Relaxed Stability Criteria for Time-Delay Systems:A Novel Quadratic Function Convex Approximation Approach

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.

Distributed Stochastic Optimization with Compression for Non-Strongly Convex Objectives

We are investigating the distributed optimization problem,where a network of nodes works together to minimize a global objective that is a finite sum of their stored local functions.Since nodes exchange optimization parameters through the wireless network,large-scale training models can create communication bottlenecks,resulting in slower training times.To address this issue,CHOCO-SGD was proposed,which allows compressing information with arbitrary precision without reducing the convergence rate for strongly convex objective functions.Nevertheless,most convex functions are not strongly convex(such as logistic regression or Lasso),which raises the question of whether this algorithm can be applied to non-strongly convex functions.In this paper,we provide the first theoretical analysis of the convergence rate of CHOCO-SGD on non-strongly convex objectives.We derive a sufficient condition,which limits the fidelity of compression,to guarantee convergence.Moreover,our analysis demonstrates that within the fidelity threshold,this algorithm can significantly reduce transmission burden while maintaining the same convergence rate order as its no-compression equivalent.Numerical experiments further validate the theoretical findings by demonstrating that CHOCO-SGD improves communication efficiency and keeps the same convergence rate order simultaneously.And experiments also show that the algorithm fails to converge with low compression fidelity and in time-varying topologies.Overall,our study offers valuable insights into the potential applicability of CHOCO-SGD for non-strongly convex objectives.Additionally,we provide practical guidelines for researchers seeking to utilize this algorithm in real-world scenarios.

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

ON α-CONVEX FUNCTIONS WITH MISSING COEFFICIENTS

Let Jn(α,A,B),α≥0,-1≤B<A≤1,n≥1,denote the class of functions f(z)=z+∑k=n+1^∞αkZ^k which are analytic in E={z:|z|<1} and satisfy the conditions f(z)f′(z)/z≠0 and (1-α)zf′(z)/f(z)+α(1+zf″(z)/f′(z))-<1+Az/1+Bz for z∈E.In this paper we obtain incluion relations,distortion properties and estimates of |αn+2-λα^2n+1| for the class Jn(α,A,B),where λ is complex.

THE SCHUR HARMONIC CONVEXITY FOR A CLASS OF SYMMETRIC FUNCTIONS

In this article, we prove that the symmetric function Fn（x,r）=∑i1＋i2＋……in=r（x1（i1x2^i2……xn^in）1/r is Schur harmonic convex for x ∈ R＋n and r ∈N -=（1, 2, 3,...} As its applications, some analytic inequalities are established.

SOME INEQUALITIES OF HERMITE-HADAMARD TYPE FOR s-CONVEX FUNCTIONS

In this paper several inequalities of the left-hand side of Hermite-Hadamard’s inequality are obtained for s-convex functions.

Differentiability of Convex Functions in Locally Convex Spaces（Ⅰ）──Continuous Convex Functions and Gateaux Differentiability

The aim of this paper is to investigate the differentiability(Gateaux differentiabllity and subdifferentiability) of continuous convex functions on locally convex spaces and to study the behaviour of some important results for this research area in locally convex spaces.

THE HADAMARD INEQUALITY FOR CONVEX FUNCTION VIA FRACTIONAL INTEGRALS

In this paper, we establish several inequalities for some differantiable mappings that are connected with the Riemann-Liouville fractional integrals. The analysis used in the proofs is fairly elementary.

DIFFERENTIABILITY OF CONVEX FUNCTIONS AND ASPLUND SPACES

Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.

CERTAIN SUFFICIENT CONDITIONS FOR STARLIKENESS AND CONVEXITY OF MEROMORPHICALLY MULTIVALENT FUNCTIONS

In this paper we derive certain sufficient conditions for starlikeness and convexity of order α of meromorphically multivalent functions in the punctured unit disk.

Third Hankel Determinant for the Inverse of Starlike and Convex Functions

Denote S to be the class of functions which are analytic,normalized and univalent in the open unit disk U={z:|z|<1}.The important subclasses of S are the class of starlike and convex functions,which we denote by S and C.In this paper,we obtain the third Hankel determinant for the inverse of functions f(z)=z+∞Σn=2 anz^n belonging to S^*and C.

APPROXIMATE CONVEXITY AND CONCAVITY OFGENERALIZED GROTZSCH RING FUNCTION

In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa （r） by studying the monotonieity,convexity or concavity of certain composites of μa（r） are obtained.

THE FEKETE-SZEG PROBLEM FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO THE KOEBE FUNCTION

An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.

Inequalities for Tφ-convex functions

In this paper, the Tφ-convex functions were introduced as a generalizations of convex functions. Then the characteristics of the Tφ-convex functions were discussed. Furthermore, some new inequalities for the Tφ-convex functions were derived.

Convexity for New Integral Operator on k-uniformly p-valent a-convex Functions of Complex Order

In this paper, we obtain the convexity of new general integral operator on some classes of k-uniformly p-valent a-convex functions of complex order. These results extend some known theorems.

Several Hermite-Hadamard Type Inequalities for Harmonically Convex Functions in the Second Sense with Applications

In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Finally, some applications to special mean are shown.

Symmetry Function Induced by a Convex Body

This paper shows some properties of symmetry function induced by a convex body in a normal linear space. Some relationships between symmetry function induced by a convex body and Minkowski functional of the convex body are presented.

Protein-Protein Interaction Extraction Based on Convex Combination Kernel Function

Owing to the effect of classified models was different in Protein-Protein Interaction(PPI) extraction, which was made by different single kernel functions, and only using single kernel function hardly trained the optimal classified model to extract PPI, this paper presents a strategy to find the optimal kernel function from a kernel function set. The strategy is that in the kernel function set which consists of different single kernel functions, endlessly finding the last two kernel functions on the performance in PPI extraction, using their optimal kernel function to replace them, until there is only one kernel function and it’s the final optimal kernel function. Finally, extracting PPI using the classified model made by this kernel function. This paper conducted the PPI extraction experiment on AIMed corpus, the experimental result shows that the optimal convex combination kernel function this paper presents can effectively improve the extraction performance than single kernel function, and it gets the best precision which reaches 65.0 among the similar PPI extraction systems.

Generalizations of Hermite-Hadamard Type Inequalities Involving S-convex Functions

Some new Hermite-Hadamard type's integral equations and inequalities are established. The results in [3] and [6] which refined the upper bound of distance between the middle and left of the typical Hermite-Hadamard's integral inequality are generalized.

Integral Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are <i>s</i>-Convex

In the paper, the authors find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex and apply these inequalities to discover inequalities for special means.