Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Basic math problem with Integrating Factors: Differential Equations I was watching a youtube tutorial on Integrating Factors and I'm lost in a part where the derivative of: xy' + 1y becomes xy.
Please I need some clarification on that part.
Secondly.
I was watching a another youtube tutorial on Integrating Factors and ... | Suppose you have
$$xy'+1y=0$$
Notice that we have
this is that we have $$\frac{d}{dx}(xy)=x\frac{dy}{dx}+\frac{dx}{dx}y=xy'+y$$
by product rule.
Similarly, for $$\frac{dy}{dx}+\frac{2x}{1+x^2}y=0$$
By multiplying integrating factor of $(1+x^2)$, we have
$$(1+x^2) \frac{dy}{dx}+2xy=0$$
which can be written as
$$(1+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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intersection of all neighborhoods of a point in zariski topology. Consider the affine space $A^n$ with the Zariski topology,
V a variety (with the induced topology) and let $P\in V$ a point. Let B be the set of all neighborhoods of the point P in V. Is it true that $P=\bigcap\limits_{U_i\in B} U_i$?
Obviously this is ... | Follows from the fact that $\Bbb A^n$ is $T_1$ and that subspace of $T_1$ space is $T_1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Show that each connected component of a topological space $X$ is connected.
Show that each connected component of a topological space $X$ is connected.
Here is my proof
My Proof: Choose a connected component $$C = [x] = \{y \in X \mid \exists \text{ a connected set containing both $x$ and $y$}\}$$
Now for each $y \in... | I don't think it matters too much that the sets $C_y$ may not be unique. The proof rests on the notion that a union of connected sets with common intersection is connected, which seems plausible (I haven't tried to prove it though). If that isn't an established proposition in your text though, I think it should be pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Determining the null space of the matrix
Determine the null space of the matrix:$$\begin{bmatrix} 1 & -1 \\ 2 & 3 \\ 1 & 1 \end{bmatrix}$$
My try:
$$\begin{bmatrix} 1 & -1 \\ 2 & 3 \\ 1 & 1 \end{bmatrix}_{R_2\rightarrow R_2-2R_1\\R_3\rightarrow R_3-R_1}$$
$$\begin{bmatrix} 1 & -1 \\ 0 & 5 \\ 0 & 2 \end{bmatrix}_{R_3\... | Recall that by definition the nullspace is the subspace of all vectors $\vec x$ such that $A\vec x=\vec 0$ and in that case we have ony the trivial solution $(x_1,x_2)=(0,0)$ then $Null(A)=\{\vec 0\}$.
Notably to solve $Ax=0$ we can proceed by RREF to obtain
$$\begin{bmatrix} 1 & -1 \\ 2 & 3 \\ 1 & 1 \end{bmatrix}\to
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to write a polynomial in mod p Consider the polynomial $ \ x^3 - 3\ x^2 +2\ x -1$
How can this polynomial be written in mod 3 ?
What confuses me is that I thought we don't change any power . If we have to also consider the powers then will the leading coefficient be equal to zero then.
|
What confuses me is that I thought we don't change any power . If we have to also consider the powers then will the leading coefficient be equal to zero then.
Your first thought is correct: we don't change any power. Even though $3$ is in the exponent, we don't take the exponents mod 3, but only the coefficients.
So,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there a fundamental mathematical function that requires 3 inputs or more? So a mathematical operation can be represented as a function that maps inputs to outputs. For example "sin(x)" is a function that maps 1 input to 1 output, and "a + b" maps 2 inputs to 1 output. My question is is there a function that requires... | Thanks for your answers and comments! A conditional statement is to me a satisfactory answer. In order for a conditional statement to make sense it requires (a) the condition, (b) the yield if the condition evaluates to true, and (c) the yield if the condition evaluates to false. To the best of my knowledge all three c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2877962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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show that $\operatorname{rank}(g\circ f) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$ Let $E$ a vector space and $\dim(E)=n$
and let $f,g \in L(E)$
show that $\operatorname{rank}(f\circ g) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$
I can see that $\operatorname{Ker}(g) \subset \operatorname{Ker}(f\... | The reverse inequality is true. To see it, apply the rank-nullity theorem twice. $\DeclareMathOperator{\Im}{Im}$
Observe in the first place that $$\DeclareMathOperator{\rk}{rank}\rk(g\circ f)=\rk\Bigl(g_{\,\bigm\vert_{\,\scriptstyle\Im f}}\Bigr)\quad\text{and}\quad
\ker\Bigl(g_{\,\bigm\vert_{\,\scriptstyle\Im f}}\Bigr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2878056",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Uniform Convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n} x^n$ Prove that $\sum_{n=1}^\infty \dfrac{(-1)^n}{n} x^n$ converges uniformely on $[0,1]$.
Using the Leibiz criteria, I could show, that the series converges for $x\in[0,1]$. But how can I show the unifom convergence? I know about the Weierstrass M-test... | The Dini Theorem: If each $f_n:[0,1]\to \Bbb R$ is continuous and if $(f_n(x))_n$ converges monotonically to $f(x)$ for each $x\in [0,1],$ and if $f$ is continuous, then $f_n\to f$ uniformly on $[0,1].$
Apply this, first to $(g_n)_n,$ where $g_n(x)=\sum_{j=1}^{2n}(-1)^jx^j/j,$ and second to $(h_n)_n$ where $h_n(x)=\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2878157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of n terms of this series $\frac{1}{1.3} + \frac{2}{1.3.5} +\frac {3}{1.3.5.7} + \frac{4}{1.3.5.7.9}........ n $ Terms.
I Know the answer to this problem but I couldn't find any proper way to actually solve this question. I thought the denominators were the product of n odd natural numbers. So I wrote the nth te... | The $n$th term has $n$ as its denominator, and the odd factors in $(2n+1)!$ as its numerator. Or, if we multiply both by the even factors, which are $\prod_{i=1}^n 2i=n!2^n$, the denominator becomes $(2n+1)!$. The $n$th term is therefore $\dfrac{n! n2^n}{(2n+1)!}$. (The formula you've obtained is also correct, by the w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2878281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$ABC$ is a triangle there are points $D, E$ and $F$ on sides $AC, AB$ and $BC$ respectively if $AF = 4$ and $BD = 12$ find the minimum value of $EC$ Does this use any specific inequality?
here's my approach:
since a line interior of a triange drawn from a vertice cannot exceed the neighbouring sides we can set an ineq... | It’s simple if you allow points A&D, C&F, and B&E to coincide. Then, a variation of (assuming angle BAC is not right) the Pythagorean theorem will yield [a variation of] $(EC)^2=(AF)^2+(BD)^2$. Whatever the approach, the minimum length of EC will require the segments AF, BD, and EC each to be the entire length of thei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2878385",
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Prove by induction that $n^2 + n + 1 \forall n\geq 1$ given the following recurrence relation My question is as follows:
Consider the following recurrence relation:
$a_{n} = a_{n-1}+2n$, with $a_{1}=3$
Prove by induction that
$a_{n}=n^{2}+n+1 \forall n\geq 1$
I have no idea how to even approach this question. I am qui... | Induction is usually achieved in three steps.
1) Initialization. You just need to check that your property is valid for the lowest integer required (here $n = 1$). So what you need to check is that indeed, $a_1 = 1^2 + 1 + 1$.
2) Heredity. This step is usually the most difficult, and you have to be careful when writing... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Quantifying a free variable in an example from "How To Prove It" by Velleman This is an example from How To Prove It by Daniel J. Velleman (2nd Ed., p. 71):
Example 2.2.3. Analyze the logical forms of the following statements.
*
*Statements about the natural numbers. The universe of discourse is $\mathbb N$.
... | You shouldn't confuse variables (the only ones that can be quantified) with individual constants.
The logical form of the sentence "$x$ is a perfect square" is indeed $\exists y (x = y^2)$. In this formula, $x$ plays the role of an individual constant, not of a variable, because it refers to one specific individual in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2878577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the coefficient of $a$ in a quadratic equation?
If a root of the equation $$3x^2 + 4x + 12a + 9ax = 0$$ is greater than 6, then the correct statement of the coefficient $a$ is:
a) $a = 2$
b) $a> -2$
c) $a = -2$
d) $a <-2$
e) $-2\le a\le 2$
What I did was solve it as if a root was exactly $6$, and I find t... | Hint
\begin{align} 3x^2+4x+12a+9ax = 0&\iff 3x^2+(4+9a)x+12a= 0\\ &\iff x=\dfrac{-4-9a\pm \sqrt{(4+9a)^2-144a}}{6}.\end{align} So,
$$\dfrac{-4-9a\pm \sqrt{(4+9a)^2-144a}}{6}\ge 6 \iff -4-9a\pm \sqrt{(4+9a)^2-144a}\ge 36. $$
That is
$$-4-9a\pm \sqrt{16+72a -63a^2}\ge 36$$
So, you can check what is the correct answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2878680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Need help finding a sum I found this problem that I'm not sure how to solve. I would appreciate if anyone could point me to the right direction.
I need to find the following sum:
$\sum_{i+j+k=7} (-1)^i(-1)^j\frac{7!}{i!j!k!}$ where $i,j,k$ are elements of $\mathbb{N_0}$
There must be a much better way to solve this oth... | There is no $(-1)^k$ because $x_3$ is going to be $1$: $$-1=((-1)+(-1)+1)^7=\sum_{i+j+k=7} (-1)^i(-1)^j(1)^k\frac{7!}{i!j!k!}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2878780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to construct a matrix given the null basis of A? Construct a $4\times4$ matrix $A$ such that $\{(1,2,3,4),(1,1,2,2)\}$ is a basis of $N(A)$.
So I know that $A$ will have two pivot columns and two free columns, but beyond this I'm not sure how to approach/solve.
| The row space of a matrix is the orthogonal complement of its null space. So, you can construct the required matrix by finding a basis for this orthogonal complement. In this case, this will give you two of the rows, and the other two rows can be any linear combinations of those two rows, including rows of all zeros. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Discriminant (in the context of PDE classification): $b^2 - 4ac$ or $b^2 - ac$? I'm reading two textbooks on partial differential equations. In their respective sections on classification of PDEs (hyperbolic, parabolic, elliptical), they differ in what they describe as being the discriminant. One textbook says that the... | A few points -
*) If we are discussing the second order linear PDE of two independent variables, which say is of the form - $Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$, where $A$, $B$, $C$, $D$, $E$, $F$, $G$ are functions of $x$ and $y$ and could be constants, then we must be careful about the coefficients ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's the equation of the graph $y=x^3-x^2+x-2$ after it is reflected in the x axis, and then the y axis as well? I don't fully understand the steps to reach the answer.
I've plotted the graph and I found out that the graph is an odd graph and that after reflecting in both axes, the only thing that changes on the gra... | 1)Reflection about $x-$axis:
$(x,y) \rightarrow (x,-y)$;
2)Reflection about $y$-axis:
$(u,v) \rightarrow (-u,v)$;
Combining:
3) $(x,y) \rightarrow (x,-y) \rightarrow (-x,-y)$.
Given : $y=f(x) =x^3-x^2+x-2$;
We have $(x,f(x)) \rightarrow (-x,-f(x))$, the reflected graph .
$(-x)$ is mapped to $-f(x)$, or
$x = -(-x)$ is m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that if $f$ differentiable in $x^*$ and $f(x^*)=0$, then $\liminf_{x\to x^*}\frac{|f(x)|}{||x-x^*||}=0$ if $n>1$ Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be differentiable at $x^*$ and $f(x^*)=0$. Prove that if $n>1$, then $$\liminf_{x\to x^*}\frac{|f(x)|}{||x-x^*||}=0.$$
Is this true for $n=1$?
I know that because... | Why the difference between the cases $n=1$ and $n>1$?
For $n=1$, the kernel of a linear map can be reduced to the zero vector. This isn't the case for $n>1$.
The result is true for $n>1$
In that case $L$ is a linear form and its kernel is not reduced to the zero vector. Take $a \in \ker L \setminus \{0\}$. For $m \in \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Ask about global maximum and global minimum? The temperature distribution in a metal rod given by the following function of the position $x \in \mathbb{R}$: $$T(x) = \frac{1 + 2x}{2 + x^2}$$
What is the maximal and minimal temperature in the metal rod?
$T'(x) = 0$ when $x = 1$ or $x = -2$. But I can't calculate the glo... | Assuming your work on the derivative is correct (I haven't checked), since $T$ is continuous:
\begin{align}
T(1)&=1\\
T(-2)&=-\frac{1}{2}\\
T(-\infty)&=T(+\infty)=0
\end{align}
So there you have: global max at $x=1$ and global min at $x=-2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to argue that a series that isn't a power series is analytic? Let $f(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{1-z^{n}}$. Then $f$ converges in the unit disc. I want to show that it is analytic in this region as well, but since it is not a power series I don't have any theorems to apply. My only thought was to try and ... | Let $C$ be a compact subset of the open unit disk and let $M=\sup_{z\in C}|z|$. Then $M<1$ and, for each $n\in\mathbb N$,$$\left|\frac{z^n}{1-z^n}\right|=\frac{|z|^n}{|1-z^n|}\leqslant\frac{M^n}{1-|z|^n}\leqslant\frac{M^n}{1-M^n}.$$Since the series $\sum_{n=1}^\infty\frac{M^n}{1-M^n}$ converges, your series converges u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does it mean for points to be picked uniformly and independently? I saw a which said that points were picked uniformly and independently. I have a feeling this is important for the solution but I am not sure what they mean by uniformly and independently. Any help would be appreciated. I will post the original ques... | Sometimes a picture helps more than words and formulas, at least for the term "uniform":
"Independent" means that a single one of the random points does not care where the other points are, but its position is picked independently of the others, as if they were not there. E.g. to be not independently chosen, a point c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879582",
"timestamp": "2023-03-29T00:00:00",
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Why is the following statement impossible for a sequence $\{p_n\}$?
A sequence is said to be convergent if there is a point to which it converges. A convergent sequence cannot converge to two distinct limits. For if $\{p_n\}$ were to converge to both $p$ and $q$ with $p \ne q$, then we could choose spherical neighborh... | It is impossible to converge to two different points by definition of convergence described by @copper.hat on the comment above. You cannot have only finite number of values outside two disjoint neighborhoods
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Given $k$ points in $n$-dimensional space, is there always a continuous $n-1$ surface that can divide the points into two arbitrary groups? Say we have $k$ points in set $P\mid x_i\in\mathbb{Z}^n$, such that $k=\mathcal{O}(n!)$. We now arbitrarily divide the points into two sets, $A$, $B$. Note that $A\cup B = P$ and $... | Solve the poisson equation,
$\Delta u = f$
with $f$ consisting of delta function sources at points in $A$ and sinks at points in $B$. The zero set of the solution $u$ separates the points. One should be able to prove that either this zero set is smooth, or the zero set associated with arbitrarily small perturbations of... | {
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"url": "https://math.stackexchange.com/questions/2879778",
"timestamp": "2023-03-29T00:00:00",
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Proving $f'$=$g'$ has some c such that $g=f+c$ Suppose that $f$ and $g$ are differentiable functions on $(a,b)$ and suppose that $g'(x)=f'(x)$ for all $x \in (a,b)$. Prove that there is some $c \in \mathbb{R}$ such that $g(x) = f(x)+c$.
So far, I started with this:
Let $h'(x)=f'(x)-g'(x)=0$, then MVT implies $\exists$ ... | You say that
Let $h'(x)=f'(x)-g'(x)=0$, then MVT implies $\exists$ c $\in
> \mathbb{R}$ such that $h'(c) = \frac{h(b)-h(a)}{b-a} =0$. Then
$h'(c)=0 \implies h(c)=c.$
This is not correct. If $h'(c)=0$ then you have $h(b)=h(a).$
But you are in the correct way. Instead of $a,b$ consider $x,y\in [a,b], x\ne y.$ Then ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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When "If $A$ is true then $B$ is true", is it valid to assert that "If $B$ is false, $A$ must also be false"? If it is given that:
"People that ride buses, also ride planes"
then is the statement
"people that don't ride planes, also don't ride buses"
necessarily true?
I don't think so, but the explanation to a pr... | What you are asking is does:
A ⟹ B (A implies B)
mean
Not B ⟹ Not A (The converse of B implies the converse of A)?
The answer is yes.
In fact this is the basis of what is known as a Proof By Contradition in Maths.
The way I explain it to my A-Level students (18 year old Mathematicians in the UK) that meet this for th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2879999",
"timestamp": "2023-03-29T00:00:00",
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Zeroth-homology of a complex of $n$ connected components I am new at Algebraic topology and am reading Basic Concepts of Algebraic Topology of Croom.
I have a question. In Theorem 2.4/page 25, it states that if $K$ is a complex with $n$ connected components, then $H_0(K)$ is isomorphic to $\mathbb{Z}^n$.
Anyway, in the... | The boundary of a $1$-chain is a linear combination of
boundaries of $1$-simplex. A $1$-simplex in $K$ has endpoints in the
same component of $K$, so in the same $K_i$. If $\partial(c)=\sum_v r_v
\langle v\rangle$ then the sum of the $r_v$
over the vertices $v$ in the same $K_i$ is zero. In your example this sum
is $g_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2880114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Try to learn more about linear operator,subspace and dimension Prove that for every subspace $F$ and every linear transformation $L$ of linear space $V$ and $\dim V=n$ is $\dim L(F)+\dim\ker L=\dim(F+\ker L)$
I know that $F$ and $\ker L$ are subspaces of the same vector space, so we have $\dim(F+\ker L)=\dim F +\dim\ke... | This is a consequence of the isomorphism theorems for modules (in particular, vector spaces).
We have $$ L(F) \simeq \frac{F}{F\cap kerL} \simeq \frac{F+kerL}{kerL}.$$
Computing the dimensions we have $$dimL(F) = dim(F+kerL) - dimKerL. $$
| {
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"url": "https://math.stackexchange.com/questions/2880438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Expected least distance between closest two points out of $n$ drawn from a distribution Suppose I draw $n$ points in $\mathbb{R}$ from a distribution $p$. What is the expected least distance between two of the points drawn? I am particularly interested in the uniform distribution $\texttt{Unif}(a,b)$.
I'd be interested... | With the uniform distribution, this can be discretized nicely, and then it leads to a standard combinatorial problem. For simplicity, I am working on the interval $[0,1]$.
So divide the interval into $N$ equal subintervals ($N$ is large). Then we pick $n$ of these (the points picked are identified by the small interva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2880670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How does one create a polynomial equation based on graphic features? So as many of you know, graphs of polynomial equations with degree greater than 2 have what are known (to me) as local minima and local maxima (the point(s) on its graph where the derivative of the function is zero, the point to which output values co... | Taking the points with the maximum and minimum, which are sufficient to define the cubic, we can just write the cubic as $y=ax^3+bx^2+cx+d$ and plug in the data we have.
$$5=a3^3+b3^2+c3+d\\2=a6^3+b6^2+c6+d\\0=3a3^2+2b3+c\\0=3a6^2+3b6+c$$
This is four equations in four unknowns which can be solved by the usual techniqu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are there any 2 primitive pythagorean triples who share a common leg? So is it possible for:
$\gcd(a,b,c)=1$
$a^2+b^2=c^2$
and
$\gcd(a,d,e)=1$
$a^2+d^2=e^2$
?
| This system of equations:
$$\left\{\begin{aligned}&a^2+b^2=c^2\\&z^2+b^2=x^2\end{aligned}\right.$$
Solutions have the form:
$$b=4tkp^2s^2$$
$$a=2(t^2-k^2)p^2s^2$$
$$z=4k^2s^4-t^2p^4$$
$$c=2(t^2+k^2)p^2s^2$$
$$x=4k^2s^4+t^2p^4$$
$t,k,p,s$ - integers asked us.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2880890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Prove that $a_n
So I got this question and I was able to do up to (III) (a) however part b has been pretty difficult for me to complete and I can't seem to get it right . Any help ?
I tried considering $a_n - a_{n+1}$ and I got the required proof but I don't feel as though this method is logical enough
$a_n - a_{n+1}... | Guide:
*
*Prove that $a_n>0$ if $a_1>0$.
*Compute $a_{n+1}-a_n$ in terms of $a_n$, your goal is to check that the expression is positive.
*check that the denominator is positive.
*Show that the numerator is positive using the property that $0<a_n<2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\log_a(\log_a x^n)$ if $n=a^2$ ; $x= e^2$ My brother challenged me to solve this problem. Trying since 2 days. I came up with $a^{a^y}= x^n$ assuming $y$ is $\log_a(\log_a x^n)$. There's no solution available on net as well. If someone can solve it, it would be of great help! Thanks
Trial 1:
$\Rightarrow\log_a(\... | First notice that your expression is
$$
\log_a(n\log_a x)=\log_a n+\log_a\log_ax
$$
For $n=a^2$ you have $\log_an=\log_a(a^2)=2$; for $x=e^2$,
$$
\log_a\log_a x=\log_a(2\log_a e)=\log_a2+\log_a\log_ae
$$
You could observe that
$$
\log_a2=\frac{\log 2}{\log a}
$$
and
$$
\log_ae=\frac{1}{\log a}
$$
so
$$
\log_a\log_ae=\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2881103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Associative laws with negation Is it possible to simplify a statement like the one below with the associative law despite the negation? I can't seem to find a law that outlines this. The associative law is the closest I could find.
\begin{align}
\ (\lnot p \land \lnot q) \ \lor \ q
\end{align}
If the above is possible... | Associative low holds only when three formulas are put together by two occurrences of the same connective $\land$ or $\lor$. For instance:
\begin{align}
(\lnot p \land \lnot q) \land q &\equiv \lnot p \land (\lnot q \land q) &&\text{or} & (\lnot p \lor \lnot q) \lor q &\equiv \lnot p \lor (\lnot q \lor q)
\end{align}
S... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve: $(x+1)^3y''+3(x+1)^2y'+(x+1)y=6\log(x+1)$ Solve: $(x+1)^3y''+3(x+1)^2y'+(x+1)y=6\log(x+1)$
Is my solution correct:
Answer given in the book : $y(x+1)=c_1+c_2\log(x+1)+\log3(x+1)$
For my later reference: link wolfram alpha
| Robert pointed out your mistake..
Here is another approach
$$(x+1)^3y''+3(x+1)^2y'+(x+1)y=6\log(x+1)$$
divide by $x+1$
$$(x+1)^2y''+3(x+1)y'+y=6\frac {\log(x+1)}{x+1}$$
on the left there is a derivative
$$((x+1)^2y')'+((x+1)y)'=6\frac {\log(x+1)}{x+1}$$
Integrate
$$(x+1)^2y'+(x+1)y=6\int \frac {\log(x+1)}{x+1} dx$$
$$(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2881378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Can any metric on $\Bbb R^n$ be bounded above and below for any other metric? Let $d_1(x,y)$ and $d_2(x,y)$ be any two metrics on $\mathbb{R}^n$. Can it be shown that,
$$c\cdot d_2(x,y) \le d_1(x,y) \le C\cdot d_2(x,y)$$
for all $x,y \in \mathbb{R}^n$ for some fixed positive constants $c,C$? If not, under what conditio... | If $d_1$ is the usual metric and $d_2$ is the discrete metric, then there are no such constants.
However, your statement is true if the metrics are induced by norms in $\mathbb{R}^n$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $\Omega\subset\mathbb{R}^n$ is convex and $f$ is differentiable, then $f(x)-f(y)\geq f'(y)(x-y),\;\forall \;x,y\in \Omega$ Let $\Omega$ be a convex set in $\Bbb{R}^n$. We say that that $f:\Omega\to \Bbb{R}$ is convex if $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y),\;\forall\;0\leq t\leq 1,\;\&\;\forall\;x,y\in \Omega.$$ I wan... | Clearly You have $$\frac{f(tx+(1-t)y)-f(y)}{t}\leq f(x)-f(y)$$
and for $t\rightarrow 0$ the LHS numerator and denominator tend to zero so that De L´Hospital applies and the partial derivative $\partial/\partial t$ of the numerator is $$f^´(tx+(1-t)y)(x-y)$$ and that of the denominator $1$ and now take $t\rightarrow 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2881581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Writing a Cartesian Equation for a plane. A plane $\pi_2$ intersects $\pi_1$: $4x-2y+7z-3=0$ at right angles. Two points lie on $\pi_2$: $A(3,2,0)$ and $B(2,-2,1)$. Write a cartesian equation for $\pi_2$.
I know that the normals of these two planes must be perpendicular since the planes are perpendicular. So if $n_2=(... | The plane $\pi_2$ has a cartesian equation of the type $ax+by+cz=d$. And you know that $(3,2,0),(2,-2,1)\in\pi_2(\iff3a+2b=2a-2b+c=d)$. Furthermore, $\pi_1$ and $\pi_2$ intersect at right angles, which is equivalent to the assertion$$(a,b,c).(4,-2,7)=0(\iff4a-2b+7c=0).$$Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2881686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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On the proof of Weils hyperelliptic theorem Let $p$ be a large prime. Consider $F_p$
Theorem:
Let $P$ be an element in $F_p[x]$ of degree $k$, assume that $P$ is not a constant multiple of a square. Then the number of solutions $(x,y)$ in $(F_p)^2$ to $y^2 = P(x)$ is $p+O_k(\sqrt p)$
A proof of this result can be found... | Yeah it's actually stupid:
We want the derivative to also be of the form with $P^((p-1)/2)$, and a way to promise that is multiply by $P^l$ and give copies of it when taking the derivative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2881792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which of the following intervals contains integers satisfying the following three congruences: Question: Which of the following intervals contains integers satisfying the following three congruences:
$x\equiv 2\pmod 5, x\equiv 3\pmod 7$ and $x\equiv 4\pmod {11}$,
(i) $[401,600]$, (ii) $[601,800]$, (iii) $[801,1000]$, a... | The point is that there is no reason to focus on the smallest non negative solution.
Adding/subtracting an arbitrary amount of times the number $385 $ will yield other solutions. In fact, all of them, by the Chinese Remainder Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2881895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Optimal number of elements in a subset The following problem appears in many Combinatorics books:
Prove that among 70 positive integers less than or equal to 200 there are two whose difference is 4,5, or 9. The solution is straightforward using Pigeonhole principle. I was wondering is 70 the optimal number? I have tri... | Your proof looks O.K. to me. Anyway, your result is correct. Some comments:
*
*Your introduction is slightly misleading. If you have quoted the problem correctly from the (unnamed) combinatorics books, then you have improved the number $70$ to $65,$ not $64.$ You have proved that, among $65$ positive integers less t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2881997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inscribed trapezoids problem Let $ABCD$ be inscribed trapezoid with $(AB) \parallel (CD)$ and let $P$ be the point where its diagonals meet.
The circumcircle of $\triangle APB$ meets line $(BC)$ (again) at $X$.
$Y$ is a point on $(AX)$ such that $(DY)\parallel (BC)$.
Prove that $\angle YDA = 2 \angle YCA $.
Any hints ... | Some hints:
*
*Prove that $ADPY$ is cyclic.
*Prove that $PC=PY$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2882130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to calculate (x,y) position of number in square I am looking for a way to determine the X & Y position of a number to draw following square:
1 2 5 10 17
3 4 6 11
7 8 9 12
13 14 15 16
What kind of algorithm / formula can I use ?
I have tried rounding the square root of the number to determine ... | Here is an algorithm (written for Matlab) that works:
n = 5;
x = zeros(n*n,1);
y = zeros(n*n,1);
x(1) = 1; % set the first 3 points
y(1) = 1;
x(2) = 2;
y(2) = 1;
x(3) = 1;
y(3) = 2;
for i = 4:n*n
if (x(i-1) == y(i-1)) % jump all the way down (and 1 to the right)
x(i) = x(i-1)+1;
y(i) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2882220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Finding the Jordan Form of a transformation defined by $T(X)=AX$ when $A,X \in M_{4\times 4}(\mathbb C)$
Given $A=\left(\matrix{0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&-1}\right)$ and define $T(X)=AX$. when $A,X \in M_{4\times 4}(\mathbb C)$
Find the Jordan Form of T
I found that the minimal polynomial is $(t-1)(t+1)$ there... | I identify a $4\times 4$ matrix by a vector of length $16$ by reading it from top to bottom, from left to right. So the first four coordinates are from the first column, etc.
Then $T$ is diagonal in the last $8$ coordinates: identical on coordinates $9-12$, and negative of the identity on $13-16$, so that part is cove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2882345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence in probability, mean and almost surely $X$, $Z_n$, $Y_n$ are independent random variables, where X is integrable, $Y_n$ has Bernoulli distribution $b(1,n^{-2})$ , $Z_n$ has Poisson disribution with parameter $n^2$. I need to check convergence of $V_n = Y_nZ_n + (1-Y_n)X$ to $X$ in mean, probability and a.s.... | Observe that for $\epsilon>0$: $$\{|V_n-X|>\epsilon\}\subseteq\{V_n\neq X\}\subseteq\{Y_n\neq 0\}$$so that:$$\mathsf P(|V_n-X|>\epsilon)\leq\mathsf P(V_n\neq X)\leq\mathsf P(Y_n\neq 0)=n^{-2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2882413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is combination of elementary row operation considered elementary row operation? I came across this question in my homework if $R_1-R_2-R_3$ considered elementary row operation. My opinion is that it should not be an elementary row operation since it contains three rows which violates the rule "Add a multiple of one row... | I wouldn't consider it an elementary row operation. In much the same way, when we write 'transposition' (for instace), we mean a very specific kind of permutation, and the result of composing two transpositions is not a transposition itself.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2882524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that, $\left\lfloor{\frac{x}{n}}\right\rfloor=\left\lfloor{\frac{\lfloor{x}\rfloor}{n}}\right\rfloor$ where $n \in{\mathbb{N}}$
Prove that
$$\left\lfloor{\frac{x}{n}}\right\rfloor=\left\lfloor{\frac{\lfloor{x}\rfloor}{n}}\right\rfloor,$$ where $n \in{\mathbb{N}}.$
My Attempt:
Let $x=nt$.
Then, I need to prove,
... | It is actually much simpler.
See, $\left\lfloor \frac xn\right\rfloor$ is the unique integer $l$ such that $l \leq \frac xn \leq l+1$, or in other words, $nl \leq x \leq nl+n$.
Note that $nl$ is an integer less than or equal to $x$, and $\lfloor x \rfloor$ is the greatest such integer, so $\lfloor x \rfloor \geq nl$. O... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $ab = e$, then $ba = e$? $x\cdot (a\cdot b) = x\cdot e$
$(x\cdot a)\cdot b = x$
$e\cdot b = x$
$b = x$
Are my steps correct?
What I wanted to prove is that if $ab = e$, then $ba = e$
$x$ is inverse of $a$ and $e$ is identity element .
| I don't see why you need to introduce $x$ for $a^{-1}$. Since $ab=e$ we have
\begin{align*}
e & = a^{-1}ea\\
& = a^{-1}(ab)a\\
& = ba
\end{align*}
by multiplying
by $a^{-1}$ from the left and then by $a$ from the right. So conjugating $ab=e$ gives $ba=e$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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Shortest distance between two non-intersecting differentiable curves is along their common normal I want to formally prove the following property:
The shortest distance between two differentiable non-intersecting curves is along their common normal.
I looked at the discussion on this on Quora, (Link: https://www.quora.... | Suppose $a(s),b(t)$ are two curves in $\mathbb R^2$ with parameter interval $(0,1).$ Assume $a'(s),b'(t)$ never vanish, otherwise normal vectors make no sense. Define
$$f(s,t)= |a(s)-b(t)|^2.$$
Suppose $f(s_0,t_0)>0$ is the minimum value of $f$ (hence $\sqrt {f(s_0,t_0)}$ is the distance between the two curves). Then $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2882830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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How can I plot a straight line which is in normal form? Take a look at this link.
Suppose, I have the following equation of a straight line:
$$x \cos\theta+ y \sin \theta=p$$
If we rearrange the equation as follows:
$$y = \frac{p - x\cos\theta}{\sin\theta}$$
we are again stuck with infinite values of $x$. So, what is ... | The normal form can be intepreted as follow, let
*
*$p$ the distance between the line and the origin
*$n=(\cos \theta, \sin \theta)$ the normal unitary vector to the line
therefore for any point $OP=(x,y)$ on the line by dot product we have
$$OP\cdot n=(x,y)\cdot (\cos \theta, \sin \theta)=x\cos \theta+y\sin \theta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Differential forms and partial derivatives If I have a vector valued function $f(\vec R)$ and I imagine that $R$ is also a function of time, so that $R(t)$ Then I'm struggling to prove that:
$$ df \left({\partial R \over \partial t}\right) = {df \over dt}$$
Intuitively, it makes some sense, that if df is a differential... | The function $f$ takes in a point $(x,y,z)$ in space and outputs a real number $f(x,y,z)$; provided that it is regular enough, you can write by the total differential theorem that
$$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz \tag1$$
On the other hand, $\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2883046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Problem regarding convergency: $\sum_{k=0}^{\infty}\frac{x^k}{k!} \to e^x$
Question. Let $X$ be the space of all polynomials in one variable,with real coefficients. If $p=a_0+a_1x+\dots+a_nx^n\in X$, define $$|p|=|a_0|+|a_1|+\dots+|a_n|,$$ which gives the metric $d(p,q)=|p-q|$ on $X$.
Does the metric space $X$ is com... | The statement that $p_n$ converges to $e^{x}$ and hence it does not converge in the given space is just suppose to be a hint and not a complete proof. No norm is used in saying that $p_n$ converges to $e^{x}$. You have to give a rigorous proof along the following lines: if a sequence of polynomials converges in the g... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do you explain to a 5th grader why division by zero is meaningless? I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing division by zero?
| You shouldn't try to do that. Instead make counter question.
"What should it be, then?" and let them think about it.
(Lengthy) justification: There are many important concepts in math you can come up with if you start experimenting with multiplication. Take for example area of a rectangle. You multiply the sides. Are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2883450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "120",
"answer_count": 21,
"answer_id": 1
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Average of Vectors squared minus the square of the averaged vectors I have already asked this question in Physics, but as someone pointed out it is more appropriate to place it here.
Say I've obtained a series of 3D vectors, (x, y, z), with a different vector at different points in time. I want to obtain the second mom... | The average of the square is the average of the square of the components by linearity of expectations:
$$
\langle\mathbf M^2\rangle=\langle M_x^2+M_y^2+M_z^2\rangle=\langle M_x^2\rangle+\langle M_y^2\rangle+\langle M_z^2\rangle
$$
The square of the average can be decomposed into components, and no cross terms will surv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2883653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the $n$-th derivative of $f(x)=\frac{x}{\sqrt{1-x}}$ Find the $n$-th derivative of
$$f(x)=\frac{x}{\sqrt{1-x}}$$
First I just calculated the first, second and 3-th, 4-th derivatives and now I want to summarize the general formula. But it seems too complicated. Then I want to use binomial theorem or Taylor expansio... | We have $f(x)=g(x)h(x)$ where $g(x)=x$ and $h(x)=(1-x)^{-1/2}$. Hence using Leibniz's rule we have
$$f^{(n)}(x)=\sum_{k=0}^n{n\choose k}h^{(k)}(x)g^{(n-k)}(x)={n\choose n}h^{(n)}(x)g(x)+{n\choose n-1}h^{(n-1)}(x)g'(x) \quad(*)$$
It remains to compute
$$h^{(n)}(x)=\frac{1}{2}\frac{3}{2}\cdots \frac{2n-1}{2}(1-x)^{(2n-3)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2883765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Stuck with boundary value PDE problem The last time I posted this I got many down votes and I don't know why. Maybe because I forgot to include my work?
I got the PDE $\dfrac{\partial^2 u}{\partial x^2} -2 \dfrac{\partial^2 u}{\partial x \, \partial y} - 3\dfrac{\partial^2 u}{\partial y^2} = 0$
General solution found t... | You have $F_1(x) + F_2(3x) = x$, differentiating by $x$ we'll get $F_1'(x)+3F_2'(3x)=1$. From the last boundary condition we have $-F_1'(x)+F_2'(3x)=0$. Adding two equations together, we get $4F_2'(3x)=1$. Now we can integrate and find $F_2$: $4\int{F_2'(3x)}dx=\int{dx}=x+c$; $F_2(x)=\frac{x}{12}+c$. Now substitute $F_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Convergence of $\sum \frac{\left|\sin\left(\left(1-\frac{1}{n}\right)^n-\frac{1}{e}\right)\right|^{\alpha}}{e-\left(1+\frac{1}{n}\right)^n}$ Study the convergence of the following series as $\alpha \in \mathbb{R}$
$$\sum_{n=1}^{\infty}\frac{\left|\sin\left(\left(1-\frac{1}{n}\right)^n-\frac{1}{e}\right)\right|^{\alpha}... | Hint for the denominator, you can write:
\begin{align*}
\left(1+\frac{1}{n}\right)^n
&= \exp\left( n \log\left(1+\frac{1}{n}\right)\right) \\
&= \exp\left( n\left(\frac{1}{n} - \frac{1}{2 n^2} + o\left(\frac{1}{n^2} \right)\right)\right) \quad (n\rightarrow \infty) \\
&= e^1\left( \exp\left( - \frac{1}{2 n} + o\left(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can variance and mean of brownian motion be stated I have seen in the construction of brownian motion that the Its mean is 0 and variance t. i.e
$E(B(t))=0$ , $Var(B(t))=t$ Why in a problem sheet does it state $B_{j}(t)$ for $j=1,...,n$ are brownian motions with variances $\sigma^{2}_{j}$. How can these be brownian... | If $B(t)$ is a standard Brownian motion, then indeed $\mathbb E[B(t)]=0$ and $\mathsf{Var}(B(t))=t$ for $t\geqslant0$. Let $X(t) = \sigma B(t) + \mu t$ for some $\mu\in\mathbb R$ and $\sigma>0$. Then $X(t)\sim\mathcal N(\mu t, \sigma^2 t)$, and is called a Brownian motion with drift $\mu$ and volatility (or variance) $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\limsup |C_n|^{\frac{1}{n}} = \limsup |C_{n+j}|^{\frac{1}{n}}? $ where $j\geq0 $ $C_n$ is a sequence of complex numbers.
I would like to prove that the radius of convergence of a differentiated power series is the same as the original series.
i.e. $\limsup |(n+j)(\cdots)(n+1)C_{n+j}|^{\frac{1}{n}} = \limsup |C_n|^{... | I'll assume $j=1$ (the case of arbitrary $j$ is analogous, and actually directly addressed by the proposition in the block below).
The question is then equivalent to asking whether, for a positive sequence $a_n$,
$$\limsup a_n^{1/(n-1)}=\limsup a_n^{1/n}.$$
By putting the left side as $(a_n^{1/n})^{(n/(n-1))}$, we have... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluation of the limit $\lim_{q\rightarrow 1} \frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$ Given the Euler function $\phi(q)=\prod_{n = 1}^{\infty}(1-q^{n})$ which is a modular form where $q=\exp(2\pi i \tau)$, $|q|\lt1$
Then what is the limit $\lim_{q\rightarrow 1}\frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$ ?
| When $q\to 1^-$, $\tau \to 0$, hence $-1/\tau \to i\infty$,
$$\begin{aligned}\frac{{{\phi ^5}{{(q)}_\infty }}}{{\phi {{({q^5})}_\infty }}} = \frac{{{\eta ^5}(\tau )}}{{\eta (5\tau )}} &= \frac{{{{\left( {\frac{{ - 1}}{{\tau i}}} \right)}^{5/2}}{\eta ^5}(\frac{{ - 1}}{\tau })}}{{{{\left( {\frac{{ - 1}}{{5\tau i}}} \rig... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Let $f : [1,2] → \mathbb {R}$ be defined by $f(x) = x$. Prove that $f$ is Riemann integrable and compute $\int_1^2f(x)dx$ Let $f : [1,2] → \mathbb {R}$ be defined by $f(x) = x$. Prove that $f$ is Riemann integrable and
compute $$\int_1^2f(x)dx$$ as the limit of upper (and lower) sums.
I tried solving this but I don't k... | Yes, your proof is fine, everything is o.k.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find Laplace transform of $(sin^2 2t)$ Find Laplace transform of $(\sin^2 2t)$
How do I go about this ?
do I spilt them up like
$ L ( \sin 2t) \cdot L (\sin 2t) $ ?
| Given $\sin^22t$
$$\sin^22t=\dfrac{1-\cos4t}{2}$$
$$L(\sin^22t)=\dfrac{L(1)-L(\cos4t)}{2}=\dfrac{1}{2s}-\dfrac12\left(\dfrac{s}{s^2+16}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2884718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence of sequence of functions in metric spaces Let $f_n(x)=\frac{nx}{nx+1}$
$a)$ Show that in $C_{[0,2]}$ $$\lim_{n \to \infty}{f_n(x)}=1$$
$b)$ Does $f_n$ converge in $C_{[0,1]}?$
Here is my attempt:
$a)$ We need to show that $d(f_n,f)<\epsilon$ where $f$ is the constant function $f(x)=1$. Following previous ... | Note that for $x\neq 0$ we have that
$$
\lim_{n\to \infty}f_n(x)=\lim_{n\to \infty}\frac{x}{x+n^{-1}}=1
$$
and clearly $f_n(0)\to 0$ as $n\to \infty$. To note that the convergence is not uniform observe that
$$
f_n(1/n)=\frac{1}{2}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2884872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving lim inf means "all but a finite number" for measure theory Full disclosure: a philosophical logic person trying to learn math. Here is the problem from Gallant(1997), a measure-theoretic intro to econometrics:
Let $F_i$ where $i = 1, 2, ...$ be an infinite sequence of events from
the sample space $\Omega$. L... | If $\omega\in F$, then $\omega$ is in all but a finite number of the events $F_i$. Therefore, there is a natural number $N$ such that $\omega$ is in every $F_i$ when $i\geqslant N$. So, $\omega$ belongs to each$$\bigcap_{i=k}^\infty F_i$$as long as $k\geqslant N$ and therefore$$\omega\in\bigcup_{k=1}^\infty\bigcap_{i=k... | {
"language": "en",
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"source": "stackexchange",
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Properties of $\{x\in X\mid f(x)=||f||\}$ Let $X$ be a normed space, $f\in X^*\setminus\{0\}$ (the continuous dual), $E:=\{x\in X\mid f(x)=\|f\|\}$. Prove that $E$ is a nonempty closed set and that $\inf \{\|x\|\mid x\in E\}=1$.
I have no idea how to prove that $E$ is non empty and that $\inf \{\|x\|\mid x\in E\}=1$.
$... | Since $f \neq 0$ we have $f(x^*) \neq 0$ for some $x^*$, so then $f({\|f\| \over f(x^*)} x^*) = \|f\|$ so $E$ is not empty.
Since $f$ is continuous, $f^{-1} ( \{ \|f\| \})$ is closed.
Note that $f(x) = \|f\|$ for all $x \in E$. Since $\|f\| = f(x) \le \|f\| \|x\|$ we see that $\|x\| \ge 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2885070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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$xy'-y=0$ when $x<0$
How to solve $xy'-y=0$ when $x<0$?
It seem to be a simple equation, but it is confusing when $x<0$
$$\frac{y'}{y}=\frac{1}{x}$$
Now to integrate the both sides must I assume that $y$ is positive then
$$\ln y =- \ln(-x)+c$$so
$$y(x)=e^{-\ln(-x)}$$
Am I right? what if the assumption on $y$ is f... | Without log function
$$y'x-y=0$$
for $x \ne 0$
$$\frac {(y'x-y)}{x^2}=0$$
$$\left(\frac yx \right)'=0$$
Integrate
$$\frac yx= K$$
$$y(x)=Kx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2885150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof by induction that you can order natural numbers where the average isn't between any pair of numbers Consider a list of natural numbers like
$$1, 2, 3, \dots, n$$
Prove using strong induction that you can order the list for any Natural n, in a way where if you pick any pair of numbers, the average of the numbers ... | The method you mentioned is for all intents and purposes a complete solution. Here is way to write it as a proof by strong induction.
The base case $n=1$ is obvious.
For any $n>1$, $d=\lceil n/2\rceil$ and let $e=\lfloor n/2\rfloor$. Note $d,e<n$. Also, $d$ is the number of odd numbers in $\{1,2,\dots,n\}$, while $e$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2885266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does there exist a right triangle with area 7 and perimeter 12? This question is really trivial. I can prove that there is no right triangle with area 7 and perimeter 12, but what I do is solve the following system: if $a$, $b$ and $c$ are, respectively, the two legs and hypotenuse of such a triangle, then
$$a^2 + b^2 ... | I'm not sure if this counts as a "simple answer", but let $x$ be the length of one leg of a right triangle of area $7$; then the other leg is $\frac{14}x$, and the hypotenuse is $\sqrt{x^2 + \left(\frac{14}x\right)^2}$, so the perimeter is given by the function
$$P(x) = x + \frac{14}x + \sqrt{x^2 + \left(\frac{14}x\rig... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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How to find point in Descartes Folium with slope of -1/3? I stumbled upon a problem in my calculus book that asked to find the point in $x^3 + y^3 = 3xy$ that had a slope perpendicualr to $y = 3x + 1$ and also was in the first quadrant.
I began by getting the derivative of the equation and got $\dfrac{x^2-y}{x-y^2}$.
... | Let's find a parametric equation of the curve defined by $x^3+y^3=3xy$.
Let $y=xt$, then
$$x^3(1+t^3)=3x^2t$$
Hence
$$x=\frac{3t}{1+t^3}$$
$$y=\frac{3t^2}{1+t^3}$$
Now, the tangent vector is given by
$$x'(t)=\frac{3(1+t^3)-3t(3t^2)}{(1+t^3)^2}=\frac{3-6t^3}{(1+t^3)^2}$$
$$y'(t)=\frac{6t(1+t^3)-3t^2(3t^2)}{(1+t^3)^2}=\f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Counting problem - fault in my reasoning. The problem is as follows:
The dean of science wants to select a committee consisting of mathematicians and physicists to discuss a new curriculum. There are $15$ mathematicians and $20$ physicists at the faculty; how many possible committees of $8$ members are there, if there ... | You don't have $8!$ possible permutations. For instance, the first person you chose can never be a mathematician and the second, third, fourth, fifth and sixth persons you chose can never be physicists.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2885617",
"timestamp": "2023-03-29T00:00:00",
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Does this mathematical technique have a formal name? And why does it work? When I split a number in the powers of 2. I am able to make any combination of any number that is less than it by taking each number of the series only once.
For example:
$7=1+2+4$
I can construct any number less than or equal to seven using th... | Countrary to most of your feedback, this is not at all binary representation, because you split from the small end and not from the big end.
For binary representation to represent 7 you start with:
*
*8 no it's too big
*4, ok it fits, 7-4=3 left,
*2, ok it fits 3-2=1 left
*1, ok it fits 1-1=0 left, done!
What... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If H(b) $\leq$ b then $\int_0^k (v-b)h(b) - [1-H(b)]db \leq \int_0^k (v-b) - [1-b]db $? $H$ is a CDF over $[0,1]$
I want to prove that if $H(b) \leq b$ then :
$$\int_0^k (v-b)h(b) - [1-H(b)]db \leq \int_0^k (v-b) - [1-b]db $$ for all $k \in [0,1]$
I think this is true, but a rigorous proof does seem difficult to me. Ca... | I assume $h(b)=H'(b)$ is a density, $v$ is a constant and $H(b) \leq b \quad\forall b$
\begin{align}
&H(b) \leq b \implies H(b)-1 \leq b-1 \implies 1-H(b)\geq1-b \implies -(1-H(b))\leq-(1-b)
\end{align}
Substituting in the equation you get the following inequality.
$\int_0^k (v-b)h(b) - [1-H(b)]db \leq \int_0^k (v-b)h... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why should the error of a local linearization for $x$ near $a$ be small relative to $(x-a)$? I am reading a calculus book. It says
The fact that the graph of $f$ looks like a line as we zoom in means that not only is $E(x)$ small for $x$ near $a$, but also that $E(x)$ is small relative to $(x−a)$
where $E(x) = f(x) -... | I'm going to focus on the intuition. I make no claims that my answer is rigorous.
Write $$E(x) = f(x)-f(a)-f^{\prime}(a)(x-a)\text{.}$$
$f^{\prime}(a)$ is the "slope" of $f$ at $x = a$. Make sure you understand that derivatives are just slopes. For sake of convenience, let's call this $m$.
Way back in your Pre-Calculus... | {
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"timestamp": "2023-03-29T00:00:00",
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Reduction Formula for $I_n=\int \frac{dx}{(a+b \cos x)^n}$ Reduction Formula for $$I_n=\int \frac{dx}{(a+b \cos x)^n}$$
I considered $$I_{n-1}=\int \frac{(a+b \cos x)dx}{(a+b \cos x)^n}=aI_n+b\int \frac{\cos x\:dx}{(a+b\cos x)^n}$$
Let $$J_n=\int \frac{\cos x\:dx}{(a+b\cos x)^n}$$ using parts for $J_n$ we get
$$J_n=\fr... | This is not a full answer the the original question but instead evidence that any such answer will not lead to the underlying problem at hand: calculating the given integral.
Mathematica yields:
$$\frac{\csc (x) \sqrt{\frac{b (\cos (x)+1)}{b-a}} \sqrt{\frac{b-b \cos (x)}{a+b}} (a+b
\cos (x))^{1-n} F_1\left(1-n;\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2886043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Rectangle on sphere Is it true that every set of positive measure on a 2-sphere contains four points which form a rectangle? Note that I am not asking about if there exists a "filled-in" rectangle, or even a "rectangle" in the product of measurable sets sense, but simply four points which form a rectangle in $\mathbb{R... | Suppose $X$ is a subset of the sphere $S$ and measure $\mu(X)>0$. It is straightforward to show that one can find a point $x\in S$ and a disk $D(x,r)\subseteq S$ centered at $x$ of some radius $r>0$ such that
$$\dfrac{\mu(X \cap D(x,r))}{\mu(D(x,r)} \approx 1.$$
Project the set $X\cap D(x,r)$ on the plane through the... | {
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"timestamp": "2023-03-29T00:00:00",
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If $S(n)$ is an odd integer, what is the sum of all possible $\frac1n?$ If $n$ is a positive integer, let $S(n)$ be the sum of all the positive divisors of $n$.
If $S(n)$ is an odd integer, what is the sum of all possible $\frac1n?$
The function $S$ is multiplicative and so, if we have the prime factorisation $n = p_1... | $S(n)$ is multiplicative, so $S(n) = \prod_{i=1}^m S(p_i^{a_i})$. If any of the $S(p_i^{a_i})$ are even, the whole product will be even. So what is the condition on $S(p^a)$ being even for some prime $p$ and natural number $a$? What does that tell you about the set of numbers $n$ for which $S(n)$ is odd?
Once you've fi... | {
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"timestamp": "2023-03-29T00:00:00",
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Solutions of a differential equation I'm trying to solve the following differential equation and I'm stuck at what it appears to be simple calculations. I'm terribly sorry if this turns out to be really simple.
$(1)$ $X(f)=2f$
where $X=x_1^2 \frac \partial {\partial x_1}-x_2^2 \frac \partial {\partial x_2}$ in $\Bbb R^... | $$x^2 \frac {\partial f}{\partial x}(x,y)- y^2 \frac {\partial f}{\partial y}(x,y)=2f(x,y).$$
Search for the general solution (without taking account of the boundary condition) with the method of characteristics :
The characteristic ODEs are :
$$\frac{dx}{x^2}=\frac{dy}{-y^2}=\frac{df}{2f}$$
A first characteristic equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2886403",
"timestamp": "2023-03-29T00:00:00",
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Find a simple formula of k numbers which gives output a number not among these k numbers. This question just came to my mind while thinking. I just wanted to know if it is even possible. Anyway, the question goes as follows.
You are given k numbers $a_{i}$ (k < n) from 1 to n.
Your job is to give a simple formula repre... | You can do $f(K) = \min\{\{1,\dots,n\} \setminus K\} $ where $K=\{a_1,\dots\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2886496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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calculating a higher order derivative My task is to find the values $f^{(2017)}(0)$ and $f^{(2018)}(0)$ for $f(x)=\frac{\arccos(x)}{\sqrt{1-x^2}}$.
Basically, it's about finding the $n^{th}$ derivative of $f$.
So I noticed if I let $g(x)=arccos(x)$, then $f(x)=-g(x)\cdot g'(x)$. I was able to prove by induction that f... | Let $ n\in\mathbb{N} $, if $ f $ is a $ \mathcal{C}^{n+1} $ function on $ \left(-1,1\right) $, then we can use Leibniz formula for the $ n $-th derivative of the product :
$$ \left(\forall x\in\left(-1,1\right)\right),\ \left(ff'\right)^{\left(n\right)}\left(x\right)=\sum_{k=0}^{n}{\binom{n}{k}f^{\left(k\right)}\left(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2886652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that $u\times v+v\times w+w\times u=\textbf{0}$ implies that $u,v,w$ are linearly dependent I have what I think might be a solution, but I'm not sure it's formal enough. I begin by noting that $u,v,w$ are linearly dependent iff. they lie on the same plane. Then I construct the following chain of equivalences:
\beg... | Note that $$v \times w + w \times u + u \times v =v \times w - u \times w + u \times v $$
$$=(v-u) \times (w-v)$$
Thus $$ v \times w + w \times u + u \times v = 0$$
$$\implies (v-u) \times (w-v)=0$$
Which implies $u$,$v$,and $w$ are linearly dependent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2886758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Definition of orthogonal subspace In the book I'm using to study there is this definition: given a vectorial space $E$ with an internal product:
*
*given $u \in E$, the set of all vectors in $E$ that is orthogonal to $u$ is a subspace of $E$;
*given a subset $M$ of $E$, the set of all v in E such that v is orthogon... | As my analysis prof used to say "you can define whatever you want as long as you are consistent" ;) note that the orthogonal set does not change, when you replace $M$ by its span. This means, you can take the vector space generated by $M$ and everything reduces to the case you are familiar with.
Added: We want to prove... | {
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Conjecture that $ \frac{\gcd(a+b,ab)}{\gcd(a,b)} \mid \gcd(a,b)$ I have discovered some exercise type conjectures which I can't prove and this is one of them:
Given positive integers $a,b$, then
$$ \frac{\gcd(a+b,ab)}{\gcd(a,b)}\ \bigg|\ \gcd(a,b)$$
Can this be proved or disproved?
From time to time, when testing ... | Write $\gcd(a,b)=d$, then $a=da',b=db'$ and thus $\frac{\gcd(a+b,ab)}{d}=\gcd(a'+b',a'b'd)$ where $\gcd(a',b')=1$. Notice now that $\gcd(a'+b',a'b')=1$ since $a'(a'+b')-a'b'={a'}^2$ and thus $\gcd(a'+b',a'b')|{a'}^2 \implies \gcd(a'+b',a'b')|a'$ and $\gcd(a'+b',a'b')|a'+b' \implies \gcd(a'+b',a'b')|a',b' \implies \gcd(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Derivative of inverse function proof verification Can someone verify whether my attempt to prove this theorem is correct?
Notice that I use a generalized definition of derivative:
Let $f: E \subseteq \mathbb{R} \to \mathbb{R}$ be a function. Let $p$ both a point and a limit point of $E$. Then we define the derivative ... | Your work will be done faster if not make use of the First Principle of Derivative.
We know that $f^{-1}(f(x))=x$, differentiate it combining with Chain Derivative, we have $$\begin{align}(f^{-1}(f(x)))'&=1\\f^{-1}{'}(f(x))f'(x)&=1\\\therefore f^{-1}{'}(f(x))&=\dfrac1{f'(x)}\end{align}$$
I am not too used in rigorous ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Existence of a transitive model is strictly stronger than consistency? It seems like we should be able to prove that the existence of a transitive model for ZFC is strictly stronger than Con(ZFC), but I can't find anything saying so / giving an argument for it. Is there a standard way of demonstrating this?
An example ... | Any model of ZFC+Con(ZFC)+$\neg$Con(ZFC+Con(ZFC)) should do. (This theory is, by the second incompleteness theorem, consistent if ZFC+Con(ZFC) is).
Because the model believes Con(ZFC) and also believes that no model of ZFC can believe Con(ZFC), the only models of ZFC it can know will be ones that it considers to have n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Properties of locally compact metric spaces.
Let $(X,d)$ be a locally compact metric space. Then for each $x \in X$ $\exists$ $\epsilon_x > 0$ such that $B[x;\epsilon_x] = \{y \in X : d(x,y) \leq \epsilon_x \}$ is compact.
How do I proceed to prove it? Please help me in this regard.
Thank you very much.
| Pretty simple!
Let us choose $x \in X$ arbitrarily. Since $X$ is locally compact so $\exists$ a compact neighbourhood $C_x$ of $x$ in $X$ i.e. $\exists$ $\epsilon >0$ such that $x \in B(x;\epsilon) \subset C_x$. Consider $0 < \epsilon_x < \epsilon$ then clearly $B[x;\epsilon_x] \subset B(x;\epsilon) \subset C_x$. Now $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Checking Continuity of functions Consider a function $\sqrt {x-1}$+$\sqrt{1-x}$.From here we can see that domain of the function is just 1 and range is 0.Still the function is continious at x=1 even though RHL and LHL limit doesn't exist. Can you guys tell me how this is possible. What is the criterion for checking con... | What is your definition for $l=\lim\limits_{x\to 1}f(x)$ if it exists?
If this is it:
$$\forall\varepsilon>0,\exists\delta>0,\forall x\in D_f,|x-1|<\delta\implies|f(x)-f(1)|<\varepsilon$$
then take ANY $\varepsilon>0$ and ANY $\delta>0$. $D_f=\{1\}$ so $|x-1|=0$ for any $x\in D_f$ and thus $f(x)=f(1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Does the series for $\cos(x)/x$ converges? The sequence of
$$
a_x ={\cos (x)\over x}
$$
does converge to zero.
As a result, intuitively
$$
\sum_{x=1}^\infty {\cos (x)\over x}
$$
should also converge right? But I've been told that the series diverges. This shouldn't be true... right?
| It is not at all intuitive to me that the series ought to converge simply because the terms go to zero. For example $\sum_{n=1}^{\infty} \frac1n$ is well known to diverge even though $\frac1n\to 0$. Or, as an even easier example, consider
$$ 1 + \underbrace{\frac12+ \frac12}_{2\text{ halves}} +
\underbrace{\frac13 + \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
Prove the inequality Using Mean Value theorem Show that : $$x < \log\Bigl(\frac{1}{1-x}\Bigr) < \frac{x}{1 - x}\,;$$
If $$0 < x < 1$$
Solution:
If $$f(x) = \log\Bigl(\frac{1}{1 - x}\Bigr)$$
Then the function is continuous in [0, x]
And also differentiable in (0, x)
So We can apply Lagrange's MVT on f(x).
Fine!
So $... | Hint:
The inequalities should be reversed.
Start from the well-known inequality
$$\log u\le u-1\quad\text{for all }u>0\qquad(< \text{ if } u\ne1),$$
and make the relevant substitutions.
Some details:
First set $\;u=\dfrac1{1-x}$ (which is positive since $0<x<1$). You obtain instantly
$$\log\frac1{1-x} <\frac1{1-x}-1=\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does $f(x,y)$ mean? I know from the chapter "functions" that $f(x)$ is a function of $x$ and to roughly put it, it maps $x$ values to another set called co-domain where all the $y$ values are.
But I also sometimes see $f(x,y)$ on internet. I can guess that it means some expression in $x$ and $y$.
I'm not familiar ... | Here is a short answer to your questions:
1) The function can be called a bivariate function; it is a function that depends on two variables $x$ and $y$ that may assume different domains. The function is defined on the union of those domains. An example is
$$ f(x,y) : = x^2 + y^2$$
If you fix $x$ to any value say $\ba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Analogues of the elementary symmetric polynomials for the alternating group In the case of three variables, the elementary symmetric polynomials are
$$ \begin{align}
e_1(X_1,X_2,X_3)&:=X_1+X_2+X_3, \\
e_2(X_1,X_2,X_3)&:=X_1 X_2+X_1 X_3+X_2 X_3, \\
e_3(X_1,X_2,X_3)&:=X_1 X_2 X_3. \\
\end{align}$$
Knowledge of the value... | Here’s one way to do it: to get a list of polynomials for any $n$, start with the list of elementary symmetric polynomials on $n$ variables and add in the Vandermonde polynomial on $n$ variables.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2887951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Are two real, two variable polynomials, satisfying the Cauchy-Riemann equations, a complex polynomial? Let $u, v \in \mathbb{R}[x,y]$ satisfying $u_{x} = v_{y}$ and $u_{y} = -v_{x}$ everywhere in $\mathbb{C}$. Is the function $f(x + iy) = u(x,y) + iv(x,y)$ a polynomial in the variable $z = x + iy$?
I really don't know... | Assuming that (as in the title) $u$ and $v$ are polynomial functions, then yes, it is true. The function $f$ is holomorphic and therefore analytic. And, since $u$ and $v$ are polynomial functions $f^{(n)}=0$ is $n$ is large enough. Therefore, $f$ is polynomial too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
G be a finite group and H be a non trivial proper subgroup of index 3, then G is not simple. If the index of a subgroup is the smallest prime dividing the order of group then that subgroup is normal, with this, I am done with the case order of group is odd. How to proceed when order is even.
| In general: if $G$ is a finite non-abelian simple group and $H$ is a subgroup, then $|G:H| \geq 5$.
Proof (Sketch) Assume $|G:H| \leq 4$. Let $G$ act by left multiplication on the left cosets of $H$. The kernel of this action is $core_G(H)=1$, since $G$ is simple. Hence $G$ can homomorphically be embedded in $S_{|G:H|}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
When is the dirac delta function taken as 1 and when it is taken as infinity I understand that the dirac delta function basically describes a pulse of area one, if the pulse is very narrow, the height will be infinity. However, Im confused because sometimes, people consider the dirac delta function as being one at t=0 ... | As you say, the "value" of the delta "function" is infinite at $0$ and the "area under it" is $1$. I am not aware of any time we take the value to be $1$. The unit impulse refers to the total impulse delivered, which is the area under the force-time curve.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
An example that $f,g$ is Riemann integrable on $[a,b]$, the range of $f$ is $[a,b]$, but $g\circ f$ is not Riemann integrable. I can easily find an example that $f,g$ is Riemann integrable but $g\circ f$ is not integrable, put $f(x)=R(x)=\begin{cases} \dfrac{1}{q}, & x=\dfrac{p}{q},\\ 0,& x\in\mathbf{Q}^{C}\end{cases},... | Why not define:
$$f : [-1, 1] \to [-1, 1] : x \mapsto \begin{cases} 2x + 1 & \text{if } -1 \le x < 0 \\ \frac{1}{q} & \text{if } x \in [0, 1] \text{ and } x = \frac{p}{q} \text{ with } p, q \in \mathbb{Z} \text{ and } \operatorname{gcd}(p, q) = 1 \\ 0 & \text{if } x \in [0, 1] \setminus \mathbb{Q}\end{cases},$$
and $g$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Need help understanding the derivation of Hydrostatic Equilibrium in a star First and foremost I am sorry for this; in order to make my questions clear I must first upload the lecture notes from my institution $^\zeta$ for the derivation of Hydrostatic Equilibrium:
$^\zeta$ Lecture notes courtesy of Imperial Colle... | In the star, light does not travel far before being scattered. Its pressure acts in all directions, not just outward.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$f(x) \equiv 1$ mod $(x-1)$ and $f(x) \equiv 0$ mod $(x-3)$ then is there any $f(x)$? Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying
$f(x) \equiv 1$ mod $(x-1)$
$f(x) \equiv 0$ mod $(x-3)$
Which of the following statements are true?
a) $S$ is empty .
b) $S$ is a singleton.
c)$S$ is a f... | We have that
$$f(x)\equiv 1\bmod{(x-1)}\iff \exists a(x)|f(x)=(x-1)a(x)+1.$$
$$f(x)\equiv 0\bmod{(x-3)}\iff \exists b(x)|f(x)=(x-3)b(x).$$
So, if $f$ exists then it must be
$$(x-1)a(x)+1=(x-3)b(x).$$ We have for $x=1:$
$$1=-2b(1)\implies b(1)=-\dfrac 12,$$ which is not possible, since $b(x)$ is a polynomial with integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 3
} |
Strange isomorphism: $R/(A + B) \cong (R/B)/\bar{A}$. $R$ is a ring and $A, B$ are ideals of $R.$
I was playing around with some stuff and I was wondering if $R/(A + B) \cong (R/B)/\bar{A}.$ for $\bar{A}$ being the image of $A$ under $R \rightarrow R/B.$ I'm pretty sure I have a proof for it. I will post a 'proof' of ... | This is a consequence of the second isomoprhism theorem (on wikipedia is anyway refered as the third): $B \leq A+B \leq R$ implies that
$$
R/(A+B) \cong (R/B)/((A+B)/B)
$$
Where is not hard to prove that $(A+B)/B$ is the image of $A$ through thr map $\pi : R \to R / B$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Derive the following identity $1^2+2^2+ \ldots + n^2 = \frac{n(n+1)(2n+1)}{6}$.
Count the elements of the following set
$$A=\{(x,y,z): 1\leq x,y,z \leq n+1, z>\max\{x,y\}\}.
$$
From this derive the following identity:
$$1^2+2^2+ \ldots + n^2 = \frac{n(n+1)(2n+1)}{6}.$$
In the same manner find the formula for... | We can prove it by induction:
Step 1: check it for base case (n=1)
If we replace $n$ by one, we get: $$ 1^2=\frac{1\times 2 \times 3}{6}$$
which holds $\checkmark$
Step 2: assume it is true for $n=k$ and check if holds for $n=k+1$
If it holds for $n=k$, then we have:
$$1^2+2^2+…+k^2=\frac{k(k+1)(2k+1)}{6}$$
Now we show... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Can there be other multiplications on $\mathbb{R}^2$ making it a ring? Together with addition $+ : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$
$$(x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1 + y_2)$$
the multiplication $\cdot : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$
$$ r_1 e ^{i\phi_1} \cdot r... | Yes, there are. One is the component-wise multiplication:
$$
(x_1, y_1)\cdot (x_2, y_2) = (x_1x_2, y_1y_2)
$$
Another (if you don't require rings to have a multiplicative unit) is the trivial multiplication
$$
(x_1, y_1)\cdot (x_2, y_2) = (0,0)
$$
and I am sure there are many others.
As a side-note, you could have some... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Convergence of $\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$ Study the convergence of the series as $a > 0$
$$\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$$
In the t... | Hint First you can write :
$$\left(\frac{n}{n^2+1}\right)^{k(n)} = \exp \left(k(n)\log\left(\frac{n}{n^2+1}\right)\right) $$
And you can try to approximate $k(n)$ and $\log\left(\frac{n}{n^2+1}\right)$ as $n \rightarrow \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2888870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Chromatic number in a union of planar graphs I am trying to solve the following problem:
Let $G_1, G_2, \dots,G_{100}$ be $100$ planar graphs on the same vertex set $V$ , with edge sets $E_1, E_2,...,E_{100}$, respectively, and consider the graph $G = (V, \bigcup_{i=1}^{100}E_i)$ which is the union of the graphs $G_1,... | I think I have the answer.
the $G_1, ..., G_{100}$ are planar graphs on the same set of vertices, thus |V| is the same for all $G_i$.
We know $|E(G_i)|≤3|V|−$6 $\forall i $, since they are all planar thus $ |\bigcup_{i=1}^{100}E_i| \leq |\sum_{i=1}^{100} E_i| ≤100 (3|V|-6)= 300|V|-600$
For convenience let $ E = \bigc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
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