0
votes
0answers
4 views

Is the length of one cathetus in a right triangle with equal catheti 2?

Look at your watch now. Now try to remember that time. By the time you have finished this sentence that time has probably increased by a couple seconds. Now you may be asking yourself "is this ...
1
vote
0answers
5 views

Find the “surface vertices” of a collection of points.

I am currently doing some experiments in order to simulate liquids. I have a collection of 3D points that interact with each other to form a body of water. I would like to form a mesh from these ...
2
votes
2answers
27 views

Wouldn't each addition take time $O(n)$?

I am going over the asymptotic runtime of regular matrix multiplication. Here is a lecture slide I am referencing(too much to type out, shown below), from Algorithms Everything makes sense up ...
0
votes
0answers
10 views

Integration in complex measure

Let $v$ be a complex measure in $(X,M)$. Then $L^{1}(v)=L^{1}(|v|)$. I have made: $L^1(v)\subset L^1(|v|)$?. Let $g\in L^1(v)$ As $v<<|v|$ and $|v|$ is finite measure, then for chain rule, ...
0
votes
0answers
6 views

Expressing the Solution to a System of Differential Equations

My professor wrote the solution to a system as $$X = C_1 \begin{bmatrix}1 \\2 \end{bmatrix} e^{\lambda_1t} + C_2 \begin{bmatrix}3 \\4 \end{bmatrix} e^{\lambda_2t}$$ Where the column vectors are the ...
0
votes
1answer
11 views

How do I interpret this operation?

This question has to do with operations and exploring their characteristics. I have just learned how to extract info from an operations table (what is the identity, inverse, etc.), but this question ...
0
votes
1answer
16 views

Seeking Recommendation on Pre-Calculus Textbooks!

S.E. advisers, I wrote this email because I am seeking a recommendation on selecting the pre-calculus textbooks. I have been studying the real analysis and number theory, and I felt that I need to ...
-1
votes
0answers
16 views

seperable differential equation question

b) $(2xy^3)dx + (3x2y^2 + y^4)dy = 0$ c) $(2xy^3)dx + (3x2y^2 + y^2)dy = 0$ I know that $c$ is a separable differential equation but $b$ is not. Why? The only difference is the power of the ...
1
vote
0answers
13 views

Roll one ellipse on another: Locus of center ever a circle?

Let $E_1$ be an ellipse fixed in the plane. Let $E_2$ be a second, possibly different ellipse, which rolls around without slippage outside $E_1$, touching perimeter-to-perimeter. Let $c_2(t)$ be the ...
2
votes
5answers
31 views

Formula for $r+2r^2+3r^3+…+nr^n$

Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I ...
0
votes
0answers
5 views

Is this an instance of the base-rate fallacy?

The following line of probability reasoning is supposedly fallacious, and is an instance of the base-rate fallacy. The argument is that $(1)-(3)$ don't give us enough reason to conclude that $(C)$. ...
2
votes
4answers
30 views

What is 3^43 mod 33?

I just took math final and one of the question was Find $3^{43}\bmod{33}$. So, I used Euler's function; $\phi(33)=20$. $3^{20}\equiv 1\pmod{\!33}$ By using this fact, I got $27$. One thing ...
0
votes
0answers
4 views

Stiefel-Whitney Numbers of $\mathbb{R}P^2\times \mathbb{R}P^2$

I'd like to calculate the Stiefel-Whitney numbers of $\mathbb{R}P^2\times\mathbb{R}P^2,$ but don't know how to. My first instinct was to say that the tangent bundle is isomorphic to the product of ...
0
votes
0answers
9 views

If K $\subset \mathbb{R}$ is compact prove that {fn} converges uniformly to f on K.

Suppose that we have a sequence of functions{f$_{n}$} that converges uniformly to a function f on any (a,b) $\subset \mathbb{R}$. If K $\subset \mathbb{R}$ is compact prove that {f$_{n}$} converges ...
-7
votes
0answers
30 views

Is the only way to go physics is a very goodeth?

Here's a qestion. If the Physics and I chemistry is the best of all the all is and what is this the your phone and is a great way for of to a the first place time half the things time people are just ...
1
vote
0answers
6 views

Start to Proof of bernoulli polynomials and sums

I need help starting this proof: For all integers k,l,m>=0 and not all equal to 0, (3.7) It says that that comparing the above equation (3.7) with the one discussed earlier in the paper(3.6) (shown ...
0
votes
0answers
10 views

Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
2
votes
2answers
22 views

Question on $\mathbb{K}$ notation

In a lot of paper and book $\mathbb{K}$ means $\mathbb{R}$ or $\mathbb{C}$. I know that $\mathbb{R}$ comes from the word real, and $\mathbb{C}$ from the word complex. But what about $\mathbb{K}$? ...
4
votes
2answers
19 views

Does $\chi(g^{-1})=\overline{\chi (g)}$ hold for infinite groups

Let $\chi$ be the character of some representation $\rho:G \to GL(M)$ over $\mathbb C$. Suppose $G$ is a group, then $\forall g \in G$ of finite order $n$, $ \chi(g^{-1})=\overline{\chi (g)}$ ...
0
votes
0answers
3 views

Vertex invariants based on finding minimal combined shortest paths

A possible vertex invariant for a vertex v is v's smallest n-neighbourhood consisting of the induced subgraph rooted in v of all vertices n edges away from v. Question 1: I'm wondering if this ...
0
votes
2answers
14 views

Proving De Morgan's law with the minus sign

So I know how to prove De Morgan's Law in this form: $A\cap (B\cup C)^{c}$, what I'm trying to do for practice is prove it in the slightly different notation: $A- (B\cup C)$ I get everything except I ...
4
votes
3answers
55 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
0
votes
1answer
10 views

Prove that if $\{1^5,2^5,\ldots, (pq)^5\}$ is a complete residue system mod $pq$, then $\{1^5,2^5,\ldots,p^5\}$ is too, mod $p$

$p,q\ge 2$ are coprime positive integers. Prove that if $\{1^5,2^5,\ldots, (pq)^5\}$ is a complete residue system mod $pq$, then $\{1^5,2^5,\ldots,p^5\}$ is a complete residue system mod $p$. ...
2
votes
1answer
21 views

Computation of a Riemann-Stieltjes integral

How do you compute the following? $$\int_{-\pi/4}^{\pi/4} f(x) \, dg(x) $$ Given $$f(x) = \begin{cases} \dfrac{\sin^4(x)}{\cos(x)} & x \in [0,\infty)\\[6pt] ...
2
votes
1answer
20 views

$ \lim_{t\to 0} \int_{|x|>\epsilon} \frac{e^{-\frac{x^2}{2t}}}{\sqrt{2 \pi t}} dx=0 $

I want to prove that: \begin{equation} \lim_{t\to 0} \int_{|x|>\epsilon} \frac{e^{-\frac{x^2}{2t}}}{\sqrt{2 \pi t}} dx=0, \end{equation} for any $\epsilon >0$ I've shown using polar ...
0
votes
1answer
17 views

Let $p$ be an odd prime, $D$ be an integer not divisible by $p$. show that $x^2 - y^2 = D (\text{mod } p)$ has $(p-1) $solutions

Let $p$ be an odd prime, $D$ be an integer not divisible by $p$. Show that $$ x^2 - y^2 = D \bmod p $$ has $p-1$ solutions Can somebody help with this problem? Thank you!
1
vote
1answer
33 views

Definitions from topology

I'm reading some papers on the unknotting problem in Knot theory and am running into some notation I don't know (my exposure to topology is minimal, but I have seen it in Analysis courses, Algebra, ...
0
votes
1answer
16 views

Inner product space and orthogonality proof.

Why does this automatically mean that the sets are orthogonal? I am a little confused about this? How would I necessarily prove also?
0
votes
0answers
6 views

Inequality for Ratio of Hardy-Littlewood Maximal Function over Balls and Cubes

Let $M$ denote the centered Hardy-Littlewood maximal function using balls, and let $M_{c}$ denote the centered Hardy-Littlewood maximal function using cubes. Exercise 2.13 in L. Grafakos, ...
0
votes
0answers
7 views

Indexing for partitions

I'm using spatial hashing for broad-phase collision detection and I'm trying to squeeze some more performance out of it. Currently, it creates a new hashset for every partition which works well ...
1
vote
0answers
4 views

Find a Conformal a Map

Please help me find a conformal map of the set $ A = \left \{\; z: \; |z-1| < \sqrt{2} \; and \; |z+1| < \sqrt{2} \; \right \}$ one-to-one onto the open first quadrant. First, I noticed that ...
0
votes
2answers
15 views

Showing $u_1, u_2, u_3$ is basis

Let $\{v_1, v_2, v_3\}$ be a basis for a vector space $V$. I want to show that $\{u1, u2, u3\}$ is also a basis where $u1 = v1, u2 = v1 + v2$ and $u3 = v1 + v2 + v3$ I wanted to use the standard ...
0
votes
1answer
10 views

Cylinder with an volume of 400

What radius and height do I need to make a cylinder with a volume of 400 cm cubed?
0
votes
1answer
33 views

Proving $\frac{\sin\pi z}{\pi z}=\prod_{n=1}^{\infty}\Big(1-\frac{z^{2}}{n^{2}}\Big)$

I apoligize if this has been answered already; the quick searches I've done have proven fruitless. I'm given that $\displaystyle\pi\cot(\pi z)=\frac{1}{z}+\sum_{1}^{\infty}\frac{2z}{z^{2}-n^{2}}$ ...
0
votes
1answer
14 views

difference between normal and diameter in circle.

A line through the centre of the circle meet the circle at two points is called a)normal b)tangent c)secant d)diameter I am pretty sure that the answer is diameter but my notes say the answer is ...
-1
votes
1answer
8 views

Area between curve and $x$ value

$R$ is the first quadrant region enclosed by the $x$-axis, the curve $y = 2x + b$, and the line $x = b$, where $b > 0$. Find the value of $b$ so that the area of the region $R$ is $288$ square ...
0
votes
2answers
27 views

Is every diagonal matrix the product of 3 matrices, $P^{-1}AP$, and why?

In trying to figure out which matrices are diagonalizable, why does my textbook pursue the topic of similar matrices? It says that "an $n \times n$ matrix A is diagonalizable when $A$ is similar to a ...
0
votes
1answer
15 views

Why is the number of subsets equal to the number of relations?

In this question I don't understand why the number of subsets is equal to the number of relations. Any help is welcome.
0
votes
0answers
3 views

recurrence tree final step - binary search

Starting with the base case and recursive case run times as follows: 􏰀 t(N) = 1 , if N = 1 t(N)= 1+t(N/2) ,ifN > 1 At the end of my tree I have ...
1
vote
3answers
11 views

What does the notation $[T]_{B^\prime \to B}$ mean?

Let $T:P_2 \to P_1$ be defined by $T(p(x))=p'(x) + p''(x)$ and let $B = \{1,x,x^2\} \text{ and } B'=\{1,x\}$. Find $[T]_{B\prime \to B}$ I do not understand the notation used when saying ...
0
votes
1answer
12 views

Distribution of Normal distribution

suppose $X \sim Normal(\mu, \sigma^{2})$. What is the distribution of $Y := N(X)$? where $N$ denotes Standard Normal Cumulative Distribution Function? e.g. in a special case when $\mu = 0$ and $\sigma ...
1
vote
2answers
24 views

Orthonormal basis proof.

Let $\beta=(v_1,\ldots,v_n)$ be an orthonormal basis for $V$. Show that for any $x,y\in V$, $$\langle x,y\rangle=\sum_{i=1}^n \langle x,v_i\rangle \overline{\langle y,v_i\rangle}$$ How ...
0
votes
1answer
14 views

Solving quadratic or higher degree congruence with very large modulus.

Is there any general way to solve a polynomial congruence with a very large modulus? An example could be $$ x^2-377x+1\equiv 0 \pmod {8683317618811886495518194401279999999 } $$ or $$ ...
0
votes
0answers
12 views

Find the maximum of a |cos(z)|

How do you find the maximum of the complex function $|\cos{z}|$ on $[0,2\pi]\times[0,2\pi]$. I believe I'm to use the maximum modulus principle, since the function is entire. I'm just having problems ...
0
votes
1answer
12 views

Subset of a normal space

Given X a normal space, and a subset $A \subset X$ not closed. Does it imply A is not normal? I understand it does not, Can someone provide me a counterexample?
1
vote
0answers
7 views

Finding potential of a given vector field

I am trying to solve the following problem: Let $ \textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2} $. Find an expression for a potential for $ \textbf{F}$. Find an expression also for ...
-4
votes
1answer
17 views

question that i have [on hold]

find a value of a in [0,90] that satisfies the given statement sec a=1.529096
1
vote
1answer
19 views

Fundamental solution and Green's function

I am currently dealing with Poisson's equation $- \Delta u= f $ on some open domain $U$ and $u =g$ on the boundary $\partial U.$ Now a fundamental solution is a solution to $- \Delta u(x) = ...
0
votes
0answers
13 views

Hausdorff measure of a subset of $\mathbb R^3$

Let $f \in L^1_{\text{loc}}(\mathbb R^3)$. We define $A \subset \mathbb R^3$ as $$ A := \left\{ x \in \mathbb R^3 \, : \, \limsup_{r \to 0} \frac 1 r \int_{\mathbb B(x,r)} \vert f(y) \vert \, \mathrm ...
2
votes
0answers
10 views

are connnected components of this scheme irreducible?

So I have a normal surface (ie, all components are dimension 2) $X$ which is smooth and affine over a dedekind domain $R$. Suppose $R'$ is also a dedekind domain and $Spec(R')\rightarrow Spec(R)$ is ...

15 30 50 per page