Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
In how many different ways can we place N identical rooks on a chess board WITHOUT CORNERS so that no two of them attack each other (for N > 3)? I'm trying, unsuccessfully, to figure it out for a moment now
For N = 8, the chessboard would look like:
Just for reference here is a similar question without the corner res... | This problem can be conveniently solved using rook polynomials. We can pairwise exchange rows and columns without changing the number of possible arrangements of non-attacking rooks. An equivalent board is shown below.
The rook polynomial of the four forb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proving the truth of an inequality (by finding a value) as part of proving a question in homework I came across the following logarithmic inequality:
$$ \log ^4n<n^{1-\epsilon}$$ I need to find a value of epsilon that the inequality will continue to be true. I checked with wolfy and the value is around 0.3 (can be less... | Hint: instead of worrying about limits at infinity, find where the function $$f(x) = \frac{\log~x}{x^\alpha}$$ takes on its maximum value.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is $f'(x)=0$ true at a cusp? If I am given a graph that has a cusp and I am asked to find every point $x$ where $f'(x)=0$ is satisfied, does this include the cusp? I know that when the derivative of a function equals zero this means that there is a horizontal tangent at that point, I also know that the derivative does ... | Regards Jotam. If a derivative does not exist at the cusp, say at point $x_o$, then it means that $f'(x_o)$ does not exist, it does not have any unique value. The limits in both direction are different :
$$ \lim_{h \rightarrow 0} \frac{ f(x_o + h) - f(x_o) }{h} \ne lim_{h \rightarrow 0^{-}} \frac{ f(x_o + h) - f(x_o)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
From the given figure, prove that $[\triangle ABC]=[\triangle CDE]$ From the given figure, prove that $\triangle ABC=\triangle CDE$ in area.
In $\triangle ABC$ and $\triangle CDE$
1. $AB=CD$
2. $BC=DE$
3.??
| It is easy to see with a picture; simply stick the triangles together by a side that is the same length, and see (where the two green sides are the same and the two blue sides are the same):
It is now trivial that the area is equal, since their base and height are equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What makes a function not defined? I have started studying precalculus and would then start up with calculus. While studying about functions I wondered whether this function would be defined at $a$ or not. Take a look at it. $$ f(x) = \frac{(x-a)(x-b)(x-c)...(x-n)}{(x-a)} $$
Here if we will simplify it further then the... | A function $f$ is a special asymmetric relation between two sets $A$ and $B$, represented by $f:A\to B$. The relation consist in that for every element of $A$ (called the domain of $f$) exists a unique element in $B$ (called the codomain of $f$) defined by the function $f$.
In your example, if the codomain of $f$ is $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Show such a function has a maximum Let $f:[0, \infty)$ be a continuous function.
$f(0) = 1 $ and $\forall x \in [0, \infty)$ $f(x)\leq \frac{x+2}{x+1}$
Show that $f$ gets a maximal value in $[0, \infty)$.
My intuition:
if $f(0)$ is the maximum i'm done if not the function you showed me converges to $1$ I want to show ... | (1). If $f(x)\leq 1$ for all $x\in [0.\infty)$ then $1=f(0)=\max_{x\geq 0}f(x).$
(2). If $f(x_0)>1$ for some $x_0>0$ then there exists $x_1>x_0$ such that $$\forall x\geq x_1\;( f(x)<\frac {1}{2}(1+f(x_0))$$ because $\lim_{x\to \infty}\sup_{y\geq x}f(y)\leq 1.$
There exists $x_2\in [0,x_1]$ with $f(x_2)=\max \{f(x): x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Find the number of ways you can put fruits into drawers We have 5 fruits and 3 drawers. I need to find the number of ways I can put the fruits in them when there will be at least one fruit in the lowest drawer and the first and the second fruit won't be in the same drawer.
I tried to do it by the complementary way, I t... | 5 fruit in any of 3 drawers.
$3^5$
Fruit 2 not in the same drawer as fruit 1
$3^4\cdot 2=162$
minus the cases where all of the fruit are in the top two drawers.
$-2^4\cdot 1=16$
$162-16 = 146$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Series Convergence / Divergence: $\sum \frac{3^{1/\sqrt{n}}-1}{n}$ Does the following series converge or diverge:
$$\sum_{n=1}^\infty \frac{3^{1/\sqrt{n}}-1}{n}$$
Many thanks!
| Using MVT $3^x-1=3^{\epsilon}\ln{3}\cdot (x-0),\epsilon \in (0,x)$ or
$$0<\frac{3^{\frac{1}{\sqrt{n}}}-1}{n}=3^{\epsilon}\ln{3}\frac{1}{n\sqrt{n}}<3\ln{3}\frac{1}{n\sqrt{n}}$$
because $\epsilon \in (0,\frac{1}{\sqrt{n}})$ or $0<\epsilon<1$ and $f(x)=3^x$ is ascending. As a result:
$$\sum_{n=1}^{\infty}\frac{3^{\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Smallest odd-deficient odd number and its family. Let $o$ be an odd positive integer. We say that $o$ is deficient if its sum of positive divisors denoted by $\sigma(o)$ satisfies the inequality $\sigma(o)<2o$. In this case $\sigma(o)=2o-d$ where $d$ is called the deficiency of $o$. If $d $ is odd we say thay $o$ is o... | The odd deficient numbers which have odd deficiency are the odd deficient squares. We have $d=2o-\sigma(o)$, so $d$ will be odd when $\sigma(o)$ is odd. All the divisors of $o$ are odd, so $\sigma(o)$ will be odd when $o$ has an odd number of divisors. But given a divisor $k, \frac ok$ is also a divisor so they come... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Residue for quotient of functions Let $f, g$ be holomorphic functions on a disk $\mathbb{D}(z_0,r)$ centered at $z_0$ and of radius $r>0$. Suppose $f$ has a simple zero at $z_0$. I want to find an expression for $Res(g/f, z_0)$. But I'm not sure what this expression should look like.
Here's my guess:
Since $f$ has a si... | Suppose that $z_0$ is a simple zero of the fraction
$$\mathrm {Res}(g/f,z_0) = \lim_{z\to z_0}(z-z_0) \frac {g(z)}{f(z)}=\frac {g(z_0)}{f'(z_0)} \tag{1}$$
Alternatively
Since $f$ has a simple zero at $z_0$
$$\frac{1}{f} = \frac{a_{-1}}{(z-z_0)}+\sum_{k=0}^\infty a_k (z-z_0)^k$$
Suppose that $g$ is analytic in a nbhd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Understanding the cohomology ring of the Grassmannian Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one understands the cohomology of the Grassmannian.
I know that the Gras... | The two texts I have most often been referred to for studying the cohomology of the Grassmannian are Young Tableaux by Fulton and Symmetric Functions, Schubert Polynomials, and Degeneracy Loci by Manivel. From my perspective, Manivel takes a fairly combinatorial perspective while Part III of Fulton gives an excellent t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to find constant term in two quadratic equations Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - x + p=0$ and let $\gamma$ and $\delta$ be the roots of the equation $x^2 -4x+q=0$. If $\alpha , \beta , \gamma , \delta$ are in Geometric progression then what is the value of $p$ and $q$?
My approach:
From... | let $$\beta=\alpha y$$,$$\gamma=\alpha y^2$$,$$\delta=\alpha y^3$$ then we get from the first equation
$$\alpha^2-\alpha=\alpha^2y^2-\alpha y$$
from here we get $$\alpha=\frac{1}{1+y}$$
analogously we get from the second equation:$$\alpha=\frac{4}{y^2(1+y)}$$ combining these equations we have $$y^2=4$$ can you finish n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find interval solutions of $y'=2y\cos^2t-\sin t$ I need to find interval solutions of following equation:
$y'=2y\cos^2t-\sin t$
So it looks like odrinary linear equation, so we can solve it first, assuming that $\sin t=0$, and obtain $$y=ce^{\sin t\cos t+t}$$ Then assume it is solution of "full" equation and find $c$, ... | You have the equation of the form $y'+p(x)y=q(x)$ It can be solved by a sum of the solution to the homogeneous equation and the particular solution given by integrating factor. The homogeneous solution you have just obtained
$$\int\frac{dy_{h}}{y_{h}}=2\int\cos^{2}(t)dt$$
$$y_{h}(t)=c_{1}e^{(\frac{1}{2}\sin(2t)+t)}$$
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Countability of Gaussian Integers I am attempting to show that Gaussian Integers are countable.
Is it valid to map $a + bi$ to an ordered pair $(a, b)$ and then map this to the set of rationals $a/b$ ?
I am unsure if this works since $a/b$ is not defined at $b = 0$ and am unsure of a different way to go about this. An... | We have $\mathbb{Z}[i]=\{a+bi\mid a,b\in \mathbb{Z}\}$ and thus a bijection to pairs $(a,b)\in \mathbb{Z}\times \mathbb{Z}$. Since the product of two countable sets is countable, this is countable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Differentiating $ x^{a}y^{b} = c $, in its simplest form. $$ x^{a}y^{b} = c, $$
where a, b and c are constants. My attempts so far
$$ \frac{dy}{dx} = ax^{a - 1}by^{b - 1}$$
$$ \frac{d^2y}{dx^2} = (a^2 - a)x^{a-2}(b^2 - b)y^{b - 2} $$
I think that these first and second derivatives are correct, however my issue is, are... | Hint:
$$\frac{d}{dx}f(x)g(x)=f'(x)g(x)+f(x)g'(x)$$
Then differentiate the $\frac{d}{dx}$ similarly.
Assuming you are not dealing with implicit differentiation otherwise you may need The Chain Rule as well as The Product Rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Using Weierstrass approximation to show $f(x) = 0.$
(i). Let $f: [0,1] \to \mathbb{R}$ be continuous such that
$$\int_{0}^{1} x^kf(x)dx = 0, \ \forall k \geq 5.$$
Show that $f(x) = 0$, for all $x \in [0,1]$.
(ii). Let $f: [0,1] \to \mathbb{R}$ be continuous such that
$$\int_{0}^{1} x^{2k}f(x)dx = 0, \ \forall... | I cannot really follow what you are trying to do in your argument, nor where your inequalities come from. The way it is usually done is that because you can approximate $x^5f(x)$ uniformly with polynomials, you get get $(x^5f(x))^2=0$, and the $f(x)=0$ by continuity. In more detail: let $\{p_n\}$ be polynomials with $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
A book on Banach space valued random variable. I have recently become interested in probability theory that take place on a Banach space setting. What are some good books for a beginner like me?
The topic that I am especially interested in is Banach space valued $L^p(\Omega;X)$, i.e. the space of all measurable functio... | I suggest Martingales in Banach spaces by Gilles Pisier. A very preliminary version of the book (242 pages vs 580 in final version) is available on the author's website.
The book begins with an introduction to Banach-space-valued $L^p$ spaces (i.e., Lebesgue-Bochner spaces). It's not long but clearly written and hits ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2202978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How do you construct a function that is continuous over $(0,1)$ whose image is the entire real line? How do you construct a continuous function over the interval $(0,1)$ whose image is the entire real line?
When I first saw this problem, I thought $\frac{1}{x(x-1)}$ might work since it is continuous on $(0,1)$, but whe... | Keeping in mind that (cumulative) distribution functions in statistics take the real line to the unit interval, inverse cdfs for variables defined over the whole real line are a rich and convenient source of these.
An example of this would be the $\log(\frac{x}{1-x})$, which is the inverse cdf for the standard logistic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2203081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 8,
"answer_id": 5
} |
Could you help with the concept of ratio and income/expenditure? Incomes of $A$ and $B$ are in the ratio $4:5$ and expenditures are also in the ratio $4:5$. Who saves more?
Options:
I) A
II) B
III) both save equally
IV) cannot be determined on the basis of the information provided
We've tried solving this by takin... | You've started in the right direction, but you've kept your equations separate; the key part is figuring out how to combine the equations you've gotten to represent the information you've been given.
In this case, you have $$Inc(A)=4x,\quad Inc(B)=5x,\quad Exp(A)=4y, \quad Exp(B)=5y.$$
Alright, but the problem is askin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2203210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
a submoudule can be both essential and superfluous enter image description here
I am reading Rings and Categories of Modules by Frank W.Aderson,on 73 pages,I can't understand the statement in the picture.I can't found a submodule is both essential and superfluous.I hope someone can help me,thanks!
| Your excerpt from Anderson and Fuller consists of two sentences: the first one is the claim and the second one is a module in which every nontrivial submodule is an example. So it is hard to understand why you are asking unless you simply don't understand how to find a single submodule of $\mathbb Z_{p^\infty}$. If th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2203424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What does this graph look like? $y = \log_x{2}$ The equation is $y = \log_x{2}$, where x is the variable and the base of the logarithm. What does the graph look like?
In general, what does $y = \log_x{k}$ look like, where k is some real constant?
I cannot plug this into online graphers like fooplot.com because they don... | $$y(x)=\ln_x(2)=\frac{\ln(2)}{\ln(x)} \qquad \text{and more generally}\quad \ln_x(k)=\frac{\ln(k)}{\ln(x)} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2203536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
A one-tape Turing machine which operates in linear time can only recognize a regular language
Show that A one-tape Turing machine which operates in linear time can
only recognize a regular language
I have no idea how to solve it. Can you give me show how to solve it ? I am beginner at this subject so I ask for an i... | The proof is not straightforward. This article Theory of one-tape linear-time Turing machines describes how it was proved.
I quote them here :
Hennie [18] made the first major contribution to the theory of one-tape linear-time Turing machines in the mid 1960s. He demonstrated that no one-tape linear-time deterministic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2203692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to find the invertible elements of $\Bbb{Q}[X]$ mod $X^2$
I am looking for some hints for finding all the invertible elements of $\Bbb{Q}[X]$ mod $X^2$.
Thank you very much in advance.
| The ring $\mathbb{Q}[X]/(X^2)$ is local, that is, it has a unique maximal ideal.
An element of the maximal ideal is not invertible (prove it). What about the elements not in the maximal ideal?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2203920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Definition of the gamma function for non-integer negative values The gamma function is defined as
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$
for $x>0$.
Through integration by parts, it can be shown that for $x>0$,
$$\Gamma(x)=\frac{1}{x}\Gamma(x+1).$$
Now, my textbook says we can use this definition to define $\Gamma... | To add on the other answers :
This is one of the 1st example of analytic continuation. It is clear that $$\Gamma(z) = \frac{\Gamma(z+1)}{z}=\frac{\Gamma(z+2)}{z(z+1)}=\frac{\Gamma(z+n+1)}{z(z+1)\ldots(z+n)}$$ makes $\Gamma(z)$ well-defined for $z \in \mathbb{C}, -z \not \in \mathbb{N}$.
But it is not so obvious (wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 0
} |
Name of Octonions With Biquaternion Coefficients? Ordinary biquaternions are quaternions $(\mathbb{H})$ whose coefficients are complex $(\mathbb{C})$. What is the name, analogous to "biquaternions", for octonions $(\mathbb{O})$ whose coefficients are ordinary biquaternions?
If there is no such name, what is the most... | The two things you are talking about are $\mathbb H \otimes \mathbb C$ and $\mathbb O \otimes (\mathbb H \otimes \mathbb C)$ (tensor products over $\mathbb R$). I would call the former the complexified quaternions or the complex quaternion algebra (there is a unique quaternion algebra over $\mathbb C$ up to isomorphis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Zero sets intersect in a line A line in $\mathbb{P}^3$ through points $[u_1:u_2:u_3:u_4]$ and $[v_1:v_2:v_3:v_4]$ is $\{[ku_1+lv_1:\dots:ku_3+lv_3]\}$ where either $k\ne 0$ or $l\ne 0$. But there's something worrying me, and it comes from the following problem.
Let $Z_1=Z(X_1^2-X_0X_2)$, $Z_2=Z(X_1X_2-X_0X_3)$, $Z_3=Z(... | So...you're right in how you're thinking about this problem. (I actually just did this problem last semester in my first alg. geo. class in grad. school and pulled up my solution set.) Ok so you have the line given by $[0:0:1:a]$ union with a point at infinity, $[0:1:0:0]$, a point lying in the set $Z_1\cap Z_2$. Sort ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Assume that $T$ is a linear transformation $T: V → ℝ$ and $T(\mathbf{v}_1)=1$ , $T(\mathbf{v}_2)=-1$, find $T(3\mathbf{v}_1-5\mathbf{v}_2)$ Assume that $T$ is a linear transformation $T: V\to\mathbb{R}$
$T(\mathbf{v}_1)=1$, $T(\mathbf{v}_2)=-1$
find $T(3\mathbf{v}_1-5\mathbf{v}_2)$
Not sure how to go about this quest... | As $T$ is a linear transformation, we have:
$$T(3v_1-5v_2)=T(3v_1)-T(5v_2)=3T(v_1)-5T(v_2)=3\cdot1-5\cdot-1=3+5=8$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I rotate a line segment in a specific point on the line? If I have two points $A$ and $B$, which $AB$ is a line segment, how can I rotate the $AB$ in another Point $C$ which is a point on the line $AB$.
Thank you in advance.
| Using complex numbers:
In the complex, multiplying by $e^{i\theta}$ amounts to a rotation around the origin by the angle $\theta$. To rotate around another point, translate so that the center moves to the origin, rotate and translate back.
Hence, for any point $a$ in the plane
$$a'=(a-c)e^{i\theta}+c.$$
This expands as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding the cluster points of sequences So, I've been doing these questions and I've done 3 out of 4 (I think :/) but I'm not sure how to do the last one. Here's the question:
Find the cluster points and name one convergent sub-sequence of each of the following sequences. For this problem you don't need to prove your s... | Your answers to $a$ and $b$ are correct, good job.
In the answer to $c$, $5$ is not a cluster point. Which subsequence converges to $5$? Remember : the cluster points of a sequence do not change if we remove finitely many terms from the start of the sequence, because the definition of a cluster point (or limit point) g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why is $5^{n+1}+2\cdot 3^n+1$ a multiple of $8$ for every natural number $n$? I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is
$$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of... | HINT:
If $a_n=(2b+1)^n$
$$a_{m+2}-a_m=8(2b+1)^m\cdot\dfrac{b(b+1)}2$$ which is multiple of $8$ as $b(b+1)$ is even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2204834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Finding $ \int \frac{5x^2-x-4}{x^5+x^4+1}dx$ Finding $\displaystyle \int \frac{5x^2-x-4}{x^5+x^4+1}dx$
Attempt : $\displaystyle I = \int\frac{5x^2-x-4}{x^5+x^4+1}dx = \int\frac{5x^2-x-4}{(x^2+x+1)(x^3-x+1)}dx$
because $\omega,\omega$ are the roots of $x^5+x^4+1 = 0$
so one factor is $(x-\omega)(x-\omega^2) = (x^2+x+1)$... | HINT:Using partial fraction decomposition
$$\frac{5x^2-x-4}{(x^2+x+1)(x^3-x+1)}=\frac{Ax+B}{x^2+x+1}+\frac{Cx^2+Dx+E}{x^3-x+1}$$
$${5x^2-x-4}=(Ax+B)(x^3-x+1)+(Cx^2+Dx+E)(x^2+x+1)$$
Solving gives..
$$\frac{5x^2-x-4}{(x^2+x+1)(x^3-x+1)}=\frac{-3x-3}{x^2+x+1}+\frac{3x^2-1}{x^3-x+1}$$
Another hint:
Maybe at some point you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is it possible to argue that $\nabla F(x^{*})\neq \textbf{0}$? Let $F:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be differentiable. For every affine subset $L$ of $\mathbb{R}^{n}$ of the form
\begin{eqnarray}
L=\{{x}\in \mathbb{R}^{n}:A{x}=b\}
\end{eqnarray}
for some $m\times n$ matrix $A$ having full rank and $b\in \mathbb... | It is possible that $\nabla F(\mathbf x^*)=0$ for $\mathbf x^*\ne \mathbf x^0$.
Example: take $f(t)=(t-1)^3$ and consider $F(x,y)=f(x^2+y^2)$. Basically, the graph of $F$ is the rotation of the following curve
The level sets of $F$ are circles, hence, the minimum on any line is unique (intersections of circles and tan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding the asymptotes of an integral I need to find the asymptote of $$\int\limits_{0}^{\infty} \frac{\ln(1 + \frac{x}{\sqrt{n}})}{x + x^3} dx$$
I've taken $b_{n} = n^{-1/2}$ which reduces the problem to finding,
$$\lim_{n\to\infty}\ \sqrt{n}\int\limits_{0}^{\infty} \frac{\ln(1 + \frac{x}{\sqrt{n}})}{x + x^3} dx$$ but... | I would use squeezing, with the inequalities
$$
\frac{x}{x+1}<\ln(1+x)<x.
$$
Inserting and calculating, you will find that
$$
\frac{\sqrt{n}\pi-\ln n}{2+2n}\leq \int_0^{+\infty}\frac{\ln(1+x/\sqrt{n})}{x+x^3}\,dx\leq \frac{1}{\sqrt{n}}\frac{\pi}{2}.
$$
It follows that
$$
\lim_{n\to+\infty}\sqrt{n}\int_0^{+\infty}\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find the integral $\int_0^1 \frac{x^{1/2}}{1+x^{1/3}}dx$? How to find $$\int_0^1 \frac{x^{1/2}}{1+x^{1/3}}dx$$ ?
My attempt: I made $t^6=x$ and got $\displaystyle \int_0^1 \frac{6t^8}{1+t^2}dt$ and got stuck.
| Try writing the numerator as $$6t^8 + 6t^6 - 6t^6 - 6t^4 + 6t^4 + 6t^2 - 6t^2 - 6 + 6.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
For sheaves, $f_*g_*=(fg)_*$ is equality, but $g^*f^* \cong (fg)^*$ only canonical isomorphism I wonder why do one cares that pushforward of quasicoherent sheaves satisfies equality $f_*g_*=(fg)_*$, but for pullback there is only canonical isomorphism $g^*f^* \cong (fg)^*$?
I believe that the fact follows because $A \o... | It seems to me that the question essentially contains its answer. One would care about the differences, if one cares to know what things are!
Having the equalities $(f\circ g)_\ast=f_\ast\circ g_\ast$ and ${id_X}_\ast=id_{\mathbf{QCoh}(X)}$, for every composable pair of morphisms $f$ and $g$ in $\mathbf{Sch}/S$, and f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is the Law of Quadratic Reciprocity necessary or just convenient for calculation? I'm rather confused. Based on my understanding, you can solve a quadratic in some ring if the discriminant is a square in that ring. So if I have:
$$ax^2 + bx + c \equiv 0 \pmod{n}$$
Then all I need to do is determine is where $b^2 - 4ac$... | It just speeds up the calculation, but it speeds up calculation rather dramatically. To compute $\left(\frac ap\right)$ by checking each value $0 \le b < p$ to see if $a \equiv b^2 \pmod p$, we need $O(p)$ steps.
On the other hand, we have an $O(\log a \log p)$ algorithm by doing two things: using quadratic reciprocit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Having trouble understanding Taylor Series I'm having trouble interpreting the Taylor series formula. The nth term of the Taylor series looks like the nth integral of f(x). Is this correct? If so, I don't quite understand the meaning of the nth integral, and how it is able approximate f(x) at higher values of n.
Edit:... | From a purely symbol manipulation point of view, you can easily obtain the Taylor series formula in the following way.
Start by assuming
$$f(x) \; = \; a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 + a_6x^6 + \; \cdots $$
Plugging $x=0$ into this tells us that $f(0) = a_0.$
Now differentiate both sides, assuming we ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Semantic Consequence Definition "What is the difference between
⊨
(semantic consequence) and
⊢
(syntactic consequence)?" was a question that has been posted, but I am wanting a more specific answer. For example, this video explains what a syntactic consequence is. After watching this video, it is obvious that we say
... | $\vdash$ is used to make statement about formal proof systems, which include rules of inference, that say:
"If you have a (or two) statement(s) that look like such-and-so, then you can write down a new statement that looks like this-and-that".
For example, many formal proof systems include the following rule of infere... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Apparently sometimes $1/2 < 1/4$? My son brought this home today from his 3rd-grade class. It is from an official Montgomery County, Maryland mathematics assessment test:
True or false? $1/2$ is always greater than $1/4$.
Official answer: false
Where has he gone wrong?
Addendum, at the risk of making the post no long... | 'What about this?' is ABSURD. Fractions are real numbers and $1/2$ is NOT smaller than $1/4$. Period.
If the dumb teacher wants to compare a half of biscuit to a quarter of pizza, then they are no longer numbers, but physical quantities (masses or volumes), which have their units, and the stupid needs to consider bring... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2205890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "146",
"answer_count": 7,
"answer_id": 5
} |
How can we one show that $\sum_{i=1}^{\infty}\sum_{j=1}^{2k}(-1)^{j-1}{2\over i+j}=H_k?$ Given the double sums $(1)$
$$\sum_{i=1}^{\infty}\sum_{j=1}^{2k}(-1)^{j-1}{2\over i+j}=\color{blue}{H_k}\tag1$$
Where $H_k$ is the n-th harmonic number
How can one prove $(1)$?
Rewrite $(1)$ as
$$\sum_{i=1}^{\infty}\left({2\ove... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
For which values of $z$ does the sequence $(e^{nz})$ converges? Where $z$ is a complex number.
For which values of $z$ does the sequence $(e^{nz})$ converges? Where $z$ is a complex number.
I was able to figure it out for numbers with null imaginary part, but I get troubled when considering the imaginary part. Being ... | Put $e^z=:w$. Then $$e^{nz}=w^n$$ for all $n\geq0$. The complex sequence $\bigl(w^n\bigr)_{n\geq0}$ converges iff either $|w|<1$ or $w=1$. The first is the case if ${\rm Re}(z)<0$ and the second if $z=2k\pi i$ for some $k\in{\mathbb Z}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Differentiation of $g(x)=f(x+c)$
Suppose $g(x)=f(x+c)$.
Prove $g'(x)=f'(x+c)$
I know that $$g'(x)=\lim_{x\to 0} \frac{g(x+h)-g(x)}{h}$$ that is it please help.
| Fix $x$ and let $\tilde{x} = x+c$. Then
$$\lim_{h \to 0} \frac{g(x+h) - g(x)}{h} = \lim_{h \to 0} \frac{f((x+h)+c) - f(x+c)}{h} $$ $$= \lim_{h \to 0} \frac{f(\tilde{x}+h) - f(\tilde{x})}{h} = f'(\tilde{x}) = f'(x+c).$$
You could also have used the chain rule: letting $u(x) = x+ c$, we have that $g = f \circ u$, so that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
About irreducible representations of C* algebra I have been thinking about what can we say about decomposing any representation as a direct sum of irreducible representations. I know that every cyclic representation corresponds to a representation coming from a state, which is the essence of the GNS construction. Moreo... | It can be shown that if $A$ is separable and $\pi:A\to B(H)$ is a representation with $H$ separable, then $\pi$ is approximately unitarily equivalent to a direct sum of irreducible representations.
But the "approximate" part is essential. For instance let $A=C_r^*(\mathbb F_2)$, the reduced C$^*$-algebra of the free g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Use the method of characteristics to solve $u_t+uu_x+\frac{1}{2}u=0$.
Use the method of characteristics to solve
$$u_t+uu_x+\frac{1}{2}u=0, \quad t>0, \quad {-\infty}<x<\infty$$
$$u(x,0)=\sin(x)\quad {-\infty}<x<\infty$$
(solution may be expressed in implicit form). Show that a shock solution is possible if $t=... | The shock between states $u_L,u_R$ moves with speed $(u_L+u_R)/2$. This follows from applying conservation to an infinitesimal rectangular control volume that has part of the shock path as a diagonal. The source term has no effect on this. Past the shock formation time, the solution is double-valued, and in this region... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Open set on $E^n$ (the $n$-dimensional euclidean space) Let $A$ be a countable set of $E^n$ (the $n$-dimensional euclidean space). Show that $A$ is not an open set of $E^n$.
Definition of open set
Let $(X, \mathcal T)$ be a topological space. A subset $U \subset X$ is called an open set of $X$ if $U \in \mathcal T$. ... | For $n>1$ and non-empty $A$: Let $x=(x_1,...,x_n)\in A.$ If $A$ is open then for some $r>0$ the open ball $B(x,r)$ of radius $r$, centered at $x$, is a subset of $A.$ The real interval $(-r+x_1,r+x_1)$ is uncountable and $$A\supset B(x,r)\supset (-r+x_1,r+x_1)\times (x_2,...,x_n)$$ which is uncountable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find the domain and range of $f(x) = \sqrt {\frac{x+1}{x+2}}$
Find the domain and range of $$f(x) = \sqrt {\frac{x+1}{x+2}}$$
I got the domain $[-1, \infty)$ but the answer contains $(-\infty, -2)$ along with it. And how to calculate range?
| We must have $$\frac{x+1}{x+2}\geq 0.$$ We consider the following cases:
Case 1. Suppose that $x+1\geq 0$ and $x+2>0$. Then $x\geq -1$ and $x>-2$. Thus,
$$SS_1=[-1,\infty).$$
Case 2. Suppose that $x+1\leq 0$ and $x+2<0$. Then $x\leq -1$ and $x<-2$. Thus,
$$SS_2=[-\infty,-2).$$
Hence, domain$=SS_1\cup SS_2$
Let $y=f(x)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing that the hyperintegers are uncountable In class, we constructed the hyperintegers as follows:
Let $N$ be a normal model of the natural number with domain $\mathbb{N}$ in the language $\{0, 1, +, \cdot, <, =\} $. Also let $F$ be a fixed nonprincipal ultrafilter on $\omega$. Then we have $N^*$ as the ultrapower $... | Here's a diagonalization argument. Let $\left(\mathbf{x}^{(m)} \right)_{m \in \mathbb{N}}$ denote a sequence of natural sequences $\left(x_{n}^{(m)} \right)_{n \in \mathbb{N}} \in \mathbb{N}^{\mathbb{N}}$. Define a sequence $\left(y_n \right)_{n \in \mathbb{N}}$ by
$$y_{n} = 1 + \max_{1 \leq m \leq n} \left(x_{n}^{(m)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2206831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Showing that $\sum_{k=0}^{n}(-1)^k{n\choose k}{1\over k+1}\sum_{j=0}^{k}{H_{j+1}\over j+1}={1\over (n+1)^3}$ Consider this double sums $(1)$
$$\sum_{k=0}^{n}(-1)^k{n\choose k}{1\over k+1}\sum_{j=0}^{k}{H_{j+1}\over j+1}={1\over (n+1)^3}\tag1$$
Where $H_n$ is the n-th harmonic
An attempt:
Rewrite $(1)$ as
$$\sum_{k=... | We may notice that
$$ \sum_{n\geq 1}\frac{x^n}{n}=-\log(1-x),\qquad \sum_{n\geq 1} H_n x^n = -\frac{\log(1-x)}{1-x}\tag{1} $$
hence
$$ \sum_{n\geq 1}\frac{H_n}{n+1} x^{n}=\frac{\log^2(1-x)}{2x},\qquad \sum_{n\geq 1}\frac{H_{n+1}}{n+1} x^{n}=\frac{\log^2(1-x)+2\text{Li}_2(x)}{2x}-1 \tag{2}$$
and we may consider what the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Let $E$ be a Banach space and $A\in L(E)$ such that $\parallel A \parallel<1$. Then, $I-A$ is invertible and $(I-A)^{-1}=\sum^\infty_{n=1}A^n$. I want to prove the following:
Let $E$ be a Banach space and $A\in L(E)$ such that $\parallel A \parallel<1$. Then, $I-A$ is invertible and $(I-A)^{-1}=\sum^\infty_{n=1}A^n$.
... | Taking your notations, we already have $\sum A^n$ that converges in $L(E)$ to some element $B$ (also, note the comment that the series should begin at $n=0$).
Now, we also have the following telescopic sum:
$$
\left(\sum_{n=0}^N A^n\right) \circ (I - A) = I - A^{N+1}
$$
The left hand side converges to $B \circ (I-A)$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Compute integral $\int_0^a{f'(x)\cdot f(x)}dx$ what is the integral of the following function:
$\int_0^a{f'(x)\cdot f(x)}dx$
Not quite sure how to integrate it.
| Alternative hint: if you integrate by parts, you obtain
$$\int_0^a f'(x)f(x)\,dx = -\int_0^af(x)f'(x)\,dx + \left[f(x)f(x) \right]_0^a $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why is the moment map constant on the orbits of the action of the Lie algebra Given the action of a Lie group on a symplectic manifold, the moment map gives a mapping $\mu: M \rightarrow \mathfrak{g}^*$ to the dual of the Lie algebra $\mathfrak{g}^*$ defined by $d(\langle \mu,\eta\rangle)=i_{X_\eta}\omega$, where $X_\e... | Since the flow is Hamiltonian we have $d(\langle \mu, \eta \rangle)=i_{X_\eta}\omega=dH_\eta$ for some function $H_\nu$. The orbits of the Hamiltonian vector field $X_\eta$, occur on the levels sets of $H_\eta$ and so on the level sets of $\langle \mu, \eta \rangle$, for which $\mu$ is constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof Verification: Eigenvalues of a Complex Vector Space I think I found a solution to question 16 from chapter 5.A of Axler's Linear Algebra book which states
Suppose V is a complex vector space and $T\in L(v)$, and the matrix of T with respect to some basis of V only has real entries show that if $\lambda$ is an ei... | T has only real values => Any complex vector field is being mapped to a real vector field.
T:C^N->R^N
Any complex vector v can be written as a+bi where a & b are vectors with real components.
Tv=zv : z is the eigenvalue here
Substituting v with a+bi & a-bi and z with x&y respectively we get:
T(a+bi)=Ta+iTb=Ta=x(a+bi)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Summing sines of different frequencies Is there a general formula for solving the following equation:
$$A \sin(Bt+C) + D \sin(Et+F) = G \sin(Ht+I)$$
All constants on the left side of the equation are known (t is a variable). Is there a formula for calculating G, H and I? Is this even solvable in general?
I searched th... | I'm not sure about a general formula, but to your answer wether it is solvable in general: the function $sin(3x+4)+sin(6x+7)$ is clearly not harmonic (you can see this by simply plotting it), providing a counterexample to the statement.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Statistics Hypothesis Testing finding out test stat and critical value
For a)
$$z =\frac{ \bar{x} - \mu }{ \frac{\sigma }{ \sqrt n}}$$
I have deciphered that sample mean is $$\frac{20 + 23 + 21 + 22}{ 4} = 21.5$$
I came up with $1.29099..$ for Standard Deviation.
Sample size is $4$ since $4$ sharks
$$z =\frac{ 21.5 -... | Your test statistic should be $2.3238$. I don't know how you got the value you got. You should calculate $(21.5-20)/(1.291/\sqrt{4}) = 2.32378...$
The critical value should be $4.54$. The question suggests doing a one-tailed test (because the biologist thinks that the sharks will be longer than $20$.) There are $4-1=3$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Supremum of sets containing inequalities. Let $r \in \mathbb{R}$ be fixed. Prove that:
a) $\sup\{q \in \mathbb{Q}: q \leq r\} = r$.
b) $\inf\{q \in \mathbb{Q}: q \leq r\} = r$.
It looks so strange for me that the supremum equals the infimum, Could anyone help me or at least say that the problem contains a mistake?
| I think you should have for (b): $\inf \{q \in \mathbb{Q}: q \geq r \} = r$ as $\inf \{q \in \mathbb{Q}: q \leq r \} = -\infty$
You might even say that the latter expression is undefined, regardless there is certainly no real $r = -\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2208049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the scalars If possible, find scalars
c1, c2, and c3, so that the following is true.
$$c_1(3, 2, -5) + c_2(-3, 3, 3) + c_3(-3, 8, 1) = (3, -3, 4)$$
I have no idea where to start. I think I have to make it into rref, but I am unsure. Can someone explain how to do this. Please.
| Its possibly the question is:$$c_1(3, 2, -5) + c_2(-3, 3, 3) + c_3(-3, 8, 1) = (3, -3, 4)$$
Then Solve the system of equations:
$$3c_1-3c_2-3c_3 =3, 2c_1+3c_2+8c_3 =-3,-5c_1+3c_2+c_3 =4.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2208170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is a positive cone defined the way it is? Let $K$ be a field. Consider $P \subseteq K$ satisfying the following properties:
*
*$x \in P$ and $y \in P$ imply $x+y \in P$ and $xy \in P$
*$x \in K$ imply $x^2 \in P$
*${-1} \notin P$
We call such $P$ a prepositive cone. We further call $P$ a positive c... | Partial answer:
A set $C$ is called a “cone” with vertex at the origin if for any $x\in C$ and any scalar $a\ge0$, $ax\in C$.
Look familiar? There is a related concept of a cone for linear algebra.
A cone $C$ is a convex cone if $ax + by$ belongs to $C$, for any positive scalars $a,b$ and any vectors $x, y$ in $C$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2208278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving Linear Congruences over Cryptography I have got some doubt in cryptography mainly related to Linear Congruences.
Question
Suppose that the most common letter and the second most common letter in a long
ciphertext produced by encrypting a plaintext using an Affine Cipher
$f\left(p \right)=\left (ap + b \right... | I don't know why you're not getting $25$ and $19$, because that's what I get:
$$f(4) = 18\cdot 4 + 5 = 77 \equiv 25 \pmod{26}.$$
And
$$f(19) = 18\cdot 19 +5 = 347 = 9 \pmod{26}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2208486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Consider the six dot product of four vectors $v_1,v_2,v_3,v_4$ on $\mathbb{R}^2$. Can all of them be negative?
Consider the six dot products of four vectors $v_1,v_2,v_3,v_4$ on $\mathbb{R}^2$. Can all of them be negative?
If all dot products are negative, then the angle between each two vectors are larger than $\pi/... | You can do this with simple algebra. Can you first convince yourself that there is no loss of generality taking $v_1 = (1,0)$?
Suppose all the inner products are negative.
Write $v_2 = (a_2,b_2)$, $v_3 = (a_3,b_3)$, and $v_4 = (a_4,b_4)$. Compute the inner products with $v_1$ to find $a_2 < 0$, $a_3 < 0$, and $a_4 < 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2208716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Describing the solution to a nonlinear PDE Given the pde $$xu_x+yu_y+uu_z=0$$ where $$u(x,y,0)=xy$$ for $x>0$ and $y>0$, the solution, gotten from the method of characteristics is $$u(x,y,z) = xye^{\frac{-2x}{u}}$$
My question is, how would one describe this solution? I have no idea how to even imagine it.
Thanks.
EDIT... | It's an inavoidably implicit function. Some nice way to visualize some of its characteristics is using a 3D plotter. For a three variables function the surfaces of $u$ constant are interesting to see. It's easy to isolate $z$ as function of $x$, $y$, and $u=k$. I've used GeoGebra to plot these surfaces. It has an slide... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $x$ is not belongs to span of $y$ find a functional on X with $\phi x=1$ and $\phi y=0$? If $x$ is not belongs to span of $y$ find a linear functional on X with $\phi x=1$ and $\phi y=0$ ? where $\phi: X->F$ ,F is a field ,X is in Banach space
| Suppose $\;y\neq0\;$, then observe that $\;\{x,y\}\;$ is a linearly independent set. Complete it to a basis $\;\{x,y,x_i\}_{i\in I}\;$ of $\;X_F\;$ (you may need AC to do this. I am not sure...), and define
$$\phi x=1\;,\;\;\phi y=\phi x_i=0$$
and extend the definition by linearity.
If $\;y=0\;$ then, as $\;x\neq0\;$ ,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Torsion points of an elliptic curve over infinite extension How can I see that, if $E$ is an elliptic curve over $\mathbb{Q}$, than $$E(\overline{\mathbb{Q}})_{tors}=E(\mathbb{C})_{tors}=(\mathbb{Q}/\mathbb{Z})^2,$$ and that $E(\overline{\mathbb{Q}})/E(\overline{\mathbb{Q}})_{tors}$ has infinite rank?
| The infinitude of the rank is somewhat subtle. You can see a proof in this article of Frey and Jarden (see Section 2).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Simplest (smallest element) of non-cyclic abelian group I am looking for some examples of non-cyclic abelian groups.
I found something like order 12, or other group.
Here i am looking for simplest (smallest element) non-cylcic abelian groups.
If you know something about this please let me know.
| The smallest noncyclic group is the four element Klein four-group https://en.wikipedia.org/wiki/Klein_four-group . All finite abelian groups are products of cyclic groups. If the factors have orders that are not relatively prime the result won't be cyclic.
It doesn't make sense (in general) to ask for "small elements" ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is $\lim_{x\to\infty} xe^{-x^2}$?
What is $\lim_{x\to\infty} xe^{-x^2}$?
I am calculating $\int_0^\infty x^2e^{-x^2}\,dx$. By integration by parts, I have
$$I = -\frac{1}{2}xe^{-x^2} |_{0}^\infty+\frac{1}{2}\int_0^\infty e^{-x^2}\,dx$$
The second integration is just $\frac{\sqrt{\pi}}{2}$. Now I want to know how... | For the limit itself, it's useful to remember that exponentials will always win out over polynomials. Even for something absurd like:
$$\lim_{x\to\infty} x^{100,000}e^{-x} = 0$$
Based off of the nature of your question, I'm guessing you are currently in an introductory calculus course. You'll later see that you can exp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove that T is a linear transformation Does it matter that in the first line it's written $T(\alpha p+ \beta g)$ and not $T(\alpha p(t)+ \beta g(t))$ but at the end it is written with $\alpha T(p(t)) + \beta T(g(t)))$ with the $t$'s.
Define $T : \mathbb{P}_3 \to \mathbb{R}^4$ by $$T(p) = \begin{bmatrix}
p(-3) \\
p(-1... | It doesn't really matter. But it doesn't look very good. I would've written $\alpha T(p) + \beta T(g)$ the last time. Also, why choose the letters $p$ and $g$? Why not $f$ and $g$, or $p$ and $q$? That would make the proof look nicer, at least to my eyes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
In induced von Neumann algebra, do we have $P(PMP) =P\cdot P(M)P$ and $U(PMP) =P\cdot U(M)P$? Let $M$ be a von Neumann algebra and $P\in M'$ be a projection. Then, $PMP$ is an induced von Neumann algebra. I know that $ P Z(M) P = Z(PMP)$ and $ (PMP)' = PM'P $ (here, $Z(M)$ is the center and $M'$ is the commutant). I wo... | Edit: *This argument works when $P\in Z(M)$. Together with Proposition 5.5.5 in Kadison-Ringrose, as mentioned by @user92646, this gives the desired proof.
If $Q\in M$ is a projection, then $QP$ is also a projection, since $PQ=QP$. Conversely, if $PX\in PMP$ is a projection, we have $(PX)^2=PX$. Let $Q=PX+I-P$. Then $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Let be $a,b$ positive real numbers; $a+\frac {b} {a+\frac {b} {a+\frac {b} {\ddots }}}$ how can we prove whether this has a limit or not? Let, given example is like this
$$3+\dfrac {4} {3+\dfrac {4} {3+\dfrac {4} {\ddots }}}=?$$
I wonder if it has a limit or not, but I have a idea, If we show that this sequence monot... | One way to prove convergence is to start with the assumed limit and consider the distance to it.
As you've seen the equation for the assumed limits is a quadratic so we could instead start in the other end and write the recurrence formula:
$$x_{n+1} = f(x) = (p+q) - pq/x_n$$
where $p$ and $q$ are the assumed limits. No... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
A consequence of the Fundamental Theorem of Arithmetic. The book said that:"If $S={1,2,3,.....,200}$, then for each $x \in S$, we may write $x=2^{k}y$, with $k \geq 0$, and gcd(2,y)=1." and the book added that this result follows from the Fundamental Theorem of Arithmetic, but I do not know how?
could anyone explain th... | Write $x$ as a product of primes, where $2$ occurs exactly $k \ge 0$ times, say. Collect together the $k$ occurrences of $2$ to get $x = 2^{k} y$. Clearly $y$ is odd.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2209987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Expansion of this expression Let $x$ be a real number in $\left[0,\frac{1}{2}\right].$ It is well known that
$$\frac{1}{1-x}=\sum_{n=0}^{+\infty} x^n.$$
What is the expansion or the series of the expression $(\frac{1}{1-x})^2$?
Many thanks.
| By the Cauchy product,
$$\left(\sum_{n=0}^{\infty} x^n\right)^2=\left(\sum_{n=0}^{\infty} x^n\right)\left(\sum_{m=0}^{\infty} x^m\right)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}x^nx^m=\sum_{k=0}^{\infty}\sum_{m=0}^k x^k=\sum_{k=0}^{\infty}(k+1)x^k.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2210103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Linear Regression: units of intecepts given $x$ and $y$ as log function? Given a line $y = ax + b$, where $x = log(w)$ and $y = log(l)$. To me, the units of $b$ should be unitless as $y$ will be unitless. Is this right?
| Yes. Formally speaking, you're not taking $y = \log (l)$ but $y = \log (l/l_0)$, where $l_0$ is one of whatever your unit of measure is - the argument of the log needs to be unitless. From the fact that you've chosen $l$ and $w$, probably "length" and "width", I'm guessing you mean for $l$ and $w$ to have units of le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2210215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to find out if there 7 prime subtractors of a number I have a number and i want to find out if i can get that number by adding 7 prime numbers.How can i find that out?
Example
Number:14
Answer:2 2 2 2 2 2 2
| For n sufficiently large $n>=100$ probably less you can substract $2$ and $3$ conveniently to form odd number. Then applying the Weak Golbach Conjecture (which as far as I remember it was proved recently) it is always possible to decompose an odd number into the sum of three prime numbers.
When n is small you can do a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2210305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Linear independence of $x, x^3, |x^3|$ How do I make sure whether $x, x^3, |x^3|$ are linearly independent? I know I have to show that $$kx+lx^3+m|x^3|=0\quad\forall x\in\mathbb{R}\quad \Rightarrow k=l=m=0,$$ but I'm not sure how to do this. Should I plug in different values for $x$?
| Yes, the idea is good! For $x=1$ you get
$$
k+l+m=0
$$
For $x=-1$ you get
$$
-k-l+m=0
$$
Can you finish?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2210554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proof by induction, dont know how to represent range The question asks for me to prove the following through induction:
$1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2^n} \geq 1 + \frac{n}{2}$
This is my proof thus far:
Proving true for $n = 1$
\begin{align*}
1 + \frac{1}{2} &\geq 1 + \frac{1}{2}\\
\end{align*}
A... | $$
\underbrace{ 1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 {2^k} + \frac 1 {2^{k+1}} }_{\Large\text{This is wrong.}}
$$
$$
\overbrace{ \underbrace{ \underbrace{1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 {2^k}}_{\Large\text{The is the case }n=k.} + \frac 1 {2^k+1} + \frac 1 {2^k+2} + \frac 1 {2^k+3} + \cdots + \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2210649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
A book that includes the main types of manifolds and geometries Is there a book (or a few books) that gives the basic theory of the different types of manifolds and their geometries in an integrated manner? By the different types I primary mean C^k real manifolds, complex manifolds, real analytic and non-archimeadian (... | As far as I know, the p-adic case is never discussed in conjunction with the "standard" geometric structures. Most people who care about p-adic structures come from the algebraic geometry side and they do not care about, say, conformal structures or contact structures or foliations on manifolds. If you want the basic t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2210779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the dimension of $V$ as a vector space over $\mathbb F$ Let $\mathbb F$ be a subfield of a field $\mathbb K$ satisfying the condition that the dimension of $\mathbb K$ as a vector space over $\mathbb F$ is finite and equal to $r$. Let $V$ be a vector space of finite dimension $n>0$ over $\mathbb K$. Find the dime... | Well you gave the answer yourself. The reason that equation is true is as follows. Suppose $\{k_1,\dots,k_r\}$ is a basis for $K$ over $F$, and $\{v_1,\dots,v_n\}$ is a basis for $V$ over $K$. Then it suffices to show $\{k_iv_j : 1 \le i \le r, 1 \le j \le n\}$ is a basis for $V$ over $F$. But this is just unfolding de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Continuity of function in Sobolev W1,1(R^2) Suppose $u(x)$ is in $W^{1,1}(R^2)$ (Sobolev space on $R^2$ with 1 weak derivative and $L_1$ norm) and is locally bounded. Must $u(x)$ be continuous?
Motivation: In $R^1$, a weakly differentiable function is continuous, but that is not the case for $R^2$ because of $f(x) = |x... | As Jose27 mentioned, we look for functions of the form:
$$f(x) = \sin(|x|^{-\alpha}), \alpha > 0$$
$$\displaystyle f_{x_i} = -\alpha \cos(|x|^{-\alpha})|x|^{-\alpha-1}\frac{x_i}{|x|}$$
Similar to the argument on Evan's page 260, we can prove $f$ is weakly differentiable.
For any $\phi \in C^{\infty}_c(R^2)$
$$\int\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the limit or prove that the limit does not exist $\lim_{x \to c}x^2 + x + 1,$ for any $c \in R$
This is what I tried.
For $\epsilon > 0$ there exists $\delta >0$ such that $0< \mid x - c \mid < \delta (x \in R)$ => $\mid x^2 + x + 1 - c^2 - c - 1 \mid = \mid x^2 + x - c^2 - c \mid < \epsilon$
After that I tried... | Here is a trick:
Start by choosing $\delta \le |c|+1$, then $|x| \le |c|+|x-c| \le 2(|c|+1)$.
Then $|x^2-c^2| = |x+c||x-c| \le 3 (|c|+1) |x-c|$.
Then $|x^2+x+1 - (c^2+c+1)| \le (3|c|+2)|x-c|$.
Now choose $\delta < \min (|c|, {\epsilon \over 3|c|+2})$, then you will
obain the required bound.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A little PDE solving Guys I really don't know what to do, I apply the method of characteristics, but the system of equations it gave me looks unfamiliar and complicated. Any hint on how to approach this?
Find a solution of the first order quasi-linear partial differential equation:
$$z(x +z)\frac{\partial z}{\partial x... | The given condition implies that the curve $\Gamma = \{ (1,s^2,s) \in \mathbb{R}^3 : s \ge 0 \} $ must be contained in the graph of our solution. We apply the characteristics method and transform the initial PDE into the ODE system:
\begin{cases} \dot x = z(x+z) \\ \dot y=y(y+z) \\ \dot z=0 \end{cases}
The last equatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Navier-Stokes equation, about pressure... I'm a computer science student writting a dissertation about fluid simulation on real time applications. I'm trying to understand a few things regarding the pressure term:
1) When talking about the Helmholtz-Hodge decomposition, my books say that you can decompose any vector fi... | here is a way to see (2). assuming the fluid to be incompressible , mass conservation tells you that $div\ u = 0.$ the newtons law for inviscid fluid is $\frac{du}{dt} = -\frac 1\rho\nabla P.$ take the divergence(dot product with $\nabla$) gives you the poisson equation $\Delta P = 0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Number of q-ary strings of length m which do not contain k consecutive zeros A finite q-ary-alphabet is given $$A_q = {0,1,2,...,q-1}.$$ Now I am considering the set of all finite strings over the alphabet $A_q$.
I am interested on the number $$N(m,k)_{A_q}$$ of strings of length $m$ which do not contain $k$ consecutiv... | Let me present another approach to this interesting problem, which will allow to get a closed expression for $N(m,k,q)$.
Let's consider a word of length $m$ from the binary alphabet $\{0,X\}$, having a total of $s$ zero's.
$$
\begin{array}{*{20}c}
X &| & {0,} & {0,} & {0,} &| & X &| & 0 &| & {X,} & X &| & {0,}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Prove: $z^{12}+3z^8+101z^4+1$ has a root on the unit circle $\newcommand{\cis}{\operatorname{cis}}$>Prove that $$f(z)=z^{12}+3z^8+101z^4+1$$ has a root on the unit circle or $|z|\leq 1$
So started with looking at $$z^{12}+3z^8+101z^4+1=0$$
Therefore
$$z^{12}+3z^8+101z^4=-1$$
looking at $z=r\cis\theta$ we get
$$r^{12}\c... | You can apply Rouché's theorem. Let $f(z)=z^{12}+3z^8+101z^4+1$ and $g(z)=101z^4$.
For $|z| \leq 1$ :
$$|f(z)-g(z)|=|z^{12}+3z^8+1|\leq|z^{12}|+|3z^8|+|1|=5$$
$$|g(z)|=101$$
We can apply the theorem because $5<101$.
So $f$ has the same numbers of roots as $g$ in $\{z\in \mathbb{C};|z|\leq1\}$, so $f$ has $4$ roots in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Why solutions of 2nd order linear equations with constant coefficients goes to infinity, but solutions of 1st order only go asymptotic to 0? What I'm talking about is why does the solution of 2nd order $$y=k^2 y''$$ tends to blow up at both $x \rightarrow + \infty$ and $x \rightarrow - \infty$; but the solution of fir... | What you say is not exactly true: $y=e^{-x}$ is a solution of $y=y''$ and goes to $0$ as $x\to+\infty$.
As for the second question, an equation of the form $y''= a\,y$ represents a mechanical system in which $y''$ is the acceleration, and $a\,y$ an force that is proportional to the displacement. If $a>0$ then the force... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is this equality true? I'm working with a proof of local and global truncation errors for the euler method in numerical methods, and I can't seem to understand the equality and the less/equal seen below. At first I thought it had to do with geometrical sums, but I still don't understand it, any help would be apprec... | The formula $$\sum^i_{n=0} x^n = \frac{x^{i+1}-1}{x-1}$$ holds for any $x\ne 1$ [this is easily seen for $i=1$ and then proven by induction]. Just plug in $x=e^{Lh}$. The inequality follows since $e^{Lh} -1 \ge Lh$ and presumably $ih = t_i -a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2211959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solution for the integral $∫ \sin(4x)(1+\cos(4x))\mathrm dx$ Good day!
I'm kind of stuck in solving this integral problem. I've tried multiple times, but I'm not still not sure what is the correct way of solving.
The problem is to solve
$$\int \sin(4x)(1+\cos(4x)) \mathrm dx$$
I've tried two methods:
The first o... | Method 1 -
Let
\begin{align}
1+\cos 4x &= u\\
-\sin 4x \cdot 4\ dx &= du\\
\sin 4x\ dx &= \frac 1{-4}\ du
\end{align}
Putting these values,
$$\int \frac 1{-4} \cdot u\ du\\
\frac 1{-4} \int u\ du\\
\frac 1{-4} \cdot \frac {u^2}2 + c\\
\frac {u^2}{-8} + c$$
Put value of $u$ back.
$\frac {(1+\cos 4x)^2}{-8} + c$
Method ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find all pairs of elements such that they are not covered by any given sets Given a universe $U$ and a family sets $F$ over $U$, I wanna find all pairs $(x, y)$ where $x, y \in U$, such that $x$ and $y$ are not in any set of $F$ at the same time, i.e., $\not\exists S \in F, x\in S \wedge y\in S$.
Also, I try to avoid c... | $$(U \times U) \setminus (\bigcup_{f \in F} f \times f)$$
EDIT: the idea is to consider all pairs of points and then for each set in $F$, subtract out the pairs of points such that both belong in it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What constitutes a mathematical proof? My question is what constitutes a mathematical proof?
I ask because I am currently doing a Calculus course in University and I am constantly confused regarding what I'm allowed to assume within a proof. Here is an example:
"Prove that f(x) = sqrt(|x| + x^2) is continuous in Real n... | I've got a question for you. If we are given two functions $f$ and $g$ such that both $f$ and $g$ are continuous. Can you prove that:
1) $f+g$ or $f-g$ is continuous?
2) $f*g$ is continuous?
3) $f\circ g$ is continuous ($\circ$ is the compositions function meaning $f\circ g = f(g(x))$ )?
4) Is $f/g$ continuous if $g(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $\gcd(a,b)=9,$ then what is $\gcd(a^2,b^3)\,?$ I know that by the euclidean algorithm, I can obtain the following equations.
I tried some algebraic manipulation but I can't seem to determine, if $$\text{if }\;\gcd(a,b)=9,\text{ then what is }\gcd(a^2,b^3)\;?$$
| Note $\ (A,B) = 9\,\Rightarrow A,B = 9a,9b,\ \color{}{(a,b) = 1},\, $ so $\,\color{#c00}{(a^2,b^3) = 1}\,$ by Euler
Thus $\ (A^2,B^3) = ((9a)^2,(9b)^3) = 81(\color{#c00}{a^2},9\color{#c00}{b^3}) = 81(a^2,9) = 81(a,3)^2 = (A,27)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
What is the differential of $X'X$? Let $X = (x_{ij})$ be a square matrix with $n\times n$ variables in $\mathbf{R}$. Could you tell me the $\text{d}(X'X)$ when $X$ has a full column rank?
[Update]
Honestly speaking. What I have got is:
$$
\text{d} X'X = (\text{d} X')X + X'\text{d}X. \tag{1}
$$
But the text book of ma... | After some exploration, I intend to answer my question.
Lets assume $X$ has $m\times n$ variables where $m \leq n$.
First we have
$$
{\rm d} XX' = ({\rm d}X)X' + X({\rm d}X'). \tag{1}
$$
Then
$$
\begin{align}
{\rm d} |XX'| &= {\rm tr} \big( (XX')^\# {\rm d} (XX') \big) \\
& = {\rm tr} \bigg( (XX')^\# \big(({\rm d}X)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Reference request: Representation theory over fields of characteristic zero Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook on representation theory which works with fi... | The book "The representation theory of the symmetric group" (Encyclopedia of
Mathematics and its Applications, Vol. 16, Addison-Wesley, Reading, Mass., 1981), by Gordon James and Adalbert Kerber, develops the characteristic zero representation theory of symmetric groups over $\mathbb{Q}$. In particular it has a theorem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Consider an isosceles triangle
Consider an isosceles triangle. Let $r$ be the radius of its circumscribed circle and $p$ the radius of its inscribed circle. Prove that the distance $d$ between the centres of these two circles is $d =\sqrt {r(r-2p)}$.
I could not get any idea to solve. However I have tried to make a ... | Consider the following figure:
Using Euclid's theorem of sides in a right triangle one has
$$2r(r+d+p)=b^2=4r^2-a^2=4r^2-\left({2r\over r+d}\>p\right)^2\ .$$
It follows that
$$(r+d)^2(r+d+p)=2r\bigl((r+d)^2-p^2\bigr)=2r(r+d+p)(r+d-p)\ .$$
Removing the factor $r+d+p$ leads to
$$r^2+2rd+d^2=2r(r+d-p)\ ,$$
from which the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Strong Induction to find all possible combinations of two numbers Full disclosure, this is a homework question, so I'm only looking for hints not full solutions please.
There is a store which offers two denominations of gift certificates, \$25 and \$40. Determine the possible total amounts that can be formed using the... | First off, if:
$$S=40a+50b$$
then:
$$S+10=40(a-1)+50(b+1)$$
but only if $a\ge1$. If $a=0$ and $b\ge 3$, then:
$$S+10=40\cdot4+50(b-3)$$
So, if $S\ge150$, we can form $S+10$ if we can form $S$. Now, something similair can be done for $S+5$. After you've got that figured out, use induction to prove you can reach all mult... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2212874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Relations between the second norms of matrices Consider two arbitrary complex matrices $L$ and $M$. Let $\|L\|_2 \le \|M\|_2$. Prove that
$$
\Bigg\|\begin{bmatrix}
I & 0 \\
L & I \\
\end{bmatrix}\Bigg\|_2 \le
\Bigg\|\begin{bmatrix}
I & 0 \\
M & I \\
\end{b... | I will handle only the square matrix case here. The other cases are similar, but require some care.
Let's reframe the question: if $L$ and $M$ are such that the largest singular values satisfy $\sigma_1(L) \geq \sigma_1(M)$, then (it suffices to show that) the diagonal matrices $\Sigma_M,\Sigma_N$ of singular values a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does it make sense to take the grad of this? I have been asked to calculate the grad of a function $V(r)$, where $r$ is a position vector and $\omega$ is the angular velocity. But surely, $V(r)$ is a scalar? So how is this possible? Context: I want to show that a force is conservative with this potential, the force is ... | The gradient operator transforms a scalar field $\Phi(\vec t)$ into the vector field $\nabla \Phi(\vec r)$. It is written in Cartesian Coordinates as
$$\nabla \Phi(\vec r)=\hat x\frac{\partial \Phi(\vec r)}{\partial x}+\hat y\frac{\partial \Phi(\vec r)}{\partial y}+\hat z\frac{\partial \Phi(\vec r)}{\partial z}$$
Let ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $G_n$ contains a Hamilton cycle. this is a past paper question on my graph theory course, and I am struggling even to know when to start. The question hasn't asked me to prove anything beforehand so I don't think it warrants Dirac's Theorem or anything else. I don't know where to start!
Given $n$ is a natural... | First, it's not hard to exhibit a Hamiltonian circuit on a $2n\times 2n$ grid graph: use for the vertices $(i,j)$, $1\le i, j\le 2n$. Then the path
$$(1,1)\to (1,2n)\to (2,2n)\to (2,2)\to (3,2)\to \cdots \to (2n-1,2n)\to (2n,2n)\to (2n,1)\to (1,1)$$
is a Hamiltonian circuit. (Note, by the way, that this construction ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How can I transform these triangles to $30-60-90$ triangles? I'm trying to transform the triangles below subject to a few constraints, and cannot figure out how (algebraically) to do so. For ease of notation, I'll refer to points like $E(C)$ as $C$.
I want $CEF$ and $BEF$ to be $30-60-90$. I want to leave $E$ and $C$ ... | In general you can map $\triangle EBC$ to $\triangle EBC^\prime$ with a linear transformation $f(\mathbb{R}^2)\mapsto\mathbb{R}^2$ which leaves every point of line $EB$ fixed and maps $B$ to $C^\prime$. However, it will not map $F$ to $F^\prime$ unless $F$ is the midpoint of segment $CB$.
For each point $X$ in $\triang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $x_n, y_n \in \mathbb{R}$, $\lim x_n=a$, $\lim y_n = b$ then $|x_n-b|
Let $x_n, y_n \in \mathbb{R}$, $\lim x_n=a$, $\lim y_n = b$ then $|x_n-b|<r<|y_n-a|$ for all $n\in\mathbb N \implies r = |a-b|$.
What I intend to do is to show that $|a-b|-\epsilon<r<|a-b|+\epsilon, \forall \epsilon>0$.
For all $n>N_0$ we have,
$... | No, your calculation has a problem; it is in the very last inequality $\vert y_n -a \vert < r$ in the second chain.
The correct procedure is as follows:
We are given that $\left( x_n \right)_{n \in \mathbb{N}}$ and $\left( y_n \right)_{n \in \mathbb{N}}$ are sequences of real numbers and $a$, $b$, and $r$ are real nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show that $\sum_{n=1}^\infty \frac{n^2}{(n+1)!}=e-1$ Show that:
$$\sum^\infty_{n=1} \frac{n^2}{(n+1)!}=e-1$$
First I will re-define the sum:
$$\sum^\infty_{n=1} \frac{n^2}{(n+1)!} = \sum^\infty_{n=1} \frac{n^2-1+1}{(n+1)!} - \sum^\infty_{n=1}\frac{n-1}{n!} + \sum^\infty_{n=1} \frac{1}{(nm)!}$$
Bow I will define e:
$$e^... | $$\frac{n^2}{(n+1)!} = \frac{(n+1)(n-1) + 1}{(n+1)!} = \frac{(n-1)}{n!} + \frac{1}{(n+1)!}$$
Remembering that we're summing to infinity, evaluating the first terms and paying careful attention to the indices,
$$
\begin{align}
\sum_{n=1}^\infty \left( \frac{(n-1)}{n!} + \frac{1}{(n+1)!} \right) &= \sum_{n=2}^\infty \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
$y''+\frac{a}{y^3}=b$, with $a,b>0$. Solving a physics problem, this equation has arised:
$$y''-\frac{a}{y^3}=b$$
with $a,b>0$. Using a lagrangian I can avoid to solve this equation, but since the problem has a solution in terms of elementary functions, I want to know some way to solve this directly.
I did
$$y'y''-\fra... | Substitute $y'=p(y)$ with the new unknown function $p$ of a variable $y$. Then separate variables.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluate the integral I need some help in evaluating this integral:
$$\int {\frac{{\cos x}}{{{{\sin }^3}x + \sin x + 4}}dx} $$
I've tried using the substitution $u=\sin{x}$ but I ended up with a cubic polynomial in the denominator.
| $$\int\limits_0^\pi {\frac{{\cos x}}{{\sin {x^3} + \sin x + 4}}} dx
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Convergence of a sequence in $\mathbb R^n$ with a special property Hi folks, I'm trying to prove (or find a counterexample) of this statement:
Let $(x_k)$ be a sequence in $\mathbb R^n$
such that $(x_k - x_{k+1})\to0$,
show that $(x_k)$ converge.
| $x_n = \sum_{k=1}^n\dfrac1{k}$.
Also,
I think you should use
different letters
for the dimension
($\mathbb R^n$)
and index
($x_n - x_{n+1}$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2213786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.