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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $P(0 \le X_{1} + X_{2} \le 1)$ 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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### 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.
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### 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 ...
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### 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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### $f : \mathbb R \to [-2 , 2]$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 (...
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### 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 ...
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### 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)
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### 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 ...
$m$ 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 ...