text stringlengths 83 79.5k |
|---|
H: Use the definition of derivative to find $f'(1)$ for $f(x) = \frac{x}{\sqrt{x^2+1}}$
This is analysis.. So I am using the definition that
$$
f'(x)=\lim_{x\to c} \frac{f(x)-f(c)}{x-c}
$$
So far I have,
$$
\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{1^2+1}}}{x-1}=\lim_{x\to 1} \frac{\frac{x}{\sqrt... |
H: Are homomorphisms a finite group?
I have a question from my homework that says:
Find all the homomorphisms φ:ℂ->ℂ such that for any $x$ in ℝ: φ($x$) = $x$
I don't even know how to begin. The field of complex numbers is infinite. The number of homomorphisms must be huge, if it is even finite. How am I supposed to f... |
H: Use product rule and mathematical induction to show that $f^n$ is differentiable on $I$
Suppose that $f$ is differentiable on $I$. Use the product rule and mathematical induction to show that $f^n$ (the function f is raised to the nth power) is differentiable on $I$ for every positive integer $n$ and find a formula... |
H: $\epsilon, \epsilon/2 $ Proof
How is it proved that $$ \left[ |X_n - X|+|Y_n-Y| \geq \epsilon \right] \subset \{ \left[ |X_n -X|\geq \epsilon/2 \right] + \left[ |Y_n-Y| \geq \epsilon/2 \right] \} $$
Thank you.
AI: Intuitively, if you add two positive numbers and get at least $1$, at least one of them must be greate... |
H: Can you simplify a expression with an exponent that is divided by a number?
As the title suggests, I have
$\;a^{(b/c)}.$
Is there any way to simplify this so that there is no dividing in the exponent?
AI: You can write it as $\sqrt[c]{a^b}=\sqrt[c]{a}^b=(a^{\frac{1}{c}})^b=a^{b/c}$. There isn't really any ot... |
H: Does $H_0^1(\Omega)$ embed into $H_0^1(R^d)$?
Given a domain $\Omega$ in $\mathbb{R}^d$ and a function $f\in H_0^1(\Omega)$, the closure of the test functions on $\Omega$, does the extension of f by 0 to all of $\mathbb{R}^d$ necessarily lie in $H_0^1(\mathbb{R}^d)$?
AI: Yes. The big picture is that for we can glue... |
H: Use the definition of derivative to find $f'(x)$ for $f(x) = x^{1/2}$.
This is an analysis question so we use the definition that the derivative of a function is, $$\lim_{x\to c} \frac{f(x)-f(c)}{x-c}$$ But I'm not really sure if I should just continue to say that $x=c$ for now as the second $x$ perhaps as if $f(x)... |
H: Writing a relatively simple MATLAB function
I'm new to MATLAB and I have been asked to write a MATLAB function whose input arguments are two integers $a$ and $b$; the output is the remainder of the integer division $a/b$ if $a>=b$ or of the integer division $b/a$ if $b>a$
Can someone help me out? Thanks.
EDIT:
This... |
H: Uniform convergence of a family of functions on $(0,1)$
Let the family of functions be
$$f_n(x) = \dfrac{x}{1+nx}.$$
Is the sequence $f_n$ uniformly convergent in the interval $(0,1)$?
AI: $\frac{x}{1 + nx} = \frac{1}{\frac{1}{x} + n} \leq \frac{1}{n}$ which doesn't depend on $x$ hence your sequence converges u... |
H: Determine whether $\int _{1}^{\infty}\frac{x\sin\left(x\right)}{\sqrt{1+x^5}}\,{\rm d}x$ is convergent or divergent
My task is to determine if the following function is convergent or divergent.
$$
\int_{1}^{\infty}{x\sin\left(x\right) \over\sqrt{1 + x^{5}\,}\,}\,{\rm d}x$$
I have been trying to find a more easily i... |
H: Question involving Sylow theorems and characteristic subgroups
Let $G$ be a group of order $2\cdot 17\cdot 23$. Let $P$ be a $17$-Sylow subgroup and $Q$ a $23$-Sylow subgroup. Show that $PQ$ is normal in $G$ and that $P$ is characteristic in $PQ$.
What I know: both $P$ and $Q$ are normal in $G$ by the Sylow theor... |
H: Integral with bounded function
If $m$ is a finite measure on $X$, for which bounded measurable functions $g$ do we have the following: if $f$ is integrable on $X$ and $\int_{X}fdm = 0$ then $\int_{X} fg dm = 0$?
Maybe I'm not understanding something here, but if $g$ is bounded, let $|g| \leq M$ don't we have $$0 =(... |
H: Modern Definition of the Real Numbers
I have been told:
"The real numbers are defined to be the set of equivalence classes of pairs of rational sequences $(a_i,b_i)$, where (1) $\{a_i\}$ is increasing, (2) $\{b_i\}$ is decreasing, (3) for each $i=1,2,..., \hspace{2mm} b_i-a_i>0$, and (4) $\lim_{x \to \infty} (b_i ... |
H: On $C^0 [0, 1]$, define $f \cdot g = \int_0^1 f(x) g(x) dx$. For $f(x) = x$.
a. find $||f||$
b. find all linear polynomials that are orthogonal to $x$
Okay, so I know that
$||f|| = \sqrt(f_1^2 + f_2^2 +... + f_n^2)$
and that linear polynomials are of the form $ax + b$
I am not sure however, how to apply these to ... |
H: A Question on Digit Occurences
Here's a question I was thinking about:
For all positive integers n, list the decimal representation of the numbers 1, 2, 3, ..., n without any leading zeroes. Does there exist an n such that this list contains an equal number of each of the digits 0, 1, 2, 3, ..., 9? (For example, if... |
H: What is the additive inverse of $\langle x-2\rangle+(x^2-2)$ in $\mathbb{Q}[x]/\langle x-2\rangle$?
I'm not sure if I'm thinking about this correctly, but this is what I did:
$\langle x-2 \rangle +(x^2-2) = x^2-2$ (since here, $\langle x-2 \rangle$ is equivalent to $0$) whose additive inverse is $2-x^2$
The book (A... |
H: Why is $-\ln(\cos(x))$ equal to $\ln(\sec(x))$?
Why does the value $-\ln(|\cos(x)|)$ become $\ln(|\sec(x)|)?$
I was doing an integral and I got my final answer as that, but I don't understand how you can just send the negative sign inside and make it $\sec(x).$
AI: You have $$a^b = c \iff b = \log_a(c). $$
Use this... |
H: Can the exponentiation of an integer by a rational be a non-integer rational?
Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$.
My question is the following: what is the nature of $n^r$?
My first guess is that $n^r$ is an integer or an irrational but cannot be a no... |
H: Combinatorics homework question
The answer is 54912.
This is what I've tried so far: So first you have to pick a rank to occur 3 times so thats 13, now you gotta pick a suit that that rank has, which is now 13 * 4. Now you need to pick that same rank 2 more times. The second time it will be 13 * 3, then the third ... |
H: Finding roots of a function in an interval
Does the equation $x^3-12x+2=0$ have three solutions in the interval $[-4,4]$?
We know that this is a continuous function because it's a polynomial, and so we can use the Intermediate Value Theorem to do this problem:
If $f(x)$ is continuous on $[a,b]$, let $M$ be any... |
H: Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?
Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition:
$x=n^r$ is an irra... |
H: Let G be cyclic and let H be a subgroup of G. Show that G/H is cyclic.
Not really sure how I go about doing this. Abstract Algebra. I'm fairly certain it has to do with normal subgroups though.
AI: Here is a quick answer, and if you fill out the details you'll learn a lot from it.
1) Prove that a group $G$ is cycl... |
H: Difference between the real projective plane and the complex projective plane
Well the title says it all. If we consider the $P^2(\Bbb R)$ and the $P^2(\Bbb C)$, and we compare them, my guess is that it will be like a round $\Bbb R^2$ versus a sphere. I don't have very good geometric intuition, so I really can't pi... |
H: Surjectivity for permutation representation of a group action
I am having trouble proving that my function is surjective. Here is the problem statement:
Problem statement: Let T be the tetrahedral rotation group. Use a suitable action of T on some set, and the permutation representation of this action, to show that... |
H: Show that $|1-\varphi_X (u)|\leq E\{ |uX| \}$
Show that $|1-\exp\{ix\}|^{2}=2(1-\cos x) \leq x^{2}$ for all $x \in \mathbb{R}$. Use this to show that $|1-\varphi_X(u)|\leq E\{|uX|\}$, where $\varphi_X(u) =E\{\exp(i\langle u,X\rangle)\}$ is the characteristic function.
We were given the hint that the power series fo... |
H: Modified dirichlet function
Can Dirichlet's function be modified in such a way that it is continuous at some real number? For instance, as $xD(x)$ is continuous at $x=0$, is it possible that $(x-1)D(x)$ is continuous at $x=1$? Here $D(x)$ is the Dirichlet function.
Can it be modified to be continuous at finitely m... |
H: Orthogonal Eigen-Basis Problem
In any euclidean space, will the eigenvectors be always orthogonal ?
As a result of spectral theorem ?
AI: For an arbitrary operator, no. For instance, on finite dimensional spaces, an operator is unitarily diagonalizable (i.e. has orthonormal eigenvectors) if and only if it is norm... |
H: Solving an inequality
Name the property that justifies each statement.
If $-3x < 24$, then $x > -8$.
AI: The property is the Additive Inverse
If $a < b$ then $-a > -b$
If $a > b$ then $-a < -b$
This is really the same as multiplying by $(-1)$, and that is why it changes direction.
http://www.mathsisfun.com/algebra/... |
H: If I have the equation, $\frac{p}{K-p}=Ce^{rt}$, how can I solve for p?
There is where I'm at:
$$
\frac{p}{(K-p)}=Ce^{rt}
$$
$$
p=(K-p)(Ce^{rt})
$$
$$
p=Ce^{rt}K-Ce^{rt}p
$$
$$
p+Ce^{rt}p=Ce^{rt}K
$$
$$
p(1+Ce^{rt})=Ce^{rt}K
$$
$$
p=\frac{Ce^{rt}K}{1+Ce^{rt}}
$$
Is it possible to simplify this further?
AI: I'll inc... |
H: $2^{2x}-3 \cdot 2^{x+1}=16 \implies 2^x=-2$ or $2^x=8$
How did they get from $$2^{2x}-3 \cdot 2^{x+1}=16$$ to $2^x=-2$ or $2^x=8$?
I know that you could also write it as $2^{2x}-3 \cdot 2^{x+1}$
and $2^{2x}-6\cdot 2^x=16$
But I got stuck there...
AI: Hint : $$2^x=t\iff t^2-6t-16=0$$ |
H: Spaces sharing all higher homotopy groups
Is it possible that two topological spaces share all higher homotopy groups, but are not homeomorphic? I should note that I have not studied much in the way of the theory of higher homotopy groups; I merely know of the groups' existence, and was wondering what could be said... |
H: Variation of parameters for a linear second order nonhomogeneous equation
I'm using variation of parameters for this problem, and I'm not sure if I'm on the right track.
The question is
Find a function $v_1$ and $v_2$ such that $v_1(x)e^x+v_2(x)e^{2x}$ is a solution of $y''-3y'+2y=4x+4$ and
$v_1(x)y_1(x)=v_2(x)y_... |
H: Fourier transform formula
How to prove that $f(x)=\sum_{n=-\infty}^\infty f(n)K(x-n)$ where $K(y)=\sin\pi y/\pi y$
Here, $f$ is moderate decrease and its fourier transform is supported in $[-1/2, 1/2]$.
I show that $\hat f(k)=\sum_{n=-\infty}^\infty f(n)e^{-2\pi ink}$ but no improvement...
Plz help
AI: This formula... |
H: Simplifying logarithmic equation $0.1 = e ^{-(\ln 2)t/5700}$?
$$0.1 = e ^{-(\ln 2)t/5700}$$
How do I simplify this? I took the ln of both sides so does the $e^{\ln}$ cancel out?
$$\ln 0.1 = \ln e ^{-(\ln 2)t/5700}$$
$$-2.3 = \ln e ^{-(\ln 2)t/5700}$$
but the $\ln$ in the exponent can't just disappear?
AI: No, but u... |
H: What is the solution to this problem
I'm sorry I can't be more specific in the title.
I have this exercise I can't solve by myself, I've tried many times and I never get the right answer:
$[(\frac{1}{a^2}-b^2):(\frac{1}{a}+b)]^{-1}$
This is the answer:
$\frac{a}{1-ab}$
Any help is appreciated.
AI: Hint: The differe... |
H: Simple way to write Cumulative distribution function of $\alpha X + \beta$?
Let $X$ be a discrete random variable with Cumulative Distribution Function $F$. What is the cumulative distribution function of the random variable $\alpha X + \beta$, where $\alpha, \beta$ are constants and $\alpha \ne 0$?
Let $Y = \alpha... |
H: which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$
Which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$?
Squaring both sides will give me something but I could not go any further.
AI: As $(\sqrt7+\sqrt5)^2=12+2\sqrt{35}$ and $(\sqrt6+\sqrt6)^2=12+2\sqrt{36}$
$$(\sqrt7+\sqrt5)^2<(\sqrt6... |
H: Show $\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A,A\in M(2,\mathbb{C})$
Show $$\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A$$
for $A\in M(2,\mathbb{C})$. In addition, $\operatorname{trace}(A)=0$.
Can anyone give me a hint how this can connect with cos... |
H: Find the limit of a fraction
So far I have,
$$
\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{1^2+1}}}{x-1}=\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{2}}}{x-1}
$$
I have no idea how to keep going with this, every way I try I get stuck and can't do anything with it.
AI: More generall... |
H: Why do we stop at double cosets and do not consider triple, ... n-ple cosets?
The obvious answer seems to be that a group has only two sides, and once you're done taking quotients on the left and on the right, you can't quotient by another subgroup from another side.
But if $G$ is a group with subgroups $H$ and $K$... |
H: Probability problem with deck of cards
In a poker game each player is dealt five cards. What is probability that a player's hand has exactly four diamonds?
The answer is .011, I just don't know how to get it.
AI: There are $\binom{52}{5}$ $5$-card hands. They are all equally likely.
Now we count the hands that have... |
H: Prove that $\sin(x + \alpha)$, $\sin(x + \beta)$ and $\sin(x + \gamma)$ are linearly dependent.
Functions $f$ and $g$ are independent on an interval $D$ if $af(x) + bg(x) = 0$ implies that $a = 0$ and $b = 0$ $\forall x \in D$
let $\alpha$, $\beta$, $\gamma$ be real constants. Prove that
$\sin(x + \alpha)$, $\sin... |
H: Find orthogonal matrices such that $P^TAP$ is diagonal (dim=2)
Let
$$A = \begin{bmatrix}
-5 & 12 \\ 12 & 5
\end{bmatrix}.$$
I found the eigenvalues $13$ and $-13$, and the eigenvectors $[2,3]^T$ and $[3,-2]^T$. However, the matrix with them as columns transposed, times $A$, times that matrix is not diagonal.
I'm no... |
H: How to calculate the angle between two vectors, defined by 3 points on the earth?
I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values.
I know that there is a little prob... |
H: Taylor Series Approximation for degree k Taylor polynomial?
Let $T_k(x)$ be the degree $k$ Taylor polynomial of the function $f(x)=\sin(x)$ at $a=0$. Suppose you approximate $f(x)$ by $T_k(x)$. If $|x|\le 1$, how many terms are needed (that is, what is $k$) to obtain an error less than $\frac 1 {5040}$?
I don't r... |
H: Is sum of $\sum\limits_{n = 1}^{p}{(n(n+1)}\mod{p})= \frac{p(p-1)}{2} $ where $p \equiv 7 \mod{8}$ correct?
$\sum\limits_{n = 1}^{p-1}{(n^{2}+n)}\mod{p})= \frac{p(p-1)}{2} $ where $p \equiv 7 \mod{8}$
I am not sure if this is correct, this is a part of my homework and is having trouble with it.
AI: Hint :
$\sum\li... |
H: Converse of a well known result.
We all know the following result.
Let $\sum a_n$ is a series of alternative positive and negative terms. Consider $p_n = \frac{a_n + |a_n|}{2}$ and $q_n = \frac{a_n - |a_n|}{2}$. So $\sum p_n$ and $\sum q_n$ are series of positive and negative terms of $\sum a_n$. Now if $\sum a_n$... |
H: Prove that $N$ is normal
Let $H$ be a subgroup of $G$. Consider the set $N=\cap_{x\in G}xHx^{-1}$. Prove that $N$ is normal subgroup of $G$.
Using the fact that any (finite or infinite) intersection of subgroups is a subgroup I am able to prove $N$ is a subgroup of $G$ and even $H$.
AI: For each given $n\in N$, $... |
H: Characterization of functional monoids
I define a functional monoid to be a monoid that is isomorphic to the set of all functions from a set $S$ to itself under the operation of composition. I want to know useful necessary and sufficient conditions for a monoid to be functional. Any help would be appreciated.
AI: T... |
H: A set of point sets, of cardinality greater than that of the continuum.
What would be the examples of a Set of point sets in $\mathbb R^2$ having the cardinality greater than that of real numbers?
Any stipulations over these point sets themselves may please be specified.
EDIT: My mistake which i correct in bold... |
H: How prove this isn't exist prime number $p>7$
show that: there isn't exsit prime number $p>7$,such
$p^{12}+5039\times 5041$ the factor number is less than $120$
I think maybe use Fermat theorem? How can solve it? Thank you
AI: Hint : $5040=7!$ , $120=5!$ , and $p=2k+1$. Our expression becomes: $$p^{12}+5039\cdot5... |
H: Complex numbers equation problem
I've been having some trouble with this complex question (not my best topic), and I was wondering if I could get any hints or explainations on how to do it.
Prove that all the roots of the equation $$z^n\cos(n\alpha)+z^{n-1}\cos((n-1)\alpha)+z^{n-2}\cos((n-2)\alpha)+\cdots+z\cos(\al... |
H: Solve for a variable in the limit of integration
How do I solve for $x$ in an equation such as the following?
$$A = \int_0^x f(t) dt$$
I feel like I must of come across this at some point in my courses, but for the life of me I cannot remember. Is this something that is really easy, or just hard to do for a general... |
H: All pairs of (a,b) of positive integers satisfying a given condition.
I have to determine all pairs $(a,b)$ of positive integers satisfying the condition $a^{b^2}=b^a$. So far i have found only one, namely $(1,1)$. How can i do it? Thanks for any help.
AI: Hints:
Take logarithms
$\log_b(a)\in\mathbb Q\Leftrightarr... |
H: Normal subgroup where G has order prime
If $o(G) = p^n,$ p a prime number, and H is a subgroup of G, show that there exists an $x\in G$, but $x\notin H$ such that $x^{-1}Hx = H$
How can I prove this?
AI: You can induct on $n$ :
Now if $N_G(H) \neq H$, then any element of $N_G(H)$ will do, so assume $N_G(H) = H$. S... |
H: Largest triangle to fit in a circle will be isosceles triangle?
Largest triangle to fit in a circle will be isosceles triangle?
Or some other type?
AI: Yes, what you say is true, but you can say more than that.
Given a particular chord of a circle, you can maximize the area of the triangle by having the third verte... |
H: Help in a proof in basic Algebraic Geometry
I'm trying to understand this proof:
Theorem
Let $X$ and $Y$ be affine closed sets. If $f:X\to Y$ is a function such that $g\circ f\in k[X]$ for every $g\in K[Y]$, then $f$ is a morphism.
Proof
Suppose that $f:X\to Y$ is a function such that $g\circ f\in k[X]$, whenever $... |
H: Wreath product of finitely generated groups is finitely generated
Let $G$ and $H$ be two groups generated by finite sets $\Sigma_G$ and $\Sigma_H$, and let $W=G \wr H$ be the wreath product of $G$ and $H$. Show that $W$ is finitely generated by $\Sigma_G \times \{1 \} \cup \{1 \} \times \Sigma_H$.
I cannot prove... |
H: Question straight from the SAT
If a coordinate system is devised so that the positive y-axis makes an angle of 60 degrees with the positive x-axis, what is the distance between the points with coordinates (4,-3) and (5,1)?
I'm sure you guys can get it without the multiple choice answers.
Keep in mind the college bo... |
H: Criterion for proving flatness
I am trying to show that:
$M$ is a flat $A$ module iff for all ideals $I$ of $A$, which are finitely generated, the $A$-linear map
$I \otimes M\longrightarrow M$, taking $x \otimes y\longrightarrow xy$ is injective.
Please help.
AI: Here's a reference: Proposition 3.58 of Rotma... |
H: Rationalizing the denominator with 3 roots
Well, I can't find the example on how to solve this.
If I multiply
$$
\dfrac{2}{\sqrt[3]{9}+\sqrt[3]{15}+\sqrt[3]{25}}
$$
with
$$
\dfrac{\sqrt[3]{9}-\sqrt[3]{15}+\sqrt[3]{25}}{\sqrt[3]{9}-\sqrt[3]{15}+\sqrt[3]{25}}
$$
or similar, it just gets even more complicated, and I ... |
H: Finding vector length based on parallell and orthogonal vectors
Do anyone know a simple way of finding the length of vector a in my figure? The known values are $(x_0, y_0), (x_1, y_1), (x_2, y_2), (x_3, y_3)$. (If you look closely, you can see that $f$-vector is a force, and that I need to find the arm between for... |
H: $e^z, \ \ z \in \mathbb{C}$ isn't invertible
Could you tell me why the function $\mathbb{R}^2 \ni z \rightarrow e^z \in \mathbb{R}^2$, complex exponential, is not invertible globally. On a horizontal strip $[\ i y, \ i(y+2 \pi))$ its inverse is $\ln z$.
I found this example on this forum, but there is no explanatio... |
H: Show that $(F^n)^A$ is cyclic as an $F[x]$-module
Problem statement:
Let $A$ be an $n \times n$ matrix over the field $F$ and let $A$ have $n$ distinct eigenvalues. Show that $(F^n)^A$ is cyclic as $F[x]$-module.
I'm not sure I understand the notation $(F^n)^A$. What does this refer to?
Thanks.
AI: I'm fairly sur... |
H: Some inequality; $(\sum a_i^p)^{1/p} \leq C \sum a_i$
I want to know whether or not the following inequality $$
(\sum_{i=1}^n a_i^p)^{1/p} \leq C \sum_{i=1}^n a_i$$
is satisfied for some $C$, where $0 \leq a_i,\ 1/2 \leq 1/p$.
For some constant $C$, the above inequality hold ?
Thank you.
AI: For $x =(x_i) \in ... |
H: The limit at the end-point of an open interval, for a uniformly continuous function
If $f: (a,b)\to\mathbb R$ is uniformly continuous and $\{x_n\}$ in the domain tends to $b$, then why does $\{f(x_n)\}$ have a limit?
AI: Because $x_n$ is Cauchy sequence then $f(x_n)$ is Cauchy sequence due to uniform continuouity.... |
H: Can somebody explain why the interval $\left ( 0,1 \right )$ is not countable?
I cannot seem to understand the proof of why the interval $\left ( 0,1 \right )$ is not countable.
The proof that is written in my book using the method of Reductio ad absurdum.
It starts with the following statement:
We know that every... |
H: Let $\, g \,$ be a function defined on $(a,b)$ such that $\, a
I am stuck on the following problem that says :
What I guess option (A) is not possible. But,
I am not sure about the other options. Can someone explain? Thanks in advance for your time.
AI: If $g$ is a constant, then $g(x) = c$ for all $x\in (a,b)$... |
H: What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?
I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams?
Would I be correct in understanding that non-Euclidean Geometry is just a more ... |
H: Prove that $[a,b]$ is connected space.
Prove that $[a,b]$ is connected space.
I know that $\mathbb{R}$ with euclidean metric is connected space. I would like find surjective function $f: \mathbb{R} \rightarrow [a,b]$. Because $\mathbb{R}$ is connected and $f$ is surjective function then $[a,b]$ is also connected sp... |
H: Solving a first order non-linear ODE
I encountered a problem solving the following equation: $$t^2dy/dt+2ty-y^3=0$$
I have already tried the following steps, but to no avail:
1)Separation of variables
2.1)Integration Factor of 1 variable.
2.2)Integration Factor of 2 variables.
3)Transposing the equation and find a ... |
H: which one is larger $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$?
Which is larger? $\sqrt[n]{x+\delta}-\sqrt[n]{x}$ or $\sqrt[n]{x}-\sqrt[n]{x-\delta}$?
Algebraic justilation does not help.
AI: $\sqrt[n]{x+\delta}-\sqrt[n]{x}\ \boxed{\phantom{A} }\sqrt[n]{x}-\sqrt[n]{x-\delta}$
$\sqrt[n]{x+\... |
H: using $\sin x$ to get a function with range $[a,b]$
$\sin x$ is a nice function on $\mathbb{R}$ whose range is $[-1,1]$. can we 'adjust' it so that its range will be $[a,b]$. By 'adjusting' I mean changing the argument $x$ to some other argument which is a function of $x$ , or multiplying by a constant or adding a ... |
H: Properties of the co-countable topology on $[0,1]$
I am trying to learn topology by myself. I have the following questions and my attempts of their proofs. Since I have no one to consult with, I am posting here. If anyone check my proofs that would be great. I know it is a bit long, my apologies for that. Many than... |
H: Probability: the expected value in a dice game.
If a dice is thrown till the sum of the numbers appearing on the top face of dice exceeds or equal to 100, what is the most likely sum?
AI: This has been computed several times on the site already so let us present some of the underlying theory.
Renewal theory deals w... |
H: integrate the following expression
Question:
integrate $\sqrt{t}$
My answer:
$t^{\frac{1}{2}} = \dfrac{t^{\frac{3}{2}}}{\frac{3}{2}} +c$
correct answer:
$\frac{2}{3}.t^{\frac{3}{2}} +c$
what am I doing wrong? thank's
AI: $$\int \sqrt{t} = \int t^{\frac{1}{2}} = \frac{t^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2t^{... |
H: Is the empty function always a bijection?
Let $f_A:\emptyset\to A$ be the empty function with range $A$. The definition of a bijection as applied to this function is:
$$\forall x,y \in \emptyset (x=y \implies f_A(x)=f_A(y))$$
negating you get:
$$\exists x,y \in \emptyset (x = y\land f_A(x) \neq f_A(y))$$
Which is o... |
H: Family of sets which intersect isn't connected
Find any family of sets $A_n$ such that $A_n$ are connected sets and $A_{n+1} \subset A_n$ and $$\bigcap_{n=1}^{\infty}A_n$$ is not connected.
I tried find family of sets such that $$\bigcap_{n=1}^{\infty}A_n = (-1,0) \cup (0,1)$$ Of course it isn't connected but I do... |
H: integrate the following expression again
Question:
$x - x^2 + 1$
My answer:
$\frac{x^2}{2} - \frac{x^3}{3} + {x} + C$
Correct answer:
$\frac{x^2}{2} - \frac{x^2}{3} - \frac{x^3}{3} + x + C$
What am I doing wrong? thanks
AI: The general formula is
$$-1\neq n\in\Bbb Z \implies \int x^ndx=\frac{x^{n+1}}{n+1}+C$$
Inste... |
H: Approximating continuous functions with growth condition by lipschitz functions
I wonder about the following. Is it possible to approximate $x^p$ for $x\in\mathbb{R}_+$ with lipschitz continuous functions? If so, is it possible to approximate it in a dominating way, i.e. $f_n\to x^p, f_n$ lipschitz and $f_n\ge x^p$... |
H: Solving a probability equation.
One group of 30 people won a contest and as a reward they got a free vacation to Hawaii. The hotel they will be in has 10 rooms with 3-bed (bed for 3 people).
The question: How many ways can we deploy the students?
I used combinations because the students can change each other places... |
H: What is the difference between these two given sums?
What is the difference between this: $\sum_{i=1}^n(x^i+9x*i)$ and $\sum_{i=0}^n x[x^i+9i+9]$ ?
So far I know that the first terms are not different.
The first term of the $\sum_{i=1}^n(x^i+9x*i)$ is $\ x+9x$.
And the first term for $\sum_{i=0}^n x[x^i+9i+9]$ i... |
H: Which is larger? $20!$ or $2^{40}$?
Someone asked me this question, and it bothers the hell out of me that I can't prove either way.
I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas $2^{40}$ has only factors of 2.
Is... |
H: Help needed with complements, partition and power sets
I'm working on some tasks which is listed below, and I'm trying to figure out if I've understood partition, power set, and complements correctly.
Here are the tasks:
Assume that $\{$$1, 2, 3, 4, 5$$\}$
What is the complement of the amount $\{$$1, 2, 3$$\}$?
... |
H: Can you get any irrational number using square roots?
Given an irrational number, is it possible to represent it using only rational numbers and square roots(or any root, if that makes a difference)?
That is, can you define the irrational numbers in square roots, or is it something much deeper than that? Can pi be ... |
H: Partial derivatives of a multivariable function
$f(tx,ty)=t^5f(x,y)$ for all values of $x, y, t$ where both functions are differentiable.
Show that
$$a)\ xf_x+yf_y=5f$$
$$b)\ \ x^2f_{xx}+2xyf_{xy}+y^2f_{yy}=20f$$
Clearly, there is differentiating of the initial equation in order to get to a. And a double differen... |
H: Convert this scenario into algebra equation
Sales for the month minus the VAT @ 20% = (x).
20% of (x) is profit margin (y).
5% of (y) is commission earned (c).
How can I write an equation that demonstrates the above please?
I.e x - 20% of y - 95% of y = c. (Sorry, not explained well). Thanks.
AI: Why not : y = 0.... |
H: For which t is the matrix invertible?
$$\begin{matrix} t&a_2&0&0&\cdots&0\\
0&t&a_3&0&\cdots&0&\\
\vdots&\vdots&\ddots&&\cdots&\vdots\\
0&0&\cdots&&t&a_n\\
a_1&0&\cdots&0&\cdots&t \end{matrix}$$
For what values of t is this matrix invertible? Thanks in advance for any help!
AI: Develop the determinant along the fir... |
H: Combinatorics question about downsets
Prove that if
$\mathcal{A}$
is a downset then the average size
of sets in
$\mathcal{A}$
is at most
$\frac{n}{2}$
($\mathcal{A} ⊂ \mathcal{P}(n)$ is a
downset
if, for every
$A∈\mathcal{A}$
, every subset
of
$A$
belongs to
$\mathcal{A}$)
Have tried using induction but got stuck a... |
H: Being Symmetrical in Limit
The question is given $f_n : \mathbb{R} \to \mathbb{R}$ a sequence of symmetrical functions around a constant point $c$. Is it true that the sequence converges to a symmetrical function around $c$?
I think the question can be modified in the following way. define $g_n(x)=f_n(x+c)$. Then $... |
H: What does $|A|$ denote in set notation?
What does $|A|$ of a set $A$ denote?
Also, what does $A\leftrightarrow B$ of sets $A, B$ mean?
I encountered this in one of my textbooks which said:
Of two sets $A, B$ we know $|B|$ but $|A|$ is unknown. If we succeed
in constructing a bijection $A\leftrightarrow B$, then$... |
H: The number of cyclic subgroups of order 15 in $\mathbb{Z}_{30} \oplus \mathbb{Z}_{20}$
What is the number of cyclic subgroup of order 15 in $\mathbb{Z}_{30} \oplus \mathbb{Z}_{20}$?
I have counted the number of elements with order 15 in $\mathbb{Z}_{30} \oplus \mathbb{Z}_{20}$ that is $48$, by counting all possib... |
H: Meaning of the set $\mathbb N^\mathbb N$
I came across a question which requires one to check if there's a bijection from the set $ \mathbb N^\mathbb N$ to another set. I've never seen a set defined this way and was wondering if this was just a typo. Could anyone clarify this for me? If this is correct, then what d... |
H: last three digits of $7^{100}-3^{100}$
How can I find the las three digits of $7^{100}-3^{100}$ ? I know one way is to use $7^{100}=(10-3)^{100}=\sum_{k=0}^n{100 \choose k}10^{100-k}(-3)^k$ but I'm totally stuck...
Thanks
AI: Well, note that for $0\le k\le 97,$ we have that $1000$ is readily a factor of $\binom{10... |
H: Finding a moment generating function given E(X) and E(x^2)
I am trying to find the moment generating function.
It takes values in the set {0,1,2} with moments
E(X) = 1 and E($X^{2}$) = $ \frac 3 2 $
I know then that M'(0) = 1 and M"(0) = $\frac 3 2 $
I have read through my course notes/ textbook and have found not... |
H: Computation of integral $\int_{0}^{1}\ln(p)\ln(1-p)p^{2}\,dp$
I want to compute this integral:
\begin{equation*}
J=\int_{0}^{1}\ln(p)\ln(1-p)p^{2}dp
\end{equation*}
It will be great if you can detail the proof.
I tried to do change of variable it does not work, and also integration by part.
Thanks.
AI: Change varia... |
H: Why is $(2+\sqrt{3})^{50}$ so close to an integer?
I just worked out $(2+\sqrt{3})^{50}$ on my computer and got the answer
$39571031999226139563162735373.999999999999999999999999999974728\cdots$
Why is this so close to an integer?
AI: Let $x=(2+\sqrt{3})^{50} + (2-\sqrt{3})^{50}$
$x$ is clearly an integer, since a... |
H: Can a vector of a-a, still be used as a point of direction even if it does equate to 0?
In my maths class we had a similar problem where we had to find the solution to get to Y from X. The answer is c however I argued that it could also be a-a, because technically you can go up to B from X by 1a and then down to Y... |
H: Maximal elements in a set
Prove that if there are two maximal elements in a partially ordered set, then these maximal elements are not comparable.
I understand that I should show that if there are two maximal elements in a set, then they must be equal or not compareable. How should I write it down exactly?
Thanks
A... |
H: Proving $\mathcal P(A-B) = \mathcal P (A) -\mathcal P (B)$
I started with trying to prove $\mathcal P(A-B) \subseteq \mathcal P (A) - \mathcal P (B)$
$x \in \mathcal P (A) - \mathcal P (B)$ that mean
$x \in \mathcal P (A) \wedge x \notin \mathcal P (B)$ means
$x \in A \wedge x \notin B$ meaning
$x \in A-B$ therefo... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.