1
vote
3answers
77 views

A question regarding $\lim\limits_{n\to\infty} n(n+1)(n+2)\dots(n+r)$.

We know that $\lim\limits_{n\to\infty} \frac{n(n+1)(n+2)\dots(n+r)}{n^{r+1}}$, where $r$ is a finite positive integer, is $1$. Hence, $\lim\limits_{n\to\infty} ...
1
vote
0answers
19 views

ortogonal projection of the null vector

Let $H$ a Hilbert space, $V$ a subspace of $H$, and $P:H\to V$ a projector operator ($P^2=P$ and for any $v\in V$, $(I-P)(v)\in V^\perp$). Then $P(0_H)=0_H$. This is because $0_H=0_V\subset V$ and any ...
0
votes
1answer
25 views

What's the right way to look at this permutations problem?

I'm having trouble with this problem. It seems very simple. Here it is exactly: Obtain the number of three–letter permutations possible for the group of letters shown. S, E, V, E, N ...
8
votes
1answer
319 views

Stochastic gradient descent for convex optimization

What happens if a convex objective is optimized by stochastic gradient descent? Is a global solution achieved?
5
votes
1answer
208 views

Invariant subalgebra of a Lie algebra under an automorphism of the Dynkin diagram

Of all the automorphisms of a (finite-dimensional, semisimple) Lie algebra which induce a particular automorphism of its Dynkin diagram, is there a particular one which is "nicer" than the others? ...
3
votes
0answers
108 views

Can we have an isometric embedding of this metric space into an Hilbert space?

A metric space (from this Q&A), is defined below. I'd like to know if its possible to have an isometric embedding of this metric space into an hilbert space? As per Schoenberg theorem $-d^2(x,y)$ ...
2
votes
4answers
73 views

Limit of the function $\lim_{x\to0}\frac{\sin 2x + a\sin x}{x^3}$?

Question If for some real number $a$, $\lim_{x\to 0}\frac{\sin 2x + a\sin x}{x^3}$ exists, then the limit is equal to: Here what i have done since it is of $0/0$ form applying L' Hospital's ...
2
votes
2answers
1k views

Equation of circle tangent to y=x and x-axis with radius 5

Find the equation of a circle in the 3rd quadrant that is tangent to the line y=x and the x-axis, with a radius of 5. One way I thought of doing it was letting the center point of the circle be ...
-1
votes
1answer
63 views

Ruler equation example

I am having trouble understanding this example in my book. First let me define ruler: Let $l$ be a line in an incidence geometry. Assume that there is a distance function $d$. A function $f: l \to ...
1
vote
1answer
56 views

I have a question about properties of series.

Let $x_n$ be a sequence of real numbers. Show that if $\sum_{n=1}^{\infty}x_n $ converges, then $\lim_{n\rightarrow\infty}\sum_{k=n}^{\infty}x_k=0.$ This is what I have done. We have by the Cauchy ...
0
votes
2answers
29 views

Something related to maximal ideal

Let $M$ be a maximal ideal in the polynomial ring $\Bbb C[x]$. Prove that there is some $a \in \Bbb C$ such that $M$ is the ideal generated by $x-a$ This question I have no idea how to prove, hope ...
1
vote
1answer
58 views

Denseness of normal and non-normal numbers

How to prove that normal numbers are dense ? I have read in a book that this set has full measure. so dense. Then how to argue for the no-normal numbers. They also turn out to be dense. How to argue ...
3
votes
1answer
58 views

Showing that $|P|$ divides $|A|!$ where $P$ is a $p$-group and $A$ is maximal abelian normal

I'd appreciate verification of the following proof. This is the part following what I asked in this question. Thanks. If $A$ is as in the linked-to question above (that is, $A$ is maximal among ...
2
votes
1answer
60 views

How to prove that the operator $(\lambda I-A)^{-1}$ exists?

Let $A:H^1(\mathbb{R})\to L^2(\mathbb{R})$ be the operator given by $Aw=w_x$, where $w_x$ denotes the weak derivative of $w$. I need help to prove that $(\lambda I-A)^{-1}$ exists and is bounded for ...
3
votes
1answer
1k views

Ring of formal power series over a field is a principal ideal domain

If $K$ is a field, any non-zero ideal in the ring of formal power series $A=K[[X]]$ is of the form $AX^n$ with $n\geq 0$, so $A=K[[X]]$ is a principal ideal ring. I can't see why. Usually when we ...
3
votes
2answers
251 views

Characteristic Function and Random Variable Transformation

Let $X$ be a random variable, and let $\phi_X(t)$ be its characteristic function. Let $Y = f(X)$ be a transformation of the random variable $X$ where $f$ is increasing and one-to-one. Is there a ...
6
votes
1answer
255 views

Predicting digits in $\pi$

Is it possible to predict next digit in $\pi$ using $N$ previous digits, so on and so forth? Or is this impossible because it's irrational? Basic assumption is that the person doesn't know a ...
1
vote
1answer
35 views

Help with vectorial analysis exercise

Let $D(0,r) := \left\{ {x \in \mathbb{R}^n: \|x\| \leq r }\right\}$ and $f:D(0,r) \rightarrow \mathbb{R}$ be a continuous differentiable function in the interior of $D(0,r)$. I'm trying to show ...
0
votes
1answer
113 views

Fundamental lemma of calculus of variations, gradients

Let $D \subset \mathbb{R}^d$ be a smooth bounded domain. Let $C_c^\infty(D)$ denote smooth and compactly supported functions on $D$. Let $f \in [C_c^\infty(D)]^d$ be a smooth, compactly supported ...
0
votes
1answer
74 views

Which of the following form an ideal in this ring?

Let $C(R)$ denote the ring of all continuous real-valued functions on with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal in this ring? The ...
1
vote
3answers
166 views

convergence of a series in real analysis proof

If $\sum_{n=1}^{\infty}a_n$ converges, prove that $\sum_{n=1}^{\infty} \dfrac{n+1}{n}a_n$ converges. It's probably pretty trivial, but I have been staring at it for a while and cannot make any ...
1
vote
1answer
429 views

If augmented matrix has zero row, does it mean infinite solutions?

Given an augmented matrix in REDUCED ROW ECHELON FORM representing a general system of linear equations (does not have to homogeneous), if the augmented matrix has a zero row, does it imply that the ...
0
votes
4answers
115 views

If $(x - 2)$ is a factor of $x^3 + ax^2 -6x -4$, then find $a$.

If $(x - 2)$ is a factor of $x^3 + ax^2 - 6x - 4$, then find $a$. This is regarding polynomials. The answer is $a = 2$. Could someone please provide the working out and help me out on this please. ...
1
vote
1answer
100 views

Numerical Analysis - Richardson Extrapolation

Question: Suppose that N(h) is an approximation to $M$ for every $h > 0$ and that $M = N(h) + K_1 h + K_2 h^2 + K_3 h^3 +\cdots$, for some constants $K_1, K_2, K_3,\cdots$. Use the values $N(h), N( ...
1
vote
1answer
56 views

measurable subset of possibly non-measurable set.

Let $X$ be a Polish space. Let $T$ be a possibly uncountable set of Borel probability measures on $X$. Suppose there exists a subset $A$ of $X$ with the following property: For each $\mu\in T$, there ...
2
votes
1answer
49 views

Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$.

Question: Find the volume of the solid bounded by $z=x^2+y^2+1$ and $z=2-x^2-y^2$. Setting the 2 equations equal w.r.t. $z$, $x^2+y^2+1=2-x^2-y^2 \rightarrow x=\pm\sqrt{\frac 12-y^2}$ Therefore the ...
1
vote
2answers
65 views

Evaluating infinite summations

How would I evaluate the following summations? $$\sum_{i=0}^{\infty}\frac{i^2}{4^i} $$ $$\sum_{i=0}^{\infty}\frac{i^N}{4^i} $$
3
votes
1answer
151 views

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \pmod 6$

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \mod 6$ Let $r$ be a root, real or complex, of multiplicity 2 of $f(x)$. Then, by the ...
1
vote
1answer
478 views

Riemann integral proof $\int^b_a f(x) \, dx>0$

Prove that if $f$ is a continuous real valued function on the interval $[a,b]$ such that $f(x)\ge 0$ for all $x\in [a,b]$ and $f(x)>0$ for some $x\in[a,b]$ then $\int^b_a f(x) \, dx >0$. The ...
1
vote
0answers
101 views

Statistics - Uniform sample vs. Representative sample

I have a question concerning two different samples, with the first being more uniform that the second. a) Chance errors are likely to be smaller... using the first set of subjects using the second ...
1
vote
1answer
60 views

how to simplify the following expression: $-(4i)^3$

How to simplify the following term $$-(4i)^3$$ I have tried solving it the following way: taking the square root of $-16$ to the third power and taking the negative of that. I am getting an answer ...
1
vote
1answer
52 views

Application of l'Hospital rule

I have a smooth function $f(x)$, with $f(0)=0$ and $f \in C^{\infty}([0,1])$, on $[0,1]$ and I'm trying to prove that $\frac{f(x)}{x}$ is continuous on the complete interval and thus belongs to ...
0
votes
1answer
43 views

Coset of the Abstract index group of a Banach Algebra?

I'm studying on the book of Douglas: "Banach algebra techniques in operator theory" and there is a passage I don't understand, and I hope you can give me a hand. "A continuous function $f$ from $X$ ...
3
votes
0answers
83 views

Find n's for which $P_n$ is prime.

Consider the numbers $P_n=(3^n-1)/2$. Find $n$'s for which $P_n$ is prime. Conjecture: If $n \equiv 1 \mod 6$, and $n$ is prime, then $P_n$ is prime. I have tried proving this by contradiction but ...
2
votes
3answers
380 views

Product of disjoint cycle

I've found the question to find product cycle of let be $\phi$ = (2, 4, 9, 7,) (6, 4, 2, 5, 9) (1, 6) (3, 8, 6) in S9. I know how to express the permutation of S9 as product of disjoint cycle, like ...
1
vote
3answers
77 views

Calculate the sum of $\sum^{\infty}_{n=0} (-1)^n \frac{n+1}{3^n}$.

Currently going through old Analysis I - exams and I'm having a problem witht his one: Calculate the sum of the series: $$\sum^{\infty}_{n=0} (-1)^n \frac{n+1}{3^n}$$ What I did was firstly to ...
3
votes
1answer
83 views

Is there a name for an algebraic structure like this?

I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse. Is there something similar except with no requirement for an additive inverse. For ...
5
votes
1answer
58 views

Prove that $n$ is even and $|A| \in \{-1,1 \}$

Let $A \in M_{n} (\mathbb R)$, such that $A^2=-I_{n}$. Prove that $n$ is even and $|A| \in \{-1,1 \}$. I started by compute the determinant of both sides: $A^2=-I_{n}\Leftrightarrow$ ...
0
votes
1answer
199 views

Probability of one, or both happening

You have two flowers, a yellow one and a red one. The probability that each one is pollinated during its two week blooming period is .8. The probability that the red one is pollinated given that the ...
0
votes
1answer
45 views

Generalized Solution to DIfferential

For $x'=\sqrt{|x|}, x(0)=0$ the solution is $x=0,x=\frac{1}{4}t^2 sign(t)$. I'm supposed to generalize the solution and show that this inital-value problem has a two-parameter family of solutions. I ...
1
vote
2answers
143 views

$z^2=-3-3i$ solve for $z$

Use De Moivre's theorem to solve the equation $z^2=-3-3i$. (Give your answers in polar form) Can you please explain why there are two answers? I cannot seem to understand why. By the way, the ...
0
votes
1answer
143 views

Write this expression with series & sum in terms of a single variable

I know this is very specific, but is there a way to represent the expression $$\frac{3}{5} \sum_{n = 0}^\infty \left(\frac{2}{5}\right)^n \sum_{j = 0}^n {n \choose j} \delta_{2j - n, k}$$ in terms ...
1
vote
1answer
52 views

Compute $\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle\int_0^1\int_1^2\frac{y}{x+y^2}dydx=\int_0^1\int_1^2y(x+y^2)^{-1}dydx$ How do I integrate the ...
4
votes
1answer
42 views

Is there a name for this family of curves?

I saw a space curve defined as the following before (but I don't remember the reference): $$ \alpha_{p,q}(t)=\{\left((2+\cos pt)\cos qt,(2+\cos pt)\sin qt,\sin pt\right)|t\in{\Bbb R}\} $$ where $p$ ...
4
votes
2answers
77 views

does a prime in an extensions of integral domains remain radical?

Let $R\subset R'$ be an extension of integral domains. So we have an inclusion map $i:R\hookrightarrow R'$. Let $\mathfrak{p}\subset R$ be a prime ideal. We know that $\mathfrak{p}^e$ (generated by ...
3
votes
1answer
43 views

How to find the number of transposition

I just learning the abstract algebra now, I'm stuck to find how many transpositions can be made from $(1\ 8)(2)(3\ 6\ 4)(5\ 7)$?
1
vote
2answers
60 views

A question in linear transformation

Let $V$ be a finite dimensional vector space and $T:V \to V$ be a linear transformation. Prove there exists a linear transformation $S:V \to V$ such that $TST=T$ I feel this should not be a hard ...
0
votes
1answer
155 views

A vector is a linear combination of two other vectors. Find all scalars c.

Q: The vector $[13, -15]$ is a linear combination of the vectors $[1, 5]$ and $[3, c].$ Find all scalars c. My Approach: $$[13, -15] = x[1, 5] + y[3, c]$$ $$13 = x + 3y \to (1)$$ $$-15 = 5x + cy ...
1
vote
0answers
84 views

Continuity of $e^{-1/x^2}$

Let $$f(x)=\begin{cases}e^{-1/{x^2}} & x\neq0\\ 0 & x=0\end{cases}$$ Show $f(x)$ is continuous Show that the $n$-th derivative is continuous. I've seen this a couple of ways but I'm ...
-2
votes
2answers
69 views

Is there a way to derive the percentage to add, from the percentage to subtract?

Here is an example: 24.30 + 66.6% = 40.50 40.50 - 60% = 24.30 or (24.30 + 66.6%) - 60% = 24.30 I know if I add $66.6\%$ to $24.30$, I get $40.50$ and if I ...

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