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I am studying introductory quantum field theory and I am trying to reduce the integral over the relativistic phase space of three particles of arbitrary masses to an integral over two energies and three angles. From [this][1] source (page 3 below figure 49.2) I got the idea that perhaps the three angles could be the Eu... |
Let $V, V'$ be full flags in $\mathbb{C}^n$, let $\lambda, \mu$ be admissible partitions, and let $$\sigma_\lambda(V) = \{\Lambda \in G(k,n): \dim(\Lambda \cap V_{n-k+i-\lambda_i}) \geq i\}$$ and $$\sigma_\mu(V) = \{\Lambda \in G(k,n): \dim(\Lambda \cap V_{n-k+i-\mu_i}) \geq i\}$$ be general Schubert cycles. Then for e... |
How to induct this problem $1-x^k < (1-x)k$? |
[![enter image description here][1]][1]
Im not sure what to do exactly. i got the equation of the line to be $y=\frac{p}{\ln\left(p\right)}x+p$ is this correct?
now i need to solve for x using the above. [![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/PSIgy.png
[2]: https://i.st... |
I came across this formula for the standard error of the mean in a book and wondered how I might be misunderstanding it?
$$\widehat{SE} = \sqrt{\sum_{i=1}^n \frac{(x_i-\bar{x})^2}{n(n-1)}}$$
I would have thought it should be
$$\widehat{SE} = \frac{\sigma_x}{\sqrt{n}}$$
Could it be that the above refers th... |
I came across this formula for the standard error of the mean in a book and wondered how I might be misunderstanding it?
$$\widehat{SE} = \sqrt{\sum_{i=1}^n \frac{(x_i-\bar{x})^2}{n(n-1)}}$$
I would have thought it should be
$$\widehat{SE} = \frac{\sigma_x}{\sqrt{n}}$$
Could it be that the above refers th... |
The author of this textbook is introducing this coordinate free definition of the divergence of tensor fields. Can someone help break this down for me, I am just really confused. First of all, does "a" represent a vector field? And then the the dot product of the two vector fields given by <div(T), a> would give us a s... |
Is it true in general that if $h \in L^{1}(\mathbb{R})$ such that $\hat{h} \in L^{2}(\mathbb{R})$, where $\hat{h}$ is the Fourier transform of $f$, then in fact $h \in L^{2}(\mathbb{R})$? If so how one can go by proving it? Otherwise is there a counter-example?
Thanks. |
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable.
grp1(1M and 1W)---------------grp2(1M and 1W)-----------------grp3(1M and ... |
so I wanted to ask if there is anything that allows for this case:
I have $3$ statements $A, B$, and $C$ and I wish to change them from the form:
$(A \operatorname{xor} B)$,
$(A \operatorname{xor} C)$
Into
$A \operatorname{xor} B \operatorname{xor} C$
Is this possible and how does one do this? P... |
I know that the spectrum of the adjacency matrix enumerates the number of closed walks, and the spectrum of the reduced laplacian enumerates the number of spanning trees. Does being both adjacency and laplacian cospectral imply that the degree sequences are the same? Has anyone proved this before, or does there exist a... |
Do Adjacency and Laplacian cospectral graphs always have the same degree sequence? |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entrie... |
Consider a nice enough function $$\begin{align*}
u: & \mathbb{R}^{3}\times \mathbb{R} \to \mathbb{C}
\\ & (x, t) \to u(x,t)
\end{align*}$$Then, we use $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ as tuple. We have the following operators and notations $$\begin{align*}
J &= x + 2it \nabla && x \in \mathbb{R}^{3}
... |
Any help with proving this inequality? |
While trying to prove a particular equation using Maxwell's equations in electromagnetic theory, there is a step in my textbook that says
$$\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times \frac{\partial H}{\partial t}$$
Why is this true? Is it because of the Clairaut's theorem on partial derivatives? |
Is $\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times \frac{\partial H}{\partial t}$? |
[![enter image description here][1]][1]
I'm not sure what to do exactly. I got the equation of the line to be $y=\dfrac{p}{\ln\left(p\right)}x+p$ is this correct?
now I need to solve for x using the above. [![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/PSIgy.png
[2]: https://i.... |
[![enter image description here][1]][1]
I'm not sure what to do exactly. I got the equation of the line to be $y=\dfrac{p}{\ln\left(p\right)}x+p$ is this correct?
Now I need to solve for x using the above. [![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/PSIgy.png
[2]: https://i.... |
By monotonic function, we refer to an increasing function over $X \subseteq \mathbb{R}$. Why is the definition of monotonic function always $f(x) \geq f(y) \implies x \geq y$ instead of $x \geq y \implies f(x) \geq f(y)$?
Let's say $f$ is increasing according to the definition: $f(x) \geq f(y) \implies x \geq y$. S... |
know that the radical axis of two circles is the best alternative line to the line connecting the intersection points of two non-intersecting circles
in the same way I'm trying to find the best alternative circle for the circle passing through the vertices of a noncircular quadrilateral
I'm looking for an actual ... |
Consider the well-known [coupon collector problem][1] with $n$ potential coupons. If I collect a coupon per day, I know it will take me $n \cdot H_n$ days to finally collect all of them, where $H_n$ is the $n$-th harmonic number. For example, if the coupons are the positive integers up to 12, one potential sequence of ... |
Consider the well-known [coupon collector problem][1] with $n$ potential coupons. If I collect a coupon per day, I know it will take me $n \cdot H_n$ days to finally collect all of them, where $H_n$ is the $n$-th harmonic number. For example, if the coupons are the positive integers up to 12, one potential sequence of ... |
We have unlimited balls from $8$ colors. Balls from the same color are identical. How many options are there to arrange them in a line of $10$, so that line will include exactly $4$ different color? |
In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable.
grp$1(1M$ and $1W)$---------------grp$2(1M$ and $1W)$-----------------grp$... |
I'm posting a question because I was curious about something while studying linear algebra.
As we all know, $x = 1$ is a point in a one-dimensional coordinate system. I understand this part.
But why does $x = 1$ represent a line in two dimensions?
Is it simply defined that way?
[![enter image descriptio... |
Why does the equation $x = 1$ represent a line in a 2-dimensional coordinate system? |
Code borrowed from [here][1] (This question has been asked there before.).
A few Definitions that needed:
> Dirichlet kernel:
> $$D_N(x) = \sum_{n=-N}^Ne^{inx} = \frac{\sin\left( (N+\frac12)x \right)}{\sin(x/2)} \qquad (77)$$
> $$s_N(f; x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x - t) D_N(t) dt \qquad (78)$$
... |
The Hopf link is torus, but if I connect it with some arbitrary knot then what would the resulting link be? Is there some understanding of how it may relate to the class of the knot itself (torus/ hyperbolic/ satellite)? For example what if we take the connected sum of Hopf link with the hyperbolic figure-8 knot?
[![... |
Is the connected sum of Hopf link with any knot a Torus link? |
Code borrowed from [here][1] (This question has been asked there before.).
A few Definitions that needed:
> Dirichlet kernel:
> $$D_N(x) = \sum_{n=-N}^Ne^{inx} = \frac{\sin\left( (N+\frac12)x \right)}{\sin(x/2)} \qquad (77)$$
> $$s_N(f; x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x - t) D_N(t) dt \qquad (78)$$
... |
Let $a\neq -1,0,1$ be an integer. Write $a=(b^2c)^k$, where $b^2c$ is not a perfect power, and $c$ is squarefree. Artin's conjecture on primitive roots states that the asymptotic density of the set of prime numbers $p$ such that $a$ is a primitive root modulo $p$ is
$$
\tag{$\star$}
A(a)=\begin{cases}
\left(\displa... |
$$a(n)=a(\lceil abs(a(n-1)) \rceil\ mod\ n)) + a(\lceil abs(a(n-2)) \rceil\ mod\ n))$$
For starting values, $a(0)=a(1)=1$, the sequence has a cycle starting with $n=441329$ having a period of $63584$ (source: https://oeis.org/A330615) For starting value $a(0)=i$ and $a(1)=1$, the cycle starts at $n=35694$ and have a p... |
Is this sequence always eventually periodic regardless of starting value? |
Question: Suppose $0<a<b<c<1$, find the minimum of $\max(b-a,c-b,1-c)$ if $b \ge 2a$ or $ a+b \le 1 $.
Here is my try:
Since $ \begin{cases}
\max(b-a,c-b,1-c) \ge b-a \\
\max(b-a,c-b,1-c) \ge c-b \\
\max(b-a,c-b,1-c) \ge 1-c \\
\end{cases} $,we have $ 3 \times \max(b-a,c-b,1-c)\ge 1-a$.
Then, $$\m... |
I want to find an equivalence relation defined on all integers (that is, all of $\mathbb{Z}$) where
- The equivalence relation partitions $\mathbb{Z}$ into infinitely many equivalence classes; and
- Every equivalence class contains an infinite number of elements.
I've been thinking about this interesting questio... |
$$\sum_{n=2}^\infty \frac{1}{3n^2-2\sqrt{n}}$$
The instructions are to use a comparison test to determine convergence of the series. I thought to compare it to
$$\sum_{n=2}^\infty \frac{1}{3n^2}$$
which converges by the p-series test. But, that series is smaller than the original series, so it doesn't prove t... |
How do I determine convergence with a comparison test? |
I have been attempting to find the volume of an Elliptic truncated cone by dividing it into cross-sections of elliptical cylinders and then stacking them up. I got the idea from the integration of the truncated cone but am not able to continue with the method as there are 3 variables involved. Could someone please guid... |
How to calculate the Volume of an Elliptic Truncated Cone? |
Using the function
$$f(x) = \begin{cases}\frac{x}{4} &\text{if }|x|\le 2\\[0.3em]
\frac{|x| - 1}{x} & \text{otherwise}\end{cases}$$
one can map $\mathbf Q$ into the interval $[-1, 1]$ in a way amenable to binary search.
Suppose $x$ is the real number we want to represent as a binary search sequence. For the first... |
I have been attempting to find the volume of an Elliptic truncated cone by dividing it into cross-sections of elliptical cylinders and then stacking them up. I got the idea from the integration of the truncated cone but am not able to continue with the method as there are 3 variables involved. Could someone please guid... |
> Is the number of disjoint open intervals in the solution set of $\text{sin} \frac{1}{x}>0, 0<x<1$
> countably finite or uncountably infinite?
Use a change of variable. Let $y=\frac{1}{x}$:
$$\text{sin } y>0\implies y \in {(1,\pi) \cup (2\pi,3\pi) \cup ...}$$
Changing back to the original variable:
$$x \in {(... |
For the following question I am asked to find the radius for circle of curvature for the function:
> -0.18e^(4.88x)
I found the radius 1.681459915,
by using the formula: R =1/ρ
and this was correct *however*
when my answers for finding the centre is **incorrect** and I am unsure why.
It is close though.
... |
In exercise 1.2.6 of the french version (seems like it is also the case of the english version) of *Algebre Commutative* by Bourbaki, a function $\lambda$ was defined to assign each $R$-module the maximal length of its presentation. Here, the presentation (of length $n$) is defined to be an exact sequence $L_n \to L_{n... |
Let $(X, d)$ be a metric space.
**Question**: Is the closed ball of radius $r > 0$ centered at some $x \in X$, i.e., $\overline{B_r(x)} \subseteq X$ totally bounded?
> By definition, a set $V \subseteq X$ is totally bounded if for any $\epsilon > 0$, there is a finite covering of $V$ with balls of radius $\epsilo... |
$$a(n)=a(\lceil abs(a(n-1)) \rceil\ mod\ n)) + a(\lceil abs(a(n-2)) \rceil\ mod\ n))$$
For starting values, $a(0)=a(1)=1$, the sequence has a cycle starting with $n=441329$ having a period of $63584$ (source: https://oeis.org/A330615) For starting value $a(0)=i$ and $a(1)=1$, the cycle starts at $n=35694$ and have a p... |
The subring of $\mathbb{Q}[x]$ which contains all rational polynomials with constant term integer sits in between $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ both of which is Noetherian, but this subring is not so although it has properties very much similar to UFD. Does this has any particular interest and any generalization?... |
Let's state this question in cokernels first, and hopefully duality will take care of the kernels.
Let $0 \to P \to N \to M \to 0$ be a short exact sequence. Given free resolutions $0 \to L_n \to L_{n-1} \to \cdots \to L_0 \to N \to 0$ and $0 \to K_m \to K_{m-1} \to \cdots \to K_0 \to P \to 0$, how to construct a fr... |
Code borrowed from [here][1] (This question has been asked there before.).
A few Definitions that needed:
> Dirichlet kernel:
> $$\tag{77}D_N(x) = \sum_{n=-N}^Ne^{inx} = \frac{\sin\left( (N+\frac12)x \right)}{\sin(x/2)} $$
> $$\tag{78}s_N(f; x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x - t) D_N(t)\, dt $$
-... |
I was given this question (in German), but I'm asked to find the representation matrix in respect to the standard basis. As far as I know I should preform the operation (the function) on all the elements of the basis (here it's a polynomial basis of degree 3). Then we rewrite the solution we get (of each) with the same... |
> As we all know, $x = 1$ is a point in a one-dimensional coordinate system. I understand this part.
Is it though? What if I write $\{a\in\mathbb{R}\ |\ x=1\}$. Is that a point? No, in fact it doesn't even make much sense.
What I'm trying to say, is that the equation "$x=1$" requires a context, on its own it is ... |
I was given this question (in German), but I'm asked to find the representation matrix in respect to the standard basis. As far as I know I should preform the operation (the function) on all the elements of the basis (here it's a polynomial basis of degree 3). Then we rewrite the solution we get (of each) with the same... |
I am studying the paper [DIOPHANTINE EQUATIONS OF THE FORM
$F(X) = G(Y)$ - AN EXPOSITION][1] which discusses the result of Erdos and Selfridge.
I am unable to understand the highlighted statement ''Clearly each prime factor of each $a_i$ is less than $k$."
For example, if I take $n=49$ and $k=4$, then we have ... |
In exercise 1.2.6 of the french version (seems like it is also the case of the english version) of *Algebre Commutative* by Bourbaki, a function $\lambda$ was defined to assign each $R$-module the maximal length of its presentation. Here, the presentation (of length $n$) is defined to be an exact sequence $L_n \to L_{n... |
I have the following question here...
>Find an equation of a plane through the origin such that the intersection between the plane and the elliptical cylinder $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is a circle.
Wouldn't this simply just be $ax+by=0$??? The plane has to pass the origin, so $d=0$ and since the pla... |
Given two 2D attractors, for example two Clifford Attractors, is it possible to combine them in an attractor in 4D? For example something of the form:
$x_{n+1} = \sin(a y_n) + c \cos(a x_n)$,
$y_{n+1} = \sin(b x_n) + d \cos(b y_n)$,
$w_{n+1} = \sin(a z_n) + c \cos(a w_n)$,
$z_{n+1} = \sin(b w_n) + d \cos(b z_n)$
i... |
How can I obtain a polynomial formula for the product (i.e. union) of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A... |
Let $a\neq -1,0,1$ be an integer. Write $a=(b^2c)^k$, where $b^2c$ is not a perfect power, and $c$ is squarefree. Artin's conjecture on primitive roots states that the asymptotic density of the set of prime numbers $p$ such that $a$ is a primitive root modulo $p$ is
$$
\tag{$\star$}
A(a)=\begin{cases}
\left(\displa... |
Let $f\in L^2(\mathbb{R}^n)$ and $g\in \mathcal{S}'(\mathbb{R}^n)$, where $\mathcal{S}'$ denotes the tempered distribution space, if $f=g$ in $\mathcal{S}'$, could we conclude $g=f$ in $L^2$? |
Are the zeros same in $L^2$ and $\mathcal{S}'$? |
I want to find an equivalence relation defined on all integers (that is, all of $\mathbb{Z}$) where
- The equivalence relation partitions $\mathbb{Z}$ into infinitely many equivalence classes; and
- Every equivalence class contains an infinite number of elements.
I've been thinking about this interesting quest... |
Consider the well-known [coupon collector problem][1] with $n$ potential coupons. If I collect a coupon per day, I know it will take me $n \cdot H_n$ days to finally collect all of them, where $H_n$ is the $n$-th harmonic number. For example, if the coupons are the positive integers up to 12, one potential sequence of ... |
Consider the well-known [coupon collector problem][1] with $n$ potential coupons. If I collect a coupon per day, I know it will take me $n \cdot H_n$ days to finally collect all of them, where $H_n$ is the $n$-th harmonic number. For example, if the coupons are the positive integers up to 12, one potential sequence of ... |
When I first learnt calculus I was so surprised to learn that there is a meaningful mathematical operator $D$ that
$$
D(x^n)= n x^{n-1}.
$$
It seems to be a very random thing to multiply the exponent by the function and subtract one from the exponent to get the correct result.
Again, this sounded very strange fo... |
How can I obtain a polynomial formula for the product (i.e. union) of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A... |
# Set up
Consider the example in section 2 of [Ferguson (1982)](http://staff.ustc.edu.cn/~zwp/teach/Math-Stat/MLE2.pdf).
Let $X_1, \ldots, X_n$ be i.i.d. with a distribution which with probability $\theta$ is the $U(-1, 1)$, and with probability $(1-\theta)$ is equal to the triangular distribution on $(\theta - c(\... |
For the following question I am asked to
>find the radius for circle of curvature for the function:
$-0.18e^{4.88x}$
I found the radius 1.681459915,
by using the formula: R =1/ρ
and this was correct *however*
when my answers for finding the centre is **incorrect** and I am unsure why.
It is close tho... |
# Problem
> Let $\left(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P}\right)$ be a filtered probability space such that $(\mathcal{F}_t)_{t\geq 0}$ is a complete and right-continuous filtration and let $B=(B_t)_{t\geq 0}$ be a standard $\mathbb{R}$-valued $(\mathcal{F}_t)_{t\geq 0}$-Brownian motion. Is $S:=... |
Is $\inf\left\{t\in\left[0,1\right]\vert t+B^2_t=1\right\}$ a stopping time? |
When textbooks say $e^{i\theta} = \cos\theta + i\sin\theta$, is the $e$ actually Euler's number, 2.71828..., or is this purely a symbolic equivalent notation where the $e$ does not represent this number? If it is the actual number, why is it used rather than another, more intuitive number? Does $\theta$ have to be me... |
How exactly does Euler notation work for complex numbers? |
4y^3+7 x sin(x)+4x^4-(16 cos(y(x)) -7x-(7/4) y(x)^2) y(x)'=sinh(y(x))-e^(y(x)-x)
I have a very difficult question about differential equations! And I think it has a strange integral factor! Does anyone have an idea about the solution? I couldn't even find the integral factor. I also got help from Maple, but he didn'... |
how too solve this very hard ODE ?! |
$$4y^3+7 x \sin(x)+4x^4-(16 \cos(y(x)) -7x-(7/4) y(x)^2) y(x)'=\sinh(y(x))-e^{y(x)-x}$$
I have a very difficult question about differential equations! And I think it has a strange integral factor! Does anyone have an idea about the solution? I couldn't even find the integral factor. I also got help from Maple, but h... |
For simplicity's sake, I would say that the rain clouds are static. The question essentially becomes: *given a finite area filled with random shaded regions covering X% of the area, what is the probability that a random contiguous line Y units long will cross through any of the shaded regions?*
[![Example, circle wi... |
Given the percent rain coverage in an area, what is the probability of hitting rain after driving X distance? |
Before getting into differentials, let's talk about that calculus formula,
$V = \iint 2 \pi r\,\mathrm dr \,\mathrm dh$.
How do we know that this gives the volume of a cylinder?
I think it is easier to justify the formula
$$V = \iiint_S f(r,\theta,h)\, r\,\mathrm d\theta \,\mathrm dr \,\mathrm dh$$
where $f(r,\t... |
I was wondering if every 2-coloring of the integer set would result in some number of equidistant (equally-spaced) points. I proved that there will always be 3 equally-spaced points of the same color, and I wonder if I could locate a bound for the largest number of equidistant points I could find. (If it exists, of cou... |
When I was in high school, I discovered in an exercise that $$\boxed{\pi=12\int_{0}^{2-\sqrt3}\frac{dt}{1+t^2}}$$since $\frac{\pi}{12}=\arctan( 2-\sqrt3)$, what we were made to demonstrate with formulas with half angles.
This formula has always fascinated me and I have memorized it.
I wondered what we could do w... |
Yes, $e$ is the same value of $e$ that is called Euler's number.
There isn't any "intuitive" reason I can think of why another number would work in this equation. What kind of number could be the "intuitive" one to put in this equation?
Yes, you need to assume $\theta$ is in radians when using the formula
$e^{i... |
Yes, $e$ is the same value of $e$ that is called Euler's number.
There isn't any "intuitive" reason I can think of why another number would work in this equation. What kind of number could be the "intuitive" one to put in this equation?
In fact, the logarithm with base $e$ is called the *natural* logarithm becaus... |
There is a systematic approach to the problems of integrating $\frac{1}{p(x)}$ where $p(x)$ is a polynomial. That is to factor $p(x)$ as a product of linear and quadratic polynomials, and then rewrite $\frac{1}{p(x)}$ as a sum of partial fractions. You can find the technique in any decent calculus textbook.
For this... |
In Artin's "Galois Theory" P38, he said the function $f(x) = \frac{(x^2 - x + 1)^3}{x^2(x-1)^2}$ satisfies the properties of $f(x)=f(1-x)=f(\frac{1}{x})$. Is the function given by some rational step or just by a flash of insight？
if $f(0)$ is a number, then $f(0) = f(\frac{1}{0})$. So that the domain of definition o... |
How did Artin discovered this function? |
Let $a\neq -1,0,1$ be an integer. Write $a=(b^2c)^k$, where $b^2c$ is not a perfect power, and $c$ is squarefree. Artin's conjecture on primitive roots states that the asymptotic density of the set of prime numbers $p$ such that $a$ is a primitive root modulo $p$ is
$$
\tag{$\star$}
A(a)=\begin{cases}
\left(\displa... |
On page 187, it is stated that the theorems for the rest of the chapter are for Riemann-integrable functions. So in order to apply Theorem 8.12, which involves equation (74) in your question, you need to know that the functions $\varphi_{1}:[-\pi , \pi ]\to\mathbb{C}$ defined by $\varphi_{1}(t) := g(t)\cos (t/2)$ and $... |
Code borrowed from [here][1] (This question has been asked there before.).
A few Definitions that needed:
> Dirichlet kernel:
> $$\tag{77}D_N(x) = \sum_{n=-N}^Ne^{inx} = \frac{\sin\left( (N+\frac12)x \right)}{\sin(x/2)} $$
> $$\tag{78}s_N(f; x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x - t) D_N(t)\, dt $$
-... |
Not sure if this an appropriate venue for this question, please close question as opposed to migrating to Physics SE, because it is not appropriate there, thanks.
If we have a functional given by:
$$I=\int_1^2\bigg(L+\lambda f\bigg)\;dt,\tag{1}$$
the Euler-Lagrange equations are given by:
$${d\over dt}\bigg({\par... |
Let the digits $a, b, c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?
I figured out that they want permutations of $aaabbbccc$ such that it must contain atleast one of the foll... |
In a cards game, there are ten cards, $1$ to $10$. Two players, seated facing each other, randomly choose $5$ cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/3sRqU.png
The difference between the corresponding cards is calculated such tha... |
I am stumped at this question our instructor gave as at class. Within my understanding, this is a general form of a linear differential equation: $\frac{dy}{dx} + P(x)y = Q(x)$. However, the first term is what I'm really interested about, where two derivative operators are multiplied. Is $\frac{df(y)}{dx} \frac{dy}{dx}... |
How do you go by solving this kind of linear differential equation? df(y)/dy dy/dx + P(x)f(y) = Q(x) |
To directly addresses the last question:
> If so, why do the larger values of θ in the exponent not increase the distance the point is from the origin?
Because that's how exponentiation works with complex arguments, when you change the complex part of the number (and as $\theta$ is multiplied by $i$ that is what... |
I am stumped at this question our instructor gave as at class. This is the differential equation:
$\frac{df(y)}{dx} \frac{dy}{dx} + P(x)y = Q(x)$
Within my understanding, this is a general form of a linear differential equation: $\frac{dy}{dx} + P(x)y = Q(x)$. However, the first term is what I'm really interested... |
I am stumped at this question our instructor gave as at class. This is the differential equation:
$\frac{df(y)}{dy} \frac{dy}{dx} + P(x)y = Q(x)$
Within my understanding, this is a general form of a linear differential equation: $\frac{dy}{dx} + P(x)y = Q(x)$. However, the first term is what I'm really interested... |
When I was in high school, I discovered in an exercise that $$\boxed{\pi=12\int_{0}^{2-\sqrt3}\frac{dt}{1+t^2}}$$since $\frac{\pi}{12}=\arctan( 2-\sqrt3)$, what we were made to demonstrate with formulas with half angles.
This formula has always fascinated me and I have memorized it.
I wondered what we could do w... |
Consider the triangle formed by joining the circumcenter, the incenter and the centroid of a a triangle (is there is already a name for this triangle in literature?). Simulations show that this new triangle is always obtuse (except when it is degenerate).
[![enter image description here][1]][1]
> **Conjecture**: ... |
I am told to find the two tangent lines to the ellipse that pass through the origin, but have been stuck for far too long with my approach, hence am thinking that my approach may be flawed. Here is what I have so far:
If I interpret the ellipse as the level curve of some function $f(x,y)=1$, then I can use the fact... |
> Let $Q_c(x)=x^2+c$. Prove that if $c<\frac14$, there is a unique
> $\mu>1$ such that $Q_c$ is topologically conjugate to
> $F_\mu(x)=\mu(1-x)$ via a map of the form $h(x)=\alpha x+\beta$.
Interpretation: $h(x)$ is a linear map. $Q_c(x)$ and $F_\mu(x)$ are a quadratic and affine map respectively. The claims is th... |
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