Homology Theory for CW-Complexes

In this thesis we will have a study on homology theory of CW- complexes with an emphasis on finite-dimensional CW-complexes. We will first give a brief introduction on basic definitions and basic preliminaries of topological space and definition of CW-complexes and brief discussion on some important keywords in CW-complexes. Then certain definitions on singular homology theory of CW-complexes will be discussed. Then, we will give a brief discussion on axioms of homology theory for topological spaces and axioms of homology theory for CW-complexes. Finally, we will discuss Whitehead theorem and its proof

Diophantine equations with balancing-like sequences associated to Brocard-Ramanujan-type problem

In this paper, we deal with the Brocard-Ramanujan-type equations (A_{n_1}A_{n_2}cdots A_{n_k}pm 1=A_m) or (G_m) or (G_m^2) where ({A_n}_{ngeq0}) and ({G_m}_{mgeq 0}) are either balancing-like sequences or associated balancing-like sequences

Environmental zonation across the Australian arid region based on long-term vegetation dynamics

Zonation of landscapes is generally based on broad scale biophysical data, field surveys, imagery and expert knowledge. Such zonation represents a static view of the environment and does not reflect dynamics and function. Arid environments are however often highly dynamic, and spatial and temporal patterns may be expressed over long periods of time. These dynamics need to be understood for management. Our aim is to understand the dynamics and functional response of vegetation in the Australian arid zone, and use this to inform and potentially improve the currently employed stratification. Principal component analysis of 25 years of satellite imagery identified underlying factors influencing patterns of arid vegetation growth, and regions of similar long-term response. Dominant factors of variation were identified as the spatial distribution of total vegetation growth, seasonality of growth, magnitude of seasonal variability in growth, and regularity of variation in growth. Additional variation resulted from episodic vegetation growth of limited spatial extent and duration. Classes expressing these functional components were compared with the existing biogeographical regions, revealing agreement in some instances, and in other cases adding information previously not available. The study demonstrates a new approach to Australian landscape zonation that has potential for much wider application.E.F. Lawley, M.M. Lewis, B. Ostendor

A Study on Arithmetic Functions and Diophantine Equations Associated with Balancing and Related Sequences

A Lucas sequence is a binary recurrence sequences that includes as special cases, the Pell, the associatedPell, the balancing, the Lucasbalancing, the balancinglike and the associated balancinglike sequences. Arithmetic functions of some of these sequences result in interesting inequalities. In particular, σk(Bn) ≤ Bσk(n), where σk denotes the sum of the kth power of divisors function and Bn is the nth balancing number. Repdigits are positive integers with one distinct digit, that is, a single digit appears in the unit place, decimal place and so on. The Euler functions of Pell numbers has no repdigit with at least two digits. However, it is not known whether the Euler function of any associated Pell number is a repdigit with more than one digit. Moreover, if the Euler function of an associated Pell number is a repdigit, then its index must be odd, and the repdigit consists of the digits 4 or 8; in case it contains the digit 4, then its index is an odd prime or its square. The third term of every balancinglike sequence is one less than a square, while, by choice, the second term may or may not be one less than a square and no other term of any balancinglike sequence has this property. Furthermore, except the first and second terms, no other term of any balancinglike sequence is one more than a square. There are just 2 perfect powers in sums and differences of two balancing numbers and in any balancinglike sequence, the number of perfect squares in the sums and differences of two balancinglike numbers is always finite. BrocardRamanujan identity consists of finding positive integer solutions of the equation n! + 1 = m2 and the only known solutions are (n,m) = (4, 5), (5, 11), (7, 71). This problem can be generalized by replacing the positive integers in n! and m by balancinglike or associated balancinglike numbers. The revised problem has sometimes no solution and sometimes finitely many solutions