General Natural $$(\alpha ,\varepsilon )$$-Structures

We study in a unified way the \((\alpha ,\varepsilon )\)-structures of general natural lift type on the tangent bundle of a Riemannian manifold. We characterize the general natural \(\alpha \)-structures on the total space of the tangent bundle of a Riemannian manifold, and provide their integrability conditions (the base manifold is a space form and some involved coefficients are rational functions of the other ones). Then, we characterize the two classes (with respect to the sign of \(\alpha \varepsilon \)) of \((\alpha ,\varepsilon )\)-structures of general natural type on TM. The class \(\alpha \varepsilon =-1\) is characterized by some proportionality relations between the coefficients of the metric and those of the \(\alpha \)-structure, and in this case, the structure is almost Kählerian if and only if the first proportionality factor is the derivative of the second one. Moreover, the total space of the tangent bundle is a Kähler manifold if and only if it depends on three coefficients only (two coefficients of the integrable \(\alpha \)-structure and a proportionality factor).     

A fast meta-heuristic approach for the $$(\alpha ,\beta )-k$$-feature set problem

The feature selection problem aims to choose a subset of a given set of features that best represents the whole in a particular aspect, preserving the original semantics of the variables on the given samples and classes. In 2004, a new approach to perform feature selection was proposed. It was based on a NP-complete combinatorial optimisation problem called (\(\alpha ,\beta \))-k-feature set problem. Although effective for many practical cases, which made the approach an important feature selection tool, the only existing solution method, proposed on the original paper, was found not to work well for several instances. Our work aims to cover this gap found on the literature, quickly obtaining high quality solutions for the instances that existing approach can not solve. This work proposes a heuristic based on the greedy randomised adaptive search procedure and tabu search to address this problem; and benchmark instances to evaluate its performance. The computational results show that our method can obtain high quality solutions for both real and the proposed artificial instances and requires only a fraction of the computational resources required by the state of the art exact and heuristic approaches which use mixed integer programming models.     

$$(\vec k,\vec s)$$ -Monotone Coefficients in Spaces with Mixed Quasinorm

We estimate the sums of double series in sines and cosines with multiply monotone coefficients and other more general coefficients. We obtain necessary and sufficient conditions for membership of the sums of these series in the spaces with mixed quasinorm.     

Sharp L^{p,\infty }\rightarrow L^q Estimates for the Dyadic-Like Maximal Operators

For each \(1\le q we study the sharp versions of the \(L^{p,\infty }\rightarrow L^q\) estimates for the dyadic maximal operator on \(\mathbb {R}^n\) . Actually, this is done in the more general setting of maximal operators associated with a tree-like structure. The proof rests on a novel combination of the Bellman function technique and optimization arguments.     

A classification of totally geodesic and totally umbilical Legendrian submanifolds of $$(\kappa ,\mu )$$-spaces

We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of \((\kappa ,\mu )\)-spaces with Boeckx invariant \(I \le -1\). In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.     

Existence of optimal transport maps in very strict $$CD(K,\infty )$$ -spaces

We introduce a more restrictive version of the strict \(CD(K,\infty )\) -condition, the so-called very strict \(CD(K,\infty )\) -condition, and show the existence of optimal maps in very strict \(CD(K,\infty )\) -spaces despite the possible lack of uniqueness of optimal plans.     

Dark $$f(\mathcal{R},\varphi,\chi)$$ universe with Noether symmetry

Theoretical and Mathematical Physics - Using the Noether symmetry approach, we investigate $$f( \mathcal{R} , \varphi ,\chi)$$ theories of gravity, where $$ \mathcal{R} $$ is the scalar curvature,...     

关于超代数$(\alpha,\beta,\gamma)$-超导子的一个注记

This paper is concerned with two results for （α, β, γ）-superderivations of general superalgebras over the field of complex numbers.     

强正则 $(\alpha,\beta)$-族和平移强正则 $(\alpha,\beta)$-几何

本文给出了强正则$(\alpha,\beta)-$族的概念,它是[4]和[5]中$SPG-$族概念的推广.进一步,给出了一种用强正则 $(\alpha,\beta)-$族构造强正则$(\alpha,\beta)-$几何的方法.另外,本文还证明了由强正则$(\alpha,\beta)-$线汇构造的强正则$(\alpha,\beta)-$几何是平移强正则$(\alpha,\beta)-$几何;当$t-r>\beta$时,反之亦成立.     

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces $\varLambda^{p}_{u}(w)$ , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L ^p (u) and Muckenhoupt weights A _p , and the theory on classical Lorentz spaces Λ ^p (w) and Ariño-Muckenhoupt weights B _p .     

Approximation of Solutions of Fractional-Order Delayed Cellular Neural Network on $$\varvec{[0,\infty )}$$

In this paper, we study a class of fractional-order cellular neural network containing delay. We prove the existence and uniqueness of the equilibrium solution followed by boundedness. Based on the theory of fractional calculus, we approximate the solution of the corresponding neural network model over the interval \([0,\infty )\) using discretization method with piecewise constant arguments and variation of constants formula for fractional differential equations. Furthermore, we conclude that the solution of the fractional-delayed system can be approximated for large t by the solution of the equation with piecewise constant arguments, if the corresponding linear system is exponentially stable. At the end, we give two numerical examples to validate our theoretical findings.     

A modular-type formula for $$(x;q)_\infty $$

Let \(q=\text {e}^{2\pi i\tau }, \mathfrak {I}\tau >0\), \(x=\text {e}^{2\pi i{z}}\), \({z}\in \mathbb {C}\), and \((x;q)_\infty =\prod _{n\ge 0}(1-xq^n)\). Let \((q,x)\mapsto ({q_1},{x_1})\) be the classical modular substitution given by the relations \({q_1}=\text {e}^{-2\pi i/\tau }\) and \({x_1}=\text {e}^{2\pi i{z}/{\tau }}\). The main goal of this paper is to give a modular-type representation for the infinite product \((x;q)_\infty \), this means, to compare the function defined by \((x;q)_\infty \) with that given by \(({x_1};{q_1})_\infty \). Inspired by the work (Stieltjes in Collected Papers, Springer, New York, 1993) of Stieltjes on semi-convergent series, we are led to a “closed” analytic formula for the ratio \((x;q)_\infty /({x_1};{q_1})_\infty \) by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at \(\log q=0\). Thus, the function \((x;q)_\infty \) is linked with its modular transform \(({x_1};{q_1})_\infty \) in such an explicit manner that one can directly find the modular formulae known for Dedekind’s Eta function, Jacobi Theta function, and also for certain Lambert series. Moreover, one can remark that our results allow Ramanujan’s formula (Berndt in Ramanujan’s notebooks, Springer, New York, 1994, Entry 6’, p. 268) (see also Ramanujan in Notebook 2, Tata Institute of Fundamental Research, Bombay, 1957, p. 284) to be completed as a convergent expression for the infinite product \((x;q)_\infty \).     

关于空间$S(\Omega,\Sigma,\mu)$与$L^\beta(\Omega,\Sigma,\mu)$上的次加上半连续$\alpha$-正齐性泛函

本文得到了$S(\Omega,\Sigma,\mu)$和$L^\beta(\Omega,\Sigma,\mu)$分别不存在非零的上半连续、次加、$\alpha$-正齐性泛函(分别有本文得到了$S(\Omega,\Sigma,\mu)$和$L^\beta(\Omega,\Sigma,\mu)$分别不存在非零的上半连续、次加、$\alpha$-正齐性泛函(分别有本文得到了$S(\Omega,\Sigma,\mu)$和$L^\beta(\Omega,\Sigma,\mu)$分别不存在非零的上半连续、次加、$\alpha$-正齐性泛函(分别有本文得到了S（Ω，∑，μ）与L^β（Ω，∑，μ）分别不存在非零的上半连续、次加、α-正齐性泛函（分别有0≤α≤1和β〈α≤1）的充要条件．     

Some Ricci-flat ($$\alpha ,\beta $$)-metrics

In this paper, we study a special class of Finsler metrics, \((\alpha ,\beta )\)-metrics, defined by \(F=\alpha \phi (\beta /\alpha )\), where \(\alpha \) is a Riemannian metric and \(\beta \) is a 1-form. We find an equation that characterizes Ricci-flat \((\alpha ,\beta )\)-metrics under the condition that the length of \(\beta \) with respect to \(\alpha \) is constant.     

Sharp Weak-Type Inequalities for Analytic Functions on the unit Disc

If Q is a Steiner symmetric domain, symmetric about the y-axis,then it can be proved that for an analytic function F:D C (D= {zC: |z| < 1}), such that Im F(0) = 0, and for t> 0,the inequality holds, where denotes the Lebesgue measure of a set A T and C_Q is some constant dependent onlyon Q. The best constantC_Q is found. Some weak-type inequalitiesfollow, which generalize results of Baernstein.     

Weak-Type Inequality for Conjugate First Order Riesz-Laguerre Transforms

In this paper we introduce a conjugate class of Riesz transforms in the context of Laguerre polynomials. We prove their weak-type (1,1) and L ^p , 1<p<∞, boundedness with respect to the Laguerre measure. A similar result is known in the Hermite context, see Aimar et al. (Trans. Am. Math. Soc. 359(5), 2137–2154, 2007).     

亚BCI-代数的$(\alpha,\beta)$-软理想

首先将软集的参数集赋予亚BCI-代数, 给出了亚BCI-代数的$(\alpha,\beta)$-软理想的概念.当$U=[0,1], \alpha=U, \beta=\phi$时,相应地就得到了亚BCI-代数的犹豫模糊理想的概念.研究了亚BCI-代数的$(\alpha,\beta)$-软理想的一些重要性质.最后讨论了亚BCI-代数的$(\alpha,\beta)$-软理想的同态像和原像的性质.     

具有可反测地线的$(\alpha,\beta)$-度量

本文我们得到了$(\alpha,\beta)$-度量的测地系数$G^{i}(x,y)$和其逆$G^{i}(x,-y)$有相同Douglas曲率的充分必要条件.这个充分必要条件恰好是$(\alpha,\beta)$-度量具有可反测地线的充分必要条件.     

A Sharp Inequality for a Trigonometric Sum

We prove that the inequality $$-\frac{1}{2}\leq {\sum\limits_{k=1}^{n}} \left( \frac{{\rm cos}(2kx)}{2k - 1}+\frac{{\rm sin}((2k - 1)x)}{2k} \right)$$ holds for all natural numbers n and real numbers x with ${x \in [0, \pi]}$ . The sign of equality is valid if and only if n =  1 and x =  π /2.