数学代写:Algebra MAT062 代数代写,90高分通过

代数是数学的一个分支,它有助于以数学表达式的形式表示问题或情况。它涉及 x、y、z 等变量和加法、减法、乘法和除法等数学运算,以形成有意义的数学表达式。所有数学分支,如三角学、微积分、坐标几何,都涉及到代数的使用。代数表达式的一个简单示例是 2x + 4 = 8。

代数处理符号,这些符号在运算符的帮助下相互关联。这不仅仅是一个数学概念,而是我们所有人在日常生活中使用的技能,甚至没有意识到它。将代数理解为一个概念比解方程和找到正确答案更重要,因为它对您将来要学习或过去已经学过的所有其他数学主题都很有用。

代数的分支主要有以下几类:

预代数

将未知值表示为变量的基本方法有助于创建数学表达式。它有助于将现实生活中的问题转化为数学中的代数表达式。形成给定问题陈述的数学表达式是预代数的一部分。

初等代数

初等代数涉及解决代数表达式以获得可行的答案。在初等代数中,像 x、y 这样的简单变量以方程的形式表示。根据变量的次数,方程称为线性方程、二次方程、多项式。

抽象代数

抽象代数处理诸如群、环、向量之类的抽象概念的使用,而不是简单的数学数字系统。环是一个简单的抽象级别,通过将加法和乘法属性写在一起而找到。群论和环论是抽象代数的两个重要概念。抽象代数在计算机科学、物理学、天文学中有许多应用,并使用向量空间来表示数量。

通用代数

所有其他涉及三角学、微积分、涉及代数表达式的坐标几何的数学形式都可以被视为通用代数。在这些主题中,通用代数研究数学表达式,不涉及代数模型的研究。代数的所有其他分支都可以被视为通用代数的子集。任何现实生活中的问题都可以归类为数学的一个分支,并且可以使用抽象代数来解决。

Algebra 高分代写案例

a) You are asked to solve a particular problem concerning a simple graph with n vertices and m edges. Three different algorithms A, B, and C are available for solving this problem. In the worst case the maximum number of steps carried out by these algorithms are as follows: A = 2n2 + 10; B = 4m; C = 105 . Using big-O notation, discuss the relative benefits and drawbacks of these algorithms

b) Consider the graph G below, together with its adjacency list.

Algebra代写

Starting at v1 show the spanning trees produced by the BFS and DFS algorithms. In both cases, clearly indicate the order in which edges are added to the trees.

c) Prove whether or not the graph in Part b) is planar

d) Let Kn be a complete edge-weighted graph on n vertices. Edge weights are denoted by w(u,v) (where w(u,v) = w(v,u)). You are asked to design an algorithm that will partition the vertices of this graph into k sets such that the sum of the weights across all groups is minimised. That is, given a vertex partition {S1, S2,…, Sk} we want to minimise the cost

Algebra代写

Describe an application of the simulated annealing algorithm for this problem. In your answer be sure to describe:

· A method of generating an initial solution,

· Any neighbourhood operators,

· The acceptance criteria,

· The role of “temperature” in the algorithm,

· The differences between this method and the random descent method.

Given a simple graph, give definitions and draw examples of the following: i) a bridge, ii) a cut vertex, iii) a complete graph, iv) a complement graph, v) a simple path.

b) Given a simple graph G = (V, E), let L(v) denote a list of “permissible” colours for each vertex v in V. We are now interested in colouring the vertices of V such that

· Adjacent vertices are always assigned to different colours

· The number of different colours used across the graph is minimal

· Each vertex v is assigned to a colour appearing in its list L(v).

Prove that this problem is NP-complete / NP-hard.

Design a simple greedy algorithm for the problem stated in Part b). In cases where a vertex cannot be coloured, it is acceptable to leave it uncoloured.

d) Consider the problem of sorting a list of n integers. It is easy to check (in O(n) time) whether any arbitrary arrangement is in numerical order. On the other hand, the number of different ways of arranging n integers is O(n!). Does this mean that the sorting problem is NP-complete? In your answer, fully discuss your reasoning.

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