elasmobranchs.

The Ilısu Hydroelectric Power Plant (HEPP), situated on the Tigris river in Türkiye, began filling and forming a reservoir in 2021. In addition, there are plans for the construction of the upcoming Cizre HEPP. This situation has the potential to significantly impact the bio–ecology of fish species in the area. The research aimed to determine the length–weight relationships (LWR) and condition factors of thirteen fish species in the Tigris River: Arabibarbus grypus, Acanthobrama marmid, Alburnus sellal, Carassius gibelio, Cyprinion kais, Cyprinion macrostomus, Chondrostoma regium, Paracapoeta trutta, Capoeta umbla, Garra rufa, Squalius lepidus, Luciobarbus esocinus and Luciobarbus mystaceus, where Ilisu and Cizre dams reservoir will be formed. Length–weight relationships (LWRs) were determined using the formula W = aLb, where W represents weight and L represents length. Three types of lengths were considered: total length, fork length, and standard length. The exponent values (b) varied across the species, ranging from 2.7875 for Acanthobrama marmid to 3.2214 for Carassius gibelio. The relationships between length and weight were found to be highly significant, as indicated by the R2 values, which were all greater than 0.80, except for Alburnus sellal and Garra rufa, which had lower values below 0.80. The condition factors were calculated using Fulton’s condition factor (K) and relative condition factor (Kn). In all species, except for Arabibarbus grypus and Chondrostoma regium, the condition factors were greater than 1 (K>1). However, for Capoeta umbla and Garra rufa, the relative condition factor (Kn) was less than 1 (Kn

We study the isometry group of the Grothendieck group equipped with a bilinear asymmetric Euler form. We prove several properties of this group; in particular, we show that it is isomorphic to the direct product of by the free Abelian group of rank . We also explicitly calculate its generators for . [ABSTRACT FROM AUTHOR]

Purpose: This study aimed to investigate the length change patterns of the native deep medial collateral ligament (dMCL) and potential anteromedial reconstructions (AMs) that might be added to a reconstruction of the superficial MCL (sMCL) to better understand the control of anteromedial rotatory instability (AMRI). Methods: Insertion points of the dMCL and potential AM reconstructions were marked with pins (tibial) and eyelets (femoral) in 11 cadaveric knee specimens. Length changes between the pins and eyelets were then tested using threads in a validated kinematics rig with muscle loading of the quadriceps and iliotibial tract. Between 0° and 100° knee flexion, length change pattern of the anterior, middle and posterior part of the dMCL and simulated AM reconstructions were analysed using a rotary encoder. Isometry was tested using the total strain range (TSR). Results: The tibiofemoral distance of the anterior dMCL part lengthened with flexion (+12.7% at 100°), whereas the posterior part slackened with flexion (−12.9% at 100°). The middle part behaved almost isometrically (maximum length: +2.8% at 100°). Depending on the femoral position within the sMCL footprint, AM reconstructions resulted in an increase in length as the knee flexed when a more centred position was used, irrespective of the tibial attachment position. Femoral positioning in the posterior aspect of the sMCL footprint exhibited <4% length change and was slightly less tight in flexion (min TSR = 3.6 ± 1.5%), irrespective of the tibial attachment position. Conclusion: The length change behaviour of potential AM reconstructions in a functionally intact knee is mainly influenced by the position of the femoral attachment, with different tibial attachments having a minimal effect on length change. Surgeons performing AM reconstructions to control AMRI would be advised to choose a femoral graft position in the posterior part of the native sMCL attachment to optimise graft length change behaviour. Given the high frequency of MCL injuries, sufficient restoration of AMRI is essential in isolated and combined ligamentous knee injuries. Level of Evidence: There is no level of evidence as this study was an experimental laboratory study. [ABSTRACT FROM AUTHOR]

This paper aims to classify a certain type of three-dimensional complete non-Sasakian contact manifold with specific properties, namely Q ξ = σ ξ and admitting Ricci–Bourguignon solitons. In the case of constant σ , the paper proves that if the potential vector field of the Ricci–Bourguignon soliton is orthogonal to the Reeb vector field, then the manifold is either Einstein or locally isometric to E(1, 1). Under a similar hypothesis, the paper shows that a (κ , μ , ϑ) -contact metric manifold is locally isometric to E(1, 1). Finally, the paper considers the scenario where the potential vector is pointwise collinear with the Reeb vector field and presents some results. [ABSTRACT FROM AUTHOR]

Objective Isolated intra‐articular anterior cruciate ligament (ACL) reconstruction is not capable of restoring instability in many cases leading some to recommend concomitant anterolateral ligament (ALL) reconstruction. The satisfactory fixation site and graft length change are crucial in ligament reconstruction to restore the ALL function and avoid some unwanted graft behavior. The purpose of this investigation is to determine the optimal isometric region on the anterolateral aspect of the knee for ALL reconstruction using a three‐dimensional optical instrument and a suture similar to an intraoperative isometric test. Methods Six freshly frozen cadaveric human knees were used in this study. Data regarding the anterolateral surface were obtained using an optical measurement system to create a three‐dimensional model. Nine points were selected on the femur (F1‐F9) and tibia (Ta‐Ti) respectively. The three‐dimensional length change between each pair of tibial and femoral points was measured during passive knee flexion from 0° to 90° in 15° increments. Subsequently, five femoral points (A–E) were selected from the lateral femur, located in different areas relative to the lateral femoral epicondyle, and three tibial reference points (T1‐T3) were selected in the isometric test. The changes in the length between each pair of reference points were measured using sutures. The 95% confidence interval for the rate of length change was estimated using the mean and standard deviation of the maximum rate of length change at different flexion angles, and the data were expressed as the mean (95% confidence interval) and compared with the maximum acceptable rate of change (10%). Results The maximum acceptable change rate for ligament reconstruction is 10%, and the mean maximum rates and the 95% confidence interval (CI) of length change for the point combinations were calculated. Among all the combined points measured using the optical measurement system and the suture, the qualified point combination for reconstruction was F3 (8mm posterior and 8mm proximal to the lateral femoral epicondyle)‐Tb (8mm proximal to the midpoint between the center of Gerdy's tubercle and the fibula head), A (posterior and proximal to the lateral femoral epicondyle)‐T2 (10mm below the joint line)and A‐T3 (15 mm below the joint line). The position of F3‐Tb and A‐T2 are close to each other. Conclusion The most isometric area of the femur for ALL reconstruction was posterior and proximal to the lateral femoral epicondyle. We recommend that the initial location of the femoral point be set at 8 mm posterior and 8 mm proximal to the lateral femoral epicondyle and the tibial point at approximately 10 mm below the joint line, midway between Gerdy's tubercle and fibular head, and subsequently adjusted to the most satisfactory position according to the isometric test.

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×2k matrices M over R satisfying the conditions that its rows are linearly independent over R and MM⊤=α⊤α for an R-linearly independent vector α∈Rk. Let C be a linear code generated by such a matrix M. We prove that the dual code C⊥ of C is also a free linear code with dimension k, as well as C/Hull(C) and C⊥/Hull(C) are one-dimensional free R-modules, where Hull(C) represents the hull of C. Based on these facts, an isometry from Rx+Ry onto R2 is established, assuming that x+Hull(C) and y+Hull(C) are bases for C/Hull(C) and C⊥/Hull(C) 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 MM⊤ takes the form of α⊤α with α being a linearly independent vector in Rk, a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy MM⊤=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.

This article reviews the symmetry pattern which has rotational symmetry and translational symmetry along the -axis and -axis. The symmetry pattern is generated through the implementation of Matlab application using discrete dynamical system simulation, after analysing the requirements and determining the appropriate dynamical function. In this context, each point on the plane serves as the starting point in the iteration of the dynamical system. The colour assigned to each point is determined by the number of iterations performed. The generation of the symmetry pattern involves three convergence tests, namely the Euclidean Test, the Fractional Distance Test, and the Maximum Distance with Weighted test. Through a variety of dynamic function parameter combinations, a more visually appealing and diverse range of symmetry patterns is obtained.

Objective: Isolated intra‐articular anterior cruciate ligament (ACL) reconstruction is not capable of restoring instability in many cases leading some to recommend concomitant anterolateral ligament (ALL) reconstruction. The satisfactory fixation site and graft length change are crucial in ligament reconstruction to restore the ALL function and avoid some unwanted graft behavior. The purpose of this investigation is to determine the optimal isometric region on the anterolateral aspect of the knee for ALL reconstruction using a three‐dimensional optical instrument and a suture similar to an intraoperative isometric test. Methods: Six freshly frozen cadaveric human knees were used in this study. Data regarding the anterolateral surface were obtained using an optical measurement system to create a three‐dimensional model. Nine points were selected on the femur (F1‐F9) and tibia (Ta‐Ti) respectively. The three‐dimensional length change between each pair of tibial and femoral points was measured during passive knee flexion from 0° to 90° in 15° increments. Subsequently, five femoral points (A–E) were selected from the lateral femur, located in different areas relative to the lateral femoral epicondyle, and three tibial reference points (T1‐T3) were selected in the isometric test. The changes in the length between each pair of reference points were measured using sutures. The 95% confidence interval for the rate of length change was estimated using the mean and standard deviation of the maximum rate of length change at different flexion angles, and the data were expressed as the mean (95% confidence interval) and compared with the maximum acceptable rate of change (10%). Results: The maximum acceptable change rate for ligament reconstruction is 10%, and the mean maximum rates and the 95% confidence interval (CI) of length change for the point combinations were calculated. Among all the combined points measured using the optical measurement system and the suture, the qualified point combination for reconstruction was F3 (8mm posterior and 8mm proximal to the lateral femoral epicondyle)‐Tb (8mm proximal to the midpoint between the center of Gerdy's tubercle and the fibula head), A (posterior and proximal to the lateral femoral epicondyle)‐T2 (10mm below the joint line)and A‐T3 (15 mm below the joint line). The position of F3‐Tb and A‐T2 are close to each other. Conclusion: The most isometric area of the femur for ALL reconstruction was posterior and proximal to the lateral femoral epicondyle. We recommend that the initial location of the femoral point be set at 8 mm posterior and 8 mm proximal to the lateral femoral epicondyle and the tibial point at approximately 10 mm below the joint line, midway between Gerdy's tubercle and fibular head, and subsequently adjusted to the most satisfactory position according to the isometric test. [ABSTRACT FROM AUTHOR]

Chirality was traditionally considered a binary property of periodic lattices and crystals. However, the classes of two‐dimensional lattices modulo rigid motion form a continuous space, which was recently parametrized by three geographic‐style coordinates. The four non‐oblique Bravais classes of two‐dimensional lattices form low‐dimensional singular subspaces in the full continuous space. Now, the deviations of a lattice from its higher symmetry neighbors can be continuously quantified by real‐valued distances satisfying metric axioms. This article analyzes these and newer G‐chiral distances for millions of two‐dimensional lattices that are extracted from thousands of available two‐dimensional materials and real crystal structures in the Cambridge Structural Database. [ABSTRACT FROM AUTHOR]

The Fréchet mean generalizes the concept of a mean to a metric space setting. In this work we consider equivariant estimation of Fréchet means for parametric models on metric spaces that are Riemannian manifolds. The geometry and symmetry of such a space are partially encoded by its isometry group of distance-preserving transformations. Estimators that are equivariant under the isometry group take into account the symmetry of the metric space. For some models, there exists an optimal equivariant estimator, which will necessarily perform as well or better than other common equivariant estimators, such as the maximum likelihood estimator or the sample Fréchet mean. We derive the general form of this minimum risk equivariant estimator and in a few cases provide explicit expressions for it. A result for finding the Fréchet mean for distributions with radially decreasing densities is presented and used to find expressions for the minimum risk equivariant estimator. In some models the isometry group is not large enough relative to the parametric family of distributions for there to exist a minimum risk equivariant estimator. In such cases, we introduce an adaptive equivariant estimator that uses the data to select a submodel for which there is a minimum risk equivariant estimator. Simulation results show that the adaptive equivariant estimator performs favourably relative to alternative estimators. [ABSTRACT FROM AUTHOR]

In this paper, we study the first and second fundamental forms of surfaces, exploring their properties as they relate to measuring arc lengths and areas, and identifying isometric surfaces. These forms can be used to define the Gaussian curvature, which is, unlike the first and second fundamental forms, independent of the parametrization of the surfaces. Also, we investigate the geometric behaviour of rectifying curves on regular surfaces under conformal and isometric transformation by using the concept of fundamental forms of surfaces. [ABSTRACT FROM AUTHOR]

We investigate different possiblities of subspaces of the space ℓ ∞ in terms of whether the subspaces are polyhedral or not. We further study finite-dimensional subspaces of ℓ ∞ which are of the form ℓ ∞ n for some n ≥ 2. As an application of the results we compute the number of extreme contractions for a class of the space of bounded linear operators. In particular we find the number of extreme contractions of L (X , ℓ ∞ n) , where X is a finite-dimensional polyhedral space. [ABSTRACT FROM AUTHOR]

In this note, we present some results concerning intertwining properties of two isometries on Banach spaces. In this connection, we obtain also some Katznelson–Tzafriri type results for power bounded operators. [ABSTRACT FROM AUTHOR]

We give an example for a C0-semigroup of non-linear 0-preserving holomorphic Carathéodory isometries of the unit ball.

The question of whether there is a true isometry that approximates the ε-isometry defined on a bounded set has long interested mathematicians. The first paper on this topic was published by Fickett, whose result was subsequently greatly improved by Alestalo et al., Väisälä and Vestfrid. Recently, the authors published some papers improving the previous results. The main purpose of this paper is to improve all of the abovementioned results by utilizing the properties of the norm and inner product for Euclidean space.

The article focuses on understanding the morphology of coracoclavicular ligament attachments for anatomical reconstruction of the acromioclavicular (AC) joint. Topics include the lack of studies on the morphology of these ligaments, particularly in the Indian population; the methods used to measure anatomical parameters in cadaveric shoulders; and the identification of isometric points for tunnel placements during AC joint reconstructions.

This article describes the surjective linear isometries between spaces of p-times differentiable maps from a domain of the Euclidean space into a certain Banach space. [ABSTRACT FROM AUTHOR]

This study aimed to establish the effectiveness of slow tempo bodyweight squat combined with an isometric squat (ST-ISO), and an isometric squat alone (ISO), as a post-activation performance enhancement protocol (PAPE) for jump height improvement. The study sample consisted of 41 trained men aged 18-24. The ST-ISO group (n = 17) performed three five-second sets of the maximal voluntary back squat while pushing on an immovable bar and two sets of five repetitions of a slow-tempo (5-0-5-0) body squat immediately after isometry with a 2-m rest interval. The ISO (n = 14) group only performed isometric squats, and the control group (CG; n = 10) performed a 5-min treadmill run at 6 km/h. The countermovement jump (CMJ) height results were analyzed from the baseline and then at 3, 5, 7, and 9 min after the PAPE protocols. The statistical significance was set at p < 0.05. RM-ANOVA revealed differences in the group-minute interaction (F = 2.70; p = 0.0083; 2 = 0.1243), and post-hoc tests demonstrated a significant decrease in CMJ after 5 min in the ISO group (p < 0.0446). The performance of the ST-ISO group markedly decreased in the 3rd and 7th min after PAPE (p = 0.0137; p = 0.0424, respectively), though it improved significantly in the final minute (p < 0.0030). Chi-squared analysis revealed that the ST-ISO group peaked more frequently in the 9th min (X2 = 17.97; p = 0.0214). However, CMJ height improvement did not differ between the PAPE protocols, thus it was close to statistical significance (t = -1.82; p = 0.07; ES = 0.7). The ST-ISO protocol provided jump enhancement, though the deterioration observed in the first minutes after the protocols suggest the rest period after activity requires attention, and the methods need to be individualized. [ABSTRACT FROM AUTHOR]