Search

Relevance. One of the key points in plasty of the anterior cruciate ligament (ACL) is the isometric position of the graft, in which its tension remains the same during flexion and extension of the knee joint. However, no method has been described today for the intraoperative determination of the isometricity of the location of the femoral and tibial tunnels (for placing the graft) before their formation.Purpose of the study. To develop a method for intraoperative determination of the isometricity of the location of the femoral and tibial tunnels before their formation during ACL plasty and to study its effectiveness.Material and methods. The study included 30 patients who underwent ACL repair. For a preliminary intraoperative assessment of the isometric areas of graft fixation on the femur and tibia, the proposed original method with the use of two knot pushers and a thread passing through them. Isometry was assessed by the degree of displacement of this thread. After determining the isometric areas of fixation, the femoral and tibial tunnels were formed, and the isometric position of the graft before its fixation in the tibia was checked by the degree of displacement of the threads with which the graft was sutured relative to the aperture of the tibial tunnel.Results. The average displacement of the thread relative to the pusher of the knot in the preliminary determination of the isometric areas of fixation according to the proposed method corresponded to the displacement of the threads with which the distal end of the graft was sewn relative to the outer aperture of the tibial tunnel (this value did not exceed 2 mm on average) until the final fixation of the graft in the tibial tunnel.Conclusion. The developed method makes it possible to determine the isometric location of the femoral and tibial tunnels during arthroscopic plasty of the anterior cruciate ligament before their formation. If the location of the fixation points on the femur and tibia is determined non-isometric, it is possible to correct their position.

Growth pattern and condition factor (CF) are essential to fish resource assessment but limited to Periophthalmus gracilis--an amphibious fish living in the mudflats along the Indo-Pacific regions, including the Mekong Delta (MD), Vietnam. This study lasted from April 2020 to March 2021 to verify if their growth pattern and CF change with sex, size, season, month and site. The total length and weight of 486 individuals (236 females and 250 males) were 2.9-5.9 cm and 0.13-1.66 g, respectively. The mudskipper displayed negative allometry as the slope value (b D 2.69 ± 0.06) obtained from length and weight (LWR) was significantly less than 3 (p < 0.01), indicating that most fish specimens were caught in the immature stage. The fish growth pattern did not change with sex as both males and females displayed negative allometry but varied by size since the mudskipper showed negative allometry in the immature group and isometry in the mature group. Likewise, growth type changed with season since fish showed negative allometry in the dry season but isometry in the wet season. As the slope value (b) varied by site and month, the mudskipper displayed spatiotemporal growth patterns, ranging from negative to positive allometry. The CF was impacted by sex as this value of females (1.09 ± 0.02) was higher than that of males (0.96 ± 0.01, p < 0.01). Besides, CF was regulated by fish length since this value was higher in the mature group (1.12 ± 0.03) than in the immature group (1.01 ± 0.01, p < 0.01). Likewise, CF was affected by season as this value was higher in the wet season (1.05 ± 0.02) than in the dry season (0.99 ± 0.01, p < 0.01). Although the CF varied with site and month variables (p < 0.01), this value (1.02 ± 0.01) was generally higher than 1, showing fish adapted well to their habitat. The fish length at first capture should be increased to exploit this species sustainably. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the geometric invariant properties of a normal curve on a smooth immersed surface under conformal transformation. We obtain an invariant-sufficient condition for the conformal image of a normal curve. We also find the deviations of normal and tangential components of the normal curve under the same motion. [ABSTRACT FROM AUTHOR]

Introduction An -dimensional Riemannian manifold is said to be flat (or locally Euclidean) if locally isometric with the Euclidean space, that is, admits a covering of coordinates neighborhoods each of which is isometric with a Euclidean domain. A Riemannian manifold is flat if and only if admits a covering of coordinates neighborhoods on each of, the function is independent of . A classical result affirms that a Riemannian manifold is flat if and only if its Riemann curvature vanishes (equivalently, the sectional curvature; This is usually taken as the definition of a flat Riemannian manifold in the contexts. The universal Riemannian covering space of a complete and flat Riemannian manifold is the Euclidean space . Up to local isometry, Bieberbach proved that any compact flat Riemannian manifold, is realized as a quotient space , where is a discrete, co-compact and torsion free subgroup of the Euclidean group , cf. [2]. The only 1 dimensional complete, flat and connected manifolds are and . In 2 dimensions, the only complete, flat and connected manifolds are cylinder, Mӧbius strip, Torus and Klein bottle. In 3 dimensions, there are only 10 complete, flat and connected manifolds including 6 oriented and 4 non-oriented manifolds, cf. [7]. Likewise the Riemannian case, a Finslerian manifold is said to be flat (or locally Minkowskian) if, admits a covering of coordinates neighborhoods each of which isometric with a single Minkowski normed domain. A Finslerian manifold is flat if and only if it admits a covering of coordinates neighborhoods on each of, the function is independent of . The flag curvature of any flat Finsler manifold vanishes identically. Material and methods Thanks to the works of Bieberbach and Schoenflies, we apply an group theoretic approach to classify flat Randers manifolds. The key idea is that the isometry group of Randers manifold is a subgroup of the Euclidean group. This fact, may ease our approach to find and count discrete, co-compact and torsion frees subgroups of . First we find the Bieberbach subgroups and then, we count those that could form an isometry subgroup. Results and discussion Here, flatness of a generic Finsler manifold is aimed to be defined so that it generalizes the flatness for Riemannian manifolds. The following result outcome in dimension 2 and 3: Theorem 1. The only connected and closed -dimensional () closed flat Randers manifolds is the torus , respectively. To classify the flat Randers manifolds, we find out that the flat Randers manifolds are flat Riemannian manifolds. Besides, the isometry group of a Randers manifold is a subgroup of the isometry group . Our discussion also apply the following results: Every dimensional flat Randers manifold is itself a flat Riemannian manifolds. Every dimensional flat Randers manifold is orientable. The non-Riemannian properties for generic Finsler metrics may cause obstructions for a Finsler manifold to be falt. Conclusion The following conclusions were drawn from this research. In dimensions 2 and 3, the only connected and closed flat Randers manifolds are the tori , respectively. Every dimensional flat Randers manifold is itself a flat Riemannian manifolds../files/site1/files/62/11Abstract(1).pdf

For the general class of pseudo-Finsler spaces with (α,β)-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has a Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future-pointing time-like vectors is ensured. The identified (α,β)-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an (α,β)-metric and the isometries of the underlying pseudo-Riemannian metric a; in particular, we list all (α,β)-metrics which admit isometries that are not isometries of a.

Length-weight relationship (LWR), growth pattern and condition factor (CF) play a vital role in fish resource evaluation and management but data on this is limited for Periophthalmus variabilis. This is an amphibious fish that lives in the mudflats of the mangrove forests in the Western Pacific regions, including the Vietnamese Mekong Delta (VMD). This paper contributes to the understanding of the LWR, growth pattern and CF of P. variabilis. Fish specimens were collected by hand at four sites in the VMD from April 2020 to March 2021. The analysis of 495 individual fish (259 females and 236 males) showed that fish weight could be estimated from a given fish length due to high determination values (r2 = 0.70 − 0.97). Dusky-gilled mudskippers of the VMD exhibited positive allometry, as indicated by their larger than 3 b value (b = 3.094 ± 0.045, p = 0.04). However, the growth pattern of the mudskipper between the sex and maturation stage (immature versus mature) showed isometry. These fish displayed positive allometry in the dry season (b = 3.138 ± 0.065) (>3, p = 0.04) but isometry in the wet season (b = 3.058 ± 0.061) (≈3, p = 0.34). Fish growth ranged from isometry to positive allometry and varied by site (b = 2.850 ± 0.103–3.370 ± 0.114) and month (b = 2.668 ± 0.184–3.588 ± 0.299) based on the b value. The CF of P. variabilis was not affected by sex (p = 0.29), body size (p = 0.64) or season (p = 0.43), but was affected by site (p = 0.01) and month (p = 0.01). The CF of this species (1.05 ± 0.02) was higher than 1 (p

Fickett proved the stability of isometries on bounded subsets of R n for n ≥ 2 . Jung then improved Fickett's theorem for n ≥ 3 . In this paper, we improve Fickett's theorem for n = 2 and improve Jung's result for n = 3 , by employing a fundamental analytic method, as it can be used to explain mathematically many practical engineering problems. [ABSTRACT FROM AUTHOR]

The distribution of disease vectors such as mosquitoes is changing. Climate change, invasions and vector control strategies all alter the distribution and abundance of mosquitoes. When disease vectors undergo a range shift, so do disease burdens. Predicting such shifts is a priority to adequately prepare for disease control. Accurate predictions of distributional changes depend on how factors such as temperature and competition affect mosquito life‐history traits, particularly body size and reproduction. Direct estimates of both body size and reproduction in mosquitoes are logistically challenging and time‐consuming, so the field has long relied upon linear (isometric) conversions between wing length (a convenient proxy of size) and reproductive output. These linear transformations underlie most models projecting species' distributions and competitive interactions between native and invasive disease vectors. Using a series of meta‐analyses, we show that the relationship between wing length and fecundity are nonlinear (hyperallometric) for most mosquito species. We show that whilst most models ignore reproductive hyperallometry (with respect to wing length), doing so introduces systematic biases into estimates of population growth. In particular, failing to account for reproductive hyperallometry overestimates the effects of temperature and underestimates the effects of competition. Assuming isometry also increases the potential to misestimate the efficacy of vector control strategies by underestimating the contribution of larger females in population replenishment. Finally, failing to account for reproductive hyperallometry and variation in body size can lead to qualitative errors via the counter‐intuitive effects of Jensen's inequality. For example, if mean sizes decrease, but variance increases, then reproductive outputs may actually increase. We suggest that future disease vector models incorporate hyperallometric relationships to more accurately predict changes in mosquito distribution in response to global change. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sufficient condition for the invariance of a rectifying curve on a smooth surface immersed in Euclidean 3-space under isometry by using Darboux frame {T, P,U}. Further, we find the deviations of the position vector of a rectifying curve on the smooth surface along any tangent vector T = aϕu +bϕv with respect to the isometry. We also find the deviations of the position vector of a rectifying curve on the smooth surface along the unit normal U to the surface and along P(= U × T) with respect to the isometry. [ABSTRACT FROM AUTHOR]

A map f : H → K , where H and K are real Hilbert spaces, is called an ε -phase isometry if f (0) = 0 and min { | ‖ f (x) - f (y) ‖ - ‖ x - y ‖ | , | ‖ f (x) - f (y) ‖ - ‖ x + y ‖ | } ≤ ε , whenever x , y ∈ H . We prove the following stability property. If f is an ε -phase isometry, then there is a linear isometry V and a phase isometry γ I , where γ : H → { - 1 , 1 } is a phase mapping and I is the identity, such that ‖ V ∗ f (x) - γ (x) x ‖ ≤ 2 2 ε for all x ∈ H . If f is surjective, then V is surjective and ‖ f (x) - γ (x) V x ‖ ≤ 2 2 ε for all x ∈ H . [ABSTRACT FROM AUTHOR]

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces. [ABSTRACT FROM AUTHOR]

Introduction Hagler used a diameter norm to construct a separable Banach space X with nonseparable dual such that l1 does not embed in X. Also, Bayati Eshkaftaki considered the diameter norm on c0(I) and characterized all isometries on this space. In this article, we are going to first introduce the space B0 (R) and show that this space is Banakh space with supremum norm, and then we express the diameter norm on the space B0(R) and examine some of the characteristics of this space. We also examine the relationship between this space and Banach space c0(R) and some of the reference theorems [3] shown on space c0(R)on space B0(R). Material and methods We express some properties of the space B0(R) and define the diameter norm on this space, and then we examine the equivalence of the diameter norm with the supremum norm on this space. We define the longitudinal operator relative to the diameter norm on B0 (R) and state the conditions under which a linear operator on B0 (R)is longitudinal. In the following, we will examine the results of the length of the linear operator on the space B_0 (R). We will use the notation l∞(R) for the set of all bounded functions f : R → R. Also we denote c0(R) for the set of all functions f : R → R with lim x∈R f = 0. The space l∞(R) is a Banach space with the norm ||f||∞ = sup{|f(x)| : f ∈ B0 (R)}. We denote B0 (R) for the set of all bounded functions f : R → R with lim x→∞ f(x) = 0. This space is a Banach space with norm||⋅||∞ . For every j ∈ R, we consider the function ej (i) = {1 i = j 0 i ≠ j, on real numbers. Then each ej belongs to B0(R). We define for f ∈ B0(R) define ∥ f ∥d = diam(f) and it is called the diameter norm. We define||f|| = ||f||d + ||f||∞ for each f ∈ B0 (R) that the B0(R) with this norm is a Banach algebra. Main Results We have identified some properties of the space B0 (R) is introduced and some of its properties are examined. Then with the help of a diameter norm on the space B0(R), define a norm ||∙|| on the B0(R), and show that the space B0(R) is a Banach algebra with this norm||∙|| . Also, we got the relationship between the space B0(R)and the space c0(R) and we have obtained a few theorems in connection with isometry under ||∙||d. Conclusion The following conclusions were drawn from this research. If f ∈ B0(R) and φ: R → R be a function that lim x→∞ φ(x) = ∞, then foφ ∈ B0(R). The space c0(R) is an ideal in algebra B0(R). (B0(R)‚ ||∙||d) is a Banach space. The space B00(R) is dense in B0(R)with ||∙||d. If i‚j ∈ R and i ≠ j, then ei be orthogonal to ej . Suppose f ∈ B0 (R) and this function is continuous at point i. In this case, f is orthogonal to ei, and if f is continuous on R then f is orthogonal to each of the vectors produced by {ei}. If φ: R → R be a function that lim x→∞ φ(x) = ∞, then the operator T: B0 (R) ⟶ B0(R), T(f) = f o φ‚ is an isometry under ||∙||d, if and only if φ is a surjective function. For any polynomial Pn (x) = xn + an−1xn−1 + ⋯ + a0, where n an odd number. the operator Tnf(x) = f(Pn (x)) is an isometry under ||∙||d. Suppose T: B0 (R) ⟶ B0 (R), is an isometry under ||∙||d and i0 ∈ R. Then −1 ≤ Σj∈I−Tej(i0) ≤ 0 ≤ Σj∈I+Tej(i0) ≤ 1. where I+ = {j ∈ R: Tej (i0) > 0} and I− = {j ∈ R: Tej(i0) < 0}. Let T: B0(R) ⟶ B0(R) be an isometry under ||∙||d, j0 ∈ R and λ = sup(Tej0). Then λ = lim λj and λ − 1 = lim nj, where λj = sup (T(ej0 + ej)) and nj = inf(T(ej0 + ej)). [ABSTRACT FROM AUTHOR]

Length-weight relationship (LWR), growth pattern and condition factor (CF) play a vital role in fish resource evaluation and management but data on this is limited for Periophthalmus variabilis. This is an amphibious fish that lives in the mudflats of the mangrove forests in the Western Pacific regions, including the Vietnamese Mekong Delta (VMD). This paper contributes to the understanding of the LWR, growth pattern and CF of P. variabilis. Fish specimens were collected by hand at four sites in the VMD from April 2020 to March 2021. The analysis of 495 individual fish (259 females and 236 males) showed that fish weight could be estimated from a given fish length due to high determination values (r² = 0.70 - 0.97). Dusky-gilled mudskippers of the VMD exhibited positive allometry, as indicated by their larger than 3 b value (b = 3.094 ± 0.045, p = 0:04). However, the growth pattern of the mudskipper between the sex and maturation stage (immature versus mature) showed isometry. These fish displayed positive allometry in the dry season (b = 3.138 ± 0.065) (>3, p = 0:04) but isometry in the wet season (b = 3.058 ± 0.061) (≈3, p= 0.34). Fish growth ranged from isometry to positive allometry and varied by site (b = 2.850 ± 0.103-3.370 ± 0.114) and month (b = 2.668 ± 0.184-3.588 ± 0.299) based on the b value. The CF of P. variabilis was not affected by sex (p = 0.29), body size (p = 0.64) or season (p = 0.43), but was affected by site (p = 0:01) and month (p=0.01). The CF of this species (1.05 ± 0.02) was higher than 1 (p<0:001), indicating that the fish adapted well to their habitat. [ABSTRACT FROM AUTHOR]

We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$. We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$. We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle. [ABSTRACT FROM AUTHOR]

We characterize the left-symmetric points as well as the right-symmetric points in the sense of Birkhoff-James orthogonality, of the Banach spaces l p n (1 ≤ p ≤ ∞). As an application of our study, we produce an elementary proof of the well-known result: T is an isometry on l p n (p ≠ 1 , 2 , ∞) if and only if T is a signed permutation. This illustrates the pivotal role played by the set of left-symmetric points in determining the isometry group of a given Banach space. [ABSTRACT FROM AUTHOR]

We characterize isometric vector fields through a distance resembling the Gromov–Hausdorff one (Burago D, Burago Y, Ivanov S in A course in metric geometry. Graduate studies in mathematics, vol 33, American Mathematical Society, Providence, 2001). We then use this distance to generate a metrizable topology in the space of vector fields on closed manifolds up to isometry. We prove that the set of Killing vector fields is closed with respect to this topology. [ABSTRACT FROM AUTHOR]

The objective of this study was to analyze the acute effects of breaking up prolonged sitting with isometric exercise on the cardiovascular health of sedentary adults. This is a three-condition randomized crossover trial. The sample was comprised of 17 subjects (11 women; 29 ± 10 years old; 25,1 ± 5,1 kg/m²). The participants completed, in randomized order, three experimental conditions (control, breaks with isometric leg extension exercise, and breaks with walking), with the order of the conditions determined through simple automatic randomization. All the conditions had in common a sitting period of 3 h. During the conditions with isometric exercise and walking breaks the participants performed breaks with isometric leg extension exercise and with walking every 30 min, while in the control condition they remained seated with no breaks. Before and after this period, vascular function (primary outcome), blood pressure, and cardiac autonomic modulation (secondary outcomes) were measured. Generalized estimated equations were used to analyze the data. The results did not indicate significant interaction effects for vascular function among experimental conditions (p > 0,05 for all). We also did not find significant interaction effects for systolic or diastolic blood pressure among the conditions (p > 0,05 for all). The heart rate variability parameters did not present significant interaction effects among conditions (p > 0,05 for all). In conclusion, breaking up sitting with isometric exercise does not seem to lead to significant effects on the cardiovascular health of sedentary adults. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this study was to examine the length change patterns of the native medial structures of the knee and determine the effect on graft length change patterns for different tibial and femoral attachment points for previously described medial reconstructions. Methods: Eight cadaveric knee specimens were prepared by removing the skin and subcutaneous fat. The sartorius fascia was divided to allow clear identification of the medial ligamentous structures. Knees were then mounted in a custom-made rig and the quadriceps muscle and the iliotibial tract were loaded, using cables and hanging weights. Threads were mounted between tibial and femoral pins positioned in the anterior, middle, and posterior parts of the attachment sites of the native superficial medial collateral ligament (sMCL) and posterior oblique ligament (POL). Pins were also placed at the attachment sites relating to two commonly used medial reconstructions (Bosworth/Lind and LaPrade). Length changes between the tibiofemoral pin combinations were measured using a rotary encoder as the knee was flexed through an arc of 0–120°. Results: With knee flexion, the anterior fibres of the sMCL tightened (increased in length 7.4% ± 2.9%) whilst the posterior fibres slackened (decreased in length 8.3% ± 3.1%). All fibre regions of the POL displayed a uniform lengthening of approximately 25% between 0 and 120° knee flexion. The most isometric tibiofemoral combination was between pins placed representing the middle fibres of the sMCL (Length change = 5.4% ± 2.1% with knee flexion). The simulated sMCL reconstruction that produced the least length change was the Lind/Bosworth reconstruction with the tibial attachment at the insertion of the semitendinosus and the femoral attachment in the posterior part of the native sMCL attachment side (5.4 ± 2.2%). This appeared more isometric than using the attachment positions described for the LaPrade reconstruction (10.0 ± 4.8%). Conclusion: The complex behaviour of the native MCL could not be imitated by a single point-to-point combination and surgeons should be aware that small changes in the femoral MCL graft attachment position will significantly effect graft length change patterns. Reconstructing the sMCL with a semitendinosus autograft, left attached distally to its tibial insertion, would appear to have a minimal effect on length change compared to detaching it and using the native tibial attachment site. A POL graft must always be tensioned near extension to avoid capturing the knee or graft failure. [ABSTRACT FROM AUTHOR]

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.

In this paper, we consider the groups of isometries of metric spaces arising from finitely generated additive abelian groups. Let A be a finitely generated additive abelian group. Let R={1,ϱ} where ϱ is a reflection at the origin and T={ta:A→A,ta(x)=x+a,a∈A}. We show that (1) for any finitely generated additive abelian group A and finite generating set S with 0∉S and −S=S, the maximum subgroup of IsomX(A,S) is RT; (2) D⊴RT if and only if D≤T or D=RT′ where T′={h2:h∈T}; (3) for the vector groups over integers with finite generating set S={u∈Zn:|u|=1}, IsomX(Zn,S)=On(Z)Zn. The paper also includes a few intermediate technical results.

A Banach space X has the Mazur–Ulam property if any isometry from the unit sphere of X onto the unit sphere of any other Banach space Y extends to a linear isometry of the Banach spaces X , Y. A Banach space X is called smooth if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur–Ulam property. [ABSTRACT FROM AUTHOR]

Objectives: Some previous studies suggest that humans do not conform to geometric similarity (isometry) in anthropometric dimensions of the upper and lower limbs. Researchers often rely on a single statistical approach to the study of scaling patterns, and it is unclear whether these methods produce similar results and are equally robust. This study used one bivariate and one multivariate method to examine how linear anthropometric dimensions scale in a sample of adult humans. Materials and methods: Motion capture marker data from 104 adults of varying height and mass were used to calculate anthropometric dimensions. We analyzed scaling patterns in pooled and separate sexes with two methods: (1) bivariate log–log regression and (2) multivariate principal component analysis (PCA). We calculated 95% highest density/confidence intervals for each method and defined positive/negative allometry as estimates lying outside those intervals. Results: Results identified isometric scaling of the upper arm, thigh, and shoulder, positive allometry of the forearm and shank, and negative allometry of the pelvis in the pooled sample using both statistical methods. Patterns of allometry in the pooled sample were similar between methods but differed in magnitude. Sex‐specific results differed in both pattern and magnitude between log–log regression and PCA. Only one measurement (shoulder width) departed from isometry in the sex‐specific log–log regressions. Discussion Our findings suggest that especially in sex‐specific analyses, the pattern and magnitude of allometry are sensitive to statistical methodology. When body mass was selected as the size variable, most human linear anthropometric dimensions in this sample scaled isometrically and were therefore geometrically similar within sexes. [ABSTRACT FROM AUTHOR]

We say that a field extension L / F has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over F that become isometric (resp. similar) over L are already isometric (resp. similar) over F. The famous Artin–Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic 2. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic 2, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau's norm principle for arbitrary quadratic or bilinear forms in characteristic 2. [ABSTRACT FROM AUTHOR]

for a proper subdomain D of and for all x,y ∈ D define where the infimum is taken over all curves Cxy = γ[0, 1] in D with γ(0) = x and γ(1) = y, and Cap denotes the conformal capacity of condensers. The quantity μD is a metric if and only if the domain D has a boundary of positive conformal capacity. If Cap(∂D) > 0, we call μD the modulus metric of D. Ferrand et al. (1991) have conjectured that isometries for the modulus metric are conformal mappings. Very recently, this conjecture has been proved for n = 2 by Betsakos and Pouliasis (2019). In this paper, we prove that the conjecture is also true in higher dimensions n ⩾ 3. [ABSTRACT FROM AUTHOR]

A gauge γ in a vector space X is a distance function given by the Minkowski functional associated to a convex body K containing the origin in its interior. Thus, the outcoming concept of gauge spaces (X , γ) extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of X is even, then the fixation of a symplectic form yields an identification between X and its dual space X ∗ . The image of the polar body K ∘ ⊆ X ∗ under this identification yields a (skew-)dual gauge on X. In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur–Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality. [ABSTRACT FROM AUTHOR]

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a C2 -smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces, we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary is the identity mapping. [ABSTRACT FROM AUTHOR]

This study analysed and compared the shell shape, morphometrics and relative growth of four sympatric limpet species (Patella depressa, Patella ulyssiponensis, Patella vulgata and Siphonaria pectinata) collected at Praia da Luz in Lagos (Algarve coast – southern Portugal). Morphometric relationships were established through regression analysis between linear (shell length, width and height), ponderal (total weight), area (shell base and surface areas) and volume variables (shell internal and total volumes). Relative growth (isometry vs allometry) was analysed to assess variation in the growth rate of morphometric variables throughout the species ontogeny. In addition, morphometric indices (ellipticity, conicity, density, surface area and volumetry) were calculated to further characterize shell shape. Overall, 1482 individuals with broad size and weight ranges were analysed (P. depressa = 354; P. ulyssiponensis = 306; P. vulgata = 408; S. pectinata = 414). All regressions were highly significant (P < 0.001) and the morphometric variables were strongly correlated (r = 0.761 to 0.994). Among 28 morphometric relationships, there were 14 isometries, 13 positive allometries and only one negative allometry. The morphometric indices revealed clear morphological differences between species and were mostly size-dependent, reflecting gradual changes in shell shape during growth. The main results are compared with a compilation of analogous data reported for these limpet species throughout their distributional range. Overall, the general trends in relative growth are discussed in terms of the species life habits, main traits and functional morphology. [ABSTRACT FROM AUTHOR]

Let X and Y be normed spaces over F ∈ { R , C } and f : X → Y a surjective mapping. Suppose that | ϕ f (y) (f (x)) | = | ϕ y (x) | holds for all x , y ∈ X and all support functionals ϕ f (y) at f(y) and ϕ y at y, or equivalently, suppose that for all semi-inner products on X and Y, compatible with given norms, | [ f (x) , f (y) ] | = | [ x , y ] | holds for all x , y ∈ X . Then f = σ U , where σ : X → F is a phase function, and U : X → Y is a linear or a conjugate linear isometry. [ABSTRACT FROM AUTHOR]

In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

We obtain a Shimorin Wold-type decomposition for a doubly commuting covariant representation of a product system of C ∗ -correspondences over N 0 k . This result gives Shimorin-type decompositions of recent Wold-type decompositions by Jeu and Pinto (Adv Math 368:107–149, 2020) for the q-doubly commuting isometries and by Popescu (J Funct Anal 279:108798, 2020) for Doubly Λ -commuting row isometries. Application to the wandering subspaces of the induced representations is explored, and a version of the Beurling–Lax-type characterization is obtained to study doubly commuting invariant subspaces. [ABSTRACT FROM AUTHOR]

Background: In the literature there are divergent results as to the native MPFL length change pattern. The reason for such divergent results may be the heterogeneity of design of studies analyzing MPFL isometry. The hypothesis of this review was that studies assessing MPFL length change pattern are highly heterogenous. The aim was to present a state-of-the-art review of sources of this heterogeneity. Materials and Methods: A total of 816 records were identified through the initial search of MEDLINE and Scopus databases. After eligibility assessment, 10 original articles and five reviews were included. In the included studies, the following 15 potential sources of heterogeneity were assessed: number of patients/cadavers, age, males to females ratio (demographics), identification of measured fibers, measurement method, measurement precision, quadriceps muscle activity, iliotibial band activity, hamstrings activity (study design), patellar height, trochlear or patellar dysplasia, femoral anteversion, mechanical axis of the limb, tibial tubercle–trochlear groove distance, and condylar anteroposterior dimensions (morphology). Each variable was graded in every included article with 1 point if reported precisely and not introducing bias; or with 0 points if reported not precisely, introducing bias, or not reported at all. Results: Within original articles, the highest achieved score was 10 out of 15 possible points with mean score of 6.7, SD = 2.37, and minimum score of just 3 out of 15 points. In the demographics section, mean score was 2.4, SD = 0.8 (80% of maximum possible score of 3); in the study design section it was 3.1, SD = 1.87 (52% of maximum possible score of 6); and in the morphology section it was 1.5, SD = 1.43 (25% of maximum possible score of 6). Conclusions: There is high heterogeneity and incomplete reporting of potential sources of bias in studies assessing native MPFL length change pattern. Future investigators should be aware of the presented factors and their potential impact on MPFL isometry. All methodologic factors should be meticulously reported. Detailed description of demographic data is already a standard; however, authors should more extensively report variables concerning study design and morphology of patients' patellofemoral joint. Furthermore, future studies should try to meticulously simulate the real-life working environment of MPFL and ensure usage of proper measurement methods. [ABSTRACT FROM AUTHOR]

The circumcentered Douglas–Rachford method (C–DRM), introduced by Behling, Bello Cruz and Santos, iterates by taking the circumcenter of associated successive reflections. It is an acceleration of the well-known Douglas-Rachford method (DRM) for finding the best approximation onto the intersection of finitely many affine subspaces. Inspired by the C–DRM, we introduced the more flexible circumcentered reflection method (CRM) and circumcentered isometry method (CIM). The CIM essentially chooses the closest point to the solution among all of the points in an associated affine hull as its iterate and is a generalization of the CRM. The circumcentered–reflection method introduced by Behling, Bello Cruz and Santos to generalize the C–DRM is a special class of our CRM. We consider the CIM induced by a set of finitely many isometries for finding the best approximation onto the intersection of fixed point sets of the isometries which turns out to be an intersection of finitely many affine subspaces. We extend our previous linear convergence results on CRMs in finite-dimensional spaces from reflections to isometries. In order to better accelerate the symmetric method of alternating projections (MAP), the accelerated symmetric MAP first applies another operator to the initial point. (Similarly, to accelerate the DRM, the C–DRM first applies another operator to the initial point as well.) Motivated by these facts, we show results on the linear convergence of CIMs in Hilbert spaces with first applying another operator to the initial point. In particular, under some restrictions, our results imply that some CRMs attain the known linear convergence rate of the accelerated symmetric MAP in Hilbert spaces. We also exhibit a class of CRMs converging to the best approximation in Hilbert spaces with a convergence rate no worse than the sharp convergence rate of MAP. The fact that some CRMs attain the linear convergence rate of MAP or accelerated symmetric MAP is entirely new. [ABSTRACT FROM AUTHOR]

We describe the orbits of a cohomogeneity two Riemannian G-manifold M from topological point of view, under the conditions that G is nonsemisimple and M decomposes as a product of negatively curved Riemannian manifolds. [ABSTRACT FROM AUTHOR]

The present study was conducted in four sampling sites, from fresh to brackish waters, to contribute knowledge on the length-weight relationship, growth, and body condition factor of the commercial fish Glossogobius aureus (Gobiiformes: Gobiidae) in the Mekong Delta. Fish specimens were caught using gill nets from January to December 2020. Data analyzed results of 742 specimens (382 males and 360 females) showed that weights of males and females in various sizes, seasons, and sites are able to estimate from a fish given length due to high determination coefficients (r²> 0.8, in all cases). The growth pattern of Glossogobius aureus fluctuated monthly from negative allometry to isometry and positive allometry as its slope (b) obtained from LWRs ranged from 2.583±0.119 SE to 3.123±0.066 SE. Likewise, b of this fish changes with fish size and site, but not season and gender. Albeit the growth pattern showed intraspecific and spatio-temporal variations, this goby displayed negative allometry since b (2.859±0.030) was lower than three. Likely, fish body condition factors, ranging from 0.937±0.022 SE to 1.218±0.030 SE, varied between fish sizes, sites, and months. The present findings contributed valuable data to understand fish adaptation and fishery management. [ABSTRACT FROM AUTHOR]

Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { ‖ f (x) + f (y) ‖ , ‖ f (x) − f (y) ‖ } = { ‖ x + y ‖ , ‖ x − y ‖ } , x , y ∈ X , if and only if f is phase equivalent to a surjective linear isometry, that is, f = σ U , where U : X → Y is a surjective linear isometry and σ : X → { − 1 , 1 }. This is a Wigner's type result for real normed spaces. [ABSTRACT FROM AUTHOR]

Allometric growth reflects different allocation patterns and relationships of different components or traits of a plant and is closely related to ecosystem carbon storage. As an introduced species, the growth and carbon storage of Sonneratia apetala are still unclear. To derive allometric relationships of the mangrove S. apetala and to estimate carbon storage in mangrove ecosystems, we harvested 12 individual Sonneratia apetala trees from four different diameter classes in the Futian National Nature Reserve, Guangdong, China. Allometric growth models were fitted. The results showed that diameter at breast height (DBH) and wood density were better variables for predicting plant biomass (including above- and below-ground biomass) than plant height. There were significant power function relationships between biomass and DBH, with a mean allometric exponent of 2.22, and stem biomass accounted for 97% of the variation in S. apetala total biomass. Nearly isometric scaling relationships were developed between stem biomass and other biomass components. To better understand the carbon stocks of the S. apetala ecosystem, we categorized all trees into five age classes and quantified vegetation carbon storage. The S. apetala vegetation carbon storage ranged from 96.48 to 215.35 Mg C ha−1, and the carbon storage significantly increased with stand age. The allometric equations developed in this study are useful to estimate biomass and carbon storage of S. apetala ecosystems. [ABSTRACT FROM AUTHOR]

This study aimed to measure the reliability of a test for measuring the strength and strength imbalance of the hip abductors and adductors, using isokinetic equipment adapted for isometric testing. Thirteen healthy, physically active male individuals took part in the research. Two unilateral isometric tests were undertaken using a load cell attached to an adapted abductor bench machine: a hip abduction test and hip adduction test. Tests consisted of two maximum voluntary isometric contractions made for six seconds with a break of one minute between each. The following dynamic variables were measured: maximum force, mean force, rate of force development for each limb (right and left), and the existence of asymmetries between the limbs. For statistical analysis, the t-test, intraclass correlation coefficient (ICC), and standard error of measurement (SEM) were applied. Results: The methodology utilized for the evaluation of the hip abductors and adductors did not show reliability in most of the parameters researched, with the ICC neither sufficient or low, and the retest performance higher than the test (p < 0.05). The applied test was not reliable for assessing strength and strength imbalances of hip abductors and adductors in most of the parameters investigated. These results indicate that the hip joints, more precisely, the abductor and adductor muscles, are complex structures to be assessed. They need to be previously familiarized with the proposed exercise, as their performance does not occur habitually. It is recommended to develop new tests in order to measure hip abduction and adduction strength adding a prior familiarization procedure.

In this paper, we first define an equivalence relation on the sequence space Ε2. Then we equip the quotient set Ε2/~ with a metric d1. We also determine an isometry map between the metric spaces (Ε2/~, d1) and ([0, 1], deucl). Finally, we investigate the symmetry conditions with respect to some points on the metric space (Ε2/~, d1) and we compare truncation errors for the computations which is obtained by the metrics deucl and d1. [ABSTRACT FROM AUTHOR]

In this paper, we first define an equivalence relation on the sequence space Ε2. Then we equip the quotient set Ε2/~ with a metric d1. We also determine an isometry map between the metric spaces (Ε2/~, d1) and ([0, 1], deucl). Finally, we investigate the symmetry conditions with respect to some points on the metric space (Ε2/~, d1) and we compare truncation errors for the computations which is obtained by the metrics deucl and d1. [ABSTRACT FROM AUTHOR]

In this note we generalize the well-known Wigner's unitary-antiunitary theorem. For smooth normed spaces X and Y and a surjective mapping f : X → Y such that | [ f (x) , f (y) ] | = | [ x , y ] | , x , y ∈ X , where [ · , · ] is the unique semi-inner product, we show that f is phase equivalent to either a linear or an anti-linear surjective isometry. When X and Y are smooth real normed spaces and Y is strictly convex, we show that Wigner's theorem is equivalent to { ‖ f (x) + f (y) ‖ , ‖ f (x) - f (y) ‖ } = { ‖ x + y ‖ , ‖ x - y ‖ } , x , y ∈ X . [ABSTRACT FROM AUTHOR]

Purpose: To define the length-change patterns of the superficial medial collateral ligament (sMCL), deep MCL (dMCL), and posterior oblique ligament (POL) across knee flexion and with applied anterior and rotational loads, and to relate these findings to their functions in knee stability and to surgical repair or reconstruction. Methods: Ten cadaveric knees were mounted in a kinematics rig with loaded quadriceps, ITB, and hamstrings. Length changes of the anterior and posterior fibres of the sMCL, dMCL, and POL were recorded from 0° to 100° flexion by use of a linear displacement transducer and normalised to lengths at 0° flexion. Measurements were repeated with no external load, 90 N anterior draw force, and 5 Nm internal and 5 Nm external rotation torque applied. Results: The anterior sMCL lengthened with flexion (p < 0.01) and further lengthened by external rotation (p < 0.001). The posterior sMCL slackened with flexion (p < 0.001), but was lengthened by internal rotation (p < 0.05). External rotation lengthened the anterior dMCL fibres by 10% throughout flexion (p < 0.001). sMCL release allowed the dMCL to become taut with valgus rotation (p < 0.001). The anterior and posterior POL fibres slackened with flexion (p < 0.001), but were elongated by internal rotation (p < 0.001). Conclusion: The structures of the medial ligament complex react differently to knee flexion and applied loads. Structures attaching posterior to the medial epicondyle are taut in extension, whereas the anterior sMCL, attaching anterior to the epicondyle, is tensioned during flexion. The anterior dMCL is elongated by external rotation. These data offer the basis for MCL repair and reconstruction techniques regarding graft positioning and tensioning. [ABSTRACT FROM AUTHOR]

The Mazur-Ulam theorem for Fréchet algebras is proved. [ABSTRACT FROM AUTHOR]

We study the space of orthogonally additive n-homogeneous polynomials on C(K). There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive n-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach–Stone theorem. We conclude with a classification of the exposed points. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate a sufficient condition for the invariance of a normal curve on a smooth immersed surface under isometry. We also find the deviations of the tangential and normal components of the curve with respect to the given isometry. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces, and let ƒ : X → Y be an ε-isometry with ƒ(0) = 0 for some ε ≥ 0. In this paper, we show that for every x* ∈ X*, there exists φ ∈ Y* with ||φ|| = ||x*|| ≡ r such that || − <φ,ƒ(x)>| ≤ 3rε, ∨x ∈ X. Moreover, the constant ''3'' is the best one in general. [ABSTRACT FROM AUTHOR]