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This manuscript is concerned with the stability and synchronization for fractional-order delayed gene regulatory networks (FODGRNs) via Razumikhin approach. First of all, the existence of FODGRNs are established by using homeomorphism theory, 2-norm based on the algebraic method and Cauchy Schwartz inequality. The uniqueness of this work among the existing stability results are, the global Mittag-Leffler stability of FODGRNs is explored based on the fractional-order Lyapunov Razumikhin approach. In the meanwhile, two different controllers such as linear feedback and adaptive feedback control, are designed respectively. With the assistance of fractional Razumikhin theorem and our designed controllers, we have established the global Mittag-Leffler synchronization and adaptive synchronization for addressing master-slave systems. Finally, three numerical cases are given to justify the applicability of our stability and synchronization results.

In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.

This article discusses some interesting methods for solving inequalities. And there are some inequalities that will help with this. For example: Cauchy inequality, Cauchy-Schwarz inequality, Titu's Lemma, Holder's inequality, Chebyshev's inequality. Examples of inequalities are also consideredfor better understanding.

We present the sufficient condition for a classical two-class problem from Fisher discriminant analysis has a solution. Actually, the solution was presented up to our knowledge with a necessary condition only. We use an extended Cauchy–Schwarz inequality as a tool.

Certain subclass of p-valently analytic functions with positive coeffi- cients is introduced, and some interesting properties belonging to the this class is obtained.

This study presents major developments in relation to problem dimensionality and formulation generality, endowing the Schwarz-inequality-based approach for finite-horizon minimum effort control wit...

We give several norm and numerical radius inequalities for Hilbert space operators. These inequalities improve on some earlier related inequalities. As an application of one of our results, we give...

In this article, we use Kantorovich and Kantorovich type inequalities in order to prove some new Berezin number inequalities. Also, by using a refinement of the classical Schwarz inequality, we prove Berezin number inequalities for powers of f (A), where A is self-adjoint operator on the Hardy space H 2(D) and f is a positive continuous function. Some related questions are also discussed.

We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.

In this paper we consider two-mode superposition of truncated coherent states. Using this we construct quasi-Bell states and investigate the non-classical features of two-mode superposed truncated coherent states such as oscillatory and sub-Poissonian photon statistics and violation of Cauchy-Schwartz inequality.

We establish some new refinements to the Hölder inequality. We then apply them to provide some refinements to the extended Euler’s gamma and beta functions. As another application of our results, we give a new proof of the equivalence between the Hölder inequality and the Cauchy-Schwarz inequality.

This paper deals with the Choi’s inequality for measurable operators affiliated with a given von Neumann algebra. Some Young and Cauchy-Schwarz type inequalities for 𝜏 -measurable operators are also given.

Recently, Torra, Narukawa and Sugeno [Fuzzy Sets and Systems, 292 (2016) 364–379] obtained the Cauchy-Schwarz inequality for Choquet integral. In this paper, we first introduce a refined Holder’s inequality in Choquet calculus. Then by this inequality, we show that Cauchy-Schwarz inequality and Holder’s inequality for Choquet integral are equivalent only when the monotone measure is submodular or the integrands are comonotonic, thus closing the series of papers in this literature. Note that it is obvious that Holder’s inequality implies Cauchy-Schwarz’s inequality. But the converse is an interesting subject. This paper focuses on this subject.

We derive distance-dependent estimators for two-center and three-center electron repulsion integrals over a short-range Coulomb potential, $\textrm{erfc}(\omega r_{12})/r_{12}$. These estimators are much tighter than one based on the Schwarz inequality and can be viewed as a complement to the distance-dependent estimators for four-center short-range Coulomb integrals and for two-center and three-center full Coulomb integrals previously reported. Because the short-range Coulomb potential is commonly used in solid-state calculations, including those with the HSE functional and with our recently introduced range-separated periodic Gaussian density fitting, we test our estimators on a diverse set of periodic systems using a wide range of the range-separation parameter $\omega$. These tests demonstrate the robust tightness of our estimators, which are then used with integral screening to calculate periodic three-center short-range Coulomb integrals with linear scaling in system size., Comment: 13 pages, 8 figures (main text)

In this work, an improvement of Holder–McCarty inequality is established. Based on that, several refinements of the generalized mixed Schwarz inequality are obtained. Consequently, some new numerical radius inequalities are proved. New inequalities for numerical radius of $$n\times n$$ matrix of Hilbert space operators are proved as well. Some refinements of some earlier results were proved in literature are also given. Some of the presented results are refined and it shown to be better than earlier results were proved in literature.

A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of variables, there must be convex forms of degree as low as 4 that are not sums of squares. Remarkably, no examples are known to date. In this paper, we show that all convex forms in 4 variables and of degree 4 are sums of squares. We also show that if a conjecture of Blekherman related to the so-called Cayley-Bacharach relations is true, then the same statement holds for convex forms in 3 variables and of degree 6. These are the two minimal cases where one would have any hope of seeing convex forms that are not sums of squares (due to known obstructions). A main ingredient of the proof is the derivation of certain "generalized Cauchy-Schwarz inequalities" which could be of independent interest.

In this work, an extension of the generalized mixed Schwarz inequality is proved. A companion of the generalized mixed Schwarz inequality is established by merging both Cartesian and Polar decompositions of operators. Based on that some numerical radius inequalities are proved.

A new bound on quantum version of Wielandt inequality for positive (not necessarily completely positive) maps has been established. Also bounds for entanglement breaking and PPT channels are put forward which are better bound than the previous bounds known. We prove that a primitive positive map $\mathcal{E}$ acting on $\mathcal{M}_d$ that satisfies the Schwarz inequality becomes strictly positive after at most $2(d-1)^2$ iterations. This is to say, that after $2(d-1)^2$ iterations, such a map sends every positive semidefinite matrix to a positive definite one. This finding does not depend on the number of Kraus operators as the map may not admit any Kraus decomposition. The motivation of this work is to provide an answer to a question raised in the article \cite{Wielandt} by Sanz-Garc\'ia-Wolf and Cirac., Comment: The proof of Theorem 6.3 is modified. To appear in IEEE Transactions on Information Theory

In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of H?lder's inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. Our results are (p, q)-generalizations of some known inequalities, such as Opial-type integral inequalities and (p, q)-Wirtinger inequality.

Let $$\int _\Omega {\vert {\!\!\;\vert {\,\!A_t h\!\,}\vert \!\!\;}\vert ^2 +\vert {\!\!\;\vert {\,\!B_t^* h\!\,}\vert \!\!\;}\vert ^2}\,d\mu (t)

The optimization problem of finding minimum or maximum of the quadratic function Q(x) = xT Cx subject to the constraint xT x = 1 is considered in this paper. Using specific characteristics of this ...

Aim: the probability of correctness of the collegial decision, which is made by a majority vote of some collective (board), consisting of an odd number of members is investigated, if the probability of correctness of the individual decision of each member of Board is known. Мaterials and methods: Bernoulli scheme, asymptotic representation, estimation via geometric series, power series expansion, the formula of Wallis, a power scale of averages, average of Kolmogorov. Result: it is established, that if for each member of the board the probability of correctness of the individual decision is more than ½, then with an unlimited increase in the number of members of the Board the probability of correctness of the collegial decision tends to 1. The asymptotic representation and a number of bilateral estimates characterizing the speed of this aspiration are obtained. For heterogeneous Board (that is a Board, whose members make the right individual decision with different probability) introduced the concept of collegial average as an average characteristics, which can replaced the individual probability of each member of the board with the preservation of the probability of a collegial decision. The existence and uniqueness of the collegial average are proved. We derive a collegial inequality showing that the collegial average of some a set of numbers is not less than the geometric average of the same numbers with the equality takes place in the case and only if all the numbers are equal to each other. The collegial inequality serves as an analogue and complement to known set of inequalities establishing a connection between different averages (for example, Cauchy inequality for arithmetic average and geometric average). Conclusion: thus, the results of the study fully meet the aim of determining the probability of correctness of collegial decision taken by a majority of votes under the assumptions. As a result we obtain an asymptotic representation and bilateral estimates characterizing the rate of striving for the correct solution. For a heterogeneous board, the existence uniqueness of the concept of collegial average as an average characteristics is introduced and strictly proved, which can be replaced by an individual probability of each with preserving the probability of correctness of the collegial decision. It is established that the collegial average is not less than the geometric average. Possible applications of the results obtained can be the quantitative evaluation of election procedures and the solution of problems associated with improving the reliability of recognition of weak signals of control sensors of various transport systems, including high-speed transport systems on magnetic suspension.

New integral inequalities of Hilbert's type are introduced through Fenchel-Legendre transform. Some of these inequalities are considered as generalizations as well as improvements of some previously proved results. Applications of the extracted inequalities are presented.

The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.

A new class of positive maps is introduced. It interpolates between positive and completely positive maps. It is shown that this class gives rise to a new characterization of entangled states. Additionally, it provides a refinement of the well-known classes of entangled states characterized in terms of the Schmidt number. The analysis is illustrated with examples of qubit maps.