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Consider a number in its decimal representation that begins with an **odd number of consecutive digits of 4**, followed by a **single digit of 1**. An example of such a number would be 41, 4441, or any similar pattern extending with 4s. My question is: Can such a number ever be a **perfect square**?
To clarify, the ... |
Can $4\cdots41$ (with odd number of $4$s) be a Square Number? |
I wanted to confirm whether or not I am correctly identifying a mistake in the author-provided solution to the following problem (located within Spivak's *Calculus* 4th Ed.):
> Let $f$ be defined by: $\begin{cases} \frac{e^x-1}{x}\quad &x \neq 0\\ 1\quad & x=0\end{cases}\quad$ Find the Taylor polynomial of degree $n... |
Let $\mathbb C[S_3]$ be the group algebra of $S_3$. We have an isomorphism of $\mathbb C$-algebras
$$\mathbb C[S_3]\cong \mathbb C \times \mathbb C \times M_2(\mathbb C).$$
The existence of such an isomorphism is not difficult to show using Maschke's theorem and some other observations, as done, e.g., in [this post... |
How can you construct a right triangle, $\triangle ABC$ with a right angle in $C$, if you are given the hypotenuse, $c$, and the altitude of the point $C$?
I know it's very basic but I just can't seem to figure this out - I've tried everything I can think of, started from both $h_c$ and $c$ - but I keep getting stuc... |
I have been reading the paper "[Point Groups and Space Groups in Geometric Algebra][1]" by David Hestenes as part of my introduction seminar to GA. On page 5 the remark is made, that to prove that $C_p$ and $H_p$ are the only point groups in 2D, one would need to prove the following: Consider the possibility of a group... |
How do you prove that if $f : [a,\infty) \longrightarrow \mathbb R$ has bounded (not necessarily continuous) derivative on its domain, then it is uniformly continuous? Can you prove the same for $f : (a,\infty)$?
**Some (possibly incorrect) ideas I managed to find online:**
Given $f: [a, \infty) \rightarrow \math... |
first question here :)
I am reading an article about the Heston model and stochastic volatility, in which they state the following: "because the process for $V_t$ is independent of the brownian motion $W_t$, the distribution of $\int_u^t \sqrt{V_s} \, dW_s$ is normal with mean $0$ and variance $\int_u^t V_s \, ds$".... |
I was reading the [following](https://arxiv.org/pdf/2204.03587.pdf) paper, and was struggling to understand the final part of the proof of Lemma 5.3, where they paraphrase a proof from [this](https://storage.googleapis.com/wzukusers/user-33425553/documents/5b0993445e04cXCmxMD9/Lattice%20theory%20and%20the%20motion.pdf)... |
Consider a $C^*$-algebra $A$ and a continuous linear functional $\varphi:A\to\mathbb{C}$ with adjoint given by $\varphi^*(a):=\overline{\varphi(a^*)}$. We know that each positive (cont. lin.) functional is self-adjoint.
How would one show that each $\varphi\in A^*$ can be written as a linear combination of two self-... |
The question I am going to present may as well sound very dumb. But this is becoming a hell of a confusing thing for me.
The question is from ISI B.Math-B.Stat entrance exam 2022 UGA question paper. It is problem 29. The question included pictures, but I am going to rephrase the question in a way such that the pictu... |
Why would the triangles join up to a rhombus? |
How to solve $\int \frac{1}{\sqrt{x^2+x-1}}$? |
I just found this kind of “conjecture” recently. Define the set of positive prime numbers $P$. Suppose $$ n \in P, n\geq 3. $$ Define $S := \{1,p_1,\dots,p_{x-1}\}$ that are primes and $1$ which are smaller than $n$.
(in which $p_{x-1}$ is the prime directly before n)
1. Can you find at most 2 distinct $a, b \in S$ (... |
How to solve $\int \frac{1}{\sqrt{x^2+x-1}}dx$? |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
Consider a number in its decimal representation that begins with an 32. My question is: Can such a number ever be a **perfect square**?
Here are the specific points I'm curious about:
1. Is there a mathematical approach or theorem that directly addresses the properties of numbers with specific digit patterns... |
Can $32$ (with odd number of $4$s) be a Square Number? |
The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. [this][1] MathSE post):
\begin{align}
&\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y)
\quad \Rightarrow \quad
\phi(x) = \frac{-1}{4 \pi |x-y|} \qquad
\\
&\mathbb{R}^2: \qquad \nabla^2 \phi(x) = \delta^2(x-y)
\quad ... |
In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field configuration $\mathbf{B}$ via the [Biot-savart law][1]. If the wire is a curve $\gamma$ parametrized as $\mathbf{y}(s)$, where $s$ is the arc-length, then
$$
\mathbf{B}(\mathbf{x}) = \bet... |
We see [here][1] a proof of the random-time memoryless property $P(X>T+s|X>T)=P(X>s)$ where $E\sim Exp(\lambda)$ and $T\ge 0$ is a *continuous* random variable independent of $E$.
The proof, however, relies upon the existence of a pdf $f_T$ for $T$ (and hence the continuity of $T$); this places restrictions on the r... |
Consider a number in its decimal representation that begins with an **odd number of consecutive digits of 4**, followed by a **single digit of 1**. An example of such a number would be 41, 4441, or any similar pattern extending with 4s. My question is: Can such a number ever be a **perfect square**?
To clarify, the ... |
Can $4\cdots41$ (with odd number of $4$s) be a Square Number? |
Why do you think ‘apparently… we get a rhombus’? It is not always true.
Any proof that assumes that the triangles join to form a rhombus is wrong, because we have a counter example: take $a=b=c=d=6$. Unless we use the areas somehow, we can never assure that they will join to form a rhombus.
Thinking of a proper p... |
The phrase "prebase" is a bit unusual, most frequently one uses the word *[subbase](https://en.wikipedia.org/wiki/Subbase)*.
Given a set $X$ and a set $\Sigma$ of subsets of $X$, there is a coarsest topology $\mathcal T(\Sigma)$ containing $\Sigma$. It is called the *topology generated by $\Sigma$*. Explicitly it ca... |
**Background:** In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field $\mathbf{B}$ via the [Biot-savart law][1]. If the wire is a curve $\gamma$ parametrized as $\mathbf{y}(s)$, where $s$ is the arc-length, then
$$
\mathbf{B}(\mathbf{x}) = \b... |
How can one evaluate $\int \frac{1}{\sqrt{x^2+x-1}}dx$?
Our professor has given us this integral to think about and I've no idea what to do to solve it. I understand there's probably a substitution somewhere but i can't figure out how. The integral calculators I've used online haven't been helpful enough to explain ... |
Let $d\in\mathbb{Z}$ be an integer which is not a square (it does not have to be squarefree, though).
**Question.** Assume that $\vert d\vert\geq 3$ to avoid special cases. Is is true that $\pi=-1+\sqrt{d}$ is irreducible in $R=\mathbb{Z}[\sqrt{d}]$ ?
**Remark.** I can show this it is true in the following cases:... |
First we want to factorise the denominator, in order to set up a convenient substitution: $$\int\frac1{\sqrt{x^2+x+1}}\mathrm{d}x$$ can be rewritten as $$\frac2{\sqrt3}\int\frac1{\sqrt{(\frac2{\sqrt3}x+\frac1{\sqrt3})^2+1}\mathrm{d}x$$ by completing the square and rescaling the constant term in the denominator. Looking... |
Let $G$ be a finite group. When does the automorphism group $\text{Aut}(G)$ act transitively on a subset of $n$-tuples of generators of $G$ that multiply to 1?
I'm interested in this question in relation to the triangle groups. For example, when $G=\text{PSL}(2,7)$, there are exactly 336 triples of generators of ord... |
First we want to factorise the denominator, in order to set up a convenient substitution: $$\int\frac1{\sqrt{x^2+x+1}}\mathrm{d}x$$ can be rewritten as $$\frac2{\sqrt3}\int\frac1{\sqrt{(\frac2{\sqrt3}x+\frac1{\sqrt3})^2+1}}\mathrm{d}x$$ by completing the square and rescaling the constant term in the denominator. Lookin... |
Consider a number in its decimal representation that begins with an **odd number of My question is: Can such a number ever be a **perfect square**?
Is there a mathematical approach or theorem that directly addresses the properties of numbers with specific digit patterns in relation to being perfect squares?
2. Co... |
Can $32$ be a Square Number? |
When does $\text{Aut}(G)$ act transitively on $n$-tuples of generators of $G$ whose product is 1? |
Now I have a nonlinear elliptic PDE, if I do scaling such as $u=\lambda v$ and then get a new PDE, then I turn to study the new equation since it has a simpler form. My question is easy to understand: **if I prove that for the new PDE, all the solutions are not the local minimum of the energy functional (of the new PDE... |
Consider $T, g: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ which are bounded on any bounded interval, satisfy $$T(x)=2 T(x / 2)+g(x)$$ for every real number $x \geqslant 1$ and $g(x)=\mathcal{O}(x)$.
I would like to show $T(x)=\mathcal{O}(x \log (x))$. For this I want to show $\lim_{x\rightarrow \infty} \dfrac{T(x)}{x\lo... |
> Prove that the map $$f:(0,1)\to\mathbb{R}^2$$ $$t\mapsto (\cos 2\pi
t,\sin 2\pi t)$$ is an embedding.
PS: In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map $f:X\to Y$ between topological spaces $X, Y$ is a topological embedding if $f$ yields a hom... |
Consider a number in its decimal representation that begins with an **odd number of consecutive digits of 4**, followed by a **single digit of 1**. An example of such a number would be 41, 4441, or any similar pattern extending with 4s. My question is: Can such a number ever be a **perfect square**?
To clarify, the ... |
Can $4\cdots41$ (with odd number of $4$s) be a Square Number? |
The game setup is as follows: there is a bag with 6 matching pairs of cards (two 1s, two 2s, two 3s, two 4s, two 5s, two 6s).
We randomly draw one card at a time, but put matching cards aside as soon as they appear in our hand. The game ends and we lose if we ever hold three cards, no two of which match. What is th... |
I'm looking for a simple hyperbola-like function that has a linear like segment that can be parameterized to start at ``` a ``` and end at ```b``` (see attached illustration). Does anybody know a simple function that fullfils that? I've been trying ``` y=1/(x+1)+1/(x*a)``` so far.
https://i.stack.imgur.com/5pnlU.png |
Let $\mathbb C[S_3]$ be the group algebra of $S_3$. We have an isomorphism of $\mathbb C$-algebras
$$\mathbb C[S_3]\cong \mathbb C \times \mathbb C \times M_2(\mathbb C).$$
The existence of such an isomorphism is not difficult to show using Maschke's theorem, character theory, and some other observations, as done, ... |
In general, I do not think the statement in the title is true. But from *Galois Theories*, by Francis Borceus and George Janelidze, they claimed a similar fact without proof. I shall give sufficient background for the question before quoting his words.
Let $K\subset L$ be a Galois extension and let $\operatorname{Sp... |
You don't need to prove that the limit is finite the limit can even not exist. However, you can easily prove that the function is bounded. Indeed, for $x\in \mathbb R_{>0}$, $n = \left\lceil\log_2(x)\right\rceil$ then
$$\log_2(x) \le n \le \log_2(x) + 1 \implies x \le 2^n \le 2x \implies \frac12 \le \frac{x}{2^n} \le... |
In general, I do not think the statement in the title is true. But from *Galois Theories*, by Francis Borceus and George Janelidze, they claimed a similar fact without proof. I shall give sufficient background for the question before quoting his words.
Let $K\subset L$ be a Galois extension and let $\operatorname{Sp... |
Let $\mathbb C[S_3]$ be the group algebra of $S_3$. We have an isomorphism of $\mathbb C$-algebras
$$\mathbb C[S_3]\cong \mathbb C \times \mathbb C \times M_2(\mathbb C).$$
The existence of such an isomorphism is not difficult to show using [Maschke's theorem](https://en.wikipedia.org/wiki/Maschke%27s_theorem). Mor... |
In general, I do not think the statement in the title is true. But from *Galois Theories*, by Francis Borceus and George Janelidze, they claimed a similar fact without proof. I shall give sufficient background for the question before quoting his words.
Let $K\subset L$ be a Galois extension and let $\operatorname{Sp... |
> May someone help me understand the following inductive definition of concatenation of strings
You would have an *inductive definition* of "string", which typically has two constructors, one for the "empty" string, and one for the "compound" string with one head character followed by the rest of the string. (In fa... |
I'm trying to solve the following integral:
Integral of: 1 / (1 + ax^2) dx
Where a is some positive constant and a is not equal to 1.
We can't of courses use basic U Substitution as the derivative of 1 + ax^2 is 2ax, which isn't found elsewhere in the expression.
I tried to use fractional decomposition to int... |
How can we integrate 1 / 1 + ax^2? |
I aim to assess the soundness of my approach. In an exercise, the question is posed regarding whether $\mathbb{R}[x]$ is Noetherian, and my response is negative. To support this, I employ the definition that a ring $R$ is Noetherian if each ascending chain of ideals in $R$ eventually becomes stable.
Considering the ... |
Is the ring $\mathbb{R} [x]$ Noetherian? |
In general, I do not think the statement in the title is true. But from *Galois Theories*, by Francis Borceus and George Janelidze, they claimed a similar fact without proof. I shall give sufficient background for the question before quoting his words.
Let $K\subset L$ be a Galois extension and let $\operatorname{Sp... |
Is it possible to show
$$
\int_{0}^{1}\frac{K(k)\ln\left[\tfrac{\left ( 1+k \right)^3}{1-k} \right] }{k}
\text{d}k=\frac{\pi^3}{4}\;\;?
$$
where $K(k)$ is the complete elliptic integral of the first kind with modulus $k$. One proof is to compute the twins first:
\begin{aligned}
&\int_{0}^{1} \frac{K(k)\ln(1+k)}... |
To find the maximum of a BST I thought that it would be sufficient to start at the root node and then check if it has a right child (The left child is not relevant since every left subtree of a node has smaller entries than the node itself). Then repeat the process until there is no more right child. The final node is ... |
Rectangle $ABCD$ is tilted such that its base $AB$ makes an angle of $\theta$ with the horizontal floor. It has its vertex $A$ in contact with the horizontal floor, and the rectangle is released from this position, to fall freely and the vertex at $A$ is free to slide along the floor.
What will be the rectangle's ... |
To expand on my comment, suppose $f \in \mathbb{R}[x]$ vanishes at every point outside of a finite set $S \subset \mathbb{R}$. Then, the set of elements of $\mathbb{R}$ which are sent to zero are $f^{-1}(\{0\}) \subset \mathbb{R}$, which is a closed set. Hence, since $\mathbb{R} \setminus S$ is dense in $\mathbb{R}$, w... |
I'm trying to solve the following integral:
Integral of: $\int_{}^{}\frac{1}{1+ax^2}dx$
Where $a$ is some positive constant.
We can't of course use basic U Substitution as the derivative of $1 + ax^2$ is $2ax$, which isn't found elsewhere in the expression.
I tried to use fractional decomposition to integrate... |
Given a measure $m \in \mathcal P(\mathbb R^d)$, one can sample from $m$ using realizations of a random variable $U$ uniformly distributed on $[0,1]^d$. The procedure is called (multivariate if $d \ge 2$) inverse transformation sampling, and defines a map
$$
F : \mathcal P(\mathbb R^d) \times [0,1]^d \to \mathbb R^d
... |
I want to lower bound the following expression of choosing $d$ numbers from $[n]:=\{ 1,2,\dots,n \}$.
$$
\sum_{\{ s_{1},..,s_{d}\}\subset [n]} d!\frac{1}{s_{1}s_{2}\dots s_{d}}
$$
Here $s_{1},\dots,s_{d}$ are all distincts. I know I can upper bound this by $\left( 1+\frac{1}{2}+\dots+\frac{1}{n} \right)^{d} = H_{n... |
Before starting, I need to make a confession: up to the very end, I hoped that someone could write this answer instead of me. That's because this story deserves to be explained and shared, but even this case is all but straightforward. By "tractable", I mean that *there exists someone* who could likely pass an open boo... |
So we're looking at the ($\epsilon$, $\delta$)-definition of the limit in class and I am kind of confused, because the teacher says one thing but the books say another. Teacher says that $\lim_{x \to p} f(x) = L$ if:
$(\forall \epsilon > 0)(\exists \delta > 0)(\forall x \in \mathbb{R})(0 < |x - p| < \delta \iff |f(... |
How important is the order of the conditional in the ($\epsilon$, $\delta$)-definition of the limit? Would it matter if it is a biconditional? |
I have the equation from one of the textbooks (which is, I know, very simple):
$40x +3 = (x+14)(x+3) +(x+1)$. My first approach to the solution is as follows
\begin{equation}
40x+3=x^{2} +14x +3x +42 +x +1 \\
40x+3 = x^{2} +18x +43\\
0= x^{2} +18x + 40 \\
0 = x^{2} -22x +14 \\
(x-2)(x-20)
\end{equation}
... |
**The question**
The natural numbers $a,b,c$ are considered non-zero and distinct two by two, so that $a^3+b^3+c^3=\frac{a^2b^2}{c}+\frac{a ^2c^2}{b}+\frac{b^2c^2}{a}$
a) It is possible that one of the numbers is equal to the gemometric average of the other two?
b) Show that $a+b+c$ divides at least one of the... |
I'm new to topology and so I'm questioning myself here. But, this is what I've got:
Let $Y = \{ U \subseteq X : p \notin U \}$ so that $\tau_p = Y \cup X$. Let $x \in X$ such that $x \neq p$. Then,
$X^{\circ} = X$, $Y^{\circ} = Y$, $\emptyset^{\circ} = \emptyset$, $\{p\}^{\circ} = \emptyset$, $\{x\}^{\circ} = \{x\... |
How can we integrate $\frac{1 }{1 + ax^2}?$ |
I have the equation from one of the textbooks (which is, I know, very simple):
$40x+3 = (x+14)(x+3)+(x+1)$. My first approach to the solution is as follows
$$\begin{align*}
40x+3&=x^{2} +14x +3x +42 +x +1 \\[0.1cm]
40x+3 &= x^{2} +18x +43\\[0.1cm]
0 &= x^{2} +18x + 40 \\[0.1cm]
0 &= x^{2} -22x +14 \\[0.1cm]
... |
"Find an explicit biholomorphism between $\{ z \in \mathbb{C} | |(z)| <1 \}$ and $\{ z \in \mathbb{C} | \text{Im}(z) > 0 \}$.
I was thinking about the function $f:\{z \in \mathbb{C} | |(z)| < 1 \}->\{z \in \mathbb{C} | \text{Im}(z) > 0 \}$ given by $f(z)=\frac{iz}{z-1}$ and is inverse $g(w)=\frac{w}{w-i}$, how can ... |
The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/Generalized_continued_fraction#The_equivalence_transformation
Wikipedia says that this can be proved via induction but give... |
Suppose I have a bounded subset of the real numbers. Call this subset $A$. Assume that $A$ has nonzero measure. Can I show that there is an interval such that almost every number in that interval has to be an element of $A$? Is Lebesgue's density theorem sufficient for this? |
I have the equation from one of the textbooks (which is, I know, very simple):
$40x+3 = (x+14)(x+3)+(x+1)$. My first approach to the solution is as follows
$$\begin{align*}
40x+3&=x^{2} +14x +3x +42 +x +1 \\[0.1cm]
40x+3 &= x^{2} +18x +43\\[0.1cm]
40x &= x^{2} +18x + 40 \\[0.1cm]
0 &= x^{2} -22x +14 \\[0.1cm]... |
I have the equation from one of the textbooks (which is, I know, very simple):
$40x+3 = (x+14)(x+3)+(x+1)$. My first approach to the solution is as follows
$$\begin{align*}
40x+3&=x^{2} +14x +3x +42 +x +1 \\[0.1cm]
40x+3 &= x^{2} +18x +43\\[0.1cm]
40x &= x^{2} +18x + 40 \\[0.1cm]
0 &= x^{2} -22x +40 \\[0.1cm]... |
Let a finite group $G$ of order $n$ be given, so $G$ is isomorphic to a permutation group embedded in $S_n$. Can we always find a set $\Omega$ such that $G$ acts transitively on $\Omega$? (For example, does $G$ always act transitively on a set $\Omega$ of size $n$)? If so, is it a difficult problem to find the smallest... |
Can we choose a set to make sure the action of a permutation group transitive? |
Why do you think ‘apparently… we get a rhombus’? It is not always true.
Any proof that assumes that the triangles join to form a rhombus is wrong, because we have a counter example: take $a=b=c=d=6$. Unless we use the areas somehow, we can never assure that they will join to form a rhombus.
**The following was no... |
The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/Generalized_continued_fraction#The_equivalence_transformation
Wikipedia says that this can be proved via induction but give... |
I am working on a differential equation problem, and have ended up with the following form:
$$g(x; N) = \sum_{n=0}^{\infty} \frac{\chi_0(n)}{n!} x^n,$$
with $\chi_0(n)$ the principal Dirichlet character mod $N$.
It is clear that this is equivalent to
$$g(x; N) = \sum_{(n,\,N) \, = \, 1} \frac{x^n}{n!}.$$
... |
How to construct $\Delta ABC$, given the angle at $C$ , $\gamma$ the median drawn fom $C$, $t_c$, and the angle at the point $B$, $\beta$? |
I'm new to topology and so I'm questioning myself here. But, this is what I've got:
We have $\tau_p = \{ U \subseteq X : p \notin U \} \cup \{X\}$. Let $x \in X$ such that $x \neq p$. Then,
$X^{\circ} = X$, $\emptyset^{\circ} = \emptyset$, $\{p\}^{\circ} = \emptyset$, $\{x\}^{\circ} = \{x\}$.
Hence, the only ne... |
Suppose I have a bounded subset of the real numbers. Call this subset $A$. Assume that $A$ has nonzero measure. Can I show that there is an interval from the real numbers such that almost every number in that interval has to be an element of $A$? Is Lebesgue's density theorem sufficient for this? |
The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/Generalized_continued_fraction#The_equivalence_transformation
Wikipedia says that this can be proved via induction but give... |
Do we have
$$
7^k\mid 3^n+5^n-1\Longleftrightarrow n=t\cdot 7^{k-1}\quad\text{with}\quad t\equiv 1,5\,(\operatorname{mod}6)?
$$
The part "$\Leftarrow$" is easy. Let $a=3^{t\cdot 7^{k-1}}$, then $5^{t\cdot 7^{k-1}}\equiv a^{-1}\,(\operatorname{mod}7^k)$, and $$7^k\mid a^6-1=(a-1)(a+1)(a^2+a+1)(a^2-a+1).$$
We have ... |
### Help! My System of Linear Equations is on a Rampage - It Won't Follow the Rules! How Do I Tame These Rebel Equations?
Hello,
I'm grappling with an intriguing puzzle involving a system of two equations with two unknowns. Despite my attempts at the conventional substitution method, I find myself stuck with trivial... |
Help! My System of Linear Equations is on a Rampage - It Won't Follow the Rules! How Do I Tame These Rebel Equations? |
>Find an explicit biholomorphism between $\{ z \in \mathbb{C}\mid |(z)| <1 \}$ and $\{ z \in \mathbb{C} \mid\text{Im}(z) > 0 \}$.
I was thinking about the function $$f:\{z \in \mathbb{C} | |(z)| < 1 \}\to\{z \in \mathbb{C} | \text{Im}(z) > 0 \}$$ given by $f(z)=\frac{iz}{z-1}$ and its inverse $g(w)=\frac{w}{w-i}$. ... |
Consider a harmonic oscillator subject to a frictional force proportional to velocity: $$\ddot{x}+2\gamma\dot{x}+\omega^2x=0.$$
Here $\dot{x}$ and $\ddot{x}$ are $\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}.$ Assume $\omega>\gamma.$ Interpreting this equation as a system of equations for two functions $x\left ( t\right ) $ ... |
I came up with this problem while messing around in Desmos and was very surprised by the solution I got from [Wolfram Alpha][1]. The [integral calculator][2] is unable to solve it and WolframAlpha does not provide an indefinite integral that could hint at a possible solution.
My intuition tells me to try integration... |
I came up with what I think is the solution to exercise **11.** (v) on chapter **1** of the third edition of book *Calculus* by Michael Spivak.
----------
Find all numbers $x$ for which $|x - 1| + |x + 1| < 2$.
*Solution.* Three cases needed consideration.
First, in case $x < -1$, $|x - 1| = -(x - 1)$ while... |
**Question**
Given a sequence $a_n$ when it is possible to express such sequence as a "linear recurrence relation with not constant coefficients"? *i.e.* when
$$
a_n= \sum_{i=0}^{n-1} f_i(n) a_{i} + g(n)
$$
where
$$
f_i(n)=\frac{P_i(n)}{Q_i(n)} r^n
$$
for $P_i, Q_i$ polynomials and $r \in \mathbb{C}$ and ... |
Given a multivariate normal vector $X \sim N(0,I_d)$ (identity covariance matrix), it is well known that :$$\frac{X}{\|X\|_2}
$$
is uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb{R}^d$, where here $\|\cdot\|_2$ denotes the usual Euclidean norm.
**Question**: Is this is not the case for **a... |
I am reading Silverman's Advanced topics in the arithmetic of elliptic curves. In the context of schemes, a point is defined to be *regular or non-singular* if its tangent space has dimension equal to the Krull dimension of its local ring. Then there's this example, which **implies the existence of singular points whic... |
In the derivation of the Bernoulli-Seki formula the first step is unfolding the exponential generating function of $S_n(k)=\sum_{a=1}^k a^n$
as
$$
\sum_{n \geq 0} S_n(k) \frac{T^n}{n !}=\sum_{a=1}^k \sum_{n \geq 0} \frac{(a T)^n}{n !}=\sum_{a=1}^k e^{a T}=e^T \frac{e^{k T}-1}{e^T-1}
$$
This part is clear.
H... |
I am reading Silverman's Advanced topics in the arithmetic of elliptic curves. In the context of schemes, a point is defined to be *regular or non-singular* if its tangent space has dimension equal to the Krull dimension of its local ring. Then there's this example, which **implies the existence of singular points whic... |
i'm trying to prove that cannot exists a countable sigma algebra (not trivial).
So, the statement is the following:
Let $X$ a set and let $\mathcal{M}$ a sigma algebra on X. Show that $\mathcal{M}$ is finite or is uncountable.
_Proof_. If $X$ is finite, the thesis is obviuos because the power set of $X$ is fi... |
In an attempt to self-study, I am trying to answer the following question:
Let X be an Euclidean space. Suppose that $a \in X\setminus \{ 0 \}$ and let $\beta \in \mathbb{R}$. When is $A = \{x \in X \mid \langle x, a \rangle \neq \beta\}$ convex?
I noticed that if I choose $x, y$ such that $\langle x, a \rangle ... |
When can the complement of a hyperplane be convex? |
Suppose $N$ is a normal operator on some $\mathbb{C}$-Hilbert space $\mathcal{H}$ and let $\mathrm{E}:\mathcal{B}(\sigma(N))\to B(\mathcal{H})$ be the associated spectral measure.
Fix $\lambda_0\in\sigma(N)$ and suppose $x\in\mathrm{E}(\{\lambda_0\})=\mathrm{im}(\mathrm{E}(\{\lambda_0\}))$. How would one show that $... |
### Help! My System of Linear Equations is on a Rampage - It Won't Follow the Rules! How Do I Tame These Rebel Equations?
Hello,
I'm grappling with an intriguing puzzle involving a system of two equations with two unknowns. Despite my attempts at the conventional substitution method, I find myself stuck with trivial... |
I am reading Silverman's Advanced topics in the arithmetic of elliptic curves. In the context of schemes, a point is defined to be *regular or non-singular* if its tangent space has dimension equal to the Krull dimension of its local ring. Then there's this example, which **implies the existence of singular points whic... |
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