I am trying to understand the mathematics and methodology behind a newly published outlier detection algorithm in the Computer & Security journal. This algorithm uses heuristic-based approaches, combining density- and distance-based scoring with empirical weights. Unsupervised outlier detection (...

I am working on understanding the relationship between the solutions of two related SVM optimization problems, defined as follows: Problem 1: \begin{equation} \min \frac{1}{2} \|w\|^2 + C \sum_{i=1}^n a_i \xi_i \quad \mbox{subject to}: \quad y_i (w^T x_i + b) \ge 1 - \xi_i, \qu...

Let $B_r(x_0)\subset \mathcal{R}^n$ be an open ball with radious $r>0$ and centered at $x_0$. Let $y_0\in \partial \overline{B}_r(x_0)$. I want to argue: there exists some $\delta>0$ such that both $B_\delta(y_0)\cap B_r(x_0)$ and $B_\delta(y_0)\setminus B_r(x_0)$ are connected. Is the statement...

In Stein and Shakarchi's Fourier Analysis, they often invoke the fact that differentiation with respect to time commutes with the Fourier transform with respect to space variables. For example, on page 147 (Chapter 5, Proof of Theorem 2.1(i)), they show that $u(x, t) = \int_{-\infty}^{\infty} \h...

I have an expression that looks like $$F(x) = \int_0^\infty da\, db\, dc\, f(a)\,g(b)\,h(c)\,\delta(x-a-b-c)\,.$$ Such delta function convolution is particularly simple to think about from the Laplace (or Fourier) space. In particular, I can simplify the convolution through the transform as $$\ti...

Consider a region of 3D space where there exists a continuous and differentiable vector field: $$ \vec{A}=\left[A_1(x,y,z,t),A_2(x,y,z,t),A_3(x,y,z,t)\right] $$ In that region of space there is a particle moving with velocity: $$ \vec{v}=\left[v_1(x,y,z,t),v_2(x,y,z,t),v_3(x,y,z,t)\right] $$ My q...

Find the derivative of the solution of the equation $\ddot x = x + A \dot x^2$, with initial conditions $x(0) = 1, \dot x(0) = 0$, with respect to the parameter $A$ for $A = 0$. -- Vladimir Arnold This is a great ODE question posed by Vladimir Arnold which takes creativity and ingenuity to solv...

I am trying to solve a probelom using the recursion theorm and I have 2 questions. $Theorem:$ for any $F:V\to V$ class function there is a unique clas function $G:ON\to V$ such that $(\forall\alpha):G(\alpha)=F((G(\beta):\beta<\alpha))$. Where $V$ is the class such that $V=\left\{{x\mid x=x}\righ...

Suppose that $X_1,...X_n$ is a sample from Bernoulli distribution with parameter $W$ and prior distribution of $W$ is beta with parameters $\alpha$ and $\beta$. If the loss function is $L(w,d)=(w-d)^2$ then what is the Bayes risk, i.e. $$E\left[Var\left(W|X_1,...X_n\right)\right]$$ The posterior ...

Let $X$ and $Y$ be two topological spaces and $f:X\to Y$ a function. It is well-known that continuity of $f$ implies sequentially continuity, while the reverse is in general only true if $X$ is first countable. Now, if $U\subset\mathbb{R}^{d}$ is some open set, then one defines the space of test ...

I currently reading Topology without tears by Sidney A. Morris and I encountered a question about discrete topology: Let $X$ be an infinite space and $T$ a topology on $X$. If every infinite subset of $X$ is in $T$, prove that $T$ is the discrete topology. Clearly we have to show that every sin...