0
votes
0answers
2 views

directional derivative problem

for a point M(4,1) and a function $z = x * y^2 - (x^2/y^3)$ I was tasked with finding a directional derivative in the direction which creates a 30 degree angle with the X axis....I find it a little ...
0
votes
0answers
5 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
1
vote
0answers
9 views

I want to learn mathematics to extend myself.

I am a middle school student that is highly fascinated by mathematics and its elegance. However, I have found that to appreciate a lot of mathematics a lot of knowledge is required. Currently, I can ...
0
votes
0answers
7 views

Trace in Einstein notation

I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if: 1) You could ...
0
votes
2answers
6 views

Mistake in proof of sum of divisors function $\sigma(n)$

The proof derives the correct result, but I cannot see how the first equality is correct. To begin we use the formula $\sigma(n)=\sum_{d\mid n}d$ This is the first step in the proof: $$\sum_{1\leq ...
0
votes
0answers
4 views

Commutation of Convolution, Restriction and Differentiation

Let $B$ be the open unit ball in $\mathbb R^n$ centered at zero and let $K=\bar{B}\cap (\mathbb R^{n-1}\times\{0\})$. Suppose you are given $u\in C^{1,\alpha}(B)$ such that $u|_K=f\in C^2(K)$. For a ...
0
votes
0answers
10 views

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions :

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions : $ f(x+t,y) = f(x,y) + ty~~;~~f(x,t+y) = f(x,y) + tx~~;~~f(0,0) = K$. Then $\forall ~~x,y \in \mathbb R, ...
-2
votes
0answers
15 views

How to define the isomorphism?

TO show that $R[x]/<x-1> \cong R$ we define map: $\varphi$ : $R[x]\rightarrow R$ , defined by $\varphi(f) =f(1)$ TO show that $R[x]/<x> \cong R$ we define map: $\varphi$ : ...
0
votes
0answers
5 views

“p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
0
votes
0answers
6 views

The moduli space of stable maps

I am reading Katz' book Enumerative Geometry and String Theory. I have a few questions regarding the moduli space of degree $d$ genus $0$ stable maps into $\mathbb P^n$, denoted ...
0
votes
1answer
8 views

Changing the basis of a transformation

Given $T: \mathbb{R}^2 \to \mathbb{R}^2 : T\begin{bmatrix} x \\ y \end{bmatrix} \to \begin{bmatrix} 2x+y \\ x-3y \end{bmatrix}$ with standard basis $\mathcal{B}$ and basis ...
0
votes
0answers
7 views

How to formulate the product of two generating functions without their final terms?

I know that if we have two generating functions like so: $A(z) = \sum_{n=0}^\infty a_nz^n$ and $B(z) = \sum_{n=0}^\infty b_nz^n$ Then we can write $A(z)B(z) = \sum_{n=0}^\infty(a_0b_n + a_1b_{n-1} ...
0
votes
0answers
25 views

Please help solve this equation

$$3.8r - 0.057r^2 + 0.00038r^3 + 0.00000095r^4 = 95$$ Please help.
1
vote
0answers
7 views

Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$

Actually I need only the $res(f;0)$ where $f = e^{e^{\frac{1}{z}}}$ I thought of finding the Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$ Any other Ideas if you have ?
0
votes
0answers
3 views

Suspension flow and topological equivalence(s)

Let $M$ be a compact smooth manifold, $\tau:M\to\mathbb{R}_{\geq 0}$. Let $f:M\to M$ is a surjective piecewise-smooth map. There is a standard construction of suspension allowing to extend $f$ to a ...
0
votes
4answers
21 views

Characteristic of a field

If $K$ is an infinite field, then $Char K = 0$ but the reverse is not sure. Examples of $Char K = 0$ but not infinite field $K$?
1
vote
1answer
10 views

Composition of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a group and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
-1
votes
4answers
14 views

Why is $\lim_{x\to e^+} (\ln x)^{1/(x-e)} =e^{1/e}$

$$\lim_{x\to e^+} (\ln x)^{1/(x-e)} =e^{1/e}$$ I started by taking ln on both side, which brings the power down, by I tried using L'Hopital, but it doesn't seem to work.
0
votes
0answers
10 views

Find the minimizer of the functional

Find the minimizer of the functional $ l= \int u(t) $ with $u(1)=u(1)=0 $ subject to $g=\int $$\sqrt{1+u'(t)} dt $ I want to solve it using E-L equation first $l^*=l- \lambda g$ then i used e-l ...
0
votes
0answers
6 views

relative homotopy groups

I study relative homotopy groups and I have a question: Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which ...
0
votes
0answers
15 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
0
votes
3answers
44 views

What is $n+1$ factorial or $(n+1)!$?

I have to prove by induction but first I want to know what $(n+1)!$ is? I know that $n!=n \cdot (n-1) \cdot (n-2)...$
0
votes
0answers
7 views

Finding #Groupoid like subsets

Given $S=\{x \in \mathbb{R}: 1 \leq |x| \leq 100\}$, find all subsets $M$ of $S$ such that for all $x$, $y$ in $M$, their product $xy$ is also in $M$. My attempt: If any number with magnitude ...
5
votes
1answer
41 views

Bourbaki and set inclusion

Which notation ($\subset$ or $\subseteq$) was preferred by Bourbaki for set inclusion (not proper)? A side question: Was the notation for subset one of the many notations invented by Bourbaki?
0
votes
1answer
13 views

Poisson Probability (Shopkeeper Sales)

SOLUTIONS: (A) 0.1804 (B) 0.0166 (C) 0.3233 Mean = 2/7*5 (a) x = 3 (b) x > 5 I'm still unsure how to approach each question, because I still get the wrong answers.
1
vote
0answers
12 views

Find the equation of intersection of a torus and a circle on a plane without using iterative methods.

I have the equation of a circle on the plane (where $p_0$ is the centre, $\theta$ is the angle of the circle, and $w$ and $v$ are pair of orthogonal vectors from $p_0$ to the circle (having equal ...
0
votes
1answer
21 views

Bitwise ops - The relationship between $a$, $b$, $a \wedge b$, $a \vee b$ and $a \oplus b$

In computer programming, the term bitwise operation is used to denote the use of boolean operators (and $\wedge$, or $\vee$, exclusive or $\oplus$) on corresponding bits of two numbers. Bits, in this ...
-3
votes
0answers
13 views

How to find second and third derivative of Probit likelihood function

I am using Probit likelihood function for a classification problem. I need second and third derivatives of the log likelihood function with respect to f. The likelihood function is logΦ(y.*f). The ...
0
votes
1answer
22 views

Line Integral with Large Radicals

The integral of x^(1/33)+y^(1/27)+z^(1/39) of the line segment (161, 283, 73) to (168, 361, 145). I tried to do it on my own but my answer (-2873.78) seems extremely wrong.
-3
votes
0answers
14 views

Wilson score interval proof / derivation

I need to get from here to here Can anyone help me?
5
votes
2answers
53 views

Find the limit of $\frac{1}{n+1}-\frac{1}{n+2}+…+\frac{(-1)^{n-1}}{2n}$ as $n\to\infty$

Find the limit of $\frac{1}{n+1}-\frac{1}{n+2}+...+\frac{(-1)^{n-1}}{2n}$ as $n\to\infty$. In the previous part of the question I was asked to prove that as $n\to\infty$ ...
0
votes
0answers
9 views

Dual bundle of a stable vector bundle.

Given a vector bundle $E \to X$ over a complex curve $X$, let its rank and degree be $r$ and $d$ respectively. Then the slope of $E$ is defined to be $ \mu(E):= \frac{d}{r}. $ $E$ is defined to be ...
1
vote
1answer
23 views

If $(3)=\mathfrak p_3\mathfrak p_3'$ then we can write $\mathfrak p_3=(3,1+\sqrt{17})$

Why if $(3)=\mathfrak p_3\mathfrak p_3'$ in $\mathbb Z[\sqrt{-17}]$ then we can write $\mathfrak p_3=(3,1+\sqrt{-17})$ I saw here in the first exercise that the author already knows how to ...
1
vote
2answers
13 views

Jacobi's Derivative of the Determinant

I've been given the following theorem for the derivative of the determinant of a matrix: "Let $A\in \mathbb{R}^{n\times n}$ be a square matrix. Then the Fréchet derivative of det$: ...
0
votes
0answers
7 views

modules operation with different moduli

What value it will give $((a^b \bmod n_1)\cdot(c^d \bmod n_2))^f$; it $f$ will multiplied with both the power. There is some property of modulo operation with different moduli.
-3
votes
0answers
15 views

Which of the following is/are always correct.

Let $A$ be a $4\times7$ real matrix and $B$ be a $7\times4$ real matrix such that $AB=I_4$, where $I_4$ is the $4\times4$ identity matrix. Which of the following are is/are always true ...
1
vote
0answers
5 views

Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic number $\beta$ Pisot-like if $|\beta| > 1$ and all its conjugates lie inside the complex unit circle (here $|\cdot|$ is the usual absolute ...
0
votes
1answer
24 views

Induction proofs - methodology and examples

Forenote: I have seen many induction related questions, and very often the problem lies within the OP's lack of a proper methodology (or style) in writing the proof whereas the answers focus on the ...
0
votes
0answers
9 views

Depths of top-level multiplication algorithms

I've seen that the depth of the Cantor/Kaltofen algorithm is in $O(\log n)$. Are the operations for this complexity undifferentiated ? Or this complexity is in terms of multiplications only ?
1
vote
2answers
26 views

Evaluate if $f_{_n}$ converge uniformly or not

We have $f_n:[1,2]\to \mathbb{R},\:f_n(x)=\frac{x^n}{x^n+1}$ and we have to see if the convergence is uniform or not. From what I understand we need to prove that $\lim _{n\to \infty } ...
0
votes
0answers
17 views

Rational normal curve of degree 4

Consider the rational normal curve of degree 4. It can be realized as the intersection of a certain number of quadrics. Usually in examples we consider those quadrics to be singular. However, can at ...
0
votes
0answers
26 views

Forbidden Positions

How would you go about doing forbidden positions on a rectangular chessboard? For example, we have a chessboard with $9$ rows and $3$ columns. The first column has $2$ forbidden positions ($X$) in ...
0
votes
1answer
14 views

pdf: What is the distribution of aX when X ~ Binomial / Gaussian

Question When $X$ is distributed as binomial or Gaussian, is $aX$ equivalent to some famous distribution? Here, $a$ is a real and positive number. Background I know a general formula giving $aX$'s ...
0
votes
2answers
20 views

Suppose $a \in \mathbb{C}$, $|a| < 1$, and $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. How to prove dependence of $|f(z)|$ on $|z|$? [duplicate]

Let $a \in \mathbb{C}$, $|a| < 1$. Also let $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. I am asked to prove that $|f(z)| < 1$ if $|z| < 1$ and that $|f(z)| = 1$ if $|z| = 1$. What is a good ...
0
votes
0answers
11 views

How to compute this convolution in matlab?

The equation is as above, where $f_x$ and $f_y$ refer to axises in frequency domain, $x$ and $y$ refer to axises in space domain and $F$ refers to Fourier transform. My main problems lie on 1) how ...
1
vote
0answers
22 views

Testing integral covergence

For which $\alpha$ this integral coverge (in Riemman sense) ? $$\int_0^\infty \frac{1}{1+ \sin^2(x) \cdot x^\alpha}\, dx $$
4
votes
1answer
19 views

Short exact sequence is split iff contractible

Let $0\rightarrow A\overset{f}{\rightarrow} B \overset{g}{\rightarrow} C\rightarrow 0$ be a short exact sequence in an abelian category. I am trying to prove this SES is contractible iff it is split. ...
2
votes
1answer
37 views

If $d(x_{n+1},x_n)<\frac{1}{n+1}$ then the sequence $\{x_n\}$ is a Cauchy sequence OR not?

Let , $(X,d)$ be a metric space and $\{x_n\}$ be a sequence in $X$. We have, $$d(x_{n+p},x_n)\le d(x_{n+p},x_{n+p-1})+...+d(x_{n+1},x_n)$$ $$\le \frac{1}{n+p}+...+\frac{1}{n+2}+\frac{1}{n+1}\to ...
1
vote
1answer
17 views

Does there exist such a set?

$R$ is a set of real numbers, m is the Lebesgue measure on $R$. Does exist a nowhere dense subset $A\subseteq R $, such that $m(A)=+\infty$? I know that if $A$ is of the first category, there exist ...
0
votes
1answer
29 views

A Question about Algebraic Integers

I need to prove a lemma, which uses the following fact: If $\alpha$ is an algebraic number of degree $m$ over $\mathbb{Q}$. Define $\mu(\alpha)$ to be max$\{ |\alpha_i| \}$, where $\alpha_1=\alpha$, ...

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