Mathematics

264 KiB

Open AccessArticle

Convergence of the Quadrature-Differences Method for Singular Integro-Differential Equations on the Interval

by Alexander Fedotov

Mathematics 2014, 2(1), 53-67; https://doi.org/10.3390/math2010053 - 04 Mar 2014

Cited by 1 | Viewed by 4127

In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an [...] Read more.

In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smoothness of the coefficients and the right-hand side of the equation. Full article

(This article belongs to the Special Issue Mathematics on Partial Differential Equations)

1139 KiB

Open AccessArticle

Bounded Gaps between Products of Special Primes

by Ping Ngai Chung and Shiyu Li

Mathematics 2014, 2(1), 37-52; https://doi.org/10.3390/math2010037 - 03 Mar 2014

Cited by 1 | Viewed by 4183

Abstract

In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for [...] Read more.

In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case. Full article

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171 KiB

Open AccessArticle

Some New Integral Identities for Solenoidal Fields and Applications

by Vladimir I. Semenov

Mathematics 2014, 2(1), 29-36; https://doi.org/10.3390/math2010029 - 03 Mar 2014

Cited by 5 | Viewed by 4132

Abstract

In spaces Rⁿ, n ≥ 2, it has been proved that a solenoidal vector field and its rotor satisfy the series of new integral identities which have covariant form. The interest in them is explained by hydrodynamics problems for an ideal fluid. Full article

(This article belongs to the Special Issue Mathematics on Partial Differential Equations)

351 KiB

Open AccessArticle

On the Folded Normal Distribution

by Michail Tsagris, Christina Beneki and Hossein Hassani

Mathematics 2014, 2(1), 12-28; https://doi.org/10.3390/math2010012 - 14 Feb 2014

Cited by 82 | Viewed by 12013

Abstract

The characteristic function of the folded normal distribution and its moment function are derived. The entropy of the folded normal distribution and the Kullback–Leibler from the normal and half normal distributions are approximated using Taylor series. The accuracy of the results are also [...] Read more.

The characteristic function of the folded normal distribution and its moment function are derived. The entropy of the folded normal distribution and the Kullback–Leibler from the normal and half normal distributions are approximated using Taylor series. The accuracy of the results are also assessed using different criteria. The maximum likelihood estimates and confidence intervals for the parameters are obtained using the asymptotic theory and bootstrap method. The coverage of the confidence intervals is also examined. Full article

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224 KiB

Open AccessArticle

One-Dimensional Nonlinear Stefan Problems in Storm’s Materials

by Adriana C. Briozzo and María F. Natale

Mathematics 2014, 2(1), 1-11; https://doi.org/10.3390/math2010001 - 27 Dec 2013

Cited by 9 | Viewed by 4959

Abstract

We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x > 0; with phase change temperature T_f : We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. In the first case, we assume a [...] Read more.

We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x > 0; with phase change temperature T_f : We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. In the first case, we assume a heat flux boundary condition of the type

q(t) = \frac{q_{0}}{\sqrt{t}}

, and in the second case, we assume a temperature boundary condition T = T_s < T_fat the fixed face. Solutions of similarity type are obtained in both cases, and the equivalence of the two problems is demonstrated. We also give procedures in order to compute the explicit solution. Full article

(This article belongs to the Special Issue Mathematics on Partial Differential Equations)

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Mathematics, Volume 2, Issue 1 (March 2014) – 5 articles , Pages 1-67

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