Ternary Functions Research Articles

The Walsh transform of a class of monomial functions and cyclic codes

Let 𝔽 p be a finite field with p elements, where p is a prime. Let N ? 2 be an integer and f the least positive integer satisfying p f ? ?1 (mod N). Then we let q = p 2f and r = q m . In this paper, we study the Walsh transform of the monomial function f ( x ) = Tr r / p ( a x r ? 1 N ) $f(x)=\text {Tr}_{r/p}(ax^{\frac {r-1} N})$ for a ? 𝔽 r ? $a \in \Bbb F_{r}^{*}$ . We shall present the value distribution of the Walsh transform of f(x) and show that it takes at most min { p , N } + 1 $\min \{p, N\}+1$ distinct values. In particular, we can obtain binary functions with three-valued Walsh transform and ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.

Video Image Restoration Based on Sparse Representation

The sparse representation, as a powerful model, has been successfully used in image processing. Considering that the sparse representation can be used in static image restoration, we propose a video image restoration method on sparse representation. In our algorithm we use a ternary function to express a video image. Adaptively selecting the dictionary by using the Principal Component Analysis (PCA) strategy and getting the sparse coding coefficients by the continuous iteration, finally we get the restored video image. The restored results by our method are compared with those by the other three methods. They are the wiener filter, the blind deconvolution, and the learning model method of natural image patches [7]. In order to better assess the quality of the video image restoration we use both the subjective evaluation and the objective evaluation. Extensive experiments show that our proposed method is effective and superior to the other methods.

Clustering Text Data Streams – A Tree based Approach with Ternary Function and Ternary Feature Vector

Data is the primary concern in data mining. Data Stream Mining is gaining a lot of practical significance with the huge online data generated from Sensors, Internet Relay Chats, Twitter, Facebook, Online Bank or ATM Transactions. The primary constraint in finding the frequent patterns in data streams is to perform only one time scan of the data with limited memory and requires less processing time. The concept of dynamically changing data is becoming a key challenge, what we call as data streams. In our present work, the algorithm is based on finding frequent patterns in the data streams using a tree based approach and to continuously cluster the text data streams being generated using a new ternary similarity measure defined.

Design of CNTFET-based 2-bit ternary ALU for nanoelectronics

This article presents a hardware-efficient design of 2-bit ternary arithmetic logic unit (ALU) using carbon nanotube field-effect transistors (CNTFETs) for nanoelectronics. The proposed structure introduces a ternary adder–subtractor functional module to optimise ALU architecture. The full adder–subtractor (FAS) cell uses nearly 72% less transistors than conventional architecture, which contains separate ternary cells for addition as well as subtraction. The presented ALU also minimises ternary function expressions with utilisation of binary gates for optimisation at the circuit level, thus attaining a simple design. Hspice simulations results demonstrate that the ALU ternary circuits achieve great improvement in terms of power delay product with respect to their CMOS counterpart at 32 nm.

Novel and general carbon nanotube FET-based circuit designs to implement all of the 39 ternary functions without mathematical operations

In this paper, we propose new universal designs of ternary-valued logic (TVL) with high-speed, low-power and full swing output using carbon nanotube FETs (CNTFETs). All of the TVL functions (39 functions) can be implemented in these designs. Ternary value logic is a promising alternative to binary logic due to the reduced integrated circuit (IC) interconnects and chip area. Therefore, a universal design of TVL is a good direction for the future of FPGA design using CNTFET. In this paper, new universal designs of ternary-valued logic based on CNTFETs are proposed and compared with the existing resistive-load CNTFET universal TVL designs. Extensive simulations have been performed in HSPICE to investigate the distribution of power consumption and the delay of the CNTFET-based universal cells due to variations in the supply voltage, the diameter of the CNT, and the room temperature. Simulation results show that the proposed universal TVL designs result in significantly lower power consumption and delay compared with previous resistive-load CNTFET universal TVL implementations.

Application of four sets of tectonomagmatic discriminant function based diagrams to basic rocks from northwest Mexico

Abstract Discrimination diagrams are being widely used for the past four decades to infer the tectonomagmatic affiliation of old rocks. There are several major- and trace-element composition based bivariate, ternary and multi-element discriminant function diagrams in the literature. Recent evaluation of the performance of these diagrams indicates that four sets of recent discriminant function based diagrams, in which the tectonic boundaries are determined from probability estimates for four tectonic settings of island arc, continental rift, ocean-island and mid-ocean ridge, have shown very high success rates. In this work, we have created an extensive geochemical database for on-land and off-shore basic rocks of northwest Mexico and applied these four sets of recent and highly successful tectonomagmatic discriminant function based diagrams to infer their tectonomagmatic origin. Each of these four sets of diagrams (two for major- and two for trace-elements) contained five diagrams. All the four sets of diagrams confirmed a continental rift setting for on-land basic rocks of 13 Ma age, major-element based diagrams also indicated a rift setting. However, one set of trace-element based discrimination diagrams indicated an arc to rift transition and the other set of trace- element based discrimination diagrams for the only sample from the Comondu arc, which has the required complete dataset, indicated an arc setting. All the four sets of discrimination diagrams indicated MORB setting (success rates 90%-98% and 100% for major-element and trace-element based diagrams, respectively) for off-shore rocks (most of the samples were obtained from DSDP drilled sites in the Gulf of California and some dredged rock samples from the Gulf of California) with very high success rates. All the inferred tectonic settings from these four sets of new discrimination diagrams, in general, are in conformity with those reported in the literature based on other methods. The results show that these discrimination diagrams may successfully discriminate the original tectonic setting of comparatively younger and older on-shore rocks as well as sea-water altered deep-sea rocks and dredged material.

Construction of real digital microstructures of nanocrystalline materials considering coupled grain size and grain orientation distributions

A ternary function to describe the grain orientation distribution was proposed for the first time. Meanwhile, a systematic and novel method was further established with a special cycle adjustment algorithm to construct real digital microstructures of nanocrystalline materials considering the coupled grain size and grain orientation distributions. This method is very convenient to be used for the application of finite element methods and molecular simulations.

A Family of p-ary Binomial Bent Functions

For a prime p with p ≡ 3(mod 4) and an odd number m, the Bentness of the p-ary binomial function $f_{a,b}(x)={\\ m Tr}_{1}^n(ax^{p^m-1})+{\\ m Tr}_{1}^2(bx^{\\frac{p^n-1}{4}})$ is characterized, where n=2m, $a\\in \\bF_{p^n}^*$, and $b\\in \\bF_{p^2}^*$. The necessary and sufficient conditions of ƒa,b(x) being Bent are established respectively by an exponential sum and two sequences related to a and b. For the special case of p=3, we further characterize the Bentness of the ternary function ƒa,b(x) by the Hamming weight of a sequence.

THE COMPLEXITY OF WORD CIRCUITS

A word circuit [1] is a directed acyclic graph in which each edge holds a w-bit word (i.e., some x ∈ {0, 1}w) and each node is a gate computing some binary function g : {0, 1}w × {0, 1}w → {0, 1}w. The following problem was studied in [1]: How many binary gates are needed to compute a ternary function f : ({0, 1}w)3 → {0, 1}w. They proved that (2 + o(1))2w binary gates are enough for any ternary function, and there exists a ternary function which requires word circuits of size (1 - o(1))2w. One of the open problems in [1] is to get these bounds tight within a low order term. In this paper we solved this problem by constructing new word circuits for ternary functions of size (1 + o(1))2w. We investigate the problem in a general setting: How many k-input word gates are needed for computing an n-input word function f : ({0, 1}w)n → {0, 1}w (here n ≥ k). We show that for any fixed n, (1 - o(1))2(n - k)w basic gates are necessary and (1 + o(1))2(n - k)w gates are sufficient (assume w is sufficiently large). Since word circuit is a natural generalization of boolean circuit, we also consider the case when w is a constant and the number of inputs n is sufficiently large. We show that [Formula: see text] basic gates are necessary and sufficient in this case.

Partial Differentiation of Real Ternary Functions

Partial Differentiation of Real Ternary Functions In this article, we shall extend the result of [19] to discuss partial differentiation of real ternary functions (refer to [8] and [16] for partial differentiation).

Second-Order Partial Differentiation of Real Ternary Functions

Second-Order Partial Differentiation of Real Ternary Functions In this article, we shall extend the result of [17] to discuss second-order partial differentiation of real ternary functions (refer to [7] and [14] for partial differentiation).

Terms that are self-dual in Boolean algebras but not in MO2

An absorption law is an identity of the form p = x. The ternary function x+y+z (ring addition) in Boolean algebras satisfies three absorption laws in two variables. If a term satisfies these three identities in a variety, it is called a minority term for that variety. We construct a minority term p for orthomodular lattices such the identity \(p = \tilde{p}\) defines Boolean algebras modulo orthomodular lattices. (The dual of p is denoted by \(\tilde{p}\).) Consequently, having a unique minority term function characterizes Boolean algebras among orthomodular lattices. Our main result generalizes this example to arbitrary arity and arbitrary consistent sets of 2-variable absorption laws.

Universal ternary logic circuit design through carbon nanotube technology

Multiple-valued logic circuits can reduce the number of operations necessary to implement a particular mathematical function and further have an advantage in terms of reduced area. Implementable CNTFET circuits have operational characteristics to approach the advantage of using MVL. Conventional nanotube diameters and using CNTFET characteristics has attracted considerable attention in the ternary logic family designs. In order to describe all ternary functions, we have achieved a new CNTFET circuit design in the universal form. In this design we have neither exclusive design problems nor mathematical equations to achieve the appropriate combination of the basic operations in the MVL fields.

Quantum query complexity for qutrits

We compute lower bounds for the exact quantum query complexity of a ternary function $f$. The lower bound is of order $O\mathbf{(}{\text{log}}_{3}(n)\mathbf{)}$. In case $f$ is symmetric on a sphere then the lower bound is of order $O(\sqrt{n})$. This work is a natural continuation of the work of Beals, Buhrman, Cleve, Mosca, and de Wolf on lower limits for binary functions.

On the complexity of a cyclic shift of a set of real numbers

The complexity of implementing a cyclic shift of a 2n-tuple of real numbers by Boolean circuits over the basis consisting of a ternary choice function and all binary Boolean functions is shown to be 2nn.

Crown multiplications and a higher order Hopf construction

We describe a ternary function (multiplication) σ : S 12 × S 12 × S 12 → S 4 from the product of three 12-point crowns into a 4-point crown in the category of partially ordered sets. The ternary map illustrates a version, in the context of posets, of associative multiplication. A higher order analogue of Hopf's construction applied to σ yields a poset model of a certain homotopy class, which we identify, in π 5 ( S 3 ) .

COLUMN POLARITY MATRIX ALGORITHM FOR TERNARY FIXED POLARITY REED–MULLER EXPANSIONS

A new algorithm that efficiently calculates fixed polarity Reed–Muller (FPRM) expansions for ternary functions is presented in this paper. The algorithm starts from the truth vector representation of a ternary function and recursively generates all elements of the input function's polarity matrix. Computational costs for this algorithm are derived and they are smaller than in other existing methods for generating the complete polarity matrix when n ≤ 3.

Properties and experimental results of fastest linearly independent ternary arithmetic transforms

Two categories of fastest linearly independent ternary arithmetic transforms, which possesses forward and inverse butterfly diagrams with the lowest computational complexity have been identified and their various properties have been presented in this paper. This family is recursively defined and has consistent formulas relating forward and inverse transform matrices. Computational costs of the calculation for new transforms are also discussed. Some experimental results for standard ternary benchmark functions and comparison with multi-polarity ternary arithmetic transform are also presented.

Linearly independent ternary arithmetic helix transforms, their properties and relations

New classes of linearly independent ternary arithmetic transforms in standard algebra called ternary arithmetic helix transforms are introduced. Four types of helix transform matrices with detailed recursive equations are shown. Various properties, mutual relationships among transform matrices and spectra as well as results of helix transforms for some special cases of ternary logic functions are discussed. Computational costs for the calculation of new transforms are also presented.

Comparison of Prediction of Phase Equilibria in the Ag–In–Sb System at 200°C with Experimental Data

Binary thermodynamic data, successfully used for phase diagram calculations of binary systems Ag–In, Ag–Sb, and In–Sb, were used for prediction of phase equilibria in ternary Ag–In–Sb system at 200°C. Symmetrical Redlich–Kister–Muggianu model for ternary thermodynamic function calculation was proved to be best valid in this ternary system. Predicted equilibria were compared with experimentally (SEM, EDX) determined composition of phases in chosen alloys after long term annealing and with the results of DTA measurements.