scholarly journals The four-genus of connected sums of torus knots

2017 ◽  
Vol 164 (3) ◽  
pp. 531-550
Author(s):  
CHARLES LIVINGSTON ◽  
CORNELIA A. VAN COTT
Keyword(s):  
Linear Combinations ◽  
Torus Knots ◽  
Connected Sums ◽  
The Third

AbstractWe study the four-genus of linear combinations of torus knots:g4(aT(p, q) #-bT(p′, q′)). Fixing positivep, q, p′, andq′, our focus is on the behavior of the four-genus as a function of positiveaandb. Three types of examples are presented: in the first, for allaandbthe four-genus is completely determined by the Tristram–Levine signature function; for the second, the recently defined Upsilon function of Ozsváth–Stipsicz–Szabó determines the four-genus for allaandb; for the third, a surprising interplay between signatures and Upsilon appears.

scholarly journals Convolution operators on weighted spaces of continuous functions and supremal convolution

2019 ◽  
Vol 199 (4) ◽  
pp. 1547-1569
Author(s):  
T. Kleiner ◽  
R. Hilfer
Keyword(s):  
Continuous Functions ◽  
Weighted Spaces ◽  
Constructive Method ◽  
Convolution Operators ◽  
Linear Combinations ◽  
The Third ◽  
Bounded Set ◽  
Spaces Of Continuous Functions ◽  
Bounded Sets

AbstractThe convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 617 ◽  
Author(s):  
Dmitry Dolgy ◽  
Dae Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

2018 ◽  
Author(s):  
Dmitry Victorovich Dolgy ◽  
Dae San Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions ${}_2 F_0, {}_2 F_1$, and ${}_3 F_2$.

scholarly journals Studies in Sums of Finite Products of the Second, Third, and Fourth Kind Chebyshev Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 210
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Jongkyum Kwon
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Linearization Problem ◽  
Finite Products ◽  
Fourth Kind ◽  
Classical Linearization

In this paper, we consider three sums of finite products of Chebyshev polynomials of two different kinds, namely sums of finite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the third and fourth kind Chebyshev polynomials. As a generalization of the classical linearization problem, we represent each of such sums of finite products as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. These are done by explicit computations and the coefficients involve terminating hypergeometric functions 2 F 1 , 1 F 1 , 2 F 2 , and 4 F 3 .

scholarly journals Distinguishing the generalized knot groups of square and granny knot analogues

2019 ◽  
Vol 28 (05) ◽  
pp. 1950035
Author(s):  
Howida Al Fran ◽  
Christopher Tuffley
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Torus Knots ◽  
Connected Sums ◽  
The Difference ◽  
A Finite Group

Given a knot [Formula: see text], we may construct a group [Formula: see text] from the fundamental group of [Formula: see text] by adjoining an [Formula: see text]th root of the meridian that commutes with the corresponding longitude. For [Formula: see text] these “generalized knot groups” determine [Formula: see text] up to reflection. The second author has shown that for [Formula: see text], the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues [Formula: see text], [Formula: see text], constructed as connected sums of [Formula: see text]-torus knots of opposite or identical chiralities. More precisely, for coprime [Formula: see text] and [Formula: see text] satisfying a coprimality condition with [Formula: see text] and [Formula: see text], we construct an explicit finite group [Formula: see text] (depending on [Formula: see text], [Formula: see text] and [Formula: see text]) such that [Formula: see text] and [Formula: see text] can be distinguished by counting homomorphisms into [Formula: see text]. The coprimality condition includes all [Formula: see text] coprime to [Formula: see text]. The result shows that the difference between these two groups can be detected using a finite group.

scholarly journals Sums of finite products of Chebyshev polynomials of two different types

2021 ◽  
Vol 6 (11) ◽  
pp. 12528-12542
Author(s):  
Taekyun Kim ◽  
◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Jongkyum Kwon ◽  
...  
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Linear Combinations ◽  
The Third ◽  
Fourth Type ◽  
Different Types ◽  
Finite Products ◽  
Classical Linearization

<abstract><p>In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions $ {}_{2}F_{1} $. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.</p></abstract>

scholarly journals Representing Sums of Finite Products of Chebyshev Polynomials of Third and Fourth Kinds by Chebyshev Polynomials

2018 ◽  
Author(s):  
T. Kim ◽  
D.S. Kim ◽  
D.V. Dolgy ◽  
C.S. Ryoo
Keyword(s):  
Hypergeometric Function ◽  
Chebyshev Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Finite Products

Here we consider sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials which involve the hypergeometric function 3F2.

scholarly journals Precise Point Positioning using Triple-Frequency GPS Measurements

2014 ◽  
Vol 68 (3) ◽  
pp. 480-492 ◽  
Author(s):  
Mohamed Elsobeiey
Keyword(s):  
Precise Point Positioning ◽  
Satellite System ◽  
Global Network ◽  
Convergence Time ◽  
Linear Combinations ◽  
The Third ◽  
Triple Frequency ◽  
Estimated Parameters ◽  
Point Positioning ◽  
L5 Signal

Precise Point Positioning (PPP) performance is improving under the ongoing Global Positioning System (GPS) modernisation program. The availability of the third frequency, L5, enables triple-frequency combinations. However, to utilise the modernised L5 signal along with the existing GPS signals, P1-C5 differential code bias must be determined. In this paper, the global network of Multi-Global Navigation Satellite System Experiment (MGEX) stations was used to estimate P1-C5 satellites differential code biases $(DCB_{P1 - C5}^S )$. Mathematical background for triple-frequency linear combinations was provided along with the resultant noise and ionosphere amplification factors. Nine triple-frequency linear combinations were chosen, based on different criteria, for processing the modernised L5 signal along with the legacy GPS signals. Finally, test results using real GPS data from ten MGEX stations were provided showing the benefits of the availability of the third frequency on PPP solution convergence time and the precision of the estimated parameters. It was shown that triple-frequency combinations could improve the PPP convergence time and the precision of the estimated parameters by about 10%. These results are considered promising for using the modernised GPS signals for precise positioning applications especially when the fully modernised GPS constellation is available.

Variation in the white-toothed shrews ( Crocidura spp.) in the British Isles

1966 ◽  
Vol 164 (994) ◽  
pp. 63-74 ◽  
Keyword(s):  
Variance Components ◽  
Foramen Magnum ◽  
The Other ◽  
Canonical Variate ◽  
Common Stock ◽  
Linear Combinations ◽  
Skull Length ◽  
Lower Tooth ◽  
The Third ◽  
Canonical Variates

Ten skull characters were measured on each of 300 specimens of Crocidura suaveolens (Pallas) and 99 specimens of C. russula Hermann from five of the Scilly Isles, four of the Channel Isles and one locality on the mainland of France. No place contained both species. The characters were skull length, skull width, lengths of upper and lower tooth rows, distance between the third upper molars, distance between the upper premolars, length from the palate to the foramen magnum, the combined length in ventral aspect of the third upper incisor and canine and the mandibular height. The means of each measurement at each locality were calculated. Analyses of variance were also calculated and from these variance components within locality groups were obtained. An analysis into canonical variates was made with a view to accounting for the largest possible part of the variation between groups using a limited number of linear combinations of the original measurements. Most of the variance (82%) was contained in the first canonical variate and from the dispersion of the means of the samples the popula­tions of Crocidura separated into two main groups. One contained animals from Alderney, Guernsey and Cap Gris Nez ( C. russula ) and the other the remainder ( C. suaveolens ). The latter group subdivided, particularly with reference to the second canonical variate, into animals from Sark and Jersey and those from the Isles of Scilly. Differences between popula­tions from the Scilly Isles are very small, suggesting origin from a common stock. The shrews from Sark and Jersey differ more from each other than do any pair of Scilly Island popula­tions. The three populations of C. russula do not form as close a cluster as the Scilly Island ones. The analyses of variance agree with these findings although for certain characters highly significant differences often occur between localities in a particular group.

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