Showing papers in "Results in Mathematics in 2017"

The Local Convexity of Solving Systems of Quadratic Equations

Abstract: This paper considers the recovery of a rank r positive semidefinite matrix \({X X^T \in \mathbb{R}^{n\times n}}\) from m scalar measurements of the form \({y_i := a_i^T X X^T a_i}\) (i.e., quadratic measurements of X). Such problems arise in a variety of applications, including covariance sketching of high-dimensional data streams, quadratic regression, quantum state tomography, among others. A natural approach to this problem is to minimize the loss function \({f(U) = \sum_i (y_i - a_{i}^{T}UU^{T}a_i)^2}\) which has an entire manifold of solutions given by \({\{XO\}_{O\in\mathcal{O}_r}}\) where \({\mathcal{O}_r}\) is the orthogonal group of \({r\times r}\) orthogonal matrices; this is non-convex in the \({n\times r}\) matrix U, but methods like gradient descent are simple and easy to implement (as compared to semidefinite relaxation approaches). In this paper we show that once we have \({m \geq Cnr\,{\rm log}^2(n)}\) samples from isotropic gaussian \({a_i}\), with high probability (a) this function admits a dimension-independent region of local strong convexity on lines perpendicular to the solution manifold, and (b) with an additional polynomial factor of r samples, a simple spectral initialization will land within the region of convexity with high probability. Together, this implies that gradient descent with initialization (but no re-sampling) will converge linearly to the correct X, up to an orthogonal transformation. We believe that this general technique (local convexity reachable by spectral initialization) should prove applicable to a broader class of nonconvex optimization problems.

New Subclass of Analytic Functions in Conical Domain Associated with Ruscheweyh q-Differential Operator

Abstract: The core object of this paper is to define and study a new class of analytic functions using the Ruscheweyh q-differential operator. We also investigate a number of useful properties of this class such structural formula and coefficient estimates for functions. We consider also the Fekete–Szego problem in the class, we give some subordination results, and some other corollaries.

Geometry of Warped Product Semi-Slant Submanifolds of Nearly Kaehler Manifolds

Abstract: Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in Sahin (Geom Dedic 117:195–202, 2006), it is interesting to find their existence in a more general setting, e.g., nearly Kaehler manifolds. In this paper, we obtain a necessary and sufficient condition for a semi-slant submanifold of a nearly Kaehler manifold to be a locally warped product. Also, we establish an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. Furthermore, the equality case of the statement is also considered.

Szász–Mirakyan Type Operators Which Fix Exponentials

Abstract: In this paper, we construct a new general class of operators which have the classical Szasz Mirakyan ones as a basis, and fix the functions \(e^{ax}\) and \(e^{2ax}\) with \(a>0\). The convergence of the corresponding sequences is discussed in exponential weighted spaces, and a Voronovskaya type result is given. Also we define a new weighted modulus of smoothness and determine the approximation order of the constructed operators. Finally, we study the goodness of the estimates of our new operators via saturation results.

Saturation Classes for Max-Product Neural Network Operators Activated by Sigmoidal Functions

Abstract: In the present paper, we obtain a saturation result for the neural network (NN) operators of the max-product type. In particular, we show that any non-constant, continuous function on the interval [0, 1] cannot be approximated by the above operators $$F^{(M)}_n$$ , $$n \in \mathbb {N}^+$$ , by a rate of convergence higher than 1 / n. Moreover, since we know that any Lipschitz function f can be approximated by the NN operators with an order of approximation of 1 / n, here we are able to prove a local inverse result, in order to provide a characterization of the saturation (Favard) classes.

Solvability of a Non-local Problem with Integral Transmitting Condition for Mixed Type Equation with Caputo Fractional Derivative

Abstract: In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for a mixed parabolic–hyperbolic type equation with the Caputo fractional derivative in a domain bounded by smooth curves. A uniqueness of the solution for a formulated problem we prove using energy integral method with some modifications. The existence of solution will be proved by equivalent reduction of the studied problem into a system of second kind Fredholm integral equations.

The Ulam Stability of First Order Linear Dynamic Equations on Time Scales

Abstract: In the present paper, the Ulam stability of the first order linear dynamic equation and its adjoint equation on a time scale is established by using the integrating factor method. Meantime, two examples are provided to illustrate our main results.

Gradient Ricci Solitons with Structure of Warped Product

Abstract: In this paper we consider semi-Riemannian warped product gradient Ricci solitons. We prove that the potential function depends only on the base and the fiber is necessarily Einstein manifold. We provide all such solutions in the case of steady gradient Ricci solitons when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an (n − 1)-dimensional translation group, and the fiber is Ricci-flat.

On Recovering the Dirac Operator with an Integral Delay from the Spectrum

Abstract: An inverse spectral problem for the Dirac operator with an integral delay is studied. We show, that the considered operator can be uniquely recovered from one spectrum, provide a constructive procedure for the solution of the inverse problem, and obtain necessary and sufficient conditions for its solvability.

On Common Fixed Points in the Context of Brianciari Metric Spaces

Abstract: In this paper, we introduce the concept of generalized (\({\alpha,\psi}\))-contractions and generalized (\({\alpha,\psi}\))-Meir–Keeler-contractions in the setting of Brianciari metric spaces. We prove some common fixed point results for such contractions. An example is presented making effective the new concepts and results.

Meromorphic Solutions of Complex Differential–Difference Equations

Abstract: This paper is devoted to exploring the properties of meromorphic solutions on complex differential–difference equations using Nevanlinna theory. We state some relationships between the exponent of convergence of zeros with the order of meromorphic solutions on linear or non-linear differential–difference equations.

Approximate Controllability Results for Fractional Semilinear Integro-Differential Inclusions in Hilbert Spaces

Abstract: In this paper, we consider a class of fractional integro-differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the approximate controllability for a class of fractional semilinear integro-differential inclusions in Hilbert spaces. We use Bohnenblust–Karlin’s fixed point theorem to prove our main result. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. An example is also given to illustrate our main result.

Korovkin Type Theorems for Cheney–Sharma Operators via Summability Methods

Abstract: Korovkin type approximation theory is concerned with the convergence of the sequences of positive linear operators to the identity operator. In this paper, we deal with the Korovkin type approximation properties of the Cheney–Sharma operators by using A-statistical convergence and Abel convergence that are some well known methods of summability theory. We also study the rate of convergence. Finally, we show that the results obtained in this paper are stronger than previous ones and we support our results with particular examples and graphs.

Existence and Concentration of Solutions for the Chern–Simons–Schrödinger System with General Nonlinearity

Abstract: In this paper, we study the nonlinear Chern–Simons–Schrodinger system with an external potential. Under some suitable conditions on the potential and nonlinearity, we prove the existence, multiplicity and concentration of solutions by using variational methods.

On an Inverse Transmission Problem From Complex Eigenvalues

Abstract: We prove that for unique determination of q(x) in the boundary value problem –y′′ + q(x)y = λy, y(0) = 0, \({y(1)\cos\sqrt\lambda a=y'(1)\sin\sqrt\lambda a/\sqrt\lambda}\) for a > 1 it is sufficient to specify only a part of the spectrum with exception of infinitely many eigenvalues having the asymptotics λ k,1 = (a–1)−2 π2 k 2 + O(1), k ≥ 1, and forming McLaughlin-Polyakov’s almost real infinite subspectrum (McLaughlin and Polyakov in J Differ Equ 107:351–382, 1994). This result improves the uniqueness theorem in Aktosun et al. (Inverse Probl 27:115004, 2011).

Bounds for Radii of Starlikeness of Some $${\varvec{q}}$$ q -Bessel Functions

Abstract: In this paper the radii of starlikeness of Jackson’s second and third q-Bessel functions are considered and for each of them three different normalization are applied. By applying Euler–Rayleigh inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre–Polya class of real entire functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind.

Interaction of Codazzi Couplings with (Para-)Kähler Geometry

Abstract: We study Codazzi couplings of an affine connection $$ abla $$ with a pseudo-Riemannian metric g, a nondegenerate 2-form $$\omega $$ , and a tangent bundle isomorphism L on smooth manifolds, as an extension of their parallelism under $$ abla $$ . In the case that L is an almost complex or an almost para-complex structure and $$(g, \omega , L)$$ form a compatible triple, we show that Codazzi coupling of a torsion-free $$ abla $$ with any two of the three leads to its coupling with the remainder, which further gives rise to a (para-)Kahler structure on the manifold. This is what we call a Codazzi-(para-)Kahler structure; it is a natural generalization of special (para-)Kahler geometry, without requiring $$ abla $$ to be flat. In addition, we also prove a general result that g-conjugate, $$\omega $$ -conjugate, and L-gauge transformations of $$ abla $$ , along with identity, form an involutive Abelian group. Hence a Codazzi-(para-)Kahler manifold admits a pair of torsion-free connections compatible with the $$(g, \omega , L)$$ . Our results imply that any statistical manifold may admit a (para-)Kahler structure as long as one can find an L that is compatible to g and Codazzi coupled with $$ abla $$ .

On the Bohnenblust–Hille Inequality for Multilinear Forms

Abstract: The general versions of the Bohnenblust–Hille inequality for m-linear forms are valid for exponents \(q_{1},\ldots ,q_{m}\in [1,2]\). In this paper we show that a slightly different characterization is valid with no restrictions for the range of the parameters, i.e., for \( q_{1},\ldots ,q_{m}\in (0,\infty )\).

Functional Inequalities for the Mittag–Leffler Functions

Abstract: In this paper, some Turan-type inequalities for Mittag–Leffler functions are considered. The method is based on proving monotonicity for special ratio of sections for series of Mittag–Leffler functions. Furthermore, we deduce the Lazarevic- and Wilker-type inequalities for Mittag–Leffler functions.

Generalized Weaving Frames for Operators in Hilbert Spaces

Abstract: This paper studies weaving properties of a family of operators which are analysis and synthesis systems with frame-like properties for closed subspaces of a separable Hilbert space $$\mathcal {H}$$ , where the lower frame condition is controlled by a bounded operator on $$\mathcal {H}$$ . In short, this family of operators is called a $$\Theta $$ -g-frame, where $$\Theta $$ is a bounded operator on $$\mathcal {H}$$ . We present sufficient conditions for weaving $$\Theta $$ -g-frames in separable Hilbert spaces. A characterization of weaving $$\Theta $$ -g-frames in terms of an operator is given. It is shown that if frame bounds of frames associated with atomic spaces are positively confined, then $$\Theta $$ -g-woven frames gives ordinary weaving $$\Theta $$ -frames and vice-versa. We provide classes of operators for weaving $$\Theta $$ -g-frames.

Some Results on Rota–Baxter Monoidal Hom-Algebras

Abstract: We give constructions of Rota–Baxter monoidal Hom-(co)algebras from Hom-Hopf module (co)algebras, and then introduce the concept of Rota–Baxter monoidal Hom-bialgebras. Furthermore, we consider the relations between Rota–Baxter monoidal Hom-systems and monoidal Hom-dendriform algebras, and also derive the structures of pre-Lie Hom-(co)algebras via Rota–Baxter monoidal Hom-(co)algebras of different weight.

Statistical Relative Approximation on Modular Spaces

Abstract: In the present paper, using the concept of statistical relative convergence, we study the problem of approximation to a function by means of double sequences of positive linear operators defined on a modular space. Also, a non-trivial application is presented.

Controlled K-g-Frames in Hilbert Spaces

Abstract: Controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concept of controlled K-g-frames, and make controlled g-frames as a special case of that. Then, we discuss characterizations of controlled K-g-frames, hash out equivalent conditions of them and show that controlled K-g-frames are equivalent to K-g-frames under some conditions. Finally, we propose several methods to construct controlled K-g-frames.

A New Method for Sharpening the Bounds of Several Special Functions

Abstract: This paper presents a new method to sharpen bounds of both sinc(x) and $$\arcsin (x)$$ functions, and the inequalities in exponential form as well. It also provides a method for finding two-sided bounds, which are also unsolved in previous state-of-art references.

4-colored graphs and knot/link complements

Abstract: A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e. 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.

Direct Results for Certain Summation-Integral Type Baskakov–Szász Operators

Abstract: In the present article we define a mixed hybrid operator based on two parameters. As special case, we get the genuine hybrid operators viz. Phillips and Baskakov–Szasz type operators. We estimate moments and obtain some direct results, which include the Voronovskaja type asymptotic formula, local approximation, error estimation in terms of the modulus of continuity and weighted approximation. Furthermore, we obtain the rate of convergence for unbounded functions with derivatives of bounded variation by these operators. The paper contains also numerical examples based on Maple algorithms, which verify approximation properties of these operators.

Triviality of Compact m-Quasi-Einstein Manifolds

Abstract: The goal of this note is to show that a compact m-quasi-Einstein manifold \({(M^{n}, g, X, \lambda)}\) has the vector field X identically zero provided that \({(M^{n}, g)}\) is an Einstein manifold.

Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane

Abstract: For singular plane curves, the classical definitions of envelopes are vague. In order to define envelopes for singular plane curves, we introduce a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane and the curvature. Then we give a definition of an envelope for the one-parameter family of Legendre curves. We investigate properties of the envelopes. For instance, the envelope is also a Legendre curve. Moreover, we consider bi-Legendre curves and give a relationship between envelopes.

Bezier variant of the Bernstein–Durrmeyer type operators

Abstract: In the present paper, we introduce the Bezier-variant of Durrmeyer modification of the Bernstein operators based on a function \(\tau \), which is infinite times continuously differentiable and strictly increasing function on [0, 1] such that \(\tau (0)=0\) and \(\tau (1)=1\). We give the rate of approximation of these operators in terms of usual modulus of continuity and K-functional. Next, we establish the quantitative Voronovskaja type theorem. In the last section we obtain the rate of convergence for functions having derivative of bounded variation.

Bounds for the Gamma Function

Abstract: We improve the upper bound of the following inequalities for the gamma function $$\Gamma $$ due to H. Alzer and the author. $$\begin{aligned} \exp \left( -\frac{1}{2}\psi (x+1/3)\right)<\frac{\Gamma (x)}{x^xe^{-x}{\sqrt{2\pi }}} <\exp \left( -\frac{1}{2}\psi (x)\right) . \end{aligned}$$ We also prove the following new inequalities: for $$x\ge 1$$ $$\begin{aligned} {\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a_*\right) ^{\frac{1}{4}}<\Gamma (x+1)<{\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a^*\right) ^{\frac{1}{4}} \end{aligned}$$ with the best possible constants $$a_*=\frac{e^4}{4\pi ^2}-\frac{4}{3}=0.049653963176\ldots $$ , and $$a^*=1/18=0.055555\ldots $$ , and for $$x\ge 0$$ $$\begin{aligned} \exp \left[ x\psi \left( \frac{x}{\log (x+1)}\right) \right] \le \Gamma (x+1)\le \exp \left[ x\psi \left( \frac{x}{2}+1\right) \right] , \end{aligned}$$ where $$\psi $$ is the digamma function.