Additive Number Theory can be best described as the study of sums of sets of integers. A simple example is given two subsets A and B of a set of integers, what facts can we determine about A + B where A + B := {a + b | a ∈ A and b ∈ B}? We will state a result regarding this example shortly. We note that a very familiar problem in Number Theory, namely Lagrange’s theorem that every nonnegative i...

Let M be a tracial von Neumann algebra and A be a weakly dense unital C-subalgebra of M . We say that a set X is a W -generating set for M if the von Neumann algebra generated by X is M and that X is a C-generating set for A if the unital C-algebra generated by X is A. For any finite W generating set X for M we show that δ0(X) ≤ sup{δ0(Y ) : Y is a finite C-generating set for A}. It follows tha...

Distances in the well known fuzzy c-means algorithm of Bezdek (1973) are measured by the squared Euclidean distance. Other distances have been used as well in fuzzy clustering. For example, Jajuga (1991) proposed to use the L1-distance and Bobrowski and Bezdek (1991) also used the L∞-distance. For the more general case of Minkowski distance and the case of using a root of the squared Minkowski ...

In this work, a novel multi-objective design optimization procedure is presented for the Minkowski Reflectarray RA s using a complete 3-D CST Microwave Studio MWSbased Multilayer Perceptron Neural Network MLP NN model including the substrate constant εr with a hybrid Genetic GA and Nelder-Mead NM algorithm. The MLP NN model provides an accurate and fast model and establishes the reflection phas...

Let P,Q ⊆ R be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ Rλ×2, r = [y ], and b ∈ R. Define the constrained Minkowski sum (P ⊕Q)Ar≥b as the multiset {(p + q)|p ∈ P, q ∈ Q,A(p + q) ≥ b}. Given P , Q, Ar ≥ b, an objective function f : R → R, and a positive integer k, the Minkowski Sum Selection Problem is to find the k largest objective value among all objec...

A molecular dynamics simulation of the demixing process of a binary complex plasma is analyzed and the role of distinct interaction potentials is discussed by using morphological Minkowski tensor analysis of the minority phase domain growth in a demixing simulated binary complex plasma. These Minkowski tensor methods are compared with previous results that utilized a power spectrum method based...

In this paper we have studied the nature of kinematical and dynamical laws in κ-Minkowski spacetime from a new perspective: the canonical phase space approach. We have introduced a new form of κ-Minkowski phase space algebra from which we recover the κ-extended finite Lorentz transformations derived in [13]. This is a particular form of a Deformed Special Relativity model that admits a modified...

This is my final project in Math 689 course taught by Prof. Michael Anshelevich. The topic is to report/summarize about the article “A Free Entropy Dimension Lemma” by Kenley Jung in [2] (Pacific Journal of Mathematics 2003).

submitted at the Oberwolfach Conference “Combinatorial Convexity and Algebraic Geometry” 26.10–01.11, 1997 Throughout, we fix the notation M := Z and MR := R . Given convex lattice polytopes P, P ′ ⊂ MR, we have (M ∩ P ) + (M ∩ P ) ⊂ M ∩ (P + P ), where P + P ′ is the Minkowski sum of P and P , while the left hand side means {m+m | m ∈ M ∩ P,m ∈ M ∩ P }. Problem 1 For convex lattice polytopes P...