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4 views

If $x^2 - y^2 = 1995$, prove that there are no whole numbers x and y divisible by 3

I'm quite new to this. As in the title I would like to ask if my reasoning here is correct and if my answer would be considered acceptable. If x or y is divisible by 3, but not both, then: $x \equiv \...
1
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0answers
25 views

Gambler's Ruin with 2 hitting points

In the Gamblers' ruin, suppose the Gambler starts with wealth $2n$. He will stop when he either has a total of $4n$, or $n$ wealth. Find the probability he hits $4n$. The proability of winning a ...
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0answers
3 views

prove that $P( min(X,Y,Z)>t \mid X<Y<Z ) = P( min(X,Y,Z)>t )$ **conditions apply

let $X,Y $ and $Z$ are independent exponentials with parameters $\lambda_1,\lambda_2$ and $ \lambda_3$ respectively. let A,B and C be independent poisson processes with arrival rates $\lambda_1,\...
3
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1answer
58 views

Set of Points in a Polygon Closer to Center than Vertices?

Let $P_n$ be the regular convex $n$-gon centered at $p_0$ with $n$ vertices $p_1, p_2, ..., p_n$ and $(n>2)$. Let $S_n$ be the set of all points "$s$" within the region bounded by $P_n$ where: $...
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0answers
11 views

Universal property of localization of algebras

I have the following question. It is well known that if we have a commutative ring $R$ and a multiplicative set $S\subset R$, then there exists a ring $S^{-1}R$ and a morphism $\pi:R\rightarrow S^{-1}...
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1answer
12 views

Verifying Stokes theorem using line integral

$$F(x, y, z) =(xyz)~\hat{i}+(y)~\hat{j}+(z) \hat{k}$$ S:6x+6y+z=12, first octant Verify stokes theorem by evaluating both double integral and Line integral. My work: I have calculated the ...
1
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1answer
13 views

Find the edge chromatic number of $K_n$ when $n$ is a positive integer.

Find the edge chromatic number of $K_n$ when $n$ is a positive integer. Note: $K_n$ is the complete graph on $n$ vertices。 I draw for $n = 1,2,3,4,5,6$ cases but have no general solution or proof ...
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3answers
13 views

Show that $\exp \biggl(\sum_{n=1}^\infty (-1)^{n-1} \frac{z^n}{n} \biggr) = 1+z \ $ for $\ \lvert z \rvert < 1$

I am working on the following exercise: Show that $$\exp \biggl(\sum_{n=1}^\infty (-1)^{n-1} \frac{z^n}{n} \biggr) = 1+z$$ for all $z \in \mathbb{C}$ with $\lvert z \rvert < 1$. Hint: $\...
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2answers
1k views

Find the marginal distributions (PDFs) of a multivariate normal distribution

Let $\mathbf{x}\in\Bbb{R}^n$ be a multi-variate normal vector with mean $\bar{\mathbf{x}}\in\Bbb{R}^n$ and covariance matrix $\Sigma\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes the space of all $...
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0answers
9 views

Cardinality of the set of all binary grids

Let $A$ be a set of all infinite binary grids (every field of each grid is occupied by either 1 or 0 and cardinality of both $X$ and $Y$ axes is $\aleph_{0}$ ). What is the cardinality of $A$? Is it ...
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0answers
7 views

Relation between forward operator and backward operator

I cannot found it.Please do the answer. and send. And Newton's Interpolation Formula: Difference between the forward and the backward formula. I was taught that the forward formula should be used when ...
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0answers
7 views

How to find the height which a ball will bounce after a collision with ground if the upward force is known?

The problem is as follows: From a height of $5\,m$ with respect to the ground a sphere is released. The time elapsed in the contact with the ground is $1\,ms$ and magnitude of the average ...
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0answers
4 views

Cat and mouse with points on a sphere

I'm looking for a function that maps a point p0 on the unit-sphere to another point p1 on the unit-sphere, but with some specific rules. (It doesn't matter if the points are expressed as two angles or ...
0
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0answers
33 views

Why are health/epidemiology measures usually multiplied by 1,000, 10,000, 100,000, etc.?

I have found various and seemingly related explanations: Multiplying by 10,000 [patient days or patients] standardizes the rate so it can be compared to other hospitals / populations that may have ...
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2answers
28 views

Neighborhood of $a$ when $f'(a)=0$

I have a function $f$ that I know is continuous and differentiable everywhere. I was reading the following theorem: "Suppose the function $f$ is defined on a neighborhood of $x=a$ with $f'(a)>0$....
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0answers
7 views

Algorithm for decomposing an element of a free lie algebra in terms of Hall basis

Let $L(x,y)$ be the free Lie algebra generated by the Hall basis. We assume $deg(x) =2$ and $deg(y) =2$ and for any $\alpha, \beta \in L(x,y)$ we have $deg([\alpha, \beta])= deg(\alpha)+deg(\beta)-1.$...
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1answer
29 views

New wrong recurrence formula for Bell numbers

Bell numbers are the numbers counting the total partitions on a set with $n$ distinct elements. Explanation: Consider a set like $A:=\left\{x_{1},x_{2},...,x_{n}\right\}$ A partial equivalence ...
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0answers
5 views

Splitting field's Galois group of $p(X)=X^4+(t^3+1)$ over $\mathbb{Q}(t)$ where $t$ transcendent over $\mathbb{Q}$

I would like to describe $$G=\text{Gal}(X^4+(t^3+1)/\mathbb{Q}(t)).$$ However, I have serious problems always when I am working with polynomials in let's say $K(t)[x]$. In my mind, things do not work ...
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0answers
11 views

Determining How Change in New Data Effects Hyperparameters in Gaussian Process with Squared Exponential Kernel

I want to determine how the inclusion of new data effects hyperparameters of the Gaussian Process kernel. For reference assuming square exponential kernels as provided here: $$K(x,x') = \sigma^2\exp\...
1
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1answer
19 views

Is a unity ring with characteristic $2$ always commutative?

Is a unity ring with characteristic $2$ always commutative? I believe it is not, but I cannot find a counterexample.
1
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0answers
15 views

A combinatorial identity

There is a well known identity due to Chu-Shih-Chieh asserting that: $$\sum_{j=0}^m \binom{r+j}{j} = \binom{m+r+1}{r+1}$$ For some proofs see here. I need to prove a kind of generalization, namely: ...
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1answer
16 views

Convex hull of path in $\mathbb R^2$ is the set of convex combination of 2 points of the path

I fell onto this post https://mathoverflow.net/questions/77379/convex-hull-of-path-connected-sets. The first answer is interesting and I can't find a simple argument to show that for $\mathbb R^2$ ...
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1answer
7 views

Limits of a monotone function

Let f: $\Bbb R \mapsto \Bbb R $ be a continuous function, we know that f is strictly decreasing. Can we directly say that $\lim_{x\to\infty} f = -\infty$ ?
0
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2answers
38 views

What is the derivative of $\cos(x² + 1)$?

Derivative is to be found out by using the first principle. What I did was that after applying the first principle, I applied the trigonometric identity $$ \cos A-\cos B = -2\sin(( A+B)/2) \sin((A-B)/...
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1answer
55 views

Examples of textbook definition: Chaper 5, Warner.

This is a question about a specific definition and its examples. The following definition is taken from Warner's Differentiable Manifolds and Lie groups, chapter $5$: Definition: Let $S$ and $X$ ...
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1answer
1k views

Is this algorithm for 3D spherical interpolation correct?

I am attempting to write a spherical interpolation algorithm for for the application of smooth 3D animation in a game. The scripting language that the game engine uses is Lua. It is often easier for ...
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0answers
8 views

$P(0 \le X_{1} + X_{2} \le 1)$ given the joint pdf of $(X_{1},X_{2})$

I'm trying to go through examples through my notes and I don't know how to set up the double integral to solve b) of this example:
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0answers
20 views

Differentation of loss function for Netflix Prize

Check out the function here I want to differentiate this function with respect to b and c i.e., dL/db_i and dL/dc_j. Help appreciated.
1
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2answers
265 views

Conjugate diameters of ellipse

How to find the length of major and minor axis of ellipse given the length of two conjugate diameters and the angle between them? I am aware about how to construct the ellipse using the above given ...
0
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0answers
10 views

What is the value of the following series [duplicate]

$\sum_{n\geq1}\frac{2^{n-1}}{1+a^{2n-1}}, a > 0$ I was considering the use of some power series, but to no end as the denominator of the fraction does no simply after derivation.
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0answers
6 views

$f : \mathbb R \to [-2 , 2]$ with $(f(0))^2 + (f'(0))^2 =85$ then there exists $x \in (-4 , 4)$ such that $f(x) +f''(x) = 0$ and $f'(x) \neq 0$.

For every twice differentiable function $f : \mathbb R \to [-2 , 2]$ with $(f(0))^2 + (f'(0))^2 =85$ then there exists $x \in (-4 , 4)$ such that $f(x) +f''(x) = 0$ and $f'(x) \neq 0$. I was trying ...
0
votes
2answers
17 views

How to define an antidiagonal positive definite matrix with a given structure?

Let us assume that I have a matrix $D\in\Re^{2N\times 2N}$ with the following structure: $$ D=\begin{bmatrix} 0 & A \\ A^T & 0 \\ \end{bmatrix} \quad $$ where $A \in\Re^{N\times N}$. Is it ...
-1
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0answers
7 views

MATLAB simulation (singularity )

I am simulating a program in Matlab and function is like $$ f(r,λ)=f_1(x,λ)\left( |f_2(x,λ))|^2 + |f_3(x,λ)|^2 \right) $$ where $f_2(x,λ)$ is a function with denominator which contain imaginary ...
0
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2answers
70 views

Find the length of the the green line

How can you find the length of the green line? The blue lines have a length of 8. Right angles are marked. (Diagram not to scale) EDIT: Here's a second diagram (The green lines are not the same ...
0
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2answers
16 views

Is there a continuous bijection?

I know there is a bijection from $(0,1) $ to $(0,1] $ Is there a function $f:(0,1) \to (0,1] $ which is a continuous surjection ? and $f:(0,1) \to (0,1] $ a continuous bijection ?
6
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5answers
507 views

Proof that base -2 with binary digits can form every integer

Basically the question is proving that you can create all integers with binary but instead using $-2$ as the base to be able to create negative integers. Exact question: Prove that every integer (...
1
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1answer
107 views

root of an odd degree polynomial

Im asked to prove that a polynomial of an odd degree has a root. Im going to use IVT to prove this, but I'm wondering if I can use the assumption that an odd degree polynomial has either two cases: $$...
-1
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0answers
44 views
+100

matrix for rotation and translation along it's local axis

I came across a particular situation when I would like to rotate an object at the origin and then translate it along its local axis.(everything here is for 2D). The transformation required is shown as:...
5
votes
3answers
2k views

Use the Residue Theorem to evaluate the integral:

$$\int_{0}^{∞} \frac{\sqrt{x}}{x^2+2x+5} dx$$ I'm thinking of using the "keyhole" contour, but I'm not sure how to proceed from there. Please help! Thanks!
1
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0answers
18 views
+50

Rayleigh quotient on circular region of radius 2

I' m struggling with the following problem. We have the eigenvalue problem: $$u'' + \lambda u = 0$$ with associated boundary condition: $$u' + 3u = 0$$ Now by using the Rayleigh quotient for $0 \leq ...
0
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2answers
26 views

Cauchy-Riemann Equations + Locally Invertible implies non-zero derivative

Suppose $C^1$ $f:\mathbb{R}^2 \to \mathbb{R}^2$ satisfies the Cuachy-Riemann equations $\frac{\partial f_1}{\partial x} = \frac{\partial f_2}{\partial y}, \frac{\partial f_1}{\partial y} = -\frac{\...
5
votes
2answers
124 views

Evaluate $\int_{0}^{\infty} \frac{x^2+x+1}{x^6+x^4+1} dx$

$$\int_{0}^{\infty} \frac{x^2+x+1}{x^6+x^4+1} dx $$ Wolfram says $1.80276\ldots $ Calculating this seems complicated to me because the residues are pretty hard to find and I have tried the infinite ...
0
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0answers
11 views

Holomorphic functions which converge pointwise but not almost uniformly

I'm looking for an example of a sequence of functions $f_n$ which are holomorphic in the open unit disk $D(0,1)$ and converge pointwise for every $z \in D(0,1)$, but are not almost uniformly ...
0
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1answer
61 views

Gödel's Incompleteness Theorem and the Millennial Problems

Disclaimer: I am not an expert in logic nor in mathematics in general, so please feel free to correct me in any of my assumptions or statements below. Gödel's Incompleteness Theorem states that "any ...
1
vote
1answer
202 views

Show that $\int_a^b cf = c \int_a^b f$

Let be $f$ and $cf$ integrable functions on the interval $[a,b ]$ where $c$ is a constant with $c<0$. Show that: $$\int_a^b cf = c \int_a^b f.$$ I know that there is a different approach where ...
1
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0answers
14 views

Subnormality of normalizers

Let $G$ be a finite group. $N$ is a non-abelian minimal normal subgroup of $G$ and $P$ is a nontrivial Sylow $p$-subgroup of $N$. Any minimal normal subgroup of $G$ is a characteristically simple ...
-1
votes
1answer
39 views

Can you list all the finite series that can be solved in a closed form?

I'm interested to know all the finite series that can be solved in a closed form (e.g. the geometric series)
0
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0answers
14 views

prove that $S$ is a $C_1$ curve given by

Let $P:R^3\to R$ be a $C^2$ function and let $\nabla_{xy}P(x,y,z)$ $=(\partial_xP(x,y,z), \partial_yP(x,y,z))$ Consider a set S given by $S={(x,y,z) : \nabla xyP(x,y,z)=(0,0)}$. Suppose that Gaussian ...
0
votes
1answer
28 views

Found some contradiction in wikipedia about topological space

Given X = {1, 2, 3, 4}, the collection τ = {{}, {1}, {1, 2, 3, 4}} of three subsets of X forms a topology of X according to the openset definition. Then {1} is neighbourhood of element "1", because "...
0
votes
1answer
15 views

Proof of the converse intercept theorem

$m$ and $n$ are given lines. $p$ and $q$ are two intersecting lines that intercept $m$ and $n$ at $C,D$ and $A,B$. I want to show that $m\parallel n$ iff $\dfrac{OC}{CD}=\dfrac{OA}{AB}=k$. I am ...

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