Quadratic Functional Research Articles

Mean-square Stabilization of Invariant Manifolds for SDEs

We consider systems of Ito's stochastic differential equations with smooth compact invariant manifolds. The problem addressed is an exponential mean square (EMS) stabilization of these manifolds. The necessary and sufficient conditions of the stabilizability are derived on the base of the spectral criterion of the EMS-stability of invariant manifolds. We suggest methods for the design of the feedback stabilizing regulator for SDEs. Parametrical criteria of the stochastic stabilizability for limit cycles and tori are given. These criteria reduce the stabilization problem to the minimization of quadratic functionals. An analysis of the minimization problem of the quadratic functional for the case of the cycle of 2D stochastic system is presented in detail. Constructiveness of the elaborated theory is demonstrated for the stabilization of stochastically forced cycles of the Hopf system.

Restricted linear constrained minimization of quadratic functionals

In this work, a linearly constrained minimization of a positive semidefinite quadratic functional is examined. We propose two different approaches to this problem. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive semidefinite operator related to the functional, and considering as constraint a singular operator. The difference between the proposed approaches for the minimization and previous work on this problem is that it is considered for all vectors belonging to the least squares solutions set, or to the vectors perpendicular to the kernel of the related operator or matrix.

Fixed Point And Generalized Hyers-ulam-rassias Stability Of A Quadratic Functional Equation

In this paper, using the fixed point alternative approach, we investigate the Hyers Ulam-Rassias stability of the following functional equation \[f(2x-yi)+f(x-2yi)=4f(x-yi)+f(x)-f(y)\] in Banach spaces.

Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression

We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional $Q$, the null hypothesis states that the regression function $f$ satisfies the constraint $Q[f]=0$, while the alternative corresponds to the functions for which $Q[f]$ is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form and check that our conditions are fulfilled in the particular case of ellipsoids corresponding to anisotropic Sobolev classes. In this case, we present a closed form of the minimax rate and the separation constant. On the other hand, minimax rates for quadratic functionals which are neither positive nor negative makes appear two different regimes: “regular” and “irregular”. In the “regular case, the minimax rate is equal to $n^{-1/4}$ while in the “irregular” case, the rate depends on the smoothness class and is slower than in the “regular” case. We apply this to the problem of testing the equality of Sobolev norms of two functions observed in noisy environments.

Computational Results on Quadratic Functional Model for the Tokens of Nuclear Safety

In this work, Nuclear Reactor safety was modeled inform of quadratic functional. The nuclear tokens are structured and used as elements of the control matrix operator in our quadratic functional. The numerical results obtained through Conjugate Gradient Method (CGM) algorithm identify the optimal level of safety required for Nuclear Reactor construction at any particular situation.

Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

We obtain the general solution of the generalized mixed additive and quadratic functional equationfx+my+fx−my=2fx−2m2fy+m2f2y,mis even;fx+y+fx−y−2m2−1fy+m2−1f2y,mis odd, for a positive integerm. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces whenmis an even positive integer orm=3.

Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory

A continuum model based on the nonlocal theory of elasticity is developed for buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates under uniform in-plane compression. The nanoplate is considered to be rested on two-parameter Winkler-Pasternak elastic foundation. The principle of virtual work is used to derive the governing vibration and stability equations. The weighted residual statements of the equations of motion are performed and the well-known Galerkin method is employed to obtain the nonlocal “Quadratic Functional” for embedded micro/nano-plates. The Ritz functions are taken to form an expression for transverse displacement which satisfies the kinematic boundary conditions. In this way, the entire nanoplate is considered as a single super-continuum element. Employing the Ritz functions eliminates the need for mesh generation and thus large number of degrees of freedom arising in discretization methods such as finite element (FE). The results show obvious dependency of critical buckling loads on the non-locality of the micro/nano elliptical plate, especially, at very small dimensions.

Fixed Point And Hyers-ulam-rassias Stability Of A Quadratic Functional Equation In Menger Probabilistic Normed Spaces

In this paper, using the fixed point alternative approach, we investigate the Hyers Ulam-Rassias stability of the quadratic functional equation \[f(x+y)+f(x-y)=2f(x)+2f(y)\] in Menger probabilistic normed spaces.

HYERS-ULAM-RASSIAS STABILITY OF FUNCTIONAL EQUATIONS IN MENGER PROBABILISTIC NORMED SPACES

We introduce the approximately quadratic functional equation in Menger probabilistic normed spaces. More precisely, we show under some suitable conditions that an approximately quadratic functional equation can be approximated by a quadratic function in above mentioned spaces. Also we consider the stability problem for approximately pexiderized functional equation in Menger probabilistic normed spaces.

О необходимом условии минимума одного квадратичного функционала с интегралом Стилтьеса

О необходимом условии минимума одного квадратичного функционала с интегралом Стилтьеса

Generalized Stability of Euler‐Lagrange Quadratic Functional Equation

The main goal of this paper is the investigation of the general solution and the generalized Hyers‐Ulam stability theorem of the following Euler‐Lagrange type quadratic functional equation f(ax + by) + af(x − by) = (a + 1)b2f(y) + a(a + 1)f(x), in (β, p)‐Banach space, where a, b are fixed rational numbers such that a ≠ −1,0 and b ≠ 0.

Fixed Points and Random Stability of a Generalized Apollonius Type Quadratic Functional Equation

AbstractUsing the fixed-point method, we prove the generalized Hyers-Ulam stability of a generalized Apollonius type quadratic functional equation in random Banach spaces.

A General System of Euler–Lagrange‐Type Quadratic Functional Equations in Menger Probabilistic Non‐Archimedean 2‐Normed Spaces

We prove the generalized Hyers‐Ulam‐Rassias stability of a general system of Euler‐Lagrange‐type quadratic functional equations in non‐Archimedean 2‐normed spaces and Menger probabilistic non‐Archimedean‐normed spaces.

Intuitionistic Fuzzy Stability of a Quadratic Functional Equation

AbstractWe consider the intuitionistic fuzzy stability of the quadratic functional equation "Equation missing" by using the fixed point alternative, where "Equation missing" is a positive integer.

A Fixed Point Approach to the Stability of Pexider Quadratic Functional Equation with Involution

We apply the fixed point method to investigate the Hyers-Ulam stability of the Pexider functional equation , for all , where E is a normed space and is an involution.

A Fixed Point Approach to the Stability of Quadratic Equations in Quasi Normed Spaces

We use the xed alternative theorem to establish Hyers{Ulam{Rassias stabil- ity of the quadratic functional equation where functions map a linear space into a complete quasi p-normed space. Moreover, we will show that the continuity behavior of an approx- imately quadratic mapping, which is controlled by a suitable continuous function, implies the continuity of a unique quadratic function, which is a good approximation to the map- ping. We also give a few applications of our results in some special cases.

Picone type identities and definiteness of quadratic functionals on time scales

In this paper we derive a new sufficient condition for the nonnegativity of time scale quadratic functionals associated to time scale symplectic systems. To establish this result, a new global Picone formula is derived. Another proof of a special case of the result is shown to be obtained via a Sturmian comparison technique. Furthermore, we derive several new Picone type identities which, in particular, do not impose a certain delta-differentiability assumption, and we survey known ones from the literature. The results in this paper complete our earlier work on the definiteness of a time scale quadratic functional in terms of its corresponding time scale symplectic system.

QUADRATIC FUNCTIONAL EQUATIONS ASSOCIATED WITH BOREL FUNCTIONS AND MODULE ACTIONS

For a Borel function ˆ :R £R !R satisfying the functional equation ˆ(s + t;u + v) + ˆ(s i t;u i v) = 2ˆ(s;u) + 2ˆ(t;v), we show that it satisfles the functional equation ˆ(s;t) = s(s i t)ˆ(1;0) + stˆ(1;1) + t(t i s)ˆ(0;1): Using this, we prove the stability of the functional equation f(ax + ay;bz + bw) + f(ax i ay;bz i bw) = 2abf(x;z) + 2abf(y;w) in Banach modules over a unital C?-algebra.

The Stability of a Quadratic Functional Equation with the Fixed Point Alternative

Lee, An and Park introduced the quadratic functional equationf(2x+y)+f(2x−y)=8f(x)+2f(y)and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation in Banach spaces.

Fuzzy Stability of the Pexiderized Quadratic Functional Equation: A Fixed Point Approach

The fixed point alternative methods are implemented to give generalized Hyers-Ulam-Rassias stability for the Pexiderized quadratic functional equation in the fuzzy version. This method introduces a metrical context and shows that the stability is related to some fixed point of a suitable operator.