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1

Some aspects of normal curve on smooth surface under isometry

Singh Kuljeet, Sharma Sandeep, Bhardwaj Arun Kumar

Demonstratio Mathematica

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2024

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Vol. 57, nr 1

art. no. 20240017

EN

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.

2

Hyers-Ulam stability of isometries on bounded domains-II

Choi Ginkyu, Jung Soon-Mo

Demonstratio Mathematica

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2023

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Vol. 56, nr 1

art. no. 20220196

EN

The question of whether there is a true isometry approximating the ε -isometry defined in the bounded subset of the n -dimensional Euclidean space has long been considered an interesting question. In 1982, Fickett published the first article on this topic, and in early 2000, Alestalo et al. and Väisälä improved Fickett’s result significantly. Recently, the second author of this article published a paper improving the previous results. The main purpose of this article is to significantly improve all of the aforementioned results by applying a basic and intuitive method.

3

Diagnostic value of the hamstring to quadriceps ratio in monitoring of the effectiveness of supervised 6-month physiotherapy in males after Anterior Cruciate Ligament Reconstruction (ACLR)

Krzemińska Katarzyna, Czamara Andrzej

Acta of Bioengineering and Biomechanics

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2020

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Vol. 22, no. 4

31--39

EN

The aim of the study was to assess the usefulness of isometric torque (IT) and peak torque (PT) of the hamstring to quadriceps muscles ratio (H/Q ratio) in monitoring the effectiveness of physiotherapy (PH) in males after ACLR. Hypothesis: The H/Q ratio is a diagnostic tool for monitoring of the effectiveness of the 6- month PH after ACLR. Methods: Twenty males 6 months after ACLR (ACLR group) and 20 male controls underwent IT and PT (60°/s and 180°/s) bilateral measurements of H and Q muscles. The IT and PT were normalized to body mass, and expressed as relative IT (RIT) and relative PT (RPT). The RIT and RPT H/Q ratios, and Limb Symmetry Index (LSI) were calculated. Results: In the ACLR group, the RIT for the H and Q, the RIT for the H/Q ratio and most of the RPT, as well as the H/Q ratio, ROM and LSI values of the operated knee, were not significantly different (NSD) than those of the non-operated side (NOS) or the control group. The between-group comparison of the H/Q ratio for RIT and RPT weren’t NSD. The isokinetic test at 180 °/s showed lower RPT, H/Q ratio and LSI values for the Q muscle than those of the NOS (p = 0.042, p = 0.001). Conclusions: The H/Q ratio, in combination with the RIT, RPT and LSI, is a useful diagnostic tool for monitoring the effectiveness of 6-month PH after ACLR. Restoring the correct H/Q ratio can reduce the risk factor for ACL graft rupture.

4

Characterizations of rectangular (para)-unitary rational functions

Alpay D., Jorgensen P., Lewkowicz I.

Opuscula Mathematica

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2016

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Vol. 36, no. 6

695--716

EN

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) through the realization matrix of Schur stable systems, (ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters, (iii) through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.

5

Isometries in Teaching Descriptive Geometry and Engineering Graphics

Ulasevich Z., Ulasevich V.

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

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2016

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Vol. 29

23--30

EN

Important basic transformations, implemented in CAD systems, are congruence transformations, so-called isometries, which preserve the distance of points. Logic of CAD software bases on the reflection, translation, rotation, and similarity. This fact is the important desideratum in the teaching of Descriptive Geometry. The paper includes a proposal for a teaching from the scope of isometries on the plane and in three-dimensional space.

PL

Ważnymi przekształceniami zaimplementowanymi w oprogramowaniu CAD, są izometrie, czyli przekształcenia zachowujące odległość punktów. Logika tych systemów opiera się gównie na pojęciu symetrii, translacji i obrotu. Ważną jeszcze rolę odgrywa podobieństwo. Uwzględnienie tego faktu w nauczaniu geometrii wykreślnej jest ważnym dezyderatem dydaktycznym. Praca zawiera propozycję dydaktyczną z zakresu zastosowania izometrii na płaszczyźnie i w przestrzeni.

6

Multiplication operators on Cesàro-Orlicz sequence spaces

Raj K., Sharma C., Pandoh S.

Fasciculi Mathematici

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2016

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Nr 57

137--145

EN

In this paper, we characterize the compact, invertible, Fredholm and closed range multiplication operators on Cesàro-Orlicz sequence spaces.

7

Multiplication operators on Cesàro sequence spaces

Komal B. S., Pandoh S., Raj K.

Demonstratio Mathematica

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2016

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Vol. 49, nr 4

430--436

EN

In this paper we characterize the compact, invertible and Fredholm multiplication operators on Cesàro sequence spaces.

8

Approximately bisectrix-orthogonality preserving mappings

Zamani A.

Commentationes Mathematicae

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2014

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Vol 54, No. 2

167--176

EN

Regarding the geometry of a real normed space X, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors x,y∈X as follows:... [formula] We study the class of linear mappings preserving the approximately bisectrix-orthogonality ε⊥W. In particular, we show that if T:X→Y is an approximate linear similarity, then xδ⊥Wy →Txθ⊥WTy(x,y∈X) for any δ∈[0,1) and certain θ≥0.

9

Major axis approach to the statistical analysis of the relative growth of Chinese mitten crab (Eriocheir sinensis) in the Odra estuary (Poland)

De Giosa M., Czerniejewski P.

Oceanological and Hydrobiological Studies

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2011

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Vol. 40, No.1

36-45

EN

A total of 379 individuals of Eriocheir sinensis (198 males and 181 females) were captured in the Odra estuary (Poland). The crabs were thawed and their carapace length (CL), the maximum carapace width (CW) and the maximum height (CH) were measured. Measurements were also taken on each claw, the claw length (CHL1, the right claw; CHL2, the left claw), the width (CHW1 and CHW2 for the right and the left claw, respectively), and the height (CHH1 and CHH2 for the right and the left claw). For each crab, the wet weight was measured for each of the following body components: the whole crab (CrWe), the carapace (CaWe), the right claw (WRC) and the left claw (WLC). For females, the relationship between CL and CW, CH and CW were isometric, and for all linear measures, the relationship with CW was positively allometric. For males only this first relationship was isometric, but others were positively allometric. The differences between relative growth parameters for males and females were statistically significant.

10

Weighted substitution operators between Lp -spaces of vector-valued functions

Raj K., Sharma S. K.

Fasciculi Mathematici

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2011

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Nr 47

63--70

EN

In this paper we characterize weighted substitution operators between Lp-spaces of vector-valued functions and also make an attempt to characterize isometry and partial isometry of these operators

11

Ergodic conditions and spectral properties for A-contractions

Suciu L., Suciu N.

Opuscula Mathematica

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2008

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Vol. 28, no. 2

195-216

EN

In this paper the canonical representation of an A-contraction T on a Hilbert space H is used to obtain some conditions concerning the concept of A-ergodicity studied in [14-17]. The regular case and the case of R(A) closed are considered, and specifically, the TT*-contractions are studied. Some spectral properties are also given for certain particular class of A-isometries.

12

On maps which preserve equality of distance in F*-spaces

Tan D.

Bulletin of the Polish Academy of Sciences. Mathematics

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2008

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Vol. 56, no 3-4

225-230

EN

We prove that every map T between two F*-spaces which preserves equality of distance and satisfies T(0) = 0 is linear.

13

Fischer-Muszély additivity on Abelian groups

Ger R.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2004

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Vol. 44, nr spec.

83-96

EN

A description of a general solution f : X -> Y mapping a commutative group (X, +) into a real normed linear space (Y, || o ||) of the functional equation [formula] is given in terms of isometrics and additive mappings. Several results describing the solutions of this equation that were obtained earlier under some alternative assumptions regarding the domains, ranges and//or by imposing some regularity upon the map f become special cases of our main result. To gain a proper proof tool we have also established an improvement of E. Berz's [4] representation theorem for sublinear functionals on Abelian groups.

14

Contractive projections onto isometric copies of L1(ν) in strictly monotone Banach lattices

Wójtowicz M.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 1

1-12

EN

Let Y be a Banach lattice and X a strictly monotone sublattice of Y. Every isometric copy of an L1-space in X is 1-complemented in Y (Theorem l). This is an extension of the classical result of Pełczyński for Y = X = L1(my), and of Dor for Y = X being q-concave. We also study the consequences of the existence of isometric copies of l1 in strictly monotone E[phi](my)-spaces, where E[phi](my) denotes the ideal of an Orlicz space L[phi](my) of the elements with absolutely continuous norm.