Ternary Set Research Articles

Ternary Sequence Set Having Periodic and Aperiodic Zero-Correlation Zone

A new class of ternary sequence with a zero-correlation zone is introduced. The proposed sequence sets have a zero-correlation zone for both periodic and aperiodic correlation functions. The proposed sequences can be constructed from a pair of Hadamard matrices of size n0 × n0 and a Hadamard matrix of size n1 × n1. The constructed sequence set consists of n0n1 ternary sequences, and the length of each sequence is n0(m+1)(n1 + 1) for a non-negative integer m. The zero-correlation zone of the proposed sequences is |τ|≤ n0m-1, where τ is the phase shift. The sequence member size of the proposed sequence set is equal to n1/n1+1 times that of the theoretical upper bound of the member size of a sequence set with a zero-correlation zone.

Arrangements of four beams for any Bravais lattice.

A single geometric model based on a new concept of a reciprocal primitive pyramid (RPP) in reciprocal space is proposed for investigation of relationships between any three-dimensional (3D) lattice and arrangements of four beams (AFBs) that produce the lattice. A ternary linear equation set, described for the one-to-one correspondence between a RPP and AFB, can readily reveal all AFBs for the same lattice (AFBSLs). Quantitative AFBs for bcc and fcc real lattices are illustrated to show that various AFBSLs can modulate the properties of a photonic bandgap (PBG) both by tuning the lattice constant and by changing the lattice-point shape. This fact may yield the appropriate AFB for a complete 3D PBG with the desired center wavelength. The nonuniqueness of AFBSLs can provide abundant choices for persons who plan interference experiments, especially for holographic fabrication of 3D photonic crystals (PCs).

The Hausdorff Dimension of the Nondifferentiability Set of a Nonsymmetric Cantor Function

Each choice of numbers a and c in the segment (0, (1/2)) produces a Cantor set C ac by recursively removing segments from the interior of the interval [0, 1] so that intervals of relative length a and c remain on the left and right sides of the removed segment, respectively. A Cantor function Φ ac is obtained from C ac in much the same way that the standard Cantor function, Φ, is obtained from the Cantor ternary set. When a = c = (1/3), C ac is the Cantor ternary set, C, and Φ ac is the standard Cantor function, Φ. The derivative of Φ is zero off C, and the upper derivative is infinite on C; the set N = {x E C the lower derivative of Φ is finite} has Hausdorff dimension [ln2/ln3] 2 . In this paper similar results are established for N ac , the nondifferentiability set of Φ ac . The Hausdorff dimension of N ac is the maximum of the real numbers satisfying the following equations: x(ln(1/c)) 2 = ln((α + c)/c) In((a/c) x + 1), and x(ln(1/α)) 2 = ln((α + c)/a) in((c/α) x + 1).

Analog calorimetry and UNIQUAC group contributions approaches to the miscibility of pvc with eva copolymers

The miscibility of blends of poly(vinyl-chloride) (PVC) with poly(ethylene-co-vinyl acetate) (EVA) was investigated through analog calorimetry and a group contribution procedure based on the UNIQUAC model. The group contribution parameters quantifying the pair interactions between the structural features of the above polymers were calculated from experimental excess enthalpies of a series of binary mixtures of chlorocompounds, esters and hydrocarbons. Enthalpy data were also collected for the ternary mixtures (2-chloropropane+ethyl acetate+n-heptane) and (2-chlorobutane + methyl acetate+n-heptane), chosen as possible models for the studied macromolecular mixtures. The miscibility window of the PVC-EVA blends is fairly predicted by the group contribution method. It is also acceptably predicted by the enthalpic behaviour of the first ternary set, but only when the latter is calculated with binary data. A slightly narrower miscibility range is predicted by the binary interaction model. The results of these procedures are compared and the higher reliability of the group contribution procedure is emphasized in terms of its capability to reproduce the exact structure of the macromolecules and the non-univocal choice of the model molecules involved in the analog calorimetry approach.

A Technique of Studying Sums of Central Cantor Sets

AbstractThis paper is concernedwith the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an M-Cantorval.

EXCESS MOLAR ENTHALPIES OF THE TERNARY SYSTEMS (ACETONITRILE+METHYL ISOBUTYRATE +TOLUENE) AND (PROPIONITRILE+METHYL ISOBUTYRATE+ETHYLBENZENE) AT 298.15 K

Excess molar enthalpies were measured at 298.15 K with a flow calorimeter built in our laboratory for the two ternary systems (acetonitrile+methyl isobutyrate+toluene) and (propionitrile+methylisobutyrate+ethylbenzene). Experimental data were nicely predicted by some empirical andsemi empirical procedures, exception made for UNIFAC group contribution method. The latter equation proved unsatisfactory for all mixtures containing methyl isobutyrate. The first ternary set proved a good model system to simulate mixing enthalpies reported in the literature for mixtures of poly(styrene-co-acrylonitrile) with polymethylmethacrylate.

An orthonormal system of functions related to the rademacher system on [0,1)

This paper presents an orthonormal system of functions defined on the interval [0,1). This system related to the construction of the Cantor ternary set and the Rademacher system [7, 8]

Quasiconformally Bi-Homogeneous Compacta in the Complex Plane

This article addresses the question of which non-empty, compact, proper subsets E of the extended complex plane C have the feature that, for some K in 1,∞, the family of K-quasiconformal self-mappings of C which leave E invariant acts transitively on the set E×Ec, where Ec is the complement of E in C. The main result in the paper asserts that the class of sets with this property comprises all one- and two-point subsets of C, all quasicircles in C and all images of the Cantor ternary set under quasiconformal self-mappings of C. It is shown that the third category includes the limit set of any non-cyclic, finitely generated Schottky group. 1991 Mathematics Subject Classification: 30C62.

A class of high-precision multiplier-free FIR filter realizations with periodically time-varying coefficients

Proposes a class of high-precision, multiplier-free realizations for FIR filters. These realizations use upsampling and downsampling in conjunction with a periodically time-varying system to achieve time-invariant, multiplier-free FIR filter operation. Nonbinary encoding schemes are used for obtaining the filter coefficients, which are periodically time-varying (PTV), i.e., they vary in a periodic fashion. Each target filter coefficient is directly mapped into a set of PTV coefficients so that the realizations are easy to obtain. The values of the PTV coefficients are restricted to either the ternary set.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Signed-digit-based multiplier-free realizations for multirate converters

We propose multiplier-free realizations for FIR multirate converters. Radix-r signed-digit number representation and periodically time-varying (PTV) coefficients are used to obtain multiplier-free realizations. The PTV coefficients, together with one hold-and-scale unit at the input and one scale-add-downsample unit at the output, are utilized effectively to distribute the required computations to the extent that coefficients are highly simplified. The coefficients in the realizations are designed to take values from either the ternary set (0, /spl plusmn/1) or the quinary set (0, /spl plusmn/1, /spl plusmn/2). Their values can be easily obtained by mapping from the target filter coefficients using a radix-r signed-digit number representation. We show that fairly high precision can be achieved by these realizations. >

Fractal neutron optics multilayers in Cantor ternary set pattern

In the present paper, it is shown that the fractal notion can be used in neutron optics. More specifically, self-similar neutron multilayer mirrors can be made according to the pattern of the Cantor ternary set. These new multilayers have a fractional Hausdorff dimension. The self-similarity gives them new reflectivity properties. Compared with periodic neutron multilayers and supermirrors, these fractal multilayers, with a large number of generations, allow high reflectivity of the order of unity at very short neutron wavelengths.

Markov's property of the Cantor ternary set

We prove that the Cantor ternary set E satisfies the classical Markov inequality (see [Ma]): for each polynomial p of degree at most n (n = 0, 1, 2,...) (M) $|p'(x)| ≤ Mn^{m} sup_{E}|p|$ for x ∈ E, where M and m are positive constants depending only on E.

An orthonormal system on the construction of the generalized cantor set

This paper presents a new complete orthonormal system of functions defined on the interval[0,1]and whose supports shrink to nothing. This system related to the construction of the Cantor ternary set. We defined the canonicalmap ξand proved the equivalence between this system and the Walsh system. The generalized Cantor set with any dissection ratio is established and the constructed system is defined in the general case.

Prediction of thermodynamic properties of ternary hydrocarbon mixtures containing olefins. I. 1-Alkenes + benzene + n-alkanes

Excess enthalpies and excess Gibbs energies (vapor-liquid equilibria) of ternary systems formed by 1-alkenes + benzene + n-alkanes are predicted from a generalized quasi-lattice group contribution theory. All the parameters were obtained by application of the same theory to the experimental values of the corresponding classes of binary systems after a thorough bibliographical search. The results compare satisfactorily with the only available experimental ternary vapor-liquid equilibrium data set found in the literature for this type of systems.

Viscosities and intradiffusion coefficients in the ternary system NaCl?MgCl2?H2O at 25�C

The intradiffusion coefficients of Na+, Cl− ions and water and the tracerdiffusion coefficients of Ca2+ ion have been measured in the ternary system NaCl−MgCl2−H2O at 25°C. The intradiffusion coefficients of Mg2+ in this system have been estimated from the corresponding Ca2+ diffusion measurements. Viscosities were measured at the same solution concentrations as were used for the diffusion experiments. Intradiffusion and tracerdiffusion coefficients in a range of temperatures from 5 to 45°C are reported for standard sea-water which is a member of the above ternary set.

A Chaotic Function whose Nonwandering Set is the Cantor Ternary Set

A Chaotic Function whose Nonwandering Set is the Cantor Ternary Set

A very sparse set of dimension 1

In July 1982, I was asked by Prof. Jorgen Hoffmann-Jorgensen to construct an uncountable compact set K in the line which was symmetric about 0 and had the property that, for all n , the set of sums of n -tuples from K has measure 0. There are two equivalent conditions: the set of such sums should never contain an interval, or K * ≠ ℝ, where K * is the subgroup of (ℝ, +) generated by K . I did so, and the set I constructed had entropy dimension 0 (and thus also Hausdorff dimension 0). Hoffmann-Jorgensen showed that every set of entropy dimension 0 would exhibit the same behaviour. However, I did not believe that the essence of the example lay in its dimension, and I here modify my construction so that the set K has dimension 1 (and thus also entropy dimension 1), while K * ≠ ℝ, as before. By contrast, the Cantor ternary set has dimension log 3 (2), but the set of differences is the interval [ –1, 1], so that it does generate ℝ. It follows that the property under consideration is arithmetical rather than dimensional.

A chaotic function whose nonwandering set is the Cantor ternary set

We introduce a continuous function from [0, 1] onto itself whose nonwandering set is the Cantor ternary set C C . This function is also chaotic in the sense of Li and Yorke, but with the property that almost all points of [0, 1] are eventually fixed. Those points of C C which are periodic, eventually periodic, asymptotically periodic or recurrent are also explicitly identified.

A Class of Distribution-Free Detectors for Pulse Amplitude Modulated Signals

A class of distribution-free detectors for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> -ary pulse amplitude modulated digital signals in the presence of certain signal dependent interference and additive noise of unknown distribution is formulated, evaluated in terms of probability of error and compared to a matched filter detector in Gaussian noise. The noise distribution is unknown except that it satisfies certain broad assumptions and that its value at the median is known. A decision method based on the value of a test statistic is developed. An analytical expression for the average probability of error and a method of calculating the optimum decision thresholds that minimize this probability of error are presented. The probability of error is independent of the form of the noise distribution but depends on the value of the noise density function at the median. An example of a ternary signal set in white Gaussian noise is worked out for illustration. The optimum decision thresholds are calculated and the resulting average probability of error is plotted as a function of signal-to-noise ratio. Average probability of error of a matched filter detector is also evaluated under similar conditions and a comparison is made with that of the proposed detector.

The midpoint set of a cantor set

A non-endpoint of the Cantor ternary set is any Cantor point which is not an endpoint of one of the remaining closed intervals obtained in the usual construction process of the Cantor ternary set in the unit interval. It is shown that the set of points in the unit interval which are not midway between two distinct Cantor ternary points is precisely the set of Cantor nonendpoints. It is also shown that the generalized Cantor setCλ, for1/3&lt;λ&lt;1, has void intersection with its set of midpoints obtained from distinct members ofCλ.