Elementary Inequality Research Articles

The bounds on the solutions of certain two-dimensional delay dynamic systems on time scales

By means of a new linear Gronwall–Bellman inequality on time scales and an elementary inequality, the bounds on the solutions of a class of new nonlinear two-dimensional dynamic systems on time scales are obtained.

Boundedness and exponential stability for nonautonomous FCNNs with distributed delays and reaction–diffusion terms

In this paper, the boundedness and exponential stability for nonautonomous fuzzy cellular neural networks (FCNNs) with distributed delays and reaction–diffusion terms are investigated. By constructing suitable Lyapunov functional, introducing ingeniously some real parameters and applying the elementary inequality, we establish a series of criteria on the boundedness and globally exponential stability of solutions. In these criteria, we do not require that the response functions are bounded and differentiable. Finally, an example is given to show the effectiveness of the obtained results.

Shifting Inequality and Recovery of Sparse Signals

In this paper, we present a concise and coherent analysis of the constrained ?1 minimization method for stable recovering of high-dimensional sparse signals both in the noiseless case and noisy case. The analysis is surprisingly simple and elementary, while leads to strong results. In particular, it is shown that the sparse recovery problem can be solved via ?1 minimization under weaker conditions than what is known in the literature. A key technical tool is an elementary inequality, called Shifting Inequality, which, for a given nonnegative decreasing sequence, bounds the ?2 norm of a subsequence in terms of the ?1 norm of another subsequence by shifting the elements to the upper end.

Generalization of an elementary inequality in Fourier analysis

The inequality \( \mathop {\sup }\limits_{n \geqslant 1} \left| {\sum\limits_{k = 1}^n {\frac{{\sin kx}} {k}} } \right| \leqslant 3\sqrt \pi , \) plays an important role in Fourier analysis and approximation theory. It has recently been generalized by Telyakovskii and Leindler. This paper further generalizes and improves their results by introducing a new class of sequences called γ-piecewise bounded variation sequence (γ-PBVS).

Zero and coefficient inequalities for stable polynomials

In this paper an elementary inequality and Cardan-Vi?te formulae are used to obtain some inequalities involving the zeros and coefficients of stable polynomials with complex coefficients.

A One-Sided Version of a Theorem of Kantorovich

Using elementary differential inequality methods, we prove an improvement of a theorem of Kantorovich concerning solutions of nonlinear equations in Banach spaces.

New algebraic conditions for global exponential stability of delayed recurrent neural networks

The global exponential stability is further discussed for a class of delayed recurrent neural networks with Lipschitz-continuous activation functions. By constructing new Lyapunov functional and applying an elementary inequality technique, a set of new conditions with less restriction and less conservativeness are proposed for determining global exponential stability of the delayed neural network model with more general activation functions. The proposed results improve and generalize some previous reports in the literature. Several examples are also given to illustrate the validity and advantages of the new criteria.

An analysis of periodic solutions of bi-directional associative memory networks with time-varying delays

In this Letter, several sufficient conditions are derived for the existence and uniqueness of periodic oscillatory solution for bi-directional associative memory (BAM) networks with time-varying delays by employing a new Lyapunov functional and an elementary inequality, and all other solutions of the BAM networks converge exponentially to the unique periodic solution. These criteria are presented in terms of system parameters and have important leading significance in the design and applications of periodic neural circuits for delayed BAM. As an illustration, two numerical examples are worked out using the results obtained.

Upper Bounds on the Sum of Principal Divisors of an Integer

A prime-power is any integer of the form p, where p is a prime and a is a positive integer. Two prime-powers are independent if they are powers of different primes. The Fundamental Theorem of Arithmetic amounts to the assertion that every positive integer N is the product of a unique set of independent prime-powers, which we call the principal divisors of N. For example, 3 and 4 are the principal divisors of 12, while 2, 5 and 9 are the principal divisors of 90. The case N = 1 fits this description, using the convention that an empty set has product equal to 1 (and sum equal to 0). In a recent invited lecture, Brian Alspach noted [1]: Any odd integer N > 15 that is not a prime-power is greater than twice the sum of its principal divisors. For instance, 21 is more than twice 3 plus 7, and 35 is almost three times 5 plus 7, but 15 falls just short of twice 3 plus 5. Alspach asked for a nice (elegant and satisfying) proof of this observation, which he used in his lecture to prove a result about cyclic decomposition of graphs. Responding to the challenge, we prove Alspach's observation by a very elementary argument. You, the reader, must be the judge of whether our proof qualifies as nice. We also show how the same line of reasoning leads to several stronger yet equally elegant upper bounds on sums of principal divisors. Perhaps surprisingly, our methods will not focus on properties of integers. Rather, we consider properties of finite sequences of positive real numbers, and use a classical elementary inequality between the product and sum of any such sequence. But first, let us put Alspach's observation in its number theoretic context.

Global exponential stability and existence of periodic solutions of CNNs with delays

In this Letter, we establish general sufficient conditions for global exponential stability and existence of periodic solutions of a class of cellular neural networks (CNNs) with delays. The key to proving the sufficient conditions is the construction of a new Lyapunov functional. An elementary inequality, which may be of independent interest, has been employed in the proof. Checking the sufficient conditions is often reduced to checking some algebraic relations among certain set of parameter. Our sufficient conditions recover the known results in literature as special cases. Finally, we give two examples to illustrate the usage of our main results.

An elementary approach to naturality, predictability, and the fundamental theorem of local martingales

We give non-technical proofs of the predictability of natural processes and the Fundamental Theorem of Local Martingales. Some arguments are shortened through a novel use of an elementary inequality known as “Chebyshev's Other Inequality”.

Carleman's inequality: history and new generalizations

We do not believe there has been hitherto a systematic survey of Carleman's inequality and its history. Moreover, after Redheffer's 1967 paper on recurrent inequalities showed how Carleman's inequality could be obtained from a more general yet quite elementary inequality, there has been the possibility of producing very simple proofs of even stronger weighted generalizations and analogues of Carleman's inequality. Our aim here is to provide both the survey and the simple proofs. In particular, we give a weighted "Lorentz" analogue of Carleman's inequality.

On the Exponential Stability and Periodic Solutions of Delayed Cellular Neural Networks

A set of criteria is presented for the global exponential stability and the existence of periodic solutions of delayed cellular neural networks (DCNNs) by constructing suitable Lyapunov functionals, introducing many parameters and combining with the elementary inequality technique. These criteria have important leading significance in the design and applications of globally stable DCNNs and periodic oscillatory DCNNs. In addition, earlier results are extended and improved; other results are contained. Two examples are given to illustrate the theory.

Exponential stability and periodic solutions of delayed cellular neural networks

A set of criteria are presented for the global exponential stability and the existence of periodic solutions of delayed cellular neural networks ( DCNNs) by constructing suitable Lyapunov functionals . introducing many parameters q ij * , r ij * , q ij , r ij ∈ R and w i > 0 ( i , j = 1 , 2 , ... , n ) and combining them with the elementary inequality 2 ab ≦ a 2 + b 2 technique. These criteria have important significance in the design and applications of globally stable DCNNs and periodic oscillatory DCNNs. In addition, the results in literature are extended and improved. Two examples are given to illustrate the theory.

Smarandache reciprocal function and an elementary inequality

The Smarandache Function is defined as Sen) = k . Where k is the smallest integer such that n divides k!

On an Inequality for the Sum of Infimums of Functions

In this paper we prove some known and new inequalities using an elementary inequality and some basic facts from differential calculus.

Hölder Means, Lehmer Means, andx−1 log cosh x

The monotonicity of the two variable Hölder mean is a consequence of a certain inequality for logcoshx. This in turn is shown to be a limiting case of an inequality not involving any transcendental functions. The proof of this elementary inequality is based on the preliminary result thatH2n+1≤LnwhereHnandLnare the Hölder and Lehmer means of ordern. Some further inequalities for logcoshxare also proved.

Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential

In [FS1] we announced a precise asymptotic formula for the ground-state energy of a non-relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas-Fermi potential VTF = -y(ar) / r, a > 0, where y(r) is defined as the solution of i y''(x) = x-1/2y3/2(x), i y(0) = 1, i y(8) = 0.

A useful elementary correlation inequality

An inequality is given that enables one to estimate the probability of a conjunction by the product of the probabilities when the events have low correlation.

Rates of Poisson convergence for some coverage and urn problems using coupling

Bounds on the rate of convergence measured by the variation distance are obtained for the number of large spacings and for two occupancy problems connected with multinomial and Pólya sampling. The bounds are derived by imbedding techniques together with the elementary coupling inequality.