In this text, we focus on operations of arity 2, 1, and 0. – For n = 2, f : A → A is a binary operation and is usually written in infix notation, using a binary operation symbol like ·, ∗, or +. Hence, instead of f(a1, a2) we write a1fa2. – For n = 1, f : A→ A is a unary operation. – For n = 0, f : A → A is a nullary operation or a constant. An algebra (or an algebraic structure) is a set A, th...

Consider lattice paths in Z 2 with three step types: the up diagonal (1; 1), the down diagonal (1; ?1), and the double horizontal (2; 0). For n 1, let S n denote the set of such paths running from (0; 0) to (2n; 0) and remaining strictly above the x-axis except initially and terminally. It is well known that the cardinalities, r n = jS n j, are the large Schrr oder numbers. We use lattice paths...

<abstract><p>The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by ^0R_{\alpha}(G) is the sum items (d_{v})^{\alpha} over all vertices v\in V_G where \alpha a pertinently chosen real number. In this paper, we obtain sharp upper and lower bounds on ^0R_{\alpha} trees with given domination number \gamma for \alpha\in(-\infty, 0)\cup(1, \infty) \alpha\in(0, 1) r...

Consider lattice paths in Z2 with three step types: the up diagonal (1, 1), the down diagonal (1,−1), and the double horizontal (2, 0). For n ≥ 1, let Sn denote the set of such paths running from (0, 0) to (2n, 0) and remaining strictly above the x-axis except initially and terminally. It is well known that the cardinalities, rn = |Sn|, are the large Schröder numbers. We use lattice paths to in...

Infinite-dimensional optimization requires – among other things – many results from functional analysis. In this script basics from functional analytic theory is reviewed. The purpose of this work is to give a summary of important facts needed to work in our research group. 1. Functional Analysis – Results and Definitions If M is a set and M1 ⊂ M , the symbol M \M1 represents the complement of ...