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By using $(71)$ and $(70)$ identities of Mathworld's [hypergeometric function][1] page,
$${_2}F{_1}(1,1;\tfrac12;\tfrac12)\stackrel{(71)}{=}2\,{_2}F{_1}(1,-\tfrac12;\tfrac12;-1)\stackrel{(70)}{=}2\left(\tfrac{\sqrt\pi}2\tfrac{\Gamma(\tfrac12)}{\Gamma(1)}+1\right)=\pi+2.$$
But, WolframAlpha calculates it as $\tfrac{\... |
The question defines the '$\le$' operation on the cardinality of arbitrary sets A, and B as follows:
Let $\mathcal F$ be defined as the family of all injections from subsets of A onto B. Since $\mathcal F$ can be considered a family of subsets of the cartesian product of A and B, $A \times B$, it can be partially orde... |
The following is a follow-up to this [previous question](https://math.stackexchange.com/questions/4861221/show-that-partial-mu-phi-ast-a-mu-phi-a-mu-phi-ast-partial-mu-phi-a-mu).
>Consider the following Lagrangian:
>$$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\nu-\parti... |
Let $\xi$ be a real random variable. We say that $\xi$ $\in$ $\phi _2$ when $\exists \lambda >0$ such that $\mathbb{E}\left(exp\left(\frac{\xi^2}{\lambda^2}\right)\right) \le e$.
**My problem:** Prove that if $\xi$ is a bounded random variable then $\xi$ is $\phi _2$.
Although I tried to find a solution, it didn... |
The question defines the '$\le$' operation on the cardinality of arbitrary sets A, and B as follows:
Let $\mathcal F$ be defined as the family of all injections from subsets of A onto B. Since $\mathcal F$ can be considered a family of subsets of the cartesian product of A and B, $A \times B$, it can be partially orde... |
Does a closed form expression exist for
$\\\sum_{k=1}^{n}\left(\sqrt{1+\frac{1}{k}}-\sqrt{1
-\frac{1}{k}}\right)$
I obtained
$0.97423066\ln(n)+1.2019463$
using log regression and it works very well for my (physics) problems but I was wondering if a better solution exists. Is the Euler-Maclaurin method... |
I've been studying about Simpson's paradox, and encountered this question:
[![Question][1]][1]
Why must the positive value be less than 2.7%?
According to the explanation, people can move between groups, which generated a total increase of 0.8% across those years.
Imagine however one person in every group, ex... |
Prove that the function is not non-expansive.
T is non-expansive if
$$ \forall x,y \hspace{3mm} ||Tx-Ty|| \leq ||x-y|| $$
$T(x)= \frac{x sin(x)}{2} \hspace{2mm} 0\leq x \leq \pi $ and $0$ elsewhere. I want to prove this without taking derivatives. My attempt:
\begin{align}
|\frac{x sin(x)}{2}-\frac{y sin(y)... |
Let $n > 1$ be an integer not a multiple of $10$.
Is there a short proof that $n = 5$ is the only solution to $n^n \equiv n^{n^n} \pmod {10^{n-1}}$, given the fact that $5^{5^5} \equiv 3125 \pmod{10^4}$ (and even $3125 \equiv 5^{5^5} \pmod{10^5}$)?
P.S.
This question is related to my research on the congruence spe... |
So the theorem is as follows
> Let $f$ be holomorphic in the ring $A=\{z\in\mathbb{C}\,|\,r_1<|z-z_0|<r_2\}$. Also assume that
$$
f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n+\sum_{n=1}^\infty \frac{b_n}{(z-z_0)^n}\,\,\forall z\in A,
$$
where both series converge independently. Then
$$
a_n=\frac{1}{2\pi}\int_{\gam... |
The commutative field $K$ can be considered a **vector space** in itself.
I define a series of properties common to fields and vector spaces and then define each of these separately and guide how I am understanding both concepts. **Some properties of algebraic structures common to vector spaces and fields are:**
... |
What does it mean that a **field** can be a **vector space** in itself? |
What does it mean that a field can be a vector space in itself? |
Let $n > 1$ be an integer not a multiple of $10$.
Is there a short proof that $n = 5$ is the only solution to $n^n \equiv n^{n^n} \pmod {10^{n-1}}$, given the fact that $5^{5^5} \equiv 3125 \pmod{10^4}$ (and even $3125 \equiv 5^{5^5} \pmod{10^5}$)?
P.S.
This question is related to my research on the congruence spe... |
How can one numerically calculate the partial derivatives of a convolution function, particularly when the closed-form or analytical expressions of the derivatives are not readily available? I am dealing with the convolution of two functions, where at least one does not have an easily computable derivative in the convo... |
The algorithm you found is correct. Here's how I think about it.
First, consider a line that passes through the origin, with homogeneous coordinates $l = (\cos \theta, \sin \theta, 0)$. What is the distance from this line to the point $p = (p_x, p_y, 1)$? Since the origin is on the line, this is simply the (absolute... |
I am studying the Scattering theory by John R. Taylor but I met a mathematic problem in Section 11-h.
$\phi_{lp}(r)$ is the solution of radial Schoedinger equation:
$$
\Big[\frac{d^2}{dr^2}-\frac{l(l+1)}{r^2}+p^2\Big]\phi_{lp}(r)=U(r)\phi_{lp}(r)
$$
with the boundary condition: $\phi_{lp}(0)=0.$ The corresp... |
How to solve out Green's function? |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
Informally, the silhouette of a 3D shape is a viewpoint-dependent 2D projection of it. You might imagine looking at several silhouettes and attempting to construct the overall shape. My question is theoretical: if you have access to the set of all silhouettes of a convex figure —unordered, that is to say, you kn... |
Are convex objects determined by their silhouettes? |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
> **Cantor Bendixson derivatives**
As Noah mentioned in the comments, the Cantor-Bendixson derivative is a very interesting (to me, at least) example of such a function. Let $K$ be a topological space. For a subset $L$ of $K$, let $L'$ denote the set of members of $L$ which are not relatively isolated in $L$. This i... |
Informally, the silhouette of a 3D shape is a viewpoint-dependent 2D projection of it. You might imagine looking at several silhouettes and attempting to construct the overall shape. My question is theoretical: if you have access to the set of all silhouettes of a convex figure —unordered, that is to say, you kn... |
I can provide a partial and unfinished answer, when one of the transformations, say $T$, is injective. Then there exists a linear transformation $T^{-1}:T(X)\to X$ such that $T^{-1}Tx=x$ for each $x\in X$. Now let $V$ be any subspace of $X$. Then $T(V)=S(V)$ iff $S(V)\subset T(X)$ and $T^{-1}S(V)=V$. That is we have to... |
In a 2015 paper titled *Lecture on the abc conjecture and some of its consequences* by Michel Waldschmidt it is stated that:
It can be easily seen that the $abc$-conjecture is equivalent to the following statement:
For each $\epsilon > 0$, there exists $\kappa(\epsilon)$ such that, for any abc triple $(a, b, c)... |
I'm new to this subject, and I'm a bit confused about how the metric changing corresponds to the manifold itself changing. For example, in [Topping's][1] note it says:
> A simple example of a Ricci flow is that starting from a round sphere. This
will evolve by shrinking homothetically to a point in finite time.
... |
Ricci flow on a manifold: How does the manifold itself change? |
> **Cantor Bendixson derivatives**
As Noah mentioned in the comments, the Cantor-Bendixson derivative is a very interesting (to me, at least) example of such a function. Let $K$ be a topological space. For a subset $L$ of $K$, let $L'$ denote the set of members of $L$ which are not relatively isolated in $L$. This i... |
To answer your question, I think it's worth asking why we want the ground model $M$ to be able to describe the forcing relation in the first place. One justification for this is that we as mathematicians cannot perform infinite tasks without assistance from the assumed existence of a model in the first place. It's simi... |
Informally, the silhouette of a 3D shape is a viewpoint-dependent 2D projection of it. You might imagine looking at several silhouettes and attempting to construct the overall shape. My question is theoretical: if you have access to the set of all silhouettes of a convex figure —unordered, that is to say, you kn... |
In the Q. 4265006 (https://math.stackexchange.com/questions/4265006/l2-convergence-for-a-simple-function-approximation) at his question related to solve "Problem 2.19." (probable copied from a manual/book)
received the following solution:
Hint: If $g \in L^{2}(0,1)$ and $g$ is orthogonal to $I_{j,n}$ for all $n$ ... |
Is the set of linear combinations of the functions $I_{j,n}$, $n \in \mathbb N$, dense in $L^{2}$? |
In a 2015 paper titled *Lecture on the abc conjecture and some of its consequences* by Michel Waldschmidt it is stated that:
It can be easily seen that the $abc$-conjecture is equivalent to the following statement:
For each $\epsilon > 0$, there exists $\kappa(\epsilon)$ such that, for any abc triple $(a, b, c)... |
Let $X_k = (X_{k, t})_{t\geq 0}$ a Gaussian process for each $k\in [m]$, and $a\in\mathbf{R}^m$. Is $Y = (Y_t)_{t\geq 0} = (\sum_{k = 1}^m a_k X_{k,t})_{t\geq 0}$ still a Gaussian process? Each $X_k$ is defined on the same probability space, and I supposedly need to use the fact that a linear transformation of a Gaussi... |
Is the linear combination of Gaussian processes also a Gaussian process? |
How to determine that $\sin\frac{x}{x+y}$ homogeneous or not? |
How can one numerically calculate the partial derivatives of a convolution function, particularly when the closed-form or analytical expressions of the derivatives are not readily available? |
I'm learning stochastic calculus right now at school and I'm coming across multiplication rules such as $$dW_t \cdot dW_t = dt$$ and I don't really quite understand what they meant. I tried looking up but I couldn't find any resources that explains this properly either. Originally, I thought it's supposed to mean that ... |
What does it mean by multiplying two differentials in stochastic calculus? |
The question is "Find the exact value of $\sqrt{97+56\sqrt{3}}$ ". It's from some regional contest back in 2013 and the answer is $7+4\sqrt{3}$. Can someone explain how they can reach the answer other than bashing in perfect squares for 97? I know you put $97+56\sqrt{3}$ into the form $a^2+2ab+b^2$ but how? |
In the context of Computer Science, I am trying to calculate the maximum depth difference between leaf nodes in any existing AVL-Trees of height $h$. I don't think any knowledge of AVL trees is needed, since the following terms are the stand-alone mathematical relations resulting from them.
After some reflections, t... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
First time I looked up this problem, I say yes but now I'm confused.
Consider the following first order differential equation, $$\frac{dy}{dx}=\sin\frac{x}{x+y}.$$
Substitute $v=y/x$ then it leads to the first order separable equation. Is this wrong?
------------------------------------------------------------... |
How to determine that $\frac{dy}{dx}=\sin\frac{x}{x+y}$ is homogeneous? |
Suppose we have $\phi : \mathbb{Z}$ x $\mathbb{Z} \rightarrow \mathbb{Z}$ x $\mathbb{Z}$ , where $\phi$ is a ring homomorphism. Then it must satisfy the following 2 conditions:
for all $a,b \in \mathbb{Z}$ x $\mathbb{Z}$
1) $\phi (a+b) =\phi (a) + \phi (b)$
2) $\phi (a\cdot b) =\phi (a)\cdot\phi (b)$
Then for ... |
I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me):
We define the exponential as follows:
$$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$
For whole number inputs of $t$, this corresponds to raising $e$ to that power.
I think $\exp(t)... |
Assume that a 52-card deck is distributed among 4 players, each with 13.
What is the probability that two specific cards, say the Ace of Hearts and Ace of Diamonds to be in my hand?
I have two approach in solving this, each giving a different answer.
**The first approach** to consider the sample space to be th... |
There are 5 people sitting on 5 chairs arranged in a straight line, all facing north. Everyone gets up and can do only one of the following things at a time.
(i) Sit back again in their original chair.
(ii)Turn their chair 180◦ to face south and then sit on it.
(iii) Sit on the chair next to their original ... |
> **Cantor Bendixson derivatives**
As Noah mentioned in the comments, the Cantor-Bendixson derivative is a very interesting (to me, at least) example of such a function. Let $K$ be a topological space. For a subset $L$ of $K$, let $L'$ denote the set of members of $L$ which are not relatively isolated in $L$. This i... |
In $(365 \choose #people)\dfrac{365^{#people}}$
this counts no repeats for the probability of none assuming Jan, Feb, March is the same thing as Feb, March Jan.
If
>We interpret the question as asking for the probability that **precisely two** people share a birthday.
>Imagine lining up our $n\ge 3$... |
Where in the original statement of the birthday problem is the order people of assign matter in the numerator of the probability? |
I was reading a proof on the lower bound of the mixing time of the lazy $(p − q)$-biased walk, which remains in its current position with probability 1/2, moves clockwise with probability $p/2$, and moves counterclockwise with probability $q/2$. Here $p + q = 1$. The proof goes as follows
[. And here's the thing about chain complex.
> **(Page 104 in [this file](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf))**
> That being given, let ${ε}^q_{i,j}$ be a number which is equal t... |
Is anyone able to construct an easy-to-follow proof that the inverse of a nonsingular lower triangular matrix is also lower triangular? I have been attempting to construct a proof by contradiction to no avail. Is this the right way to go about it? If not, could someone show a way to prove it? |
I am self studying number theory and have come across a problem involving Fermat's Little Theorem that I cannot seem to solve. The question asks to find the least remainder of $71^{71} mod \ 17$.
I started by noticing that $71^{71}=71^{(17-1)^{4}}71^{7}$, which, by Fermat's, means my answer is congruent to $71^{7} ... |
Does a closed form expression exist for
$$\sum_{k=1}^{n}\left(\sqrt{1+\frac{1}{k}}-\sqrt{1
-\frac{1}{k}}\right)$$
I obtained
$$0.97423066\ln(n)+1.2019463$$
using log regression and it works very well for my (physics) problems but I was wondering if a better solution exists. Is the Euler-Maclaurin meth... |
[This paper (pages 19-22)][1] offers an algorithm for deciding which player will win Hackenbush when the position is a parted jungle. The key step seems to be very similar to the Ford-Fulkerson algorithm, but only cares about the number of tracks produced, not how many edges they contain, let alone capacities. From t... |
Should the maximal flow through a parted jungle uniquely prescribe its atomic weight? |
I am wondering if $\int_{|z|<\frac{1}{n}} \frac{1}{|z|^{2(d-\alpha)}} \to 0$ as $n \to \infty$ for $z \in \mathbb{R}^d$. It is known that $\int_{\mathbb{R}^d} \frac{1}{|z|^{d-\alpha}}$ is Lebesgue integrable whenever $\alpha > 0$, i.e. when $d-\alpha < d$. I'm not even certain whether $\int_{|z|<\frac{1}{n}} \frac{1}{|... |
I am self studying number theory and have come across a problem involving Fermat's Little Theorem that I cannot seem to solve. The question asks to find the least remainder of $71^{71} mod \ 17$.
I started by noticing that $71^{71}=71^{(17-1)4} \times 71^{7}$, which, by Fermat's, means my answer is congruent to $71... |
Modern reference on PA degrees? |
I want to generate the trajectory of point M in space, such that the sum of its distances to two other points, P and Q in space, remains constant, that is, |PM| + |MQ| = constant. Therefore, I believe the trajectory of this point should be an ellipsoid with PQ as its foci and rotating about the PQ axis. However, I wrot... |
I am self studying number theory and have come across a problem involving Fermat's Little Theorem that I cannot seem to solve. The question asks to find the least remainder of $71^{71} \pmod {17}$.
I started by noticing that $71^{71}=71^{(17-1)4} \times 71^{7}$, which, by Fermat's, means my answer is congruent to $... |
Given $n, h\in\Bbb{N}$, let $f(n, h)$ be the number of rooted trees with $n$ vertices and height $h$. How can we compute $f$?
Let's denote by $r$ the root of a tree. Here, the height of a tree is the maximum distance between $r$ and one of it's vertices.
What I've noticed so far:
- It's easy to see that $h = 1... |
Could someone please give me a simple explanation of when it is valid to interchange the order of Intergration and summation when the summation is an infante summation?
for some context during lecture the other day my current prof interchanged the order, and I asked if that were valid here and he said he couldn't t... |
dumbed down explanation of Fubini's theorem? |
Is there an isomorphism from a smooth vector bundle $E$ and its double dual bundle $E^{**}$ which doesn't depend on a bundle metric on $E$? |
I am trying to determine the mean squared displacement $\langle r^2\rangle$ as a function of time for a discrete random walk process on a triangular grid, where each step is of size $\ell$ over a time $\tau$. For each step, there is one of six possible points to move to, with a probability of $p_n$ to move at an angle ... |
What is the correct MSD function for a biased random walk on a triangular grid in terms of step probabilities? |
I was reading "Elementary Linear Algebra Applications" by Howard Anton, and in section 4.6, Change of Basis, it talks about finding the transition matrix, $P$, from an old basis $B$ to to another new basis $B'$. I understand that if the transition matrix satisfies $P_{B \to B'} B = B'$, then the the transition matrix c... |
If $(f \circ g)(x)=h(x)$ and $(g \circ f)(x)=h^{-1}(x)$, is it necessary that $f(x)$ and $g(x)$ are inverses of each other? If not, what extra condition (other than just directly saying $h(x)=x$) must be imposed on $h(x)$ for them to be inverses? Assume that $f(x)$ and $g(x)$ are neither the identity function(s) nor eq... |
How to prove this lemma 1? |
I'm solving the Laplace equation on a two dimensional square domain. The problem is defined as:
$$
u_{xx}+u_{yy} = 0 \hskip{0.5cm}\text{for} \hskip{0.5cm} 0 \leq x \leq 1, 0 \leq y \leq 1
$$
with the following boundary conditions:
$$
u(0,y)= 0, \hskip{0.5cm} u(1,y)= 0, \hskip{0.5cm} u(x,1)= 0, \hskip{0.5cm} u(x,0... |
*Edit: In this answer I'm assuming the well-order of the reals is fixed at the outset and not somehow "chosen randomly", though, as commenter on another answer has woken me up to, you phrased the question as if were part of a random process. This would be a whole other can of worms (What distribution does this come fro... |
Is there a finite group of order greater than $1$, where all the elements are commutators?
I thought about this question once, but I couldn't think of a single example. Obviously, such a group must be perfect. Among simple finite groups, I believe there are no such groups.
Perhaps the following fact can be used s... |
Suppose we have to find the integral of the function $\sqrt{\frac{x}{1-x}}$. To solve this, I set $\sqrt{x} = -\sin\theta$, where theta belongs to [0, -π/2) which implies that $x = \sin^2\theta$. The differential $dx$ can then be expressed as $dx = 2\sin\theta\cos\theta d\theta$.So
$$\int \sqrt{\frac{x}{1-x}} \, dx$... |
The $\int_{0}^{1}x^a\,dx$ seems to equal $\frac{1}{a+1}$. I just came across this and thought it must be known why this is.
See the the table below:
![Desmos Graph][1]
Why are these nice fractions showing up out of nowhere; what does the area under $F(x)=x^a$ between $x=0$ and $x=1$ have to do with these nice f... |
Why does $\int_{0}^{1}x^1\,dx=\frac{1}{2}$, and $\int_{0}^{1}x^2\,dx=\frac{1}{3}$, and $\int_{0}^{1}x^3\,dx=\frac{1}{4}$, and so on? |
Given two strings `a` and `b` such that `a+b = b+a`, what can we infer about such strings? Its like an abstract question where we don't know anything else but want to know about the properties of such strings.
Example:
```
a = ABC
b = ABCABC
a+b = ABCABCABC
b+a = ABCABCABC
```
I think the following holds true... |
Suppose we have to find the integral of the function $\sqrt{\frac{x}{1-x}}$. To solve this, I set $\sqrt{x} = -\sin\theta$, where theta belongs to [0, -π/2) which implies that $x = \sin^2\theta$. The differential $dx$ can then be expressed as $dx = 2\sin\theta\cos\theta d\theta$.So
$$\int \sqrt{\frac{x}{1-x}} \, dx$... |
Is there a finite group of order greater than $1$, where all the elements are commutators?
I thought about this question once, but I couldn't think of a single example. Obviously, such a group must be perfect.
Perhaps the following fact can be used somehow:
$g \in G$ is commutator iff $$\sum\limits_{\chi \in I... |
From set theory we know $\text{Im}(gf)=g(\text{Im}(f))$. I was wondering how to make this work with an arbitrary abelian category. I know that every morphism $f:X\to Y$ in any abelian category has a unique (up to unique isomorphism) decomposition through the image object $\text{Im}(f)=\text{ker}\text{coker}(f)\cong \te... |
I have a 2-D Poisson Equation with homogeneous boundary conditions, and I used an eigenfunction along x, and a Fourier sine series expands the solution to $u(x,y) = \sum_{n=1}^\infty a_n(y)\sin(\frac{n\pi}{L}x)$. Plugging this into our Poisson Equation, I get the following Green's function:
$$G(\textbf{x},\textbf{x}... |
$$
\sqrt{\frac{\sin^2 \theta}{\cos^2 \theta}}
= \left\lvert \frac{\sin \theta}{\cos \theta}\right\rvert
= - \frac{\sin \theta}{\cos \theta}
$$
since $\sin\theta \leq 0$ and $\cos\theta > 0$.
So your last integrand has the wrong sign.
You made a second sign error, since $\sin(2\theta) = -\sqrt{x(1-x)}$
but ... |
**Hint:**
Just notice that:
$$(-\dot{f} +af) = -e^{at}(e^{-at}f)'$$
Then, let us denote
$$g_1 = e^{-a_1 t}f_1$$
$$g_2 = e^{-a_2 t}f_1$$
The system of two equations is equivalent to
$$\begin{align}
&\iff \cases{a_1t+\ln(g_1)= A_{11}(a_1t + \ln(-\dot{g_1})) + A_{12}(a_2t + \ln(-\dot{g_2}))\\
a_2t+\ln(g_2)=A_{... |
Question: I'm currently delving into the realm of Markov models to understand how they can be applied to determine the cost of car insurance. Thus far, I've come across the bonus-malus system and hidden Markov chains. However, I'm curious to learn about other Markov models that could be relevant in this context. Could ... |
From set theory we know $\text{Im}(gf)=g(\text{Im}(f))$. I was wondering how to make this work with an arbitrary abelian category. I know that every morphism $f:X\to Y$ in any abelian category has a unique (up to unique isomorphism) decomposition through the image object $\text{Im}(f)=\text{ker}\text{coker}(f)\cong \te... |
What are the formulas for the circumradius, surface area and volume of each Kepler-Poinsot polyhderon based on the length of the entire edge? |
If for two strings a+b=b+a holds true, what properties can we derive about such strings? |
The principle of the theory of the tangent to a curve is that it is the limit of the secant when the two intersection points, one of them approaches the other, I mean in Calculus I. I intend to prove that for every function $f$ differentiable on an interval $I$ and a number $a$ from that interval, the equation : $f’(a)... |
**The question says to find the 4th degree polynomial with integral coefficents whose root is $\\sqrt{3}$ + $\\sqrt{2}$ **
*This is how I solved*
I assumed $ax^{4} + bx^{3} + cx^{2} + dx + e$
to be the polynomial where a,b,c,d,e are integers.
as x = $\\sqrt{3}$ + $\\sqrt{2}$ is a root of the polynomial, thus... |
I’m in middle school (6th grade) and I have a question related to the exponential function (the teacher couldn’t help me):
We define the exponential as follows:
$$ \exp(t) = \sum_{n = 0}^{\infty} \frac{t^n}{n!}$$
For whole number inputs of $t$, this corresponds to raising $e$ to that power.
I think $\exp(t)... |
**The question says to find the 4th degree polynomial with integral coefficients whose root is $\\sqrt{3}$ + $\\sqrt{2}$**
*This is how I solved*
I assumed $ax^{4} + bx^{3} + cx^{2} + dx + e$
to be the polynomial where a,b,c,d,e are integers.
as x = $\\sqrt{3}$ + $\\sqrt{2}$ is a root of the polynomia... |
Let $f: D \subset \mathbb{R} \rightarrow \mathbb{R}$ continuous with $f(D) \subset D$ and let $x_0$ be a $n$-periodic atractor point of $f$. Show that every point of the cycle $\{ x_0, f(x_0),\dots,f^{n-1}(x_0) \}$ is also an $n$-periodic atractor point of $f$.
I tried using the definition of continuity in $\mathbb{... |
How to prove this inequality based on several inequalities? |
I'm going to denote concatenation of $a$ and $b$ by $ab$, rather than $a + b$. If I want $n$ copies of $a$ in a row, I will denote it by $a^n$.
We can do a kind of Euclidean algorithm-style argument to establish that there is some atomic string $c$ such that both $a$ and $b$ are a number of copies of $c$ end-to-end.... |
**The question says to find the 4th degree polynomial with integral coefficients whose root is $\sqrt{3}$ + $\sqrt{2}$**
*This is how I solved*
I assumed $ax^{4} + bx^{3} + cx^{2} + dx + e$
to be the polynomial where a,b,c,d,e are integers.
as $x = \sqrt{3}$ + $\sqrt{2}$ is a root of the polynomial, thus put... |
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