On this thesis we present the fuzzy sets, fuzzy numbers, the fractional derivative and also we discuss the solution of the first order of fuzzy hybrid equation.
A square, upper-triangular matrix U is a Cholesky root of a matrix M provided U*U=M, where * represents the conjugate transpose. Over finite fields, as well as over the reals, it suffices for U^TU=M. In this paper, we investigate the number of such f
actorizations over the finite field with two elements, F2, and prove the existence of a rank-preserving bijection between the number of Cholesky roots of the zero matrix and the upper-triangular square roots the zero matrix.
Two articles published by Information Science discuss the derivatives of interval functions, in the sense of Svetoslav Markov. The authors of these articles tried to characterize for which functions and points such derivatives exist. Unfortunately, t
heir characterization is inaccurate. This article describes this inaccuracy and explains how it can be corrected.
The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams which are K
auffmans reduction of the braid group algebra to the Temperley-Lieb algebra. We introduce an algorithm for computing all planer diagrams in a given dimension. The algorithm can also be used to multiply planer diagrams and find their matrix representation.
For a given graph $G(V,E)$ and one of its dominating set $S$, the subgraph $Gleft[Sright]$ induced by $S$ is a called a dominating tree if $Gleft[Sright]$ is a tree. Not all graphs has a dominating tree, we will show that a graph without cut vertices
has at least one dominating tree. Analogously, if $Gleft[Sright]$ is a forest, then it is called a dominating forest. As special structures of graphs, dominating tree and dominating forest have many interesting application, and we will focus on its application on the problem of planar graph coloring.
New analytical expressions are found for the Levi-Civita symbol using the Kronecker delta symbol. The expressions are derived up to 3 dimensions, extended to higher dimensions, and confirmed in Matlab for 5 dimensions. The expressions can be re-cast
in terms of elementary and/or special functions, which lead to the conclusion that the Levi-Civita Symbol can be treated as a generalized, discrete function
We present a common ground for infinite sums, unordered sums, Riemann integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.
It is well known that the Collatz Conjecture can be reinterpreted as the Collatz Graph with root vertex 1, asking whether all positive integers are within the tree generated. It is further known that any cycle in the Collatz Graph can be represented
as a tuple, given that inputting them into a function outputs an odd positive integer; yet, it is an open question as to whether there exist any tuples not of the form $(2,2,...,2)$, thus disproving the Collatz Conjecture. In this paper, we explore a variant of the Collatz Graph, which allows the 3x+1 operation to be applied to both even and odd integers. We prove an analogous function for this variant, called the Loosened Collatz Function (LCF), and observe various properties of the LCF in relation to tuples and outputs. We then analyse data on the numbers that are in cycles and the length of tuples that represent circuits. We prove a certain underlying unique factorisation monoid structure for tuples to the LCF and provide a geometric interpretation of satisfying tuples in higher dimensions. Research into this variant of the Collatz Graph may provide reason as to why there exist no cycles in the Collatz Graph.
If we cannot obtain all terms of a series, or if we cannot sum up a series, we have to turn to the partial sum approximation which approximate a function by the first several terms of the series. However, the partial sum approximation often does not
work well for periodic functions. In the partial sum approximation of a periodic function, there exists an incorrect oscillation which cannot be eliminated by keeping more terms, especially at the domain endpoints. A famous example is the Gibbs phenomenon in the Fourier expansion. In the paper, we suggest an approach for eliminating such oscillations in the partial sum approximation of periodic functions.
Let $alpha=0.a_1a_2a_3ldots$ be an irrational number in base $b>1$, where $0leq a_i<b$. The number $alpha in (0,1)$ is a $textit{normal number}$ if every block $(a_{n+1}a_{n+2}ldots a_{n+k})$ of $k$ digits occurs with probability $1/b^k$. A condition
al proof of the normality of the real number $pi$ in base $10$ is presented in this note.
