Mean amplitudes of vibration: Pyramidal XYZ2 molecules

The theory of mean square amplitude of vibration using the symmetry co-ordinates has been applied to the pyramidal XYZ₂ molecular model. The elements of the mean square amplitude matrices and the mean amplitude quantities at 300° K. for thionyl halides have been reported.     

Mean amplitudes of vibration: Bent symmetrical XY2 molecules and groups

Applying the secular equation method of Cyvin, the mean-square amplitudes and the mean amplitudes for ten bent symmetrical XY₂ molecules and groups at 300° K. have been evaluated.     

Mean amplitudes of vibration: Some pyramidal XY3 and tetrahedral XY4 molecules

The theory of the determination of the mean amplitudes of vibration in molecules using the symmetry co-ordinates has been applied to the cases of arsenic halides belonging to the pyramidal XY₃ type and a few silicon and stannic halides of the tetrahedral XY₄ type. The mean square amplitudes and mean amplitudes of vibration have been evaluated at 298° K.     

Mean amplitudes of vibration: SeO3 and ClF3

The method of the determination of the mean amplitudes of vibration in molecules using the symmetry co-ordinates has been applied to the cases of SeO₃ ion belonging to the pyramidal XY₃ type, and ClF₃ of the planar XYZ₂ type. The mean square amplitude and mean amplitudes of vibration have been evaluated at 300° K.     

Mean amplitudes of vibration: Molecules of the linear and planar AXYZ type

Applying Cyvin’s secular equation method the mean-square amplitude quantities are calculated for molecules of the linear and planar AXYZ type belonging to the symmetry C_s. The mean amplitude values for HN₃, DN₃, HNCS, HNCO, HCOF and DCOF molecules are evaluated at 300° K.     

Mean amplitudes of vibration and coriolis coupling constants of some XY3Z type silicon compounds

A theoretical treatment of the generalised mean-square amplitudes of vibration of some silicon compounds of XY₃Z type molecules is presented. The numerical values evaluated for the parallel and perpendicular mean-square amplitudes for all the bonded and non-bonded atom pairs are tabulated. The L matrix elements and Coriolis coefficients have also been evaluated for these molecules.     

Transient amplification of maximum vibration amplitudes

Many low damped structures as turbine blades or drill strings are exposed to high dynamical loads causing high vibration amplitudes. These applications comprise sub-critical eigenfrequencies. Hereby, the lower eigenfrequencies have to be passed before reaching the operating point. Most investigations of vibration amplitudes caused by a resonance passage deal with the computation of single degree of freedom systems. Thereby, it has been shown that the stationary vibration response provides the highest possible amplitude. Further it can be stated that the maximum vibration response of the resonance passage decreases with an increasing sweep velocity [3]. Isolated modes of linear systems can be represented by single degree of freedom systems. Subsequently a mode shape can be described by the multiplication of the amplification function of the mode and the belonging eigenvector. There are only some recent works that deal with resonance passages of vicinal modes, e. g. [1]. In this paper the resonance passage of a three dimensional system with nearby modes is studied. To calculate the transient vibration response an analytical approach is used. It is shown that the maximum amplitude of the stationary vibration response is not the upper limit for the maximum amplitude of the resonance passage. Thus, the maximum amplitude may rise while the sweep velocity increases. Hence, regarding a multi degree of freedom system the maximum amplitude of the resonance passage can exceed the maximum amplitude of the stationary vibration response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Properties of Bent Functions

First, this paper discusses and sums up some properties of a pair of functions p（x）, q（x） that makes （y ＋ 1）p（x） ＋ yq（x） into a bent function. Then it discusses the properties of bent functions. Also, the upper and lower bounds of the number of bent functions on GF（2）2k are discussed.     

Poisson structure of a periodic classical XYZ -chain

Classical variables ( -variables) are constructed for a periodic classical XYZ -chain (of the discrete Landau-Lifshitz equation). The connection of the quantum method of the inverse problem and the theory of finite-zone integration is discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 154–180, 1986.     

Bent functions with stronger nonlinear properties: K-bent functions

We introduce the notion of k-bent function, i.e., a Boolean functionwith evennumber m of variables υ ₁, …, υ _m which can be approximated with all functions of the form 〈u, v〉 _j ⊕ a in the equally poor manner, where u ∈ ? ₂ ^m , a ∈ ?₂, and 1 ? j ? k. Here 〈·, ·〉 _j is an analog of the inner product of vectors; k changes from 1 to m/2. The operations 〈·, ·〉 _k , 1 ? k ? m/2, are defined by using the special series of binary Hadamard-like codes A _m ^k of length 2 ^m . Namely, the vectors of values for the functions 〈u, v〉 _k ⊕ a are codewords of the code A _m ^k . The codes A _m ^k are constructed using subcodes of the ?₄-linear Hadamard-like codes of length 2 ^m+1, which were classified by D. S. Krotov (2001). At that the code A _m ¹ is linear and A _m ¹ , …, A _m ^m/2 are pairwise nonequivalent. On the codewords of a code A _m ^k , we define a group operation ? coordinated with the Hamming metric. That is why we can consider k-bent functions as functions that are maximal nonlinear in k distinct senses of linearity at the same time. Bent functions in usual sense coincide with 1-bent functions. For k > ? ? 1, the class of k-bent functions is a proper subclass of the class of ?-bent functions. In the paper, we give methods for constructing k-bent functions and study their properties. It is shown that there exist k-bent functions with an arbitrary algebraic degree d, where 2 ? d ? max {2, m/2 ? k + 1}. Given an arbitrary k, we define the subset \(\mathfrak{F}_m^k \mathcal{F}_m^k \) of the set \(\mathfrak{F}_m^k \mathcal{F}_m^k \) \(\mathcal{A}_m^k \mathcal{B}_m^k \) of all Boolean functions in m variables with the following property: on this subset k-bent functions and 1-bent functions coincide.     

Constructions of p-ary Quadratic Bent Functions

Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies the constructions of bent functions having the form for odd n and for even n, over the finite field of odd characteristic p, where . Based on the irreducibility of some polynomials on , we focus on characterizing the bent functions for n=p ^v q ^r and n=2p ^v q ^r , where is an odd prime and p a primitive root modulo q ². Moreover, the enumerations of those functions are also considered. Partially supported by the NSF of China under Grants No. 60603012 and No. 60573053.     

Potential constants,mean amplitudes of vibration,and thermodynamic properties of ethylene oxide,ethylene oxide-d4 and ethylene sulphide

A normal co-ordinate analysis of ethylene oxide, ethylene oxide-d₄ and ethylene sulphide is carried out and the potential energy constants are determined using both the most general quadratic potential function and Urey-Bradley force field. Using the potential energy constants obtained by the former method the mean amplitudes of vibration of these molecules are calculated by the secular equation method. The thermodynamic properties of these molecules are reported for the ideal gaseous state at one atmosphere pressure for eleven temperatures from 100° K. to 1000° K.     

A Characterization of Binary Bent Functions

A recent paper by Carlet introduces a general class of binary bent functions on (GF(2))ⁿ(neven) whose elements are expressed by means of characteristic functions (indicators) of (n/2)-dimensional vector-subspaces of (GF(2))ⁿ. An extended version of this class is introduced in the same paper; it is conjectured that this version is equal to the whole class of bent functions. In the present paper, we prove that this conjecture is true.     

Non-Existence of Some Generalized Bent Functions

Several new results on the non-existence of some generalized bent functions are proved by using properties of the decomposition law of primes in cyclotomic fields and properties of the solutions of some special Diophantine equations.     

Practical analyticity of scattering amplitudes

Institute of Nuclear Research, USSR Academy of Sciences; All-Union Scientific-Research Institute of Physicotechnical and Radio Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 3, pp. 416–422, June, 1989.     

Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}

Bent and almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZ_p²{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.