Abstract: In this paper, the authors present some new characterizations of the Musielak–Orlicz–Sobolev spaces with even smoothness order via ball averages and their derivatives on the radius. Consequently, as special examples of the Musielak–Orlicz–Sobolev spaces studied in this paper, the corresponding characterizations for some weighted Sobolev spaces, Orlicz–Sobolev spaces and variable Sobolev spaces are also obtained. Since these characterizations depend only on ball averages and their derivatives on the radius, they provide some possible ways to introduce the corresponding function spaces on any metric measure space. PubDate: 2019-10-19

Abstract: We investigate an inverse problem referring to roulettes in normed planes, thus generalizing analogous results of Bloom and Whitt on the Euclidean subcase. More precisely, we prove that a given curve can be traced by rolling another curve along a line if two natural conditions are satisfied. Our access involves details from a metric theory of trigonometric functions, which was recently developed for normed planes. Based on this, our approach differs from other ones in the literature. PubDate: 2019-09-26

Abstract: The author investigates the dependence of solvability of homogeneous Riemann boundary-value problem on arcs on a spectral parameter. PubDate: 2019-09-25

Abstract: In this paper we show several sufficient conditions for close-to-convex functions to be strongly starlike of some order. The results continue the line of study from the first author’s paper on the order of strong starlikeness of strongly convex functions, (Nunokawa in Proc Japan Acad Ser A 69(7):234–237, 1993). Also it appears an small improvement of a certain classical results of Ch. Pommerenke. As an application, we also derive estimates for the radii of star-likeness for close-to-convex functions. PubDate: 2019-09-13

Abstract: In this paper, we study abundant exact solutions including the lump and interaction solutions to the (2 + 1)-dimensional Yu–Toda–Sasa–Fukuyama equation. With symbolic computation, lump solutions and the interaction solutions are generated directly based on the Hirota bilinear formulation. Analyticity and well-definedness is guaranteed through some conditions posed on the parameters. With special choices of the involved parameters, the interaction phenomena are simulated and discussed. We find the lump moves from one hump to the other hump of the two-soliton, while the lump separates from the hump of the one-soliton. PubDate: 2019-09-03

Abstract: In this addendum to the paper On Bernstein’s inequality for polynomials [Anal. Math. Phys. online 20 March 2019], we rectify the beginning of Section 5 where we mentioned a proof of Mahler’s result, i.e. the case p=0 in Bernstein’s inequality, using subharmonicity. In particular, we take into account a reference that we previously missed, and that Paul Nevai, whom we thank, has very recently brought to our attention. PubDate: 2019-09-01

Abstract: The present paper deals with the approximation properties of the bivariate operators which are the combination of Bernstein–Chlodowsky operators and the Szász–Kantorovich type operators. We investigate the degree of approximation of the bivariate operators for continuous functions in the weighted space of polynomial growth. Further, we introduce the Generalized Boolean Sum (GBS) of these bivariate Chlodowsky–Szász–Kantorovich type operators and examine the order of approximation in the Bögel space of continuous functions by means of the Lipschitz class and mixed modulus of smoothness. Besides this, we compare the rate of convergence of the Chlodowsky–Szász–Kantorovich type operators and the associated GBS operators by numerical examples and tables using Maple algorithms. It turns out that the GBS operator converges faster to the function than the original operator. PubDate: 2019-09-01

Abstract: In this work, we consider the existence of solution to the following fractional advection–dispersion equation 0.1 $$\begin{aligned} -\frac{d}{dt} \left( p {_{-\infty }}I_{t}^{\beta }(u'(t)) + q\; {_{t}}I_{\infty }^{\beta }(u'(t))\right) + b(t)u = f(t, u(t)),\;\;t\in \mathbb {R}\end{aligned}$$ where \(\beta \in (0,1)\) , \(_{-\infty }I_{t}^{\beta }\) and \(_{t}I_{\infty }^{\beta }\) denote left and right Liouville–Weyl fractional integrals of order \(\beta \) respectively, \(0<p=1-q<1\) , \(f:\mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}\) and \(b:\mathbb {R} \rightarrow \mathbb {R}^{+}\) are continuous functions. Due to the general assumption on the constant p and q, the problem (0.1) does not have a variational structure. Despite that, here we study it performing variational methods, combining with an iterative technique, and give an existence criteria of solution for the problem (0.1) under suitable assumptions. PubDate: 2019-09-01

Abstract: Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N=\mathbb {R}^{N_1} \times \mathbb {R}^{N_2}\) with \(N_1, N_2 \ge 1\) , and \(N(s) = N_1 + (1+s)N_2\) be the homogeneous dimension of \(\mathbb {R}^N\) for \(s \ge 0\) . In this paper, we prove the existence and uniqueness of boundary blow-up solutions to the following semilinear degenerate elliptic equation $$\begin{aligned} {\left\{ \begin{array}{ll} G_s u = { x ^{2s}} u^p_+ \; &{}\text { in } \Omega ,\\ u(z)\rightarrow +\infty \; &{}\text { as } {d}(z) \rightarrow 0, \end{array}\right. } \end{aligned}$$ where \(u_+ = \max \{u,0\}\) , \(1<p<{{N(s)+2s} \over {N(s)-2}}\) , and d(z) denotes the Grushin distance from z to the boundary of \(\Omega \) . Here \(G_s\) is the Grushin operator of the form $$\begin{aligned} G_s u= \Delta _x u + x ^{2s}\Delta _y u, \; s\ge 0. \end{aligned}$$ It is worth noticing that our results do not require any assumption on the smoothness of the domain \(\Omega \) , and when \(s=0\) , we cover the previous results for the Laplace operator \(\Delta \) . PubDate: 2019-09-01

Abstract: In this paper, we obtain some normality criteria for families of meromorphic functions, which improve and generalize related results of Gu, Yang–Chang, Schwick, and Datt–Kumar. PubDate: 2019-09-01

Abstract: In this paper, we investigate pseudo Q-symmetric spacetimes \((PQS)_{4}\) . At first, we prove that a \((PQS)_{4}\) spacetime is a quasi-Einstein spacetime. Then we investigate perfect fluid \((PQS)_{4}\) spacetimes and interesting properties are pointed out. From a result of Mantica and Suh (Int J Geom Methods Mod Phys 10:1350013, 2013) we have shown that \((PQS)_{4}\) spacetime is the Robertson-Walker spacetime. Further, it is shown that a \((PQS)_{4}\) spacetime with cyclic parallel Ricci tensor is an Einstein spacetime. Finally, we construct an example of a \((PQS)_{4}\) spacetime. PubDate: 2019-09-01

Abstract: The famous Koebe \(\frac{1}{4}\) theorem deals with univalent (i.e., injective) analytic functions f on the unit disk \({\mathbb {D}}\) . It states that if f is normalized so that \(f(0)=0\) and \(f'(0)=1\) , then the image \(f({\mathbb {D}})\) contains the disk of radius \(\frac{1}{4}\) about the origin, the value \(\frac{1}{4}\) being best possible. Now suppose f is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises' And for which polynomials is it attained' A plausible conjecture is stated, and the case of small degrees is settled. PubDate: 2019-09-01

Abstract: For a bounded function \(\varphi \) on the unit circle \({\mathbb {T}}\) , let \(T_{\varphi }\) be the associated Toeplitz operator on the Hardy space \(H^2\) . Assume that the kernel $$\begin{aligned} K_2(\varphi ):=\left\{ f\in H^2:\,T_{\varphi } f=0\right\} \end{aligned}$$ is nontrivial. Given a unit-norm function f in \(K_2(\varphi )\) , we ask whether an identity of the form \( f ^2=\frac{1}{2}\left( f_1 ^2+ f_2 ^2\right) \) may hold a.e. on \({\mathbb {T}}\) for some \(f_1,f_2\in K_2(\varphi )\) , both of norm 1 and such that \( f_1 \ne f_2 \) on a set of positive measure. We then show that such a decomposition is possible if and only if either f or \(\overline{z\varphi f}\) has a nonconstant inner factor. The proof relies on an intrinsic characterization of the moduli of functions in \(K_2(\varphi )\) , a result which we also extend to \(K_p(\varphi )\) (the kernel of \(T_{\varphi }\) in \(H^p\) ) with \(1\le p\le \infty \) . PubDate: 2019-09-01

Abstract: We consider how m-dimensional Bakry-Émery Ricci curvature affects the geometry of the boundary \(\partial M\) . By using the Reilly’s formula with respect to f-Laplacian, geometric inequalities involving f-mean curvature are obtained. Furthermore, we also achieve the relationship between f-mean curvature of the boundary submanifold and the mean curvature of submanifold \(x:\partial M\rightarrow \mathbb {R}^N(c)\) into space form \(\mathbb {R}^N(c)\) . PubDate: 2019-09-01

Abstract: In this paper, we obtained a kind of lump solutions of (2+1)-dimensional potential Kadomstev–Petviashvili equation with the assistance of Mathematica by using the Hirota bilinear method. These resulting lump solutions contain a set of five free parameters and some contour plot with different determinant values are sequentially made to show that the corresponding lump solutions tends to zero when the determinant approaches zero. Then, a completely non-elastic interaction between a lump and a stripe of the (2+1)-dimensional potential Kadomstev–Petviashvili equation is obtained, which shows a lump solution is drowned or swallowed by a stripe soliton. PubDate: 2019-09-01

Abstract: In this paper, a partial inverse problem for the quadratic Sturm–Liouville pencil on a geometrical graph of arbitrary structure is studied. We suppose that the coefficients of differential expressions are known a priori on all the graph edges except one, and recover the coefficients on the remaining edge, using a part of the spectrum. The results of the paper are uniqueness theorems and a constructive algorithm for solving the partial inverse problem. PubDate: 2019-09-01

Abstract: Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in \(\mathrm {SL}_n(R)\) is a product of elementary matrices with entries in R. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where R is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in \(\mathrm {GL}_n(R)\) as a product of exponentials. PubDate: 2019-09-01

Abstract: We extend a classical result about weighted averages of harmonic functions to solutions of second-order strongly elliptic systems of PDE with constant coefficients in disks in the complex plane. It is well known that a non-tangential cluster set of the (harmonic) Poisson integral with a given piecewise continuous boundary function f at every point \(\zeta \) in the unit circle is the segment joining the left- and right-hand side limits of f at \(\zeta \) being taken along the unit circle. Using the recently obtained Poisson-type integral representation formula for solutions of aforementioned systems, we establish an analogous result about weighted averages for solutions of such systems. Furthermore, we illustrate the nature of the obtained results by presenting some special mappings of the unit disk by solutions with piecewise constant boundary data. PubDate: 2019-09-01

Abstract: We investigate the direct and inverse theorems for trigonometric polynomials in the Morrey space \({{\mathcal {M}}}^{p(\cdot ),\lambda (\cdot )}\) with variable exponents. For this space, we obtain estimates of the K-functional in terms of the modulus of smoothness and the Bernstein type inequality for trigonometric polynomials. PubDate: 2019-09-01