In the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. In particular, it isshown that the Cauchy-Schwarz inequality holds. Moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy Hilbert space has a complementary subspace.Finally, the notions of fuzzy bo...

The classic Poincaré inequality bounds the Lq-norm of a function f in a bounded domain Ω ⊂ Rn in terms of some Lp-norm of its gradient in Ω. We generalize this in two ways: In the first generalization we remove a set Γ from Ω and concentrate our attention on Λ = Ω \ Γ. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it m...

By the method of weight coefficients and techniques of real analysis, a Hardy-Hilbert-type inequality with a general homogeneous kernel and a best possible constant factor is given. The equivalent forms, the operator expressions with the norm, the reverses and some particular examples are also considered.

let $c$ be a nonempty closed convex subset of a real hilbert space $h$. let ${s_n}$ and ${t_n}$ be sequences of nonexpansive self-mappings of $c$, where one of them is a strongly nonexpansive sequence. k. aoyama and y. kimura introduced the iteration process $x_{n+1}=beta_nx_n+(1-beta_n)s_n(alpha_nu+(1-alpha_n)t_nx_n)$ for finding the common fixed point of ${s_n}$ and ${t_n}$, where $uin c$ is ...

Inequality constraints are often needed in optimization problems in order to deal with uncertainty. This paper introduces a simple technique that allows enforcement of inequality constraints in `1 norm problems without any modi cations to existing programs. The solution of `1 norm problems is required, for example, in implementing LAV (Least Absolute Value) state estimators in electric power sy...

Replacing the triangle inequality by ‖x + y‖ ≤ 2(‖x‖ + ‖y‖) in the definition of norm we obtain the notion of parallelogram norm. We establish that every parallelogram norm is a norm in the usual sense. ∗2000 Mathematics Subject Classification. Primary 46B20; secondary 46C05.

In this talk we deal with a more precise estimates for the matrix versions of Young, Heinz, and Hölder inequalities. First we give an improvement of the matrix Heinz inequality for the case of the Hilbert-Schmidt norm. Then, we refine matrix Young-type inequalities for the case of Hilbert-Schmidt norm, which hold under certain assumptions on positive semidefinite matrices appearing therein. Fin...

Let M be a connected, compact, orientable 3-manifold with b1(M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality ‖φ‖A ≤ ‖φ‖T between the Alexander norm on H(M,Z), defined in terms of the Alexander polynomial, and the Thurston norm, defined in terms of the Euler characteristic of embedded surfaces. (A similar result holds when b1(M) = 1.) Using this inequality w...

In this paper, an observer-based sampled-data controller of linear system is proposed for the wave energy converter. Based on the sampled-data observer, the controller is design. In the closed-loop system with controller, it obtains the norm inequality between the continuous-time state variable and the discrete-time one. Using the norm inequality, sufficient condition is derived for the asympto...