გიორგი ჭელიძე

To study intersections of embedded bounded closed sets in Banach space, a numerical parameter was introduced earlier; in a certain sense, this parameter describes the deviation of the shape of a set from that of a sphere. Critical values of this parameter for some classes of Banach spaces are determined, a new numerical parameter serving the same purpose is introduced, and the relation between the two parameters is examined.

It is well known that for the Hilbert space H the minimum value of the functional Fμ(f) = ∫ ‖f – g‖2dμ(g), f\in H, is achived at the mean of μ for any probability measure μ with strong second moment on H. We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner product that generates the norm.

We consider a real function which depends on the distances between a variable point and the points of a finite subset A of a linear normed space X. We show that X is an inner product space if this function attains its local minimum on a barycenter of points of A with well-chosen weights. Our result generalizes classical results about characterization of inner product spaces and answers a question of R. Durier, which was posed in his article [J. Math. Anal. Appl. 207 (1997) 220–239].

We study certain connections between a parameter characterizing the intersection of embedded sets and the Jung constant. We also define the critical value of the parameter for a certain product space and prove that, in reflexive spaces, the intersection of embedded sets is nonempty for the critical value of the parameter.

We construct an example of a sequence of reals Cesaro convergent to zero such that there is a permutation of the sequence having no Cesaro convergent subsequence

We prove the following result for any real normed space X with dimX ≥ 3. Let a be a fixed number from the interval (0,2). The condition ‖x+y‖ is constant for all ‖x-y‖=a, ‖x‖=‖y‖=1 implies that X is an inner product space. This proposition gives an affirmative answer to Nordlander's conjecture that is related to the modulus of convexity of X

In the present paper we prove, that in the real normed space X, having at least three dimensions, the Nordlander’s conjecture about the modulus of convexity of the space X is true, i.e. from the validity of Day’s inequality for a fixed real number from the interval (0,2), follows that X is an inner product space.

We prove analogies of the classical Gagliardo–Nirenberg inequalities when usual Lp norms are replaced by variable norms

The main result of this paper is the formulation and proof of Polya's characterization theorem of Gaussian random variables with values in quaternion algebras. We begin with some preliminary information about quaternions and quaternion random variables. More complete information about quaternions (except the items connected with the jointly quaternion systems) can be found in [N. N. Vakhania, Theory Probab. Appl., 43 (1999), pp. 99–115]. Closely related to the present paper is also the paper [N. N. Vakhania, J. Complexity, 13 (1997), pp. 480–488] in which Polya's characterization theorem for complex random variables is proved.

In the present paper we construct an example of a quaternion random variable such that Polya's type characterization theorem of Gaussian distributions does not hold. The matter is that in the linear form, consisting of the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and the other part on the left-hand side. This gives a negative answer to the question posed in [Vakhania and Chelidze, Teor. Veroyatnost. i Primenen. 54: 337–344, 2009].

We show that in any two-dimensional normed space there exists a collection of vectors such that the greedy algorithm fails to be optimal.

In the present note we show that Polya’s type characterization theorem of Gaussian distributions does not hold. This happens because in the linear form, constituted by the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and another part on the left side. This gives a negative answer to the question posed in [Vakhania, N.N., Chelidze, G.Z., Quaternion-valued Gaussian random variables, Theory of probability and its applications, 54(2009), no. 2, 337-344.]

We give an elementary proof of Skitovich-Darmois theorem for the cases of two random variables in each of the two linear forms, when the random variables take values in complex or quaternion fields. The proof is based on the reduction of Skitovich-Darmois theorem to Polya’s theorem in complex and quaternion cases respectively

To consider the intersection of embedded bounded closed sets in infinite-dimensional Banach spaces the numerical parameter was introduced earlier. In the present paper we collect some results that are known around the subject.

In this paper we concern with the question of weak independence of random elements. The case, where the random elements are Gaussian ones with values in a Banach space is considered. In this case the theory of covariance operator allows us to obtain certain (not as yet final) result.

In this paper, we obtain a result which implies, in particular, that the sum range of a special quaternion series is a closed proper subfield of the division ring of quaternions, isometrically isomorphic to the field of complex numbers

We show that if every bounded set in a Banach space has a Chebyshev center, then the intersection of nested closed bounded sets in this space is nonempty in the case of a critical parameter value. This result generalizes previously obtained sufficient conditions for the nonemptiness of the intersection in the critical case. We also answer a question posed by G. Z. Chelidze and P. L. Papini for Banach spaces satisfying the Opial condition for the weak-∗ topology.

The Method of Auxiliary Sources (MAS) is implemented to solve the problem-of EM field diffraction upon an open metallic object such as a car body. The key idea of the MAS is to present the unknown scattered field by a superposition of fundamental solutions of the appropriate wave equation Auxiliary Sources (AS), whose radiating centers are located on some auxiliary surface, shifted outside the area, where the field is to be found. In the case of semi-open objects constructed by a thin metallic surface with cavities and partitions (like the car or other means of transportation), the imaginary surfaces covering the apertures presented in the investigated object are introduced to represent the separated domains that are applicable for the MAS algorithm. The efficiency of MAS is illustrated and some resonant properties of the described objects are investigated.

A method of auxiliary sources (MAS) based method of determining the resonance frequencies of a high conductivity hollow metallic body is presented. As a working prototype a simplified closed car-shaped body is taken, and its eigenfrequencies and corresponding near fields are investigated. Numerical results are also presented.

Modern vehicles contain many sensitive electronic devices whose operation may be affected by electromagnetic pulses appearing in the car body. A car can be considered as a hollow metallic structure with partitions enclosing dielectric objects that represent human bodies and other dielectric inclusions. This system behaves as a resonator for incident EM waves and pulses. It is important to investigate the electrodynamic resonant behaviour of such a complex system to understand, predict and avoid any potential harmful influence of EM pulses on sensitive devices inside the car. Experimental measurements are expensive, and most of the numerical methods are time consuming and require significant computer resources making those investigations impractical or very slow. The method of auxiliary sources (MAS) has been implemented as a fast powerful method for simulation of diffraction processes to solve the stated problems. The computations show that this method is efficient and can be applied for real-time investigations.

We present a numerical study of EMC/EMI problems related to the resonance enhancement of the electromagnetic (EM) field inside vehicles, its influence on the passengers and on the internal sensitive electronic devices. To this problem we have applied the method of auxiliary sources (MAS) which proves to be convenient in the cases presented here. The resonance curves were obtained for various simplified models of a car body.

The paper presents approach for the electromagnetic analysis of the large semi-open structures like vehicles. It is considering the resonance enhancements of the field and characteristics of their cavities. The method of auxiliary sources (MAS) is utilized. It is an efficient computer tool for solving such problems. The EM wave source, which produced the initial excitation, incident field, may freely chosen: some inner antenna, electronic devices or some outer electromagnetic source. Developed model can consider the interaction entire structure on antenna's characteristics.