Hrodelbert
  • Member for 7 years, 10 months
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Is $\exp(x)$ the same as $e^x$?
6 votes

Although I agree with the answers already provided that in this situation (and indeed in most other ones in mathematics) there is no difference between the two notations, I would like to add the ...

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Work done by magnetic force is zero?
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4 votes

Although this might be better asked at the physics SE, the simple answer is as follows: $$W=\int \vec{F}\cdot \vec{ds} = q\int \left(\vec{B}\times \vec{v} \right)\cdot \vec{ds} = q\int \left(\vec{B}\...

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How to understand minimising length is equivalent to energy minimising?
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3 votes

Here is an idea of a proof: For any curve we can choose a reparametrisation such that its speed is constant, such that we actually have $L(\gamma)^2 = 2 E(\gamma)$ (this follows from going through ...

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Associative, but non-commutative binary operation with a identity and inverse
3 votes

Yes, matrix multiplication: given two square matrices $A,B$ with non-zero determinant (so elements of $GL(n,\mathbb{R})$) of any size, we know that $AB$ does not have to be equal to $BA$, but $A(BC) = ...

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hypothetical 4 dimensional vector space
3 votes

Yes this is possible. Simply let $a,b,c,d$ be your unit vectors and define the inner product of $a,b,c,d$ to be $0$ ( so $(a,b) = 0$ etc., but of course we still need $(a,a) = 1$). The inner product ...

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Geometric mean of 2 sets
2 votes

The simplest counter example for the case of positive integers is probably $$ A=\{1,3\} \qquad B=\{1,2,3\}, $$ both with arithmetic mean $2$. The geometric mean of $A$ is $\sqrt{3}\approx 1.73$, while ...

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Show that every non-empty set of integers that is bounded below has a minimum.
2 votes

Let $r$ be your real number and $I$ the set of integers, i.e. $$ I = \{ n\in \mathbb{Z} | n\geq r\}. $$ It is clear that $I$ is nonempty. Suppose that $I$ does not have a minimum: this means that ...

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Find a quadratic equation that approaches exponential equation.
2 votes

For small $a$, a good approximation of the term $e^{-ay}$ is given by $$e^{-ay} = 1 - ay + \frac{(-ay)^2}{2} + \mathcal{O}(a^3), $$ which, if you plug it in to the right-hand side of your equation, ...

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Norm of vector equals norm of it's basis representation
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2 votes

Only if $b$ is an orthonormal basis. In that case we have $$ ||a|| = ||\sum_{i=1}^d c_i b_i || = \sum_{i=1}^d |c_i| ||b_i|| = \sum_{i=1}^d |c_i | = ||c||. $$ If $b$ is not orthonormal, either $||b_i||...

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Finding a matrix $X$ that changes the basis(?)
2 votes

Since $X$ must be invertible, you can solve the set of equations given by $$XA = BX.$$ This gives $4$ linear equations for the $4$ unknown entries of $X$ in terms of the coefficients of $A$ and $B$. ...

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How many positive integers less than 1000 are multiples of 5 and are equal to 3 times an even number?
2 votes

As you have stated yourself, the integers we are looking for must be a multiple of $30$. Now it is simply a matter of counting all the multiples of $30$ below $1000$, which is found by dividing $1000$ ...

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$A^{T}A$ is diagonal. What can I say about $A$?
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2 votes

Concerning your second question: if $A$ is $3x3$ and satisfies $A^T A = D$ for some diagonal matrix $D$, it satisfies 6 additional constraints (namely the equations from the off-diagonal terms in $A^T ...

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Proving a function to be exponential given that it satisfies certain equations.
1 votes

Hint: consider the ratio $f/(A \exp(Bz))$ with $A$ and $B$ chosen such that $A \exp(Bz)$ has the same quasiperiods as $f$, i.e. the ratio is elliptic. Remember Liouville's theorem

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Function to express a time interval with results between 0 and 1
1 votes

The simplest approach in my opinion is to consider the function $$ w(d) =\frac{1}{d}, $$ where $d$ is the number of days difference. It is obvious that $w(1) = 1$ and that $0<w(d)<1$ for all $d\...

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Combinations for pairing groups
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1 votes

Let me first treat part one, since this is the easiest part Part 1: the number of subsets of a given set $X$ is $2^{|X|}-1$, where $|X|$ is the number of elements in $X$. I had to substract the $1$ ...

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Showing a linear form is a basis of $(\mathbb{R}[x]_2)^*$
1 votes

If a matrix is invertible, its columns seen as vectors form an independent set. Since the matrix is $d\times d$, where $d$ is the dimension of the space, this means that the columns of your matrix ...

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General Formula of the $n$th Derivative for $f(x) = xe^{2x}$
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1 votes

How about $$ A(n) = n2^{n-1} $$ and $$ B(n) = 2^n. $$ These reproduce your sequences. To prove that they work for general $n$ though, you would have to use mathematical induction.

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A basic combinatorial sum
1 votes

This sum is certainly not larger than $$ (d-1)K^{d-1}, $$ since that is the value of the summation $$ (d-1) \sum_{i_1,i_2,\cdots,i_{d-1}=1}^K 1, $$ which is certainly greater then sum you proposed. ...

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Integration of equation of motion to obtain invariant angular momentum
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1 votes

There is no univeral way to find conserved quantities of a given physical system. The most common way they are found is by looking at symmetries of the system. In this case, noticing that the ...

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Spectral representation of an analytic function
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1 votes

After some extra reading, I think I can answer this now, although not from the perspective I originally intended. We use the Plemelj-Sokhotksi formula, in the following way: in the limit $\epsilon \...

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Analytical continuation of $F(p) = \sum_{n \neq 0, n \in \mathbb{Z}} \frac{e^{ipn}}{\sinh^2\kappa n}$
1 votes

Inspired by the discussion with Kelenner, I believe I can answer my own question: We have a function $F$, which is given by a convergent series for $|$Im$(p)| <2\kappa$ (call this open ...

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Separable Differential Equation explained
1 votes

I'll post this just as an answer I suppose: $\frac{d}{dt}(1+y) = \frac{d}{dt} 1 + \frac{dy}{dt} =0 + (1+y)=1+y$ by the differential equation. Replacing $x(t)=1+y(t)$ yields a differential equation ...

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Is it possible to get the angle between two vectors in a single direction?
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1 votes

From the two vectors, define the plane spanned by these two vectors. Choose the orientation to be such that the normal vector to this plane has positive $z$-coordinate. Then set polar coordinates on ...

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Regarding Periods of Weierstrass $\wp$ function
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0 votes

The easy part was already known to the OP, but since I wrote it let me keep it here: if $u-v = \omega_i$ with $\omega_i$ one of the periods, then it follows immediately that $$ \wp(u) = \wp(\omega_i + ...

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Find the surface of fence with base located on the line $\sigma(t)=(\cos t,\sin t,0),t\in [0,\pi]$
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0 votes

Yes, your setup is correct. Note that the base of your fence is a circle, but the height of the fence is not periodic: at $0$, the height of the fence has a jump from 0 to $(2\pi)^2$, which does not ...

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How to integrate product of two functions of vectors in Spherical Coordinates?
0 votes

Heuristically, if the volume $V_b$ has spherical symmetry, we can proceed as follows: we are free to orient our original cartesian coordinates in any way we want, since the integral does not depend on ...

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Solution of a given equation.
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0 votes

No you cannot do this. The approximation you mention is only true for very small $x$ (compared to $1$), so $x <<1$. If we take $x$ to be close to one, it is easy to see that $(1-x)^{20}$ is ...

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Finding the Minimum Value
0 votes

The equation you are interested in is the following: $$ \int_{-\infty}^M N e^{-(x-\mu)^2/(2\sigma^2)} = 0.85, $$ where $N$ is a normalization such that the result of the integral is $1$ when $M\...

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Is the probability of a chain of dependent events, independent of the order in which they occur?
0 votes

I think your own explanation is already a mathematical proof that both probabilities are indeed equal. Your products of probabilities are correct and hence the result follows. Intuitively, this is due ...

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Branch points of functions defined as convolution integrals
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0 votes

After thinking some more, I think I have found an answer to my own question: Let us assume for simplicity that $f$ has exactly one pole at the origin. The case $F_{\mathbb{R}}$: it is clear that ...

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