statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Foundations of Mathematics数学基础澳洲代写代考和辅导服务！
课程介绍：
This course is a critical approach to the foundations of mathematics. In other mathematics classes, the philosophical concepts at the most basic foundations are usually treated naively. The question of what exactly a number is, or what a set or a proof or an algorithm are, is completely ignored. Some evidence that these matters are not insubstantial is that in the early twentieth century, naive attempts to address them by the great logicians of the time led to famous paradoxes and a period known as the Crisis in Foundations of Mathematics.
Show by examples that neither the assertion in lemma 6.5 .2 nor Fermat’s “Little” Theorem remains valid if we drop the assumption that $p$ is a prime. Consider a regular $p$-gon, and for a fixed $k(1 \leq k \leq p-1)$, consider all $k$-subsets of the set of its vertices. Put all these $k$-subsets into a number of boxes: We put two $k$-subsets into the same box if they can be rotated into each other. For example, all $k$-subsets consisting of $k$ consecutive vertices will belong to one and the same box. (a) Prove that if $p$ is a prime, then each box will contain exactly $p$ of these rotated copies. (b) Show by an example that (a) does not remain true if we drop the assumption that $p$ is a prime. 6.6 The Euclidean Algorithm 99 (c) Use (a) to give a new proof of Lemma
问题 2.
Imagine numbers written in base $a$, with at most $p$ digits. Put two numbers in the same box if they arike by a cyclic shift from each other. How many will be in each class? Give a new proof of Fermat’s Theorem this way. Give a third proof of Fermat’s “Little” Theorem based on Exercise 6.3.5. [Hint: Consider the product $a(2 a)(3 a) \cdots((p-1) a)$.]
问题 3.
Show that if $a$ and $b$ are positive integers with $a \mid b$, then $\operatorname{gcd}(a, b)=a$. (a) Prove that $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, b-a)$. (b) Let $r$ be the remainder if we divide $b$ by $a$. Then $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, r)$.
(a) If $a$ is even and $b$ is odd, then $\operatorname{gcd}(a, b)=\operatorname{gcd}(a / 2, b)$. (b) If both $a$ and $b$ are even, then $\operatorname{gcd}(a, b)=2 \operatorname{gcd}(a / 2, b / 2)$.
How can you express the least common multiple of two integers if you know the prime factorization of each?
Suppose that you are given two integers, and you know the prime factorization of one of them. Describe a way of computing the greatest common divisor of these numbers.
Prove that for any two integers $a$ and $b$, $$ \operatorname{gcd}(a, b) \operatorname{lcm}(a, b)=a b . $$
statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Foundations of Mathematics数学基础澳洲代写代考和辅导服务！
课程介绍：
This course is a critical approach to the foundations of mathematics. In other mathematics classes, the philosophical concepts at the most basic foundations are usually treated naively. The question of what exactly a number is, or what a set or a proof or an algorithm are, is completely ignored. Some evidence that these matters are not insubstantial is that in the early twentieth century, naive attempts to address them by the great logicians of the time led to famous paradoxes and a period known as the Crisis in Foundations of Mathematics.
Satisfaction We can already skip ahead to the semantics of first-order logic once we know what formulas are: here, the basic definition is that of a structure. For our simple language, a structure $\mathfrak{M}$ has just three components: a non-empty set $|\mathfrak{M}|$ called the domain, what a picks out in $\mathfrak{M}$, and what $P$ is true of in $\mathfrak{M}$. The object picked out by $a$ is denoted $a^{\mathfrak{M}}$ and the set of things $P$ is true of by $P^{\mathfrak{M}}$. A structure $\mathfrak{M}$ consists of just these three things: $|\mathfrak{M}|, a^{\mathfrak{M}} \in|\mathfrak{M}|$ and $P^{\mathfrak{M}} \subseteq|\mathfrak{M}|$. The general case will be more complicated, since there will be many predicate symbols and constant symbols, the constant symbols can have more than one place, and there will also be function symbols.
This is enough to give a definition of satisfaction for formulas that don’t contain variables. The idea is to give an inductive definition that mirrors the way we have defined formulas. We specify when an atomic formula is satisfied in $\mathfrak{M}$, and then when, e.g., $\neg \varphi$ is satisfied in $\mathfrak{M}$ on the basis of whether or not $\varphi$ is satisfied in $\mathfrak{M}$. E.g., we could define:
$P(a)$ is satisfied in $\mathfrak{M}$ iff $a^{\mathfrak{M}} \in P^{\mathfrak{M}}$.
$\neg \varphi$ is satisfied in $\mathfrak{M}$ iff $\varphi$ is not satisfied in $\mathfrak{M}$.
$(\varphi \wedge \psi)$ is satisfied in $\mathfrak{M}$ iff $\varphi$ is satisfied in $\mathfrak{M}$, and $\psi$ is satisfied in $\mathfrak{M}$ as well.
Let’s say that $|\mathfrak{M}|={0,1,2}, a^{\mathfrak{M}}=1$, and $P^{\mathfrak{M}}={1,2}$. This definition would tell us that $P(a)$ is satisfied in $\mathfrak{M}$ (since $a^{\mathfrak{M}}=1 \in{1,2}=P^{\mathfrak{M}}$ ). It tells
us further that $\neg P(a)$ is not satisfied in $\mathfrak{M}$, and that in turn $\neg \neg P(a)$ is and $(\neg P(a) \wedge P(a))$ is not satisfied, and so on.
The trouble comes when we want to give a definition for the quantifiers: we’d like to say something like, ” $\exists v_0 P\left(v_0\right)$ is satisfied iff $P\left(v_0\right)$ is satisfied.” But the structure $\mathfrak{M}$ doesn’t tell us what to do about variables. What we actually want to say is that $P\left(v_0\right)$ is satisfied for some value of $v_0$. To make this precise we need a way to assign elements of $|\mathfrak{M}|$ not just to a but also to $v_0$. To this end, we introduce variable assignments. A variable assignment is simply a function $s$ that maps variables to elements of $|\mathfrak{M}|$ (in our example, to one of 1,2 , or 3 ). Since we don’t know beforehand which variables might appear in a formula we can’t limit which variables $s$ assigns values to. The simple solution is to require that $s$ assigns values to all variables $v_0, v_1, \ldots$ We’ll just use only the ones we need.
statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Advanced Differential Geometry高级微分几何学澳洲代写代考和辅导服务！
课程介绍：
This is a special topics course which introduces students to the key concepts and techniques of Differential Geometry. Possible topics include:
Surfaces in Euclidean space, general differentiable manifolds, tangent spaces and vector fields, differential forms, Riemannian manifolds, Gauss-Bonnet theorem.
Note: This is an Honours Pathway course. It emphasises mathematical rigour and proof and develops the fundamental ideas of differential geometry from an abstract viewpoint.
The subset of the plane satisfying $x^{2 / 3}+y^{2 / 3}=1$ is called the astroid. Show that $\alpha(u)=\left(\cos ^3 u, \sin ^3 u\right), u \in \mathbb{R}$, is a parametrisation of the astroid. Show that the parametrisation is regular except when $u$ is an integer multiple of $\pi / 2$. Sketch the astroid and mark the singular points of the parametrisation. Find the length of the astroid between parameter values $u=0$ and $u=\pi / 2$.
A sketch of the astroid is given in Figure 1(a). It is clear that all points in the image of $\alpha$ satisfy the equation of the astroid. Conversely, if $x^{2 / 3}+y^{2 / 3}=1$, then there exists $u \in \mathbb{R}$ such that $\left(x^{1 / 3}, y^{1 / 3}\right)=(\cos u, \sin u)$. Thus every point of the astroid is in the image of $\alpha$.
问题 2.
For each positive constant $r$, the smooth curve given by $$ \boldsymbol{\alpha}(u)=(2 r \sin u-r \sin 2 u, 2 r \cos u-r \cos 2 u), \quad u \in \mathbb{R}, $$ is called an epicycloid. It is the curve traced out by a point on the circumference of a circle of radius $r$ which rolls without slipping on a circle of the same radius. Sketch the trace of the curve, and find the length of $\alpha$ between the singular points corresponding to $u=0$ and $u=2 \pi$.
A sketch of the trace of an epicycloid is given in Figure 1(b). Trigonometric identities may be used to show that $$ \alpha^{\prime}=4 r \sin (u / 2)(\sin (3 u / 2), \cos (3 u / 2)) . $$ So, for $0 \leq u \leq 2 \pi,\left|\alpha^{\prime}\right|=4 r \sin (u / 2)$, and required length is $4 r \int_0^{2 \pi} \sin (u / 2) d u=$ $16 r$.
问题 3.
For each positive constant $r$, the smooth curve given by $$ \alpha(u)=\frac{r}{\cosh u}(u \cosh u-\sinh u, 1), $$ is called a tractrix.Taking $r=1$, show that, for $u>0, \boldsymbol{\alpha}(u)$ is the curve traced out by a stone starting at $(0,1)$ on the end of a piece of rope of length 1 when the tractor on the other end of the piece of rope drives along the positive $x$-axis starting at $(0,0)$. In more mathematical terms, show that $\alpha(u)+\boldsymbol{t}(u)$ is on the positive $x$-axis for $u>0$ (and that $\alpha(0)=(0,1))$. Sketch the trace of the curve for all real values of $u$.
When $r=1$, a calculation shows that $\alpha^{\prime}=\tanh u \operatorname{sech} u(\sinh u,-1)$, so that, for $u \geq 0, t=\operatorname{sech} u(\sinh u,-1)$. It follows that $\alpha+t=(u, 0)$. A sketch of the trace of a tractrix is given in Figure 2(a).
statistics-labTM为您提供新南威尔士大学（The University of New South Wales）Abstract algebra and fundamental analysis抽象代数和基本分析澳洲代写代考和辅导服务！
课程介绍：
Mathematics went through quite a revolution around the turn of the 20th century. In particular, an axiomatic approach infiltrated the mathematical paradigm, both as a tool to ensure mathematical rigour and to abstract common principles working in a variety of different settings.
First year mathematics emphasizes computation over abstraction and rigour. Later year courses (and Pure Mathematics in general) reverse this, so students need to learn some new skills and some new ways of thinking about mathematical objects.
This course is designed to help you develop the ability to write rigorous mathematical proofs in a setting where the level of abstraction is still quite modest. As such it will serve as an excellent preparation for the third year Pure Mathematics courses.
The course consists of two halves, algebra and analysis.
澳洲大学｜MATH2701｜Abstract algebra and fundamental analysis抽象代数和基本分析 新南威尔士大学
Groups群论定义
Definition. Let $G$ be a set. A binary operation is a map of sets: $$ : G \times G \rightarrow G \text {. } $$ For ease of notation we write $(a, b)=a * b \forall a, b \in G$. Any binary operation on $G$ gives a way of combining elements. As we have seen, if $G=\mathbb{Z}$ then + and $\times$ are natural examples of binary operations. When we are talking about a set $G$, together with a fixed binary operation $*$, we often write $(G, *)$.
Fundamental Definition. A group is a set $G$, together with a binary operation *, such that the following hold:
(Associativity): $(a * b) * c=a *(b * c) \forall a, b, c \in G$.
(Existence of identity): $\exists e \in G$ such that $a * e=e * a=a \forall a \in G$.
(Existence of inverses): Given $a \in G, \exists b \in G$ such that $a * b=b * a=e$. Remarks. 1. We have seen five different examples thus far: $(\mathbb{Z},+),(\mathbb{Q},+),(\mathbb{Q} \backslash{0}, \times)$, $(\mathbb{Z} / m \mathbb{Z},+)$, and $(\mathbb{Z} / m \mathbb{Z} \backslash{[0]}, \times)$ if $m$ is prime. Another example is that of a real vector space under addition. Note that $(\mathbb{Z}, \times)$ is not a group. Also note that this gives examples of groups which are both finite and infinite. The more mathematics you learn the more you’ll see that groups are everywhere.
A set with a single element admits one possible binary operation. This makes it a group. We call this the trivial group.
A set with a binary operation is called a monoid if only the first two properties hold. From this point of view, a group is a monoid in which every element is invertible. $(\mathbb{Z}, \times)$ is a monoid but not a group.
定义 设 $G$ 是一个集合。二元运算是集合的映射： $$ : G （times） G （rightarrow） G （text） {. } $$ 为了便于记述，我们把 G$ 中的所有 a, b 写成 $(a, b)=a * b \。$G$ 上的任何二元运算都提供了一种组合元素的方法。正如我们所看到的，如果 $G=\mathbb{Z}$ 那么 + 和 $\times$ 就是二元运算的自然例子。当我们谈论一个集合 $G$ 以及一个固定的二进制运算 $$ 时，我们通常会写 $(G,)$。
基本定义。一个群是一个集合 $G$，加上一个二元运算 *，使得以下条件成立：
（关联性）： $(a * b) * c=a *(b * c) （对于 G$ 中的所有 a, b, c）。 2.（存在同一性）： 对于 G$ 中的所有 a，在 G$ 中存在 e，使得 $a * e=e * a=a. 3.（倒数的存在）： Given $a \in G, \exists b \in G$ such that $a * b=b * a=e$. 备注. 1. 到目前为止，我们已经看到了五个不同的例子： $(\mathbb{Z},+),(\mathbb{Q},+),(\mathbb{Q} \backslash{0}, \times)$, $(\mathbb{Z} / m \mathbb{Z},+)$, 以及 $(\mathbb{Z} / m \mathbb{Z} \backslash{[0]}, \times)$ 如果 $m$ 是素数的话。另一个例子是实向量空间的加法。注意 $(\mathbb{Z}, \times)$ 不是一个群。还要注意，这给出了有限群和无限群的例子。数学学得越多，你就会发现群无处不在。
similar (or related) to one another within the same group
dissimilar (or unrelated) to the objects in other groups Cluster analysis (or clustering, data segmentation, …)
Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters Unsupervised learning: no predefined classes (i.e., learning by observations vs. learning by examples: supervised) Typical applications
As a stand-alone tool to get insight into data distribution
The Orbit-Stabiliser Theorem and Sylow’s Theorem轨道稳定器定理和西洛定理
Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. We can define an equivalence relation on $S$ by $$ s \sim t \Longleftrightarrow \exists g \in G \text { such that } g(s)=t $$ Remarks. This is an equivalence relation as a consequence of the group axioms, together with the definition of an action. I leave it as an exercise to check this.
Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. Under the above equivalence relation we call the equivalence classes orbits, and we write $$ \operatorname{Orb}(s):={t \in S \mid \exists g \in G \text { such that } g(s)=t} \subset S $$ for the equivalence class containing $s \in S$. We call it the orbit of $s$. It is important to observe that $\operatorname{Orb}(s)$ is a subset of $S$ and hence is merely a set with no extra structure.
Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. We say that $G$ acts transitively on $S$ is there is only one orbit. Equivalently, $\varphi$ is transitive if given $s, t \in S$, $\exists g \in G$ such that $g(s)=t$.
An example of a transitive action is the natural action of $\Sigma(S)$ on $S$. This is clear because given any two points in a set $S$ there is always a bijection which maps one to the other. If $G$ is not the trivial group (the group with one element) then conjugation is never transitive. To see this observe that under this action $\operatorname{Orb}(e)={e}$.
Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. Let $s \in S$. We define the stabiliser subgroup of $s$ to be all elements of $G$ which fix $s$ under the action. More precisely $$ \operatorname{Stab}(s)={g \in G \mid g(s)=s} \subset G $$ For this definition to make sense we must prove that $\operatorname{Stab}(s)$ is genuinely a subgroup. Proposition. Stab(s) is a subgroup of $G$. Proof. 1. $e(s)=s \Rightarrow e \in \operatorname{Stab}(s)$
$x, y \in \operatorname{Stab}(s) \Rightarrow(x * y)(s)=x(y(s))=x(s)=s \Rightarrow x * y \in \operatorname{Stab}(s)$.
$x \in \operatorname{Stab}(s) \Rightarrow x^{-1}(s)=x^{-1}(x(s))=\left(x^{-1} * x\right)(s)=e(s)=s \Rightarrow x^{-1} \in \operatorname{Stab}(s)$ Thus we may form the left cosets of $\operatorname{Stab}(s)$ in $G$ : $$ G / \operatorname{Stab}(s):={x \operatorname{Stab}(s) \mid x \in G} . $$ Recall that these subsets of $G$ are the equivalence classes for the equivalence relation: $$ \text { Given } x, y \in G, x \sim y \Longleftrightarrow x^{-1} * y \in \operatorname{Stab}(s), $$ hence they partition $G$ into disjoint subsets. Proposition. Let $x, y \in G$ then $x \operatorname{Stab}(s)=y \operatorname{Stab}(s) \Longleftrightarrow x(s)=y(s)$.
定义 让 $(G, *)$ 是一个群，以及一个集合 $S$ 上的作用 $\varphi$。在上述等价关系下，我们称等价类为轨道，并写为 $$ \操作符名称{Orb}(s):={t \in S \mid \exists g \in G \text { such that } g(s)=t} \子集 S $$ 为在 S$ 中包含 $s 的等价类。我们称之为 $s$ 的轨道。 需要注意的是，$operatorname{Orb}(s)$ 是 $S$ 的子集，因此只是一个没有额外结构的集合。
statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Financial Mathematics金融数学澳洲代写代考和辅导服务！
课程介绍：
Algebra 1 is a foundational course in Mathematics, introducing some of the key concepts of modern algebra. The course leads on to other areas of algebra such as Galois Theory, Algebraic Topology and Algebraic Geometry. It also provides important tools for other areas such as theoretical computer science, physics and engineering.
Topics to be covered include the theory of groups and rings:
Group Theory – permutation groups; abstract groups, subgroups, cyclic and dihedral groups; homomorphisms; cosets, Lagrange’s theorem, quotient groups; group actions; Sylow theory.
Ring Theory – rings and fields, polynomial rings, factorisation; homomorphisms, factor rings.
澳洲代写｜MATH2322｜Advanced Algebra 1: Groups, Rings and Linear Algebra高等代数 1：群、环和线性代数 澳洲国立大学
Field
Information
Course Code
MATH2322
Prerequisite Courses
Not explicitly mentioned in the provided text.
Majors
Mathematics
Teachers
Not mentioned in the provided text.
Units
6 units
Advanced Algebra 1: Groups, Rings and Linear Algebra高等代数 1：群、环和线性代数问题集
问题 1.
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^2$. For the following $\mathbf{u}$ and $\mathbf{v}$ determine the angle between the vectors and label this angle and the vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^2$. (a) $\mathbf{u}=\left(\begin{array}{l}1 \ 1\end{array}\right), \mathbf{v}=\left(\begin{array}{l}0 \ 1\end{array}\right)$ (b) $\mathbf{u}=\left(\begin{array}{l}1 \ 0\end{array}\right), \mathbf{v}=\left(\begin{array}{l}0 \ 1\end{array}\right)$ (c) $\mathbf{u}=\left(\begin{array}{c}-2 \ 3\end{array}\right), \mathbf{v}=\left(\begin{array}{c}1 / 2 \ -1 / 2\end{array}\right)$
For the following vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^3$ determine the angle between them. (a) $\mathbf{u}=\left(\begin{array}{r}-1 \ 1 \ 3\end{array}\right), \mathbf{v}=\left(\begin{array}{r}3 \ -1 \ 5\end{array}\right)$ (b) $\mathbf{u}=\left(\begin{array}{l}1 \ 0 \ 0\end{array}\right), \mathbf{v}=\left(\begin{array}{r}0 \ 0 \ 15\end{array}\right)$ (c) $\mathbf{u}=\left(\begin{array}{r}-1 \ 2 \ 3\end{array}\right), \mathbf{v}=\left(\begin{array}{c}\sqrt{2} \ 1 / \sqrt{2} \ -1\end{array}\right)$
Determine the value of $k$ so that the following vectors are orthogonal to each other: (a) $\mathbf{u}=\left(\begin{array}{r}-1 \ 5 \ k\end{array}\right), \mathbf{v}=\left(\begin{array}{r}-3 \ 2 \ 7\end{array}\right)$ (b) $\mathbf{u}=\left(\begin{array}{r}2 \ -1 \ 3\end{array}\right), \mathbf{v}=\left(\begin{array}{l}3 \ 1 \ k\end{array}\right)$ (c) $\mathbf{u}=\left(\begin{array}{c}0 \ -k \ \sqrt{2}\end{array}\right), \mathbf{v}=\left(\begin{array}{c}-7 \ 5 \ k\end{array}\right)$
Determine the unit vector $\hat{\mathbf{u}}$ for each of the following vectors. (Normalize these vectors.) (a) $\mathbf{u}=\left(\begin{array}{ll}2 & 3\end{array}\right)^T$ (b) $\mathbf{u}=\left(\begin{array}{lll}1 & 2 & 3\end{array}\right)^T$ (c) $\mathbf{u}=\left(\begin{array}{lll}1 / 2 & -1 / 2 & 1 / 4\end{array}\right)^T$ (d) $\mathbf{u}=\left(\begin{array}{llll}\sqrt{2} & 2 & -\sqrt{2} & \sqrt{2}\end{array}\right)^T$ (e) $\mathbf{u}=\left(\begin{array}{lllll}-\pi / 5 & \pi & -\pi & \pi / 10 & 0\end{array}\right)^T$
Determine the value(s) of $k$ so that $\hat{\mathbf{u}}=\left(\begin{array}{c}1 / \sqrt{2} \ 1 / 2 \ k\end{array}\right)$ is a unit vector.
问题 3.
(a) Show that $\mathbf{u}=\left(\begin{array}{c}\cos (\theta) \ \sin (\theta)\end{array}\right)$ is a unit vector. (b) Plot this vector $\mathbf{u}$ in $\mathbb{R}^2$ for $\theta=\frac{\pi}{4}$. (c) Let $\mathbf{v}=\left(\begin{array}{r}\cos (\theta) \ -\sin (\theta)\end{array}\right)$ be a vector in $\mathbb{R}^2$. On the same axes plot $\mathbf{v}$ for $\theta=\frac{\pi}{4}$. (d) Determine the angle between the vectors $\mathbf{u}$ and $\mathbf{v}$.
Show that the vectors $\mathbf{u}=\left(\begin{array}{l}a \ b\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{r}-b \ a\end{array}\right)$ in $\mathbb{R}^2$ are orthogonal.
Let vectors $\mathbf{u}=\left(\begin{array}{c}\cos (A) \ \sin (A)\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}\cos (B) \ \sin (B)\end{array}\right)$ be in $\mathbb{R}^2$. Show that $$ \mathbf{u} \cdot \mathbf{v}=\cos (A-B) $$
澳洲代写｜MATH2322｜Advanced Algebra 1: Groups, Rings and Linear Algebra高等代数 1：群、环和线性代数 澳洲国立大学
statistics-labTM为您提供蒙纳士大学（Monash University）Modelling in finance金融建模澳洲代写代考和辅导服务！
课程介绍：
Topics include the development and application of financial spreadsheets, Excel and Visual Basic programming in financial modelling, modelling company financial statements, fixed income securities analysis, asset allocation and portfolio analysis, optimization using Solver, Interest rate models, option pricing models, numerical methods and risk management models.
澳洲代写｜BFW3540 ｜Modelling in finance金融建模 蒙纳士大学
Fourier Transform傅立叶变换案例
In this section, we recall the Fourier transform definition, both for notational reasons and for the reader’s convenience. The Fourier transform, for $f \in \mathcal{S}\left(\mathbb{R}^m\right)$, is denoted here as $$ \widehat{f}(\xi):=\mathcal{F}f:=\int_{\mathbb{R}^m} \mathrm{e}^{i \xi x} f(x) \mathrm{d} x, $$ where $\mathcal{S}\left(\mathbb{R}^m\right)$ is the Schwartz space of $\mathcal{C}^{\infty}\left(\mathbb{R}^m\right)$ functions of rapid decrease, see [RS75]. This is not the usual definition found in the mathematical literature. However, it is standard in probability, see [Chu01] and in the finance literature, see [CT04].
The Fourier transform is a linear bijection from $\mathcal{S}\left(\mathbb{R}^m\right)$ onto $\mathcal{S}\left(\mathbb{R}^m\right)$, whose inverse is given by the Fourier inversion formula
We also recall the Fourier transform for $f \in \mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$, where $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ is the space of tempered distributions, which is the dual of $\mathcal{S}\left(\mathbb{R}^m\right)$, the Fourier transform can be defined as $$ (\mathcal{F}[f], \varphi)=(2 \pi)^m\left(f, \mathcal{F}^{-1}[\varphi]\right) \quad \varphi \in \mathcal{S}\left(\mathbb{R}^m\right), $$ see [RR04]. This definition makes the Fourier transform in $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ an extension of the Fourier transform in $\mathcal{S}\left(\mathbb{R}^m\right)$. The Fourier transform for $L^1\left(\mathbb{R}^m\right)$ and $L^2\left(\mathbb{R}^m\right)$ are restrictions of the Fourier transform for $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$.
The Fourier transform has several useful properties. Some of them are reviewed below with the purpose of calling attention to the notation used here:
$\mathcal{F}f(x-a)=\mathrm{e}^{i a \xi} \widehat{f}(\xi)$
$(-i \xi)^\alpha \widehat{f}(\xi)=\mathcal{F}\leftD^\alpha f\right$ Some specific distributions are often used in this thesis. To present the notation, we give a brief overview of them. First, consider the Cauchy principal value $$ \begin{aligned} 1 / x: \mathcal{S}(\mathbb{R}) & \rightarrow \mathbb{R} \ f & \rightarrow(1 / x, f):=f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x, \end{aligned} $$ where $$ f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x:=\lim {\epsilon \downarrow 0}\left(\int\epsilon^{\infty} \frac{f(x)}{x} \mathrm{~d} x+\int_{-\infty}^{-\epsilon} \frac{f(x)}{x} \mathrm{~d} x\right) . $$ This defines a distribution in $\mathcal{S}^{\prime}(\mathbb{R})$.
Probability and Stochastic Processes概率与随机过程案例
In this section, we present a brief overview of the topics on probability and stochastic processes used herein. References on the subject are [CW90] and [Sat99].
In this thesis, the triple $(\Omega, \mathcal{F}, \mathbb{P})$ denotes a complete probability space, where $\Omega$ is a set of points $\omega, \mathcal{F}$ is a $\sigma$-algebra containing all $\mathbb{P}$-null sets, and $\mathbb{P}$ is a probability measure. When we say that $X$ is a random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, we mean that $X$ is real-valued function on $\Omega$, measurable with respect to $\mathcal{F}$. The characteristic function of a random variable is defined as $$ \varphi(z)=\mathbb{E}\left[\mathrm{e}^{i z X}\right] . $$ For properties of the characteristic function and a review of probability theory we refer to [CW90].
The filtered complete probability space is denoted by $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$, where, as in [Pro04], we write $\mathbb{F}$ for the filtration $\left(\mathcal{F}t\right){0 \leq t \leq \infty}$ and we assume that $\mathcal{F}_0$ contains all the $\mathbb{P}$-null sets. We use $\stackrel{\mathbb{P}}{\longrightarrow}$ to denote convergence in probability, see [CW90] and the French acronyms càdlàg (continu à droite, limité à gauche) is used to define the right continuous, left limited process, see [Pro04].
The main class of stochastic processes we are interested in this work are the Levy processes, see [Sat99] for a comprehensive treatment of the subject. We briefly review the definition of a Levy process Definition: A Levy process is a càdlàg stochastic process, $\left(X_t\right)_{t \geq 0}$, on $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ taking values in $\mathbb{R}$ and with the following properties:
Independent increments. That is, given $t_0 \leq \ldots \leq t_N$, and defined $Y_n:=X_{t_n}-X_{t_{n-1}}$ we have $\left{Y_n\right}_{n=1}^N$ independent;
Stationary increments. That is, the distribution of $X_{t+s}-X_t$ does not depend on $t$;
Stochastic continuity. That is, $$ X_{t+h} \underset{h \downarrow 0}{\stackrel{\mathbb{P}}{\longrightarrow}} X_t $$ An important stochastic process is the Brownian motion.
statistics-labTM为您提供墨尔本大学（The University of Melbourne，简称UniMelb，中文简称“墨大”）Quantum Physics量子计算澳洲代写代考和辅导服务！
课程介绍：
Quantum mechanics plays a central role in our understanding of fundamental phenomena, primarily in the microscopic domain. It lays the foundation for an understanding of atomic, molecular, condensed matter, nuclear and particle physics.
澳洲代写｜PHYC30018｜Quantum Physics量子计算 墨尔本大学
Quantum Physics量子计算问题集
问题 1. Consider a two-dimensional strip of material of length $L$ and width $W$, placed in a magnetic field perpendicular to the strip and with an electric field established in the direction of the length $L$. (a) Suppose that the resistivity matrix is given by the classical result $$ \rho=\left(\begin{array}{cc} \rho_0 & -\rho_H \ \rho_H & \rho_0 \end{array}\right) $$ where $\rho_H=B /$ nec is the Hall resistivity and $\rho_0$ is the usual Ohmic resistivity. Find the conductivity matrix, $\sigma=\rho^{-1}$. Write it in the form: $$ \sigma=\left(\begin{array}{cc} \sigma_0 & \sigma_H \ -\sigma_H & \sigma_0 \end{array}\right) . $$ What are $\sigma_0$ and $\sigma_H$ ? (b) Suppose $B=0$, so the Hall resistivity is zero. Notice that the Ohmic conductivity, $\sigma_0$, is just $1 / \rho_0$. In particular, note that $\sigma_0 \rightarrow \infty$ as $\rho_0 \rightarrow 0$. Now suppose $\rho_H \neq 0$. Show that $\sigma_0 \rightarrow 0$ as $\rho_0 \rightarrow 0$, so it is possible to have both $\sigma_0$ and $\rho_0$ equal to zero
问题 2.
This problem asks you to give a complete presentation of a calculation that is almost the same as one you saw in lecture.
Consider a constant electric field, $\vec{E}=\left(0, E_0, 0\right)$ and a constant magnetic field, $\vec{B}=\left(0,0, B_0\right)$. (a) Choose an electrostatic potential $\phi$ and a vector potential $\vec{A}$ which describe the $\vec{E}$ and $\vec{B}$ fields, and write the Hamiltonian for a charged particle of mass $m$ and charge $q$ in these fields. Assume that the particle is restricted to move in the $x y$-plane. (b) What are the allowed energies as a function of $B_0$ and $E_0$ ? Draw a figure to show how the Landau levels (energy levels when $E_0=0$ ) change as $E_0$ increases.
问题 3.
You will see the “standard presentation” of the Aharonov-Bohm effect in lecture, on the day that this problem set is due. The standard presentation has its advantages, and in particular is more general than the presentation you will work through in this problem. However, students often come away from the standard presentation of the Aharonov-Bohm effect thinking that the only way to detect this effect is to do an interference experiment. This is not true, and the purpose of this problem is to disabuse you of this misimpression before you form it.
As Griffiths explains on pages 385-387 (344-345 in 1st Ed.), the Aharonov-Bohm effect modifies the energy eigenvalues of suitably chosen quantum mechanical systems. In this problem, I ask you to work through the same physical example that Griffiths uses, but in a different fashion which makes more use of gauge invariance.
Imagine a particle constrained to move on a circle of radius $b$ (a bead on a wire ring, if you like.) Along the axis of the circle runs a solenoid of radius $a<b$, carrying a magnetic field $\vec{B}=\left(0,0, B_0\right)$. The field inside the solenoid is uniform
and the field outside the solenoid is zero. The setup is depicted in Griffiths’ Fig. 10.10. (10.12 in 1st Ed.) (a) Construct a vector potential $\vec{A}$ which describes the magnetic field (both inside and outside the solenoid) and which has the form $A_r=A_z=0$ and $A_\phi=\alpha(r)$ for some function $\alpha(r)$. I am using cylindrical coordinates $z, r$, $\phi$. (b) Since $\vec{\nabla} \times \vec{A}=0$ for $r>a$, it must be possible to write $\vec{A}=\vec{\nabla} f$ in any simply connected region in $r>a$. [This is a theorem in vector calculus.] Show that if we find such an $f$ in the region $$ r>a \text { and }-\pi+\epsilon<\phi<\pi-\epsilon, $$ then $$ f(r, \pi-\epsilon)-f(r,-\pi+\epsilon) \rightarrow \Phi \text { as } \epsilon \rightarrow 0 . $$ Here, the total magnetic flux is $\Phi=\pi a^2 B_0$. Now find an explicit form for $f$, which is a function only of $\phi$. (c) Now consider the motion of a “bead on a ring”: write the Schrödinger equation for the particle constrained to move on the circle $r=b$, using the $\vec{A}$ you found in (a). Hint: the answer is given in Griffiths. (d) Use the $f(\phi)$ found in (b) to gauge transform the Schrödinger equation for $\psi(\phi)$ within the angular region $-\pi+\epsilon<\phi<\pi-\epsilon$ to a Schrödinger equation for a free particle within this angular region. Call the original wave function $\psi(\phi)$ and the gauge-transformed wave function $\psi^{\prime}(\phi)$. (e) The original wave function $\psi$ must be single-valued for all $\phi$, in particular at $\phi=\pi$. That is, $\psi(\pi-\epsilon)-\psi(-\pi+\epsilon) \rightarrow 0$ and $\frac{\partial \psi}{\partial \phi}(\pi-\epsilon)-\frac{\partial \psi}{\partial \phi}(-\pi+\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$. What does this say about the gauge-transformed wave function? I.e., how must $\psi^{\prime}(\pi-\epsilon)$ and $\psi^{\prime}(-\pi+\epsilon)$ be related as $\epsilon \rightarrow 0$ ? [Hint: because the $f(\phi)$ is not single valued at $\phi=\pi$, the gauge transformed wave function $\psi^{\prime}(\phi)$ is not single valued there either.] (f) Solve the Schrödinger equation for $\psi^{\prime}$, and find energy eigenstates which satisfy the boundary conditions you derived in (e). What are the allowed energy eigenvalues? (g) Undo the gauge transformation, and find the energy eigenstates $\psi(\phi)$ in the original gauge. Do the energy eigenvalues in the two gauges differ? (h) Plot the energy eigenvalues as a function of the enclosed flux, $\Phi$. Show that the energy eigenvalues are periodic functions of $\Phi$ with period $\Phi_0$, where you must determine $\Phi_0$. For what values of $\Phi$ does the enclosed magnetic field have no effect on the spectrum of a particle on a ring? Show that the
Aharonov-Bohm effect can only be used to determine the fractional part of $\Phi / \Phi_0$. [Note: you have shown that even though the bead on a ring is everywhere in a region in which $\vec{B}=0$, the presence of a nonzero $\vec{A}$ affects the energy eigenvalue spectrum. However, the effect on the energy eigenvalues is determined by $\Phi$, and is therefore gauge invariant. To confirm the gauge invariance of your result, you can compare your answer for the energy eigenvalues to Griffiths’ result, obtained using a different gauge.]
statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Financial Mathematics金融数学澳洲代写代考和辅导服务！
课程介绍：
This course provides an introduction to the valuation of cash flows. Topics include: compound interest functions; valuation of annuities certain; loans repayable by instalments; comparison of value and yield of cash flow transactions; valuation of fixed interest securities, with and without tax on interest and capital gains; duration and volatility of securities; introduction to concept of immunisation and matching; consumer credit contracts.
澳洲代写｜STAT2032｜Financial Mathematics金融数学 澳洲国立大学
Field
Information
Course Code
STAT2032
Prerequisite Courses
Not explicitly mentioned in the provided text.
Majors
Actuarial Studies, Finance, Statistics
Teachers
Jacie Liu, Lucy Hu
Units
6 units
Financial Mathematics金融数学问题集
问题 1.
What annual interest rate $r$ would allow you to double your initial deposit in 6 years if interest is compounded quarterly? Continuously?
If you receive $6 \%$ interest compounded monthly, about how many years will it take for a deposit at time-0 to triple?
If you deposit $\$ 400$ at the end of each month into an account earning $8 \%$ interest compounded monthly, what is the value of the account at the end of 5 years? 10 years?
问题 2.
You deposit $\$ 700$ at the end of each month into an account earning interest at an annual rate of $r$ compounded monthly. Use a spreadsheet to find the value of $r$ that produces an account value of $\$ 50,000$ in 5 years.
You make an initial deposit of $\$ 1000$ at time-0 into an account with an annual rate of $5 \%$ compounded monthly and additionally you deposit $\$ 400$ at the end of each month. Use a spreadsheet to determine the minimum number of payments required for the account to have a value of at least $\$ 30,000$.
Suppose an account offers continuously compounded interest at an annual rate $r$ and that a deposit of size $P$ is made at the end of each month. Show that the value of the account after $n$ deposits is $$ A_n=P \frac{e^{r n / 12}-1}{e^{r / 12}-1} $$
问题 3.
Find a formula for the number $N$ of monthly withdrawals needed to draw down to zero an account initially valued at $A_0$. Use the formula to determine how many withdrawals are required to draw down to zero an account with initial value $\$ 200,000$, if the account pays $6 \%$ compounded monthly.
An account pays an annual rate of $8 \%$ percent compounded monthly. What lump sum must you deposit into the account now so that in 10 years you can begin to withdraw $\$ 4000$ each month for the next 20 years, drawing down the account to zero?
A trust fund has an initial value of $\$ 300,000$ and earns interest at an annul rate of $6 \%$, compounded monthly. If a withdrawal of $\$ 5000$ is made at the end of each month, use a spreadsheet to determine when the account will fall below $\$ 150,000$.
statistics-labTM为您悉尼大学（英语：The University of Sydney），简称悉大、USYD，简称“NCL”Business Entrepreneurship: Business Models企业创业： 商业模式澳洲代写代考和辅导服务！
课程介绍：
This unit of study aims to provide an understanding of the conservation of mass and momentum in differential forms for viscous fluid flows. It provides the foundation for advanced study of turbulence, flow around immersed bodies, open channel flow, pipe flow and pump design.
澳洲代写｜OLET1201｜Business Entrepreneurship: Business Models企业创业： 商业模式 悉尼大学
Key Information
Details
Course Number
CIVL3612
Pre-requisites
(Not mentioned in the text)
Academic Unit (Major)
Civil Engineering
Instructor
(Not mentioned in the text)
Credit Points
6
Business Models商业模式问题集
问题 1.
A common observation in big rivers or other fast-flowing bodies of water (e.g. during floods) is shown in the figures and sketch below. A fast moving stream of water that is steadily flowing along suddenly decelerates and the position of the free surface ‘jumps’ upwards. After a lot of local turbulent motion, the flow settles down again but is now steadily moving at a significantly slower speed.
We will represent the free surface height as $h(x)$ and the velocity by the function $u(x)$. The fluid has constant density $\rho$ and we will treat the problem as one-dimensional. You can assume that viscous stresses along the control surfaces of the volume shown above are negligibly small, and neglect the density of air. PART I: a) consider a streamline drawn (line $\mathrm{AB}$ in the figure) just above the smooth flat lower surface of the channel. How is the static pressure in the fluid along this line related to the height of the river? How does the static pressure vary along the line DEA?
(a): The pressure distribution on line $\mathrm{AB}$ follows the hydrostatic rule. It is true that the flow is not static but by picking an arbitrary control volume at any point on line AB (green dashed control volume in Figure 1) one can see that the balance of forces in the $y$-direction will tell us that the difference between the pressure at the bottom and the ambient pressure should balance the weight of the liquid inside the control volume. This simply implies that the static pressure on line $\mathrm{AB}$ should be equal The pressure distribution on line DEA also follows the hydrostatic change merely due to the fact that there is no curvature in the streamlines as one integrates the Euler equation normal to them and thus the only change in pressure when one moves from $\mathrm{E}$ to A will be the hydrostatic part. Ignoring the density of air one can see that the pressure is constant from D to E and then start to grow linearly with height as we move from $\mathrm{E}$ to $\mathrm{A}$. The result is shown in Figure
问题 2.
b) Using the control volume shown in the sketch develop two expressions that relate the velocity and height of the stream at station 1 and the velocity and height of the stream at station 2. Developing a table of relevant quantities along each face of the control volume ABCDEA is highly recommended! c) [2 points] Combine your expressions from (a) and (b) together to show that the speed of the river can be simply evaluated from simple measurements of the river height (e.g. using marked yardsticks attached to the channel floor): $$ u_1=\sqrt{\frac{g h_2}{2 h_1}\left(h_1+h_2\right)} $$
}(b) and (c): The selected control volume is shown in Figure 3 (dashed green line). One can subtract the ambient pressure from the entire problem and knowing that the net effect of uniform $P_a$ acting on the control volume is zero then there will be no change in the problem analysis if we only deal with gauge pressures $\left(P(x, y)-P_a\right)$Table 1 summarizes all the important parameters acting on different control surfaces for the selected control volume:
Now we can start by writing the conservation rules using the RTT. It is important to notice that due to the turbulent mixing happening in the region of the hydraulic jump, energy will not be conserved and thus either applying the conservation of energy or the Bernoulli equation will not be the right approach. If we write the conservation of mass for the selected control volume then we will have:
$$ \text { C.O.Mass: } 0=\frac{d}{d t} \int_{\text {c.v. }} \rho d V+\int_{\text {c.s. }} \rho\left(v-v_c\right) \cdot n d A $$
Knowing that the problem is steady state and using the tabulated quantities, conservation of mass can be simplified to: $\rho u_1 h_1=\rho u_2 h_2 \Rightarrow u_1 h_1=u_2 h_2$ The conservation of linear momentum in the $x$ direction can also be written in the RTT form:
$$ \text { C.O.Momentum: } \frac{1}{W} \sum F_x=\frac{d}{d t} \int_{c . v .} \rho v_x d V+\int_{\text {c.s. }} \rho v_x\left(v-v_c\right) \cdot n d A $$
where $W$ is the width into the page. The net of external forces acting in the $x$-direction on the control volume neglecting the wall shear effect is a result of pressure forces acting on the (AD) and (BC) control surfaces:
$$ \frac{1}{W} \sum F_x=\int_{A D}\left(P-P_a\right) d y-\int_{B C}\left(P-P_a\right) d y=\int_0^{h_1} \rho g y d y-\int_0^{h_2} \rho g y d y=\rho g\left(\frac{h_1^2}{2}-\frac{h_2^2}{2}\right) $$
The right hand side of the RTT for the conservation of linear momentum can also be simplified to (knowing that the problem is steady and using the tabulated identities):
$$ \text { R.H.S. of RTT for C.O. Momentum }=\rho u_2^2 h_2-\rho u_1^2 h_1 $$
thus the conservation of linear momentum implies that:
where we have used the identity $h_1^2-h_2^2=\left(h_1-h_2\right)\left(h_1+h_2\right)$.
问题 3.
A deeper question to answer is why is the water moving so fast locally to begin with. To answer this we must consider the topography of the river bed that is upstream of station 1 , as shown in the drawing below. We denote the height of the fluid stream above the river bed as $h(x)$ and the height of the riverbed by $b(x)$ : d) [1 point] Consider a slice of river $d x$ and show that conservation of mass can be written in the form: $$ u(x) \frac{d h(x)}{d x}+h(x) \frac{d u(x)}{d x}=0 $$
(d): For the selected control volume (Figure 4 ) one can easily write the conservation of mass using Taylor series to obtain expressions for $u(x+\Delta x)$ and $h(x+\Delta x)$ : $$ u(x) h(x)=u(x+\Delta x) h(x+\Delta x) \rightarrow u(x) h(x)=\left(u(x)+\frac{d u}{d x} \Delta x\right)\left(h(x)+\frac{d h}{d x} \Delta x\right) $$ which after ignoring the second order terms such $\left(\Delta x^2\right)$ it can be rewritten as: $$ \Delta x\left(u(x) \frac{d h}{d x}+h(x) \frac{d u}{d x}\right)=0 \Rightarrow u \frac{d h}{d x}+h \frac{d u}{d x}=0 $$ Another way to reach the same result is to say that since the flow is incompressible then the volumetric flow rate should remain unchanged thus $d(u h) / d x=0$ which will lead to the same result we just derived in equation (4).
Figure 4: An arbitrary control volume selected to derive the conservation of mass in the differential form.
statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Introduction to Machine Learning机器学习入门澳洲代写代考和辅导服务！
课程介绍：
Essential foundations for any machine learning application are a basic statistical analysis of the data to be processed, a solid understanding of the mathematical foundations underpinning machine learning as well as the basic classes of learning/adaptation concepts. Those foundations are bundled in this single, introductory course to machine learning in preparation for deeper explorations into the topic, but also as a standalone unit.
澳洲代写｜COMP3670｜Introduction to Machine Learning机器学习入门 澳洲国立大学
Detail
Information
Course Code
COMP3670
Prerequisite
Not mentioned in the provided text
Academic Unit
School of Computing
ANU College
ANU College of Engineering Computing & Cybernetics
Course Subject
Computer Science
Course Convener
Dr Thang Bui
Units
6 units
Binary Classification二进制分类定义
Binary Classification is probably the most frequently studied problem in machine learning and it has led to a large number of important algorithmic and theoretic developments over the past century. In its simplest form it reduces to the question: given a pattern $x$ drawn from a domain $X$, estimate which value an associated binary random variable $y \in{ \pm 1}$ will assume. For instance, given pictures of apples and oranges, we might want to state whether the object in question is an apple or an orange. Equally well, we might want to predict whether a home owner might default on his loan, given income data, his credit history, or whether a given e-mail is spam or ham. The ability to solve this basic problem already allows us to address a large variety of practical settings.
There are many variants exist with regard to the protocol in which we are required to make our estimation:
We might see a sequence of $\left(x_i, y_i\right)$ pairs for which $y_i$ needs to be estimated in an instantaneous online fashion. This is commonly referred to as online learning.
We might observe a collection $\mathbf{X}:=\left{x_1, \ldots x_m\right}$ and $\mathbf{Y}:=\left{y_1, \ldots y_m\right}$ of pairs $\left(x_i, y_i\right)$ which are then used to estimate $y$ for a (set of) so-far unseen $\mathbf{X}^{\prime}=\left{x_1^{\prime}, \ldots, x_{m^{\prime}}^{\prime}\right}$. This is commonly referred to as batch learning.
We might be allowed to know $\mathbf{X}^{\prime}$ already at the time of constructing the model. This is commonly referred to as transduction.
We might be allowed to choose $\mathbf{X}$ for the purpose of model building. This is known as active learning.
We might not have full information about $\mathbf{X}$, e.g. some of the coordinates of the $x_i$ might be missing, leading to the problem of estimation with missing variables.
The sets $\mathbf{X}$ and $\mathbf{X}^{\prime}$ might come from different data sources, leading to the problem of covariate shift correction.
We might be given observations stemming from two problems at the same time with the side information that both problems are somehow related. This is known as co-training.
Mistakes of estimation might be penalized differently depending on the type of error, e.g. when trying to distinguish diamonds from rocks a very asymmetric loss applies.
Multiclass Classification is the logical extension of binary classification. The main difference is that now y ∈ {1, . . . , n} may assume a range of different values. For instance, we might want to classify a document according to the language it was written in (English, French, German, Spanish, Hindi, Japanese, Chinese, . . . ). See Figure 1.6 for an example. The main difference to before is that the cost of error may heavily depend on the type of error we make. For instance, in the problem of assessing the risk of cancer, it makes a significant difference whether we mis-classify an early stage of cancer as healthy (in which case the patient is likely to die) or as an advanced stage of cancer (in which case the patient is likely to be inconvenienced from overly aggressive treatment). Structured Estimation goes beyond simple multiclass estimation by assuming that the labels y have some additional structure which can be used in the estimation process. For instance, y might be a path in an ontology, when attempting to classify webpages, y might be a permutation, when attempting to match objects, to perform collaborative filtering, or to rank documents in a retrieval setting. Equally well, y might be an annotation of a text, when performing named entity recognition. Each of those problems has its own properties in terms of the set of y which we might consider admissible, or how to search this space. Regression is another prototypical application. Here the goal is to estimate a real-valued variable y ∈ R given a pattern x For instance, we might want to estimate the value of a stock the next day, the yield of a semiconductor fab given the current process, the iron content of ore given mass spectroscopy measurements, or the heart rate of an athlete, given accelerometer data. One of the key issues in which regression problems differ from each other is the choice of a loss. For instance, when estimating stock values our loss for a put option will be decidedly one-sided. On the other hand, a hobby athlete might only care that our estimate of the heart rate matches the actual on average.
