# All Questions

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### Calculating arg of a function (Algebra)

I am trying to show Eq. (2.10) in the following thesis: https://document.chalmers.se/download?docid=721526871 I thought this would be easy by following the advice in the thesis and using Eq. (2.9) to ...
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### Show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$

To show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$ Let $u_r, u_{r+1}$ represent the $r^{th}$ and $(r + 1)^{th}$ terms of the expansion; then ...
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### Strings in a dictionary. A partial order, strict order, and total order?

Question: The domain is the set of all words in the Engilsh language (as defined by, say, Webster's dictionary). Word $x$ is related to word y if $x$ appears as a substring of y. For example, "ion" ...
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### derivative of gradient involving inverse of matrices

I need to take three partial derivatives of this squared mahanalobis distance with respect to these three matrices: $Q, A,$ and $S$ $$(x+Ab)^T(A^TQA+S)^{-1}(x + Ab)$$ $x$ and $b$ are vectors of ...
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### Construction of graph Laplacian

I have a weighted undirected graph, and all the edge-weights are non-negative. According to the definition of the graph Laplacian matrix, $L=D-W$. In literature, I found that $D$ is known as degree ...
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### Can every torsion-free nilpotent group be ordered?

I know that a torsion-free abelian group can be ordered and have done two proofs for that too. But the next two question that popped up in my mind were- Can every torsion-free nilpotent group be ...
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### the bump function is smooth

consider the function \begin{align*} f(x)=&\left\{ \begin{array}{ll} e^{\dfrac{-1}{x}} \quad & x>0\\ 0 \quad & x\leq0. \end{array} \right. \end{align*} I want to prove that that ...
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### $l^p$ space not having inner product

I know that $l^2$ space is a Hilbert space. But for other $l^p$ spaces, where $p\geq1$, I have to show that they do not satisfy the parallelogram equality. But, I can't find appropriate sequences ...
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### Using a markov chain to calculate the expected value of conditional/constrained choices (TopCoder PancakeStack)

I've been working on a programming challenge (link) where an expected value is calculated. Recently I learned about Markov chains and successfully applied them to solving a set of problems, but the ...
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### Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent iff $a_0,a_1,a_2,\ldots,a_n$ are nilpotent

Let $R$ be a commutative ring. Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent if and only if $a_0,a_1,a_2,\ldots,a_n$ are nilpotent. Now since $f(x)$ is nilpotent then ...
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### Proof check, showing pointwise convergence

My problem is this: For $x \in [0,\frac{\pi}{2}]$, $f_n(x) = \frac{nx}{1 + n\sin(x)}$ Find the pointwise limit of $(f_n)$ for all $x \in [0, \frac{\pi}{2}]$ I am not sure if the way I constructed ...
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### Orthogonal group acts on vector field

I recently had an exam, yesterday acctually, and there was a question that stumped me. The orthogonal group $O(n)$ acts on $\mathbb{R}^n$ by matrix multiplication, show that the orbit space is ...
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### Sum of $m\leq 300$ such that if $2013m$ divides $n^{n}-1$, then $2013m$ also divides $n-1$

Find the sum of all the integers $m$ with $1≤m≤300$ such that for any integer $n$ with $n≥2$, if $2013m$ divides $n^{n}-1$, then $2013m$ also divides $n-1$. Unfortunately I cannot think of ...
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### Calculating Percentages/Probabilities of a Specific Scenario.

I've been trying for a few hours now to wrap my brain around, what at first, seemed like a simple concept. I've tried so many different things and ways of constructing scenarios so I can figure out ...
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### Reducible cubic surface are always singular.

I want to prove that Any reducible cubic surface are always singular. A possible way may be to take a look at the intersection of the irreducible components. But I don't know how. Thanks for any ...
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### What is orthogonal group $O(1)$?

I know that $O(2)$ is the group of 2x2 orthogonal matrices, but how can we extend the meaning of group and orthogonal to $O(1)$?
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### Show that if for $a \in \mathbb{Q}$ with $0 = f(a) \in \mathbb{Z}[x]$ for a monic polynomial, then $a \in \mathbb{Z}$

I. Show that if for $a \in \mathbb{Q}$ with $0 = f(a) \in \mathbb{Z}[x]$ for a monic polynomial, then in fact $a \in \mathbb{Z}.$ II. The set of algebraic integers of $\mathbb{Q}$ is ...
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### Closed immersion and complete linear systems

Let $X$ be a local complete intersection subscheme in $\mathbb{p}^n$ for some integer $n>0$. Denote by $i:X \to \mathbb{P}^n$ the induced closed immersion, ...
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### what is the difference between Hilbert function and Hilbert-Samuel function of a ring or a module?

What is the difference between Hilbert function and Hilbert-Samuel function of a ring or an module?
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### Harmonic functions locally null on connected open set

Let $u$ be a harmonic function on $U$ connected open set of $\mathbb{R}^n$ and suppose there is a open set $V\subset U$, such that $u(x)=0$ for every $x\in V.$ Show that $u=0$ in $U$. So, I tried to ...
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### What is the volume and surface area of the 1-Sphere?

I am reading a post on here that mentioned something about the 1-sphere. I know that a 2-sphere is a circle, and 3-sphere is a volume, but what is this 1-sphere and how do you calculate the volume and ...
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### Irreducibility criteria for a class of polynomials

I would like to prove some univariate polynomials $f(x)$ are irreducible on $\mathbb{Z}[x]$. My polynomials have the following form: ...
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### Area of a triangle with one given measurement

The hypotenuse of an isosceles right triangle is 4 centimeters long. How long are its sides? Round your answers to two decimal places.
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### Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
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### $f '(x) = -f(x)$ and $f(1) = 1$, Solve for $f(2)$

I am honestly not even sure how to start this problem... My first thought was that $f(2) = 2$ ... But now I don't even know where to go from there.
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### Could someone please explain double-angle identities?

I don't understand how to do maths, mostly because I don't understand why formulae work they way they do, or the reasoning behind equations, etc. I tried to explain the $\sin(2\theta)$ double-angle ...
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### X - Y in a finite set

Question: The domain is the power set of a finite set. $X$ is related to $Y$ if $X - Y$ is not empty. Is it a partial or strict order? If it is a partial order or strict order, is it also a total ...
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### Is the left translation $T_a(x) =ax$ a homomorphism?

I apologize if this is a super basic question but I was reading Lang's undergraduate algebra book and it says that the following function is a homomorphism: $$T_a(x) = ax$$ The way I would check if ...
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### Distance from $f(0)$ to the boundary of $D$ if $f$ maps open unit disk to $D$ conformally

Let $f(z)$ be a conformal map from the open unit disk onto $D$, which is a domain. I would like to show that the distance from $f(0)$ to the boundary of $D$, denoted $\partial D$, is given by ...
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### How to solve this system of conics?

I am currently trying to figure out how to solve the following systems of conics: $\frac{(x+1)^2}{16} + \frac{(y-1)^2}{81} = 1$ $x+6=\frac{1}{4}(y-1)^2$ How would I find the four points that these ...
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### How to visualize orientation of 3d objects

The way I visualize orientations of $1$- and $2$-dimensional objects is by an ant walking along a path. For a $1$d object (like a line/ line segment/ etc), just place the ant on the line and confine ...
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### Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,…,x_n]$ is generated by $n$ linear polynomials ?

Without using Hilbert's Nullstelensatz , can we directly prove that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ is of the form $\langle x-a_1,x-a_2,...,x-a_n \rangle$ ? It is easy to prove it ...
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### Determining an inverse limit - How trivial is this?

I dont understand how the author is determining the inverse limit $\lim_{\leftarrow}U/U_n=U$? How one can see this?
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### Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
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### Is the powerset of the reals any “more uncountable” (in some sense) than the reals are?

I know that $\mathbb{N}$ is countable and has cardinality $\aleph_0$, and that $\mathbb{R}$ has cardinality $2^{\aleph_0} = \text{C}$ and is uncountable. Are sets with cardinalities greater than ...
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### Diophantine solutions to a large geometric figure

I have a related question to one I've read today, see: Integer solutions to $2x^2+5x+y^2=19$ The integers solution are part of an ellipse, with an obvious finite number of $x$. What I would like to ...