Chain models and the spectra of tridiagonal k-Toeplitz matrices

Chain models can be represented by a tridiagonal matrix with periodic entries along its diagonals. Eigenmodes of open chains are represented by spectra of such tridiagonal $k$-Toeplitz matrices, where $k$ represents length of the repeated unit, allowing for a maximum of $k$ distinct types of elements in the chain. We present an analysis that allows for generality in $k$ and values in $\mathbb{C}$ representing elements of the chain, including non-Hermitian systems. Numerical results of spectra of some special $k$-Toeplitz matrices are presented as a motivation. This is followed by analysis of a general tridiagonal $k$-Toeplitz matrix of increasing dimensions, beginning with 3-term recurrence relations between their characteristic polynomials involving a $k^{th}$ order coefficient polynomial, with the variables and coefficients in $\mathbb{C}$. The existence of limiting zeros for these polynomials and their convergence are established, and the conditioned $k^{th}$ order coefficient polynomial is shown to provide a continuous support for the limiting spectra representing modes of the chain. This analysis also includes the at most $2k$ eigenvalues outside this continuous set. It is shown that this continuous support can as well be derived using Widom's conditional theorems (and its recent extensions) for the existence of limiting spectra for block-Toeplitz operators, except in special cases. Numerical examples are used to graphically demonstrate theorems. As an addendum, we derive expressions for $O(k)$ computation of the determinant of tridiagonal $k$-Toeplitz matrices of any dimension

Sulphuric acid leaching of low/medium grade managanese ores using a novel nitrogeneous reductant-NH3NH2HSO4

Low and medium grade land as well sea based manganese ores were used for manganese extraction in H2SO4 - NH3NH2HSO4 (hydrazine sulphate) medium For land based Mn ores, only Mn recovery is important but for sea nodules which contain substantial amounts Co, Ni, and Cu, their recovery is equally important. In the present studies four samples used were: Indian ocean manganese nodules, medium and low grade Mn ores of Gujarat, and low grade Mn ore of Orissa, India. The Mn content of these ores varied from 15 to 39%. The objective of this work is to establish a reductant which can be used for leaching Mn from all types of ores. The optimum conditions established for nodules by varying parameters such as time, temperature, pulp density, H2SO4 and NH3NH2HSO4 concentrations were: pulp density 10%, time 0.5h, temperature 110ºC, NH3NH2HSO4 3.25 g/10g, H2SO4 2.0% (v/v) for 96.9% Mn, 85.25% Cu, 92.58% Ni and 76.5% Co extractions. More than 92% Mn could be leached from different types of ores by varying amount of reductant and acid concentration at 35ºC. Depending on Mn content 1.0 to 1.2 times stochiometric amount of reductant and 1.5 to 1.8 times sulphuric acid were required for >92% Mn extraction

Design of Sequential Circuit Using Quantum-Dot Cellular Automata (QCA)

Quantum dot cellular autometa presents a promissing nanoscale technology for replacement of conventional cmos based circuits.In this paper we introduce qca logic gates such has qca inverter and qca majority gate.This paper design the sequential logic gates.such as D latch,SR latch,JK latch,T flipflop,D flipflop,2 bit counter,4 bit shift register.These designs are captured and simulated using a design calld QCA designer