# All Questions

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### If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. [on hold]

If $ax^2 +bxy+cy^2+5x-2y+3$ divided by $x-y+1$ has remainder $0$, determine $a$, $b$, and $c$. I do not know how to approach this problem and would appreciate advice how to proceed.
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### Prove that the min and max of 2 continuous function are continuous

Prove that if $f$ and $g$ are continuous functions the so are min⁡{f(x),g(x)} and max⁡{f(x),g(x)} I know this is true when $f$ and $g$ are not intersect each other, then I can compare them. However, ...
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### Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
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### Representation of an invertible square matrix as a product of elementary matrices.

Suppose $A$ is an invertible $2 \times 2$ matrix. What is the smallest integer $n$ such that $A$ is a product of $n$ elementary matrices? My guess is that at most 4 elementary matrices are ...
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### Close-form solution for distribution of the stopping time for a path-dependent random process?

A time series $\{x_s\}_{s=1}^{t}$ is generated from $N(\bar{x},1/b)\ i.i.d.$. Parameter $\bar{x}$ is unknown, but we have a prior distribution $N(\phi_0,1/a)$ for it. Define conditional expectation ...
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### Formula for factorization of a Quadratic Equation?

To be clear I am looking for an equation to go from $$Ax^2 + Bx + C = 0$$ To $$(Dx + E)(Fx + G) = 0$$ And I need it to be able to be done in a computer as it will be going in my app. Thanks in ...
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### What is the difference between projected gradient descent and ordinary gradient descent?

I just read about projected gradient descent but I did not see the intuition to use Projected one instead of normal gradient descent. Would you tell me the reason and preferable situations of ...
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### Difference between Gradient Descent method and Steepest Descent

What is the difference between Gradient Descent method and Steepest Descent methods? In this book, they have come under different sections: http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf ...
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So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be: $w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t)$ $p(\mathbf{y} = 1 | \... 0answers 16 views ### Gradient descent with multiplication term Say I have the objective:$\arg \min_{R, T} \|y - RTx\|^2_2$where, R and T are matrices (not necessarily square) and y and x are known vectors. I wish to try and optimize R and T using Gradient ... 2answers 1k views ### Gradient descent with inequality constraints Suppose we are given a convex function$f(\cdot)$on$[0,1]. One wants to solve the following optimization problem: \begin{aligned} & \text{minimize} && \sum_{i=1}^n \... 1answer 58 views ### Lipschitz Number in Gradient Descent During gradient descent, if an objective function's value is greater than the previous iteration, would use of an orthogonal vector to the update vector be advantageous? Regarding trust regions, the ... 1answer 26 views ### Interpretation of points in covering spaces as homotopy classes of paths [on hold] Ifp:\widetilde{X} \to X$is a covering map,$y \in \widetilde{X}$determines a homotopy class of paths in$X$joining the base point$x_0$to the point$p(y)$. But a homotopy class of paths in$X$... 1answer 41 views ### Gradient descent with linear perturbation Given a convex, differentiable function$f$(from a Hilbert space to$\mathbb{R}$) with a minimum (say$x^*$), I know you can find$x^*$using gradient descent. Suppose now that you apply gradient ... 1answer 446 views ### Stochastic gradient descent for convex optimization What happens if a convex objective is optimized by stochastic gradient descent? Is a global solution achieved? 1answer 26 views ### Let$X$be a standard normal random variable. Then,$ P(X<0\mid |[X]| = 1)$is equal to? Let$X$be a standard normal random variable. Then,$ P(X<0\mid |[X]| = 1)$is equal to-$\frac{\Phi(1)-\frac{1}{2}}{\Phi(2)-\frac{1}{2}}\frac{\Phi(1)+\frac{1}{2}}{\Phi(2)+\frac{1}{2}}$... 1answer 148 views ### Gradient descent for periodic function Problem: minimize E=$\sum_{t=0}^T [ Y(t)-\sum_{k=0}^K(X(t+k)*cos(k*F+PHI)) ]^2$where F, PHI - has to be optimized; Y(t), X(t) 1D arrays are given; t=0...T; T=1000; k=0..K; K=10; I can use FFT ... 1answer 47 views ### Convergence of steepest gradient descent The description of gradient descent in Wikipedia says: $$x_{n+1} = x_n - \gamma_n\nabla F(x_n)$$ for$n = 0,1,2,...$Suppose that$x_n$converges to$x$. Then, is it always true that$\nabla F(x) = ...
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I would like to ask first if the second order gradient descent method is the same as the Gauss-Newton method. There is something I didn't understand. I read that with the Newton's method the step we ...
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### Implementing gradient descent based on formula

The gradient descent algorithm is given as : repeat { $$\displaystyle \theta_j := \theta_j - \frac{1}{m} \alpha \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)}) x^{(i)}_j$$ } Given these values : <...
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### How to precisely define a function that chooses randomly from a finite set?

Let $A = \{1, 2, \ldots, n\}$. I want to define a function that picks with uniform probability an element in $A$, so that $$f(A) = i \in A.$$ I don't know how to precisely define this mathematical ...
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### Gradient descent vs ternary search

Consider a strictly convex function $f: [0; 1]^n \rightarrow \mathbb{R}$. The question is why people (especially experts in machine learning) use gradient descent in order to find a global minimum of ...
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### A constrained gradient descent algorithm

I am looking for a way to find a solution to the constrained minimization problem using the gradient descent Algorithm. it follows ...
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### Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
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### Time bound for gradient descent

Have you seen any analytic bound on gradient descent (for number of iterations to achieve to $\epsilon$ error, and possibly based on the form of cost function and initial value)? Here is the problem; ...
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In order to find the local minima of a scalar function $p(x), x\in \mathbb{R}^3$, I know we can use the gradient descent method: $$x_{k+1}=x_k-\alpha_k \nabla_xp(x)$$ where $\alpha_k$ is the step size ...
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### Why does gradient descent work?

On Wikipedia, this is the following description of gradient descent: Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
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### Is it possible to solve for $x$ using the lambert W function in the expression ${\ln\left(x\right)}=(t-x)^2$?

${\ln\left(x\right)}=(t-x)^2$ $\pm\sqrt{\ln\left(x\right)}+x=t$ $\mathrm{e}^{\sqrt{\ln\left(x\right)}+x}=e^t$ And that is as close as I can get it to the form $x\mathrm{e}^x$. What do I do next? ...
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### why value of Trigonometric ratios of angle and its reference angle are same?

I'm learning Trigonometry right now with myself and at current about how to find trigonometry ratios of angles greater than $90^\circ$. I came to know that for finding trigonometric ratio of these ...
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Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and $... 0answers 16 views ### How can I use this initial condition for the heat equation How can I use the following initial condition for a partial differential equation describing heat diffusion?$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0.... 2answers 24 views ### basic Quantile proves Let this be my definition of a quantile funktion. X is a real-valued random variable. And let F be it's distribution function. then \begin{align*} F^{-1}(a):=\inf\{x\in \mathbb{R}: F(x) \ge a\}. \end{... 2answers 17 views ### Does1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$hold? I am wondering does$1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$? Even if$X_1$and$X_2$are dependent? 1answer 75 views ### Does$\mathrm A \mathrm A^T \succeq x^2 \mathrm I$imply$\frac{\mathrm A + \mathrm A^T}{2} \succeq x \mathrm I$? Let$A $be an$n \times n $matrix such that$AA^T \geq x^2I, x\geq 0 $, which means that the matrix$AA^T-x^2I$is positive semidefinite. Can we show that$(A+A^T)/2 \geq xI$? Thanks 1answer 18 views ### Why is inradius$\times$surface area equal to thrice the volume? "Inradius" means radius of largest sphere that is tangent to all faces. For example: Cube - Surface area$= 6a^2$, Inradius$= a/2$, Volume$= a^3$. Sphere - Surface area$= 4\pi r^2$, Inradius$= ...

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