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30 pages, 462 KiB

Open AccessArticle

On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

by Enrico De Micheli

Mathematics 2020, 8(2), 287; https://doi.org/10.3390/math8020287 - 20 Feb 2020

Cited by 2 | Viewed by 2942

We prove that, if the coefficients of a Fourier–Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform) of [...] Read more.

We prove that, if the coefficients of a Fourier–Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform) of the jump function across the cut. The main result of this paper is to establish the connection between the Spherical Laplace Transform and the Non-Euclidean Fourier Transform in the sense of Helgason. In this way, we find a connection between the unitary representation of

SO (3)

and the principal series of the unitary representation of

SU (1, 1)

. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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11 pages, 274 KiB

Open AccessArticle

A Deformed Wave Equation and Huygens’ Principle

by Salem Ben Saïd, Sara al-Blooshi, Maryam al-Kaabi, Aisha al-Mehrzi and Fatima al-Saeedi

Mathematics 2020, 8(1), 10; https://doi.org/10.3390/math8010010 - 19 Dec 2019

Viewed by 2150

Abstract

We consider a deformed wave equation where the Laplacian operator has been replaced by a differential-difference operator. We prove that this equation does not satisfy Huygens’ principle. Our approach is based on the representation theory of the Lie algebra [...] Read more.

We consider a deformed wave equation where the Laplacian operator has been replaced by a differential-difference operator. We prove that this equation does not satisfy Huygens’ principle. Our approach is based on the representation theory of the Lie algebra

s l (2, R)

. Full article

(This article belongs to the Special Issue Mathematical Physics II)

21 pages, 4213 KiB

Open AccessArticle

Cost-Based Optimum Design of Reinforced Concrete Retaining Walls Considering Different Methods of Bearing Capacity Computation

by Neda Moayyeri, Sadjad Gharehbaghi and Vagelis Plevris

Mathematics 2019, 7(12), 1232; https://doi.org/10.3390/math7121232 - 12 Dec 2019

Cited by 29 | Viewed by 20185

Abstract

This paper investigates the effect of computing the bearing capacity through different methods on the optimum construction cost of reinforced concrete retaining walls (RCRWs). Three well-known methods of Meyerhof, Hansen, and Vesic are used for the computation of the bearing capacity. In order [...] Read more.

This paper investigates the effect of computing the bearing capacity through different methods on the optimum construction cost of reinforced concrete retaining walls (RCRWs). Three well-known methods of Meyerhof, Hansen, and Vesic are used for the computation of the bearing capacity. In order to model and design the RCRWs, a code is developed in MATLAB. To reach a design with minimum construction cost, the design procedure is structured in the framework of an optimization problem in which the initial construction cost of the RCRW is the objective function to be minimized. The design criteria (both geotechnical and structural limitations) are considered constraints of the optimization problem. The geometrical dimensions of the wall and the amount of steel reinforcement are used as the design variables. To find the optimum solution, the particle swarm optimization (PSO) algorithm is employed. Three numerical examples with different wall heights are used to capture the effect of using different methods of bearing capacity on the optimal construction cost of the RCRWs. The results demonstrate that, in most cases, the final design based on the Meyerhof method corresponds to a lower construction cost. The research findings also reveal that the difference among the optimum costs of the methods is decreased by increasing the wall height. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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16 pages, 60309 KiB

Open AccessArticle

A Numerical Study on the Crack Development Behavior of Rock-Like Material Containing Two Intersecting Flaws

by Bing Dai, Ying Chen, Guoyan Zhao, Weizhang Liang and Hao Wu

Mathematics 2019, 7(12), 1223; https://doi.org/10.3390/math7121223 - 11 Dec 2019

Cited by 12 | Viewed by 3306

Abstract

It is quite often that rocks contain intersecting cracks. Therefore, crack behavior cannot be completely studied by only considering several isolated, single flaws. To investigate the crack behavior of rock or rock-like material containing intersecting flaws under uniaxial loading, numerical simulations were carried [...] Read more.

It is quite often that rocks contain intersecting cracks. Therefore, crack behavior cannot be completely studied by only considering several isolated, single flaws. To investigate the crack behavior of rock or rock-like material containing intersecting flaws under uniaxial loading, numerical simulations were carried out using parallel bonded-particle models containing two intersecting flaws with different inclination angles (varying β) and different intersection angles (varying αα). The crack propagation processes are analyzed and two typical patterns of linkage are observed between two intersecting flaws: (1) One-tip-linkage that contains three subtypes: Coalescence position near the tip; coalescence position at the flaw, but far away from the tip; coalescence position outside the flaw at a certain distance from the tip; and (2) two-tip-linkage with two subtypes: Straight linkage and arc linkage. The geometries of flaws influence the coalescence type. Moreover, the effects of intersection angle α and inclination angle β on the peak stress, the stress of crack initiation, and the stress of crack coalescence are also investigated in detail. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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14 pages, 1210 KiB

Open AccessArticle

On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle

by Mutaz Mohammad

Mathematics 2019, 7(10), 952; https://doi.org/10.3390/math7100952 - 12 Oct 2019

Cited by 10 | Viewed by 2766

Abstract

Gibbs effect represents the non-uniform convergence of the nth Fourier partial sums in approximating functions in the neighborhood of their non-removable discontinuities (jump discontinuities). The overshoots and undershoots cannot be removed by adding more terms in the series. This effect has been [...] Read more.

Gibbs effect represents the non-uniform convergence of the nth Fourier partial sums in approximating functions in the neighborhood of their non-removable discontinuities (jump discontinuities). The overshoots and undershoots cannot be removed by adding more terms in the series. This effect has been studied in the literature for wavelet and framelet expansions. Dual tight framelets have been proven useful in signal processing and many other applications where translation invariance, or the resulting redundancy, is very important. In this paper, we will study this effect using the dual tight framelets system. This system is generated by the mixed oblique extension principle. We investigate the existence of the Gibbs effect in the truncated expansion of a given function by using some dual tight framelets representation. We also give some examples to illustrate the results. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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8 pages, 1485 KiB

Open AccessArticle

Study on Non-Commutativity Measure of Quantum Discord

by Jin Liang and Chengwei Zhang

Mathematics 2019, 7(6), 543; https://doi.org/10.3390/math7060543 - 14 Jun 2019

Cited by 4 | Viewed by 2640

Abstract

In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their [...] Read more.

In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their non-commutativity measure of quantum discords. Furthermore, we show that the results conducted by the non-commutativity measure of quantum discord are different from those conducted by both or one of the Hilbert-Schmidt distance discord and trace distance discord. These intrinsic differences indicate that the non-commutativity measure of quantum discord is incompatible with at least one of the well-known geometric quantum discords in the quantitative and qualitative representation of quantum correlations. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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13 pages, 4203 KiB

Open AccessArticle

Preliminary Analysis of a Fully Ceramic Microencapsulated Fuel Thermal–Mechanical Performance

by Ping Chen, Suizheng Qiu, Shichao Liu, Yi Zhou, Yong Xin, Shixin Gao, Xi Qiu and Huaiyu Lu

Mathematics 2019, 7(5), 448; https://doi.org/10.3390/math7050448 - 20 May 2019

Cited by 20 | Viewed by 2984

Abstract

In this paper, a two-dimensional characteristic unit was used to simulate the thermal–mechanical performance of a fully ceramic microencapsulated (FCM) fuel pellet, and the criterion of FCM structure integrity was discussed. FCM structure integrity can be reflected though the integrity of the silicon [...] Read more.

In this paper, a two-dimensional characteristic unit was used to simulate the thermal–mechanical performance of a fully ceramic microencapsulated (FCM) fuel pellet, and the criterion of FCM structure integrity was discussed. FCM structure integrity can be reflected though the integrity of the silicon carbide (SiC) matrix or SiC layers because of the excellent fission retention capability of SiC ceramics. The maximum temperature of the SiC matrix under normal conditions of the pressure water reactor (PWR) environment was about 1390 K, which was lower than the decomposition point of SiC. The maximum hoop stress of the SiC matrix, especially the inner part, was up to about 1200 MPa, and the hoop stress of the non-fuel region part was lower than the inner part, which can be attributed to the deformation of tristructural-isotopic (TRISO) particles. The hoop stress of the SiC layers at the end of life was only about 180 MPa, which is much lower than the strength of the chemical vapor deposition (CVD)-SiC. The failure probability of the SiC layer was lower than 9 × 10⁻⁵; thus, the integrity of SiC layers and the fission retention capability were maintained. The structure integrity of FCM fuel was broken because the SiC matrix cracked. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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35 pages, 489 KiB

Open AccessArticle

Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Q p over Primes p

by Ilwoo Cho and Palle Jorgensen

Mathematics 2019, 7(2), 199; https://doi.org/10.3390/math7020199 - 19 Feb 2019

Cited by 2 | Viewed by 2234

Abstract

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space

(LS,

τ^{0})

induced by measurable functions on p-adic number fields

Q_{p}

over primes

p

. In particular, we are interested in [...] Read more.

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space

(LS,

τ^{0})

induced by measurable functions on p-adic number fields

Q_{p}

over primes

p

. In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set

R

of real numbers by defining suitable “truncated” linear functionals on

LS

. Full article

(This article belongs to the Special Issue Mathematical Physics II)

17 pages, 327 KiB

Open AccessArticle

The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

by Tongshuai Liu and Huanhe Dong

Mathematics 2019, 7(2), 170; https://doi.org/10.3390/math7020170 - 13 Feb 2019

Cited by 12 | Viewed by 2547

Abstract

In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. [...] Read more.

In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential

q (x, t)

can be represented by the solution of this Riemann-Hilbert problem. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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