news
What is a good way to think of Factor Groups?
I'm having a hard time thinking about factor groups. I just don't understand what notation like $\mathbb{Z}_{60}/\langle 12 \rangle$ means. Furt...
Apr 11, 2026
Archive: News
news
Convergence in probability implies convergence in distribution
A sequence of random variables $\{X_n\}$ converges to $X$ in probability if for any $\varepsilon > 0$, $$P(|X_n-X| \geq \varepsilon) \rightarrow 0$$ Th...
Apr 11, 2026
news
characteristic polynomial of graph
I have 2 questions about how to find the characteristic polynomial of some graphs. If G is a simple cycle with n vertices and n edges, $C_n$, I need to fi...
Apr 11, 2026
news
What is the difference between arithmetic and geometrical series?
What is the difference between arithmetic and geometrical series? Also what are they? How do they look like?
Apr 11, 2026
news
Value of $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}}$
Here is the question: $$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sq...
Apr 11, 2026
news
Notations: use of parentheses with "mod" and the "|" symbol
I'm working through a practice test with no available solutions, and I came across this question. Let $a,b,c,d,e$ be integers with $c>0$. Suppose ...
Apr 11, 2026
news
Eigenvalue degeneracy
Excuse my notation, it carries from physics. Suppose we have an operator $A$. The eigenvalue $\lambda$ is said to be $g$-fold degenerate if there exists g...
Apr 11, 2026
news
Number of ways to place 52 cards in a row the certain way
Suppose we have 52 cards: 26 black and 26 red ones. When the cards are placed in a row if between two given cards of one colour there is no cards of other...
Apr 11, 2026
news
Derivative of functions of the form $f(x)^{g(x)}$
I am trying to find the derivative of the function $h(x)=f(x)^{g(x)}$. I just wanted to be sure my derivation was correct: We proceed by using logarithmic...
Apr 11, 2026
news
Why the radius of curvature of a curve is independent of the choice of the coordinate axes.
The radius of curvature of a curve $y=f(x)$ is given by $\rho=\frac{(1+(\frac{dy}{dx})^2)^{\frac{3}{2}}}{\frac{d^2y}{dx^2}}$. I know this formula.Its deri...
Apr 11, 2026
